aa r X i v : . [ m a t h . L O ] S e p CANONICAL FRAGMENTS OF THE STRONG REFLECTIONPRINCIPLE
GUNTER FUCHS
Abstract.
For an arbitrary forcing class Γ, the Γ-fragment of Todorˇcevi´c’sstrong reflection principle
SRP is isolated in such a way that (1) the forcingaxiom for Γ implies the Γ-fragment of
SRP , (2) the stationary set preservingfragment of
SRP is the full principle
SRP , and (3) the subcomplete fragment of
SRP implies the major consequences of the subcomplete forcing axiom. Alongthe way, some hitherto unknown effects of (the subcomplete fragment of)
SRP on mutual stationarity are explored, and some limitations to the extent towhich fragments of
SRP may capture the effects of their corresponding forcingaxioms are established.
Contents
1. Introduction 12. Γ-projective stationarity and the Γ-fragment of
SRP
SRP
SRP
73. Consequences 133.1. Barwise theory and a technical lemma 133.2. Friedman’s problem, the failure of square, and
SCH CH setting: a less canonical separation 354.3. The CH setting: a canonical separation at ω Introduction
The strong reflection principle,
SRP , introduced by Todorˇcevi´c (see [6, P. 57]),follows from Martin’s Maximum, and encompasses many of the major consequencesof Martin’s Maximum: the singular cardinal hypothesis, that 2 ω = ω , that the Date : September 15, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Strong reflection principle, forcing axioms, Martin’s Maximum, sta-tionary reflection, subcomplete forcing.Support for this project was provided by PSC-CUNY Award nonstationary ideal on ω is ω -saturated, and many others; see [26, Chapter 37]for an overview. In [17], I began a detailed study of the consequences of SCFA , theforcing axiom for subcomplete forcing, with an eye to its relationship to Martin’sMaximum. Subcomplete forcing was introduced by Jensen [31], [33], and shownto be iterable with revised countable support. Since subcomplete forcing notionscannot add reals,
SCFA is compatible with CH , which sets it apart from Martin’sMaximum. In fact, Jensen [30] showed that SCFA is even compatible with ♦ , andhence does not imply that the nonstationary ideal on ω is ω -saturated. On theother hand, SCFA does have many of the major consequences of Martin’s Maximum,such as the singular cardinal hypothesis, as mentioned above.While my quest to deduce consequences of Martin’s Maximum from
SCFA (orsome related forcing principles for subcomplete forcing) has been fairly successfulin many respects, such as the failure of (weak) square principles and the reflectionof stationary sets of ordinals [20], and even the existence of well-orders of P ( ω )[19], it remained unclear until recently how to find an analog of SRP that relatesto
SCFA like
SRP relates to Martin’s Maximum. Thus, I was looking for a versionof
SRP that follows from
SCFA and that, in turn, implies the major consequencesof
SCFA .The objective of the present article is to provide such a principle. In fact, I canon-ically assign to any forcing class Γ its fragment of the strong reflection principle,which I call Γ-
SRP , in such a way that(1) the forcing axiom for Γ, FA (Γ), implies Γ- SRP ,(2) letting
SSP be the class of all stationary set preserving forcing notions,
SRP is equivalent to
SSP - SRP ,(3) letting SC be the class of all subcomplete forcing notions, SC - SRP capturesmany of the major consequences of
SCFA .While points (1) and (2) are beyond dispute, point (3) is a little vague, and Iwill give some more details on which consequences of
SCFA the principle SC - SRP captures and which it cannot. The situation will turn out to be similar to the
SRP vs. MM comparison.For the most part, I will be working with a technical simplification of the notionof subcompleteness, called ∞ -subcompleteness and introduced in Fuchs-Switzer[24]. This leads to a simplification of the adaptation of projective stationarity tothe context of this version of subcompleteness. Working with the original notionof subcompleteness would add some technicalities, but would not change muchotherwise.The paper is organized as follows. In Section 2, I give some background on thestrong reflection principle and on generalized stationarity. Given a forcing class Γ, Iintroduce the notion of Γ-projective stationarity and the Γ-fragment of SRP , and Ishow that the forcing axiom for Γ implies the Γ-fragment of
SRP , as planned. I thenreview the definitions of subcomplete and ∞ -subcomplete forcing and characterize( ∞ -)subcomplete projective stationarity combinatorially (as being “(fully) spreadout”).Section 3 is concerned with consequences of SRP , and mainly of the subcompletefragment of
SRP . The main task here is to show that certain stationary sets are notonly projective stationary, but even spread out. To this end, I give some backgroundon Barwise theory and prove a technical lemma in Subsection 3.1. Subsection 3.2then uses this in order to establish that some of the major consequences of
SRP
ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 3 already follow from the subcomplete fragment of
SRP : Friedman’s problem, thefailure of square and the singular cardinal hypothesis. In Subsection 3.3, I derivesome consequences of
SRP , and its subcomplete fragment, on mutual stationarity(which as far as I know are new even as consequences of the full
SRP ).Section 4 deals with limitations to the effects of the subcomplete fragment of
SRP on certain diagonal reflection principles for stationary sets of ordinals, and with thetask of separating it from
SCFA . In Subsection 4.1, I show that the ∞ -subcompletefragment of SRP is consistent with a failure of reflection at ω , assuming the con-sistency of an indestructible version of SRP . I also observe that, assuming theconsistency of an indestructible version of
SRP (that follows from MM ), ∞ - SC - SRP does not imply
SCFA . This follows rather directly from prior results. However theseresults don’t separate the subcomplete fragment of
SRP , together with CH , from SCFA . The last two subsections contain partial results in this direction. Subsec-tion 4.2 shows that, assuming the consistency of an indestructible version of thefragment of
SRP used to deduce the consequences in Section 3, together with CH ,that fragment with CH fails to imply a rather weak diagonal reflection principle at ω which does follow from SCFA + CH , thus separating this fragment of SRP in thepresence of CH from SCFA . Finally, in Section 4.3, I show that if the subcompleteforcing axiom up to ω (I denote this BSCFA ( ≤ ω )) is consistent, then it is consis-tent that CH and the subcomplete fragment of SRP ( ω ) hold, but BSCFA ( ≤ ω ) fails.In this last result, it matters in the proof that I deal with subcompleteness, not ∞ -subcompleteness. Along the way, I show that subcomplete forcing that preservesuncountable cofinalities is iterable with countable support.Finally, Section 5 lists some questions and open problems.2. Γ -projective stationarity and the Γ -fragment of SRP
In this section, I will give a brief introduction to the strong reflection principle,motivate how I arrive, for a forcing class Γ, at the Γ-fragment of the strong reflec-tion principle, consider a couple of examples, and then focus on the subcompletefragment of
SRP .2.1.
Some background and motivation for
SRP . Recall Friedman’s problemfrom [15]:
Definition 2.1. If γ < κ , then I write S κγ for the set of ordinals ξ < κ withcf( ξ ) = γ . Now let κ ≥ ω be an uncountable regular cardinal. Then Friedman’sProblem at κ , denoted FP κ , says that whenever S ⊆ S κω is stationary, then thereis a normal (that is, increasing and continuous) function f : ω −→ S . In otherwords, S contains a closed set of order type ω .The strong reflection principle SRP , introduced by Todorˇcevi´c (see [6] or [37] forits original formulation), can be viewed as a version of Friedman’s problem, butadapted to the generalization of stationarity due to Jech, see [27] for an overviewarticle. The role of a closed set of order type ω is taken over by the obvious analogfor the context of generalized stationarity: a continuous ∈ -chain of length ω . Ideviate slightly from the common way of presenting this. Definition 2.2.
Let κ be a regular uncountable cardinal, and let S ⊆ [ H κ ] ω bestationary. A continuous ∈ -chain through S of length λ is a sequence h X i | i < λ i of members of S , increasing with respect to ∈ , such that for every limit j < λ , X j = S i Feng & Jech [11] found an equivalent way to express Todorˇcevi´c’s original prin-ciple that is more amenable to generalization than its original formulation. Theyintroduced the following concept. Definition 2.3. Let κ be an uncountable regular cardinal. S ⊆ [ H κ ] ω is projectivestationary if for every stationary set T ⊆ ω , the set { X ∈ S | X ∩ ω ∈ T } isstationary.While I’m at it, let me introduce some terminology around generalized station-arity. Definition 2.4. Let κ be a regular cardinal, and let A ⊆ κ be unbounded. Let κ ⊆ X . Then lifting ( A, [ X ] ω ) = { x ∈ [ X ] ω | sup( x ∩ κ ) ∈ A } is the lifting of A to [ X ] ω . Now let S ⊆ [ X ] ω be stationary. If W ⊆ X ⊆ Y , thenwe define the projections of S to [ Y ] ω and [ W ] ω by S ↑ [ Y ] ω = { y ∈ [ Y ] ω | y ∩ X ∈ S } and S ↓ [ W ] ω = { x ∩ W | x ∈ S } . Thus, using this notation, and letting κ be an uncountable regular cardinal, a set S ⊆ [ H κ ] ω is projective stationary iff for every stationary T ⊆ ω , S ∩ ( T ↑ [ H κ ] ω ) isstationary. It is well-known that in the notation of the previous definition, S ↑ [ Y ] ω and S ↓ [ W ] ω are stationary.Following is the characterization of SRP , due to Feng and Jech, which I take asthe official definition. Definition 2.5. Let κ ≥ ω be regular. Then the strong reflection principle at κ , denoted SRP ( κ ), states that whenever S ⊆ [ H κ ] ω is projective stationary, thenthere is a continuous ∈ -chain of length ω through S . The strong reflection principle SRP states that SRP ( κ ) holds for every regular κ ≥ ω .Usually, the strong reflection principle is formulated so as to assert the existenceof an elementary chain of length ω through S . We’ll briefly convince ourselvesthat this version of the principle, made precise, follows from the one stated. Observation 2.6. Projective stationarity is preserved by intersections with clubs:if S ⊆ [ H κ ] ω is projective stationary, where κ is regular and uncountable, then forany club C ⊆ [ H κ ] ω , S ∩ C is also projective stationary.Proof. Let us fix S and C . Let T ⊆ ω be stationary and D ⊆ [ H κ ] ω be club.Then, since S is projective stationary, it follows that { X ∈ S | X ∩ ω ∈ T } ∩ ( C ∩ D ) = ∅ . But the set on the left is equal to { X ∈ S ∩ C | X ∩ ω ∈ T } ∩ D, and this shows that S ∩ C is projective stationary. (cid:3) In the following, for a model M of a first order language, I write |M| for itsuniverse. Further, if X ⊆ |M| , then M| X is the restriction of M to X . ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 5 Corollary 2.7. Assume SRP ( κ ) . Let S ⊆ [ H κ ] ω be projective stationary. Let M = h H κ , ∈ ↾ H κ , . . . i be a first order structure of a finite language. Then there isa continuous elementary chain hM i | i < ω i of elementary submodels of M through S , meaning that, for all i < ω , |M i | ∈ S , M i ∈ |M i +1 | , M i ≺ M i +1 and if i isa limit ordinal, then |M i | = S j
SRP is that the canonicalforcing to shoot a continuous ∈ -chain of length ω through a projective stationaryset, described in the following definition, preserves stationary subsets of ω . Definition 2.8. P S is the forcing notion consisting of continuous ∈ -chains through S of countable successor length, ordered by end-extension. For p ∈ P S , I write p = h M pi | i ≤ ℓ p i .The following fact is essentially contained in Feng & Jech [11], even though it isnot explicitly stated. Fact 2.9. Let κ be an uncountable regular cardinal, and let S ⊆ [ H κ ] ω be stationary.Then(1) for every countable ordinal α , the set of conditions p with ℓ p ≥ α is densein P S ,(2) for every a ∈ H κ , the set of conditions p such that there is an i < ℓ p with a ∈ M pi is dense in P S ,(3) P S is countably distributive.Proof. We prove clauses (1) and (2) simultaneously. Let p ∈ P S , α < ω and a ∈ H κ be given. We use Lemma 1.2 of [11], which states that if T ⊆ [ H κ ] ω is stationary,then, for every countable ordinal i , there is a continuous ∈ -chain through T oflength at least i + 1. Let T = { M ∈ S | a, M pℓ p ∈ M } . Clearly, T is stationary, soby the lemma, let h N i | i ≤ β i be a continuous elementary chain through T , with β ≥ α . Let γ = ℓ p + 1 + β + 1, and define a condition q = h M qi | i < γ i by setting M qi = M pi for i ≤ ℓ p and M qℓ p +1+ j = N j for j ≤ β . Then q ≤ p has length at least α + 1 and eventually contains a , as wished.In order to prove clause (3), we have to show that, given a sequence ~D = h D n | n < ω i of dense open subsets of P S , the intersection ∆ = T n<ω D n is dense in P S .So, fixing a condition p ∈ P S , we have to find a q ≤ p in ∆. To this end, let λ be a regular cardinal much greater than κ , say λ > | P S | , and consider the model N = h H λ , ∈ , < ∗ , S, P S , ~D, p i , where < ∗ is a well-ordering of H λ . Let M ≺ N be acountable elementary submodel with |M| ∩ H κ ∈ S . Since M is countable, we canpick a filter G which is M -generic for P S and contains p . Let ¯ q = S G . Using the Since H κ is closed under ordered pairs, it is easy to code any countable language into a finitelanguage, and it will sometimes be convenient to assume that the language at hand is finite, so Iwill usually make that assumption. GUNTER FUCHS density facts proved in (1) and (2), it follows that δ := dom(¯ q ) = M ∩ ω , and that S i<δ ¯ q ( i ) = M ∩ κ ∈ S . Thus, if we define the sequence q of length δ + 1 by setting q ( i ) = ¯ q ( i ) for i < δ and q ( δ ) = M ∩ κ , then q ∈ P S , and q extends every conditionin G . Moreover, since D n ∈ M , for each n < ω , it follows that G meets each D n ,and hence that p ≥ q ∈ ∆, as desired. (cid:3) Fact 2.10 (Feng & Jech) . Let κ ≥ ω be an uncountable regular cardinal. Then astationary set S ⊆ [ H κ ] ω is projective stationary iff P S preserves stationary subsetsof ω . For the proof of this fact, see Feng & Jech [11] – one direction is given by theproof of Theorem 1.1, and the converse is outlined in the paragraph after the proof,on page 275.2.2. Relativizing to a forcing class.Definition 2.11. I write SSP for the class of all forcing notions that preservestationary subsets of ω .With hindsight, the results in the previous subsection show that the strongreflection principle can be formulated as follows. Whenever κ ≥ ω is regular, S ⊆ [ H κ ] ω is stationary, and theforcing P S to shoot a continuous elementary chain through S is in SSP , then S already contains a continuous ∈ -chain of length ω . The advantage of this formulation is that it generalizes easily to arbitrary forcingclasses. First, let me generalize the concept of projective stationarity. Definition 2.12. Let Γ be a forcing class. Then a stationary subset S of H κ ,where κ ≥ ω is regular, is Γ -projective stationary iff P S ∈ Γ.Thus, the Feng-Jech notion of projective stationarity is the same thing as SSP -projective stationarity. Generalizing the above formulation of SRP , we arrive at: Definition 2.13. Let Γ be a forcing class. Let κ ≥ ω be regular. The Γ -fragmentof the strong reflection principle at κ , denoted Γ- SRP ( κ ), states that whenever S ⊆ [ H κ ] ω is Γ-projective stationary, then S contains a continuous chain of length ω . The Γ -fragment of the strong reflection principle, Γ- SRP , states that Γ- SRP ( κ )holds for every κ ≥ ω .The idea is that the collection of the Γ-projective stationary sets captures exactlythose sets whose instance of the strong reflection principle follows from the forcingaxiom for Γ using the simplest possible argument, namely that P S is in Γ. Let memake this precise. First, by the forcing axiom for Γ, I mean the version of Martin’sAxiom MA ω for Γ rather than the collection of all ccc partial orders. Definition 2.14. Let Γ be a class of forcing notions. The forcing axiom for Γ,denoted FA (Γ), states that whenever P is a forcing notion in Γ and h D i | i < ω i isa sequence of dense subsets of P , there is a filter F ⊆ P such that for all i < ω , F ∩ D i = ∅ .It is now easy to check that Γ- SRP behaves as claimed in the introduction. Observation 2.15. Let Γ be a forcing class. Then FA (Γ) implies Γ - SRP . ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 7 Proof. Let κ ≥ ω be regular, and let S ⊆ [ H κ ] ω be Γ-projective stationary. Then P S ∈ Γ, and, for i < ω , we can let D i be the set of conditions in P S of length atleast i . By clause (1) of Fact 2.9, D i is a dense subset of P S . So by FA (Γ), there isa filter F meeting each D i . But then S F is a continuous ∈ -chain through S . (cid:3) The utility of SRP is, of course, that it encapsulates many of the consequences ofthe forcing axiom for stationary set preserving forcing without mentioning forcing.Thus, in order to arrive at a similarly useful version of it for other forcing classes,it will be crucial to express Γ-projective stationarity in a purely combinatorial waythat does not mention Γ explicitly.As an illustration, let’s look at two examples. Example . Let Proper be the class of all proper forcing notions, and let’s con-sider the notion of projective stationarity associated to that class. It is then nothard to see that: Observation . Let κ ≥ ω be regular. Then a stationary set S ⊆ [ H κ ] ω is Proper -projective stationary iff S contains a club.Proof. For the forward direction, suppose that S is Proper -projective stationary,that is, P S is proper. One of the many characterizations of properness is thepreservation of stationary subsets of [ X ] ω , for any uncountable X . Now P S shootsa club through S , and this means that the complement [ H κ ] ω \ S could not havebeen stationary in V, since its stationarity would be killed by P S . But this meansthat S contains a club.For the converse, suppose that S contains a club C ⊆ [ H κ ] ω . Let θ be sufficientlylarge, and let M be a countable elementary submodel of h H κ , ∈ , < ∗ i , with P S , C ∈ M . Let p ∈ P S ∩ M . Let N = M pℓ q . Since M believes that C is club in H κ , it is noweasy to construct an ∈ -chain h N i | i < ω i so that N ∈ N , N i ∈ M and S i<ω N i = M ∩ H κ . It is then routine to verify that the condition q = p ⌢ ~N ⌢ ( M ∩ H κ ) is( M, P S )-generic. (cid:3) Since any club contains a continuous ∈ -chain of length ω , the proper fragmentof SRP is thus provable in ZFC . Example . For an example going in the other extreme, let Semiproper be theclass of semiproper partial orders. In [12], a set S ⊆ [ κ ] ω is defined to be spanning if for every λ ≥ κ and every club C ⊆ [ λ ] ω , there is a club D ⊆ [ λ ] ω such that forevery x ∈ D , there is a y ∈ C such that x ⊆ y and x ∩ ω = y ∩ ω and y ∩ κ ∈ S . Itis shown in [12, Theorem 4.4] that S is spanning iff P S is semiproper, that is, usingour terminology, S is spanning iff it is Semiproper -projective stationary. However,[12, Cor. 5.4] can be expressed as saying that Semiproper - SRP implies SRP , so thesemiproper fragment of SRP is equivalent to the full principle SRP .2.3. The subcomplete fragment of SRP . In the previous subsection, we haveseen that the class of all proper forcing notions is too small to be of interest, in thesense that Proper - SRP is provable in ZFC , and the class of all semiproper forcingnotions is too large to be of interest, in the sense that Semiproper - SRP is equivalentto the full principle SRP , and so is nothing new. So let us now get ready to definewhen a forcing notion is subcomplete, so that we can turn to the subcompletefragment of SRP . GUNTER FUCHS Definition 2.19 (Jensen) . A transitive model N of ZFC − is full if there is anordinal γ > L γ ( N ) | = ZFC − and N is regular in L γ ( N ), meaning thatif a ∈ N , f : a −→ N and f ∈ L γ ( N ), then ran( f ) ∈ N . A possibly nontransitivebut well-founded model of ZFC − is full if its transitive isomorph is full.The notion of fullness is central to the theory of subcomplete forcing, and so, itseems worthwhile to elaborate on it a little bit, since it is somewhat subtle. Firstoff, when I say that N is a transitive model of ZFC − , I mean that N is a modelof a countable language which may extend the language of set theory, in whichthe symbol ˙ ∈ is interpreted as the actual ∈ relation, restricted to N , and that N satisfies the usual axioms of ZFC − , with respect to its language, that is, the formulasin the axiom schemes are allowed to contain the additional symbols available in thelanguage of N . There is a subtlety in the concept of fullness, then, since whether ornot a model N is full depends on the way it is represented. For a simple example,let’s assume that N is a countable full model of ZFC − in the language of set theory.Now let us consider N to be like N , except that N has a constant symbol c a forevery a ∈ N , so that c N a = a . Clearly, N is also a model of ZFC − , and N isalso full. Now let N be like N , but equipped with constant symbols d , d , . . . ,interpreted as d N n = f ( n ), where f : ω −→ N is a bijection. In a model-theoreticsense, N and N are essentially the same, it is just that their constant symbolsare different. However, N is full, while N is not, since in L γ ( N ), the function n d N n = f ( n ) is available, and hence the fact that N is countable is revealed.Thus, in order to make sense of the definition of fullness, one has to view the model N as the triple h| N | , L , I i , where | N | is the universe of N , L is the language of N in an explicitly given G¨odelization (since this example shows that it is importantwhat the symbols in the language are), and I is the function assigning each elementof L its interpretation in N . In the context of subcomplete forcing, the model N in question will always be a model of a language with just one additional predicatesymbol (which avoids the complications just mentioned). In fact, it will alwaysbe the result of constructing relative to some set. The notation I use for relativeconstructibility follows Jensen’s conventions: for a class A , define recursively: • L [ A ] = ∅ , L A = h∅ , ∅ , ∅i , • L α +1 [ A ] = Def ( L Aα ), L Aα +1 = h L α +1 [ A ] , ∈ ↾ L α +1 [ A ] , A ∩ L α +1 [ A ] i , • for limit λ , L λ [ A ] = S α<λ L α [ A ] and L Aλ = h L λ [ A ] , ∈ ↾ L λ [ A ] , A ∩ L λ [ A ] i . Definition 2.20. The density of a poset P , denoted δ ( P ), is the least cardinal δ such that there is a dense subset of P of size δ .I can now define Jensen’s notion of subcompleteness and its simplification, ∞ -subcompleteness, introduced in [24]. Definition 2.21. A forcing notion P is subcomplete if every sufficiently large car-dinal θ verifies the subcompleteness of P , which means that P ∈ H θ , and for any ZFC − model N = L Aτ with θ < τ and H θ ⊆ N , any σ : ¯ N ≺ N such that ¯ N is countable, transitive and full and such that P , θ ∈ ran( σ ), any ¯ G ⊆ ¯ P which is¯ P -generic over ¯ N , any ¯ s ∈ ¯ N , and any ordinals ¯ λ , . . . , ¯ λ n − such that ¯ λ = On ∩ ¯ N and ¯ λ , . . . , ¯ λ n − are regular in ¯ N and greater than δ (¯ P ) ¯ N , the following holds.Letting σ ( h ¯ θ, ¯ P i ) = h θ, P i , and setting ¯ S = h ¯ s, ¯ θ, ¯ P i , there is a condition p ∈ P such Here, in the case α = 0, L [ A ] is not technically a model, because its universe is empty, sowe have to set Def ( h∅ , ∅ , ∅i ) = {∅} to make literal sense of this definition. ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 9 that whenever G ⊆ P is P -generic over V with p ∈ G , there is in V[ G ] a σ ′ : ¯ N ≺ N such that(1) σ ′ ( ¯ S ) = σ ( ¯ S ),(2) ( σ ′ )“ ¯ G ⊆ G ,(3) sup σ “¯ λ i = sup σ ′ “¯ λ i for each i < n . P is ∞ -subcomplete iff the above holds with (3) removed.I denote the classes of subcomplete and ∞ -subcomplete forcing notions by SC and ∞ - SC , respectively.It should be pointed out that full models as in the previous definition are abun-dant. For example, suppose that H θ ⊆ L [ A ], where A ⊆ L β [ A ], and let τ < τ ′ be successive cardinals in L [ A ], say, with β < τ . Then whenever X ′ ≺ L Aτ ′ and X = X ′ ∩ L τ [ A ], it follows that L Aτ | X is full.The following easy fact can be used in order to further simplify the definitionsof subcompleteness/ ∞ -subcompleteness. Fact 2.22. Let L Aτ be a model of ZFC − , and let s ∈ L τ [ A ] . Then there is a B suchthat L τ [ B ] = L τ [ A ] , L Bτ | = ZFC − and such that s is definable (without parameters)in L Bτ . Moreover, B is definable in L Aτ and A ∩ L τ [ A ] is definable in L Bτ . Inparticular, L Aτ is full iff L Bτ is.Proof. First, by replacing A with A ∩ L τ [ A ] if necessary, we may assume that A ⊆ L τ [ A ]. Second, we may assume that A ⊆ τ . That is, we may construct a set A ′ ⊆ τ such that L τ [ A ] = L τ [ A ′ ], A is definable in L A ′ τ and A ′ is definable in L Aτ . Namely, L Aτ has a definable well-order of its universe, and since it is a model of ZFC − , themonotone enumeration of L τ [ A ] according to this well-order is definable in L Aτ , andits domain is τ . Let’s call it F : τ −→ L τ [ A ]. Let R = {h α, β i | F ( α ) ∈ F ( β ) } .Then F is the Mostowski-collapse of the structure h τ, R i . Now it is easy to encode R and A as a set of ordinals, using G¨odel pairs, for example, say A ′ = {≺ , α, β ≻ | F ( α ) ∈ F ( β ) } ∪ {≺ , α ≻ | F ( α ) ∈ A } . Since A ′ is a definable class in the ZFC − -model L Aτ , it follows that L A ′ τ = ( L A ′ ) L Aτ is also a model of ZFC − , and since A ′ codes F , it follows that L τ [ A ′ ] = L τ [ A ].Moreover, A is definable in L A ′ τ , by design.It is now easy to prove the fact: we can define B = {≺ , α ≻ | α ∈ A }∪{≺ , γ ≻} ,where s is the γ -th element of L τ [ A ] in the canonical well-order. (cid:3) Of course, if any one element of L τ [ A ] can be made definable by changing A as in the previous fact, then any finitely many elements can be made definable byapplying the same method to a finite sequence listing these elements. A consequenceof this fact, or rather, its proof, is that in Definition 2.21, condition (1) is vacuous,because if P satisfies this simplified definition, in the notation of that definition,one can modify A to A ′ in such a way that the desired parameters in ¯ S becomedefinable in N ′ = L A ′ τ . Letting ¯ N = L ¯ Aτ , and ¯ N ′ = L ¯ A ′ ¯ τ (where ¯ A ′ = σ − “ A isconstructed from ¯ A the same way that A ′ is constructed from A ), it then followsthat ¯ N ′ is full and σ : ¯ N ′ ≺ N ′ . Thus, since P satisfies the simplified version ofsubcompleteness, there is a condition in P forcing the existence of an elementaryembedding σ ′ : ¯ N ′ ≺ N such that σ ′ “ ¯ G ⊆ ˙ G , where ˙ G is the canonical name forthe generic filter. But then, σ ′ : ¯ N ≺ N as well, and σ ′ must move the desiredparameters the same way σ did, since they/their preimages are definable in N ′ / ¯ N ′ . This means, in particular, that only condition (2) is really needed in the definitionof ∞ -subcompleteness.The following definition is designed to capture the concept of ∞ - SC -projectivestationarity. If N is a model and X is a subset of | N | , the universe of N , then Iwrite N | X for restriction of N to X . Definition 2.23. Let κ be an uncountable regular cardinal. A stationary set S ⊆ [ H κ ] ω is spread out if for every sufficiently large cardinal θ , whenever τ , A , X and a are such that H θ ⊆ L Aτ = N | = ZFC − , S, a, θ ∈ X , N | X ≺ N , and N | X iscountable and full, then there are a Y such that N | Y ≺ N and an isomorphism π : N | X −→ N | Y such that π ( a ) = a and Y ∩ H κ ∈ S .Using Fact 2.22 as before, one can see that the definition of being spread outcan be simplified by dropping any reference to a , since any desired parameter, oreven any finite list of such parameters, can be made definable by modifying A while preserving fullness. So S ⊆ [ H κ ] ω is spread out if for all sufficiently large θ ,whenever H θ ⊆ L Aτ = N | = ZFC − , S ∈ X , N | X ≺ N , and N | X is countable andfull, then there is a Y ∈ S ↑ [ N ] ω such that N | X ∼ = N | Y ≺ N .Thus, in the situation of the previous definition, the stationarity of S guaranteesthe existence of some elementary submodel of N in S ↑ [ N ] ω , but if S is spread out,then every elementary submodel of N has an isomorphic copy in S ↑ [ N ] ω , as longas it is full.The following theorem is the analog of Fact 2.10 for ∞ -subcompleteness, pro-viding a combinatorial characterization of ∞ - SC -projective stationarity. Theorem 2.24. Let κ be an uncountable regular cardinal, and let S ⊆ [ H κ ] ω . Then S is spread out iff S is ∞ - SC -projective stationary.Proof. For the direction from left to right, suppose S is spread out. We have toshow that P S ∈ ∞ - SC . To that end, let θ be large enough for Definition 2.23 toapply. Let N = L Aτ | = ZFC − with H θ ⊆ N , and let P S ∈ X , N | X ≺ N , X countableand full. Let a be some member of X , and let σ : | ¯ N | −→ X be the inverse ofthe Mostowski collapse of X , | ¯ N | transitive, and σ : ¯ N ≺ N . Let ¯ P ¯ S = σ − ( P S ),¯ a = σ − ( a ), and let ¯ G ⊆ ¯ P ¯ S be ¯ N -generic. Note that since P S ∈ X , S, H κ ∈ X .Let ¯ κ = σ − ( κ ). It follows from Fact 2.9 that S ¯ G is of the form h ¯ M i | i < ω ¯ N i and S i<ω ¯ N ¯ M i = H ¯ N ¯ κ . Now, since S is spread out, let π : h X, ∈i −→ h Y, ∈i bean isomorphism that fixes a, P S , with Y ∩ H κ ∈ S . Let σ ′ = π ◦ σ : ¯ N ≺ Y . Let q = h σ ′ ( ¯ M i ) | i < ω ¯ N i ⌢ ( Y ∩ H κ ). Since Y ∩ H κ ∈ S , it follows that q ∈ P S , andwhenever G ∋ q is P S -generic over V, then σ ′ “ ¯ G ⊆ G . Since σ ′ (¯ a ) = a , this showsthat P S is ∞ -subcomplete.For the converse, suppose that S is ∞ - SC -projective stationary, that is, that P S is ∞ -subcomplete. Let θ witness that P S is ∞ -subcomplete. Let N = L Aτ , X , a be asin Definition 2.23. Since S ∈ X , it follows that κ, P S ∈ X as well. Let σ : ¯ N −→ X be the inverse of the Mostowski collapse of X . Thus, σ : ¯ N ≺ N , and as usual, let ¯ a ,¯ S , ¯ κ , ¯ P ¯ S denote the preimages of a , S , κ , P S under σ . Let ¯ G ⊆ ¯ P ¯ S be an arbitrary¯ N -generic filter. By ∞ -subcompleteness of P S , let p ∈ P S force the existence ofan elementary embedding σ ′ : ¯ N −→ N with σ ′ (¯ a ) = a and ( σ ′ )“ ¯ G ⊆ ˙ G ( ˙ G beingthe canonical P S -name for the generic filter). Since P S is countably distributive byFact 2.9, it follows that there is such a σ ′ ∈ V. Let Y = ran( σ ′ ), and let G be P S -generic over V with p ∈ G . Let δ = ω ¯ N = ω ∩ X . As before, S ¯ G is of the ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 11 form h ¯ M i | i < δ i . By Fact 2.9, we have that ¯ M ∗ = S i<δ ¯ M i = H ¯ N ¯ κ . For i < δ ,let M i = σ ′ ( ¯ M i ) - note that this is the same as ( σ ′ )“ ¯ M i , as ¯ M i is countable in ¯ N .Since G contains a condition of length δ + 1, letting M δ = S i<δ M i , the sequence q = h M i | i ≤ δ i is in G . It follows that M δ ∈ S . Moreover, M δ = [ i<δ σ ′ “ ¯ M i = σ ′ “ ¯ M ∗ = σ ′ “ H ¯ N ¯ κ = Y ∩ H Nκ and so, Y ∩ H κ ∈ S . Letting π = σ ′ ◦ σ − , one sees that π : X −→ Y is anisomorphism that fixes a , thus verifying that S is spread out. (cid:3) Having a characterization of ∞ - SC -projective stationarity of course gives a char-acterization of the ∞ - SC -fragement of SRP . Theorem 2.25. For an uncountable regular cardinal κ , the principle ∞ - SC - SRP ( κ ) holds iff every spread out subset of [ H κ ] ω contains a continuous ∈ -chain of length ω . It will often be useful to work with the following seemingly weaker form of thenotion “spread out.” It will turn out to be equivalent, but it will sometimes beeasier to verify. Definition 2.26. Let κ be an uncountable regular cardinal. A stationary set S ⊆ [ H κ ] ω is weakly spread out if there is a set b such that for all sufficiently large θ , the condition described in Definition 2.23 is true of all X with b ∈ X . Observation 2.27. Let κ be an uncountable regular cardinal. A stationary set S ⊆ [ H κ ] ω is spread out iff it is weakly spread out.Proof. Of course, if S is spread out, it is also weakly spread out. For the con-verse, suppose S is weakly spread out, as witnessed by the set b , say. Then theproof of Theorem 2.24 shows that P S satisfies the definition of ∞ -subcompleteness,Definition 2.21, under the extra condition that b ∈ ran( σ ), in the notation of thatdefinition. But this implies that P S is ∞ -subcomplete, see the arguments in Jensen[33, P. 115f., in particular Lemma 2.5]. But if P S is ∞ -subcomplete, then S isspread out, by Theorem 2.24. (cid:3) I would now like to make a few simple observations on the structure of the spreadout sets and their relation to other notions of largeness of subsets of [ H κ ] ω . First,of course, being spread out is a strengthening of projective stationarity. Observation 2.28. If a stationary set S ⊆ [ H κ ] ω is spread out, then S is projectivestationary.Proof. This is because ∞ - SC ⊆ SSP . (cid:3) In particular, spread out sets are stationary. In fact, being spread out is preservedby intersecting with a club; this is the analog of Fact 2.6. Observation 2.29. Let κ be an uncountable regular cardinal, let S ⊆ [ H κ ] ω bespread out, and let C ⊆ [ H κ ] ω be club. Then S ∩ C is spread out.Proof. Let f : H <ωκ −→ H κ be such that every a ∈ H κ closed under f is in C . ByObservation 2.27, it suffices to show that S ∩ C is weakly spread out. Thus, it willbe enough to show that the condition described in Definition 2.23 is satisfied forall sufficiently large θ , assuming that f ∈ X , using the notation in the definition. Since S is spread out, there is a Y such that Y ∩ H κ in S and such that there is anisomorphism π : h X, ∈ ∩ X i −→ h Y, ∈ ∩ Y i that fixes a given a , and also f , thatis, π ( h f, a i ) = h f, a i . Since f ∈ Y , it follows that Y is closed under f , and hencethat Y ∩ H κ ∈ S ∩ C . (cid:3) Thus, ∞ - SC - SRP guarantees the existence of elementary chains through spreadout sets. Thirdly, being spread out is preserved by projections, analogous to thesituation with stationarity and projective stationarity. Observation 2.30. Let A ⊆ B ⊆ C , and let S ⊆ [ B ] ω be spread out, with S S = B .Then:(1) S ↑ C is spread out.(2) S ↓ A is spread out.Proof. We prove (1) and (2) simultaneously. Let θ be sufficiently large, and let X ≺ N = L Dτ | = ZFC − be countable and full, with S, a, A, C ∈ N (as usual, wemay require some additional parameter to be in X ). Since S is spread out, thereare a Y with N | Y ≺ N and an isomorphism π : N | X −→ N | Y that fixes a , andsuch that Y ∩ B ∈ S . But this means that Y ∩ C ∈ S ↑ C and that Y ∩ A ∈ S ↓ A ,as wished. (cid:3) Observation 2.31. If S ⊆ [ H κ ] ω contains a club, then S is spread out.Proof. Let f : [ H κ ] <ω −→ H κ be such that every x ∈ C is closed under f , thatis, f “[ x ] <ω ⊆ x . Let θ be sufficiently large that [ H κ ] ω ∈ H θ . I claim that this θ ,together with the function f , witnesses that S is weakly spread out, which impliesthat S is spread out by Observation 2.27. To see this, suppose H θ ⊆ L Aτ | = ZFC − and S, f ∈ L τ [ A ]. Let X ⊆ L τ [ A ] be countable, N | X ≺ L Aτ , and f ∈ X (and N | X is full). Since f ∈ X , X is closed under f , and hence, so is X ∩ H κ . Thus, X ∩ H κ ∈ S (so we can choose Y = X in the notation of Definition 2.21). (cid:3) Finally, an elementary argument shows that in the situation of Definition 2.21,necessarily, σ ↾ (2 ω ) ¯ N = σ ′ ↾ (2 ω ) ¯ N , see [21, Obs. 4.2], or [18, Proof of Lemma 3.22].That argument, adapted to the present context, has the following consequence.Notice the parallel to Example 2.16. Observation 2.32. Let κ ≤ ω be an uncountable regular cardinal, and let S ⊆ [ H κ ] ω . Then S is spread out iff S contains a club. Hence, ∞ - SC - SRP ( κ ) holdstrivially.Proof. Suppose S ⊆ [ H κ ] ω is spread out. Let θ be sufficiently large, and let N = L Aτ | = ZFC − with H θ ⊆ L τ [ A ], and suppose τ is an L [ A ]-cardinal. Let τ ′ = ( τ + ) L [ A ] .Let N ′ = L Aτ ′ . It is then easy to see that whenever X is countable and N ′ | X ≺ N ′ ,then, letting ˜ X = X ∩ L τ [ A ], N | ˜ X ≺ N and N | ˜ X is full. Now let C = { X ∈ [ L ωτ ′ ] | N ′ | X ≺ N } . I claim that ¯ C = C ↓ [ H κ ] ω ⊆ S . To see this, let X ∈ ¯ C . Then X = Y ∩ H κ , forsome Y ∈ C . Let ˜ Y = Y ∩ L τ [ A ], so N | Y ≺ N is full. Since S is spread out, thereis a Z such that N | Z ≺ N so that N | Y is isomorphic to N | Z and Z ∩ H κ ∈ S . Let π : N | Y −→ N | Z be this isomorphism. First, observe that π ↾ ω = id. To see this,first note that π ↾ P ( ω ) = id, and hence that P ( ω ) ∩ Y = P ( ω ) ∩ Z . Let f be the < L Aτ -least bijection between P ( ω ) and 2 ω . Clearly, f ∈ Y ∩ Z , and π ( f ) = f . It ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 13 follows that π ↾ ω = id and hence that 2 ω ∩ Y = 2 ω ∩ Z , because if α < ω and α ∈ Y ,then letting a = f − ( α ), we have that π ( α ) = π ( f ( a )) = π ( f )( π ( a )) = f ( a ) = α .But then, it follows that π ↾ H ω = id, and hence that H ω ∩ Y = H ω ∩ Z . This isbecause if a ∈ H ω ∩ X , then a can be coded by a bounded subset A of 2 ω . Then π ( A ) = A codes the same object, and so, π ( a ) is the set coded by π ( A ) = A , whichis a . In particular, X = Y ∩ H κ = Z ∩ H κ ∈ S . Thus, ¯ C = C ↓ [ H κ ] ω ⊆ S , asclaimed. Since C is a club, C ↓ [ H κ ] ω contains a club, as wished.The converse is trivial, by Observation 2.31.This proves the equivalence claimed, and of course, it is easy to construct ancontinuous ∈ -chain of length ω through S if S contains a club, so that ∞ - SC - SRP ( κ ) holds. (cid:3) In ending this section, for completeness, let me mention an obvious modifica-tion to the concept of being spread out that corresponds to subcomplete (not ∞ -subcomplete) projective stationary, as follows. Definition 2.33. Let κ be an uncountable regular cardinal. A stationary set S ⊆ [ H κ ] ω is fully spread out if for all sufficiently large θ , whenever τ , A aresuch that H θ ⊆ L Aτ | = ZFC , S, θ ∈ X ≺ L Aτ is countable and full, if a ∈ X , λ , . . . , λ n are such that λ n = sup( X ∩ On) and for every i < n , λ i is a regularcardinal in the interval (2 <κ , λ n ), then there exist π , Y such that N | Y ≺ N and π : h X, ∈ ∩ X i −→ h Y, ∈ ∩ Y i is an isomorphism such that π ( a ) = a and for i ≤ n ,sup( X ∩ λ i ) = sup( Y ∩ λ i ), and Y ∩ H κ ∈ S .A repeat of the proof of Theorem 2.24 shows that being fully spread capturessubcomplete projective stationarity: Theorem 2.34. Let κ be an uncountable regular cardinal, and let S ⊆ [ H κ ] ω . Then S is fully spread out iff S is SC -projective stationary. From here on out, to make things slightly less technical, I will for the most partfocus on spread out sets. Everything I do would also go through for fully spreadout sets, unless I explicitly say otherwise.3. Consequences In order to carry over consequences of SRP to its subcomplete fragment, I willneed to know that certain sets are not only projective stationary, but in fact spreadout. Proving that a set is spread out is generally not an easy task, but fortunately,all I need will follow from one technical lemma, which I will prove in the followingsubsection. In later subsections, I will use this in order to derive consequences con-cerning Friedman’s problem, the failure of square, the singular cardinal hypothesisand mutual stationarity.3.1. Barwise theory and a technical lemma. The proof of the main technical,but very useful lemma will emply methods of Barwise, so I will summarize whatI will need very briefly. I follow Jensen’s excellent presentation of this material in[33, p. 102 ff]. For a more detailed treatment, see Barwise [4], or Jensen’s set ofhandwritten notes [32].Recall that a structure h M, A , . . . , A n i is admissible if it is transitive and satisfies KP (which I take to include the axiom of infinity), using the predicates A , . . . , A n . For admissible M , Barwise developed an infinitary logic where the infinitary formu-las are (coded by) elements of M . Thus, infinitary conjunctions and disjunctionsare allowed, as long as they are in M , but only finite strings of quantifiers may oc-cur, and all predicate symbols are finitary. Let A be a Σ ( M ) set of such infinitaryformulas. Thus, the set of formulas A itself can be defined by a finitary first orderformula over M that is Σ . The set A may well contain formulas that are not Σ ,so it is not a Σ -theory in the usual model theoretic sense. The intuition is thatelements of M behave like finite sets in finitary logic (and hence, they are called“ M -finite”), and Σ ( M ) sets behave like recursively enumerable ones. The logiccomes with a proof theory and a model theory whose main features are:(1) The M -finiteness lemma: if a formula ϕ is provable from A , then there isa u ∈ M such that u ⊆ A and ϕ is provable from u .(2) The correctness theorem: if there is a model A with A | = A , then A isconsistent.(3) The Barwise completeness theorem: if M is countable and A is consistent,then there is a model A with A | = A . Definition 3.1. Let M be admissible. If A consists of infinitary formulas in M ,then A is a theory on M . A is an ∈ -theory on M if the language it is formulatedin contains the symbol ∈ , a constant symbol x , for every x ∈ M , and if the theorycontains the extensionality axiom, as well as the basic axiom ∀ y ( y ∈ x ⇐⇒ _ z ∈ x y = z )for every x ∈ M . It is a ZFC − -theory on M if it is an ∈ -theory on M that containsthe ZFC − axioms (viewed as a set of finitary formulas, which are also in M ).If A is an ∈ -theory on M and A is a model for A whose well-founded part is tran-sitive, then automatically, x A = x , which is why I won’t specify the interpretationof these constants by such a model. The following property is a slight weakeningof fullness, see Definition 2.19. Definition 3.2. A transitive model N of ZFC − is almost full if there is a model A of ZFC − whose well-founded part is transitive, N is an element of the well-foundedpart of A and N is regular in A , i.e., if x ∈ N , f ∈ A , and f : x −→ N , thenran( f ) ∈ N .The same comments made after Definition 2.19 apply here as well. In applica-tions, the model N will be of the form L Aτ , so that no complications arise. Definition 3.3. If N is a transitive set, then I write α ( N ) for the least α > L α ( N ) | = KP .The next lemma will be used crucially in the proof of Lemma 3.5. Transfer Lemma 3.4 ([33, p. 123, Lemma 4.5]) . Let ¯ N and N be transitive ZFC − -models. Let ¯ N be almost full and σ : ¯ N −→ Σ N be cofinal, that is, N = S ran( σ ). Then N is almost full. Further, let ¯ L be a theory in an infinitarylanguage on L α ( ¯ N ) ( ¯ N ) that has a Σ -definition in L α ( ¯ N ) ( ¯ N ) in the parameters ¯ N and p , . . . , p n ∈ ¯ N . Let L be the infinitary theory on L α ( N ) ( N ) defined over L α ( N ) ( N ) by the same Σ -formula, using the parameters N , σ ( p ) , . . . , σ ( p n ). If ¯ L is consistent, then so is L . I will denote L by σ ( L ). ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 15 Coming up is the technical lemma I need, a general version of Jensen’s [33,Lemma 6.3]. The present lemma differs from Jensen’s version in several respects.First, the formulation is different. Jensen’s lemma states that if κ > ω is regularand A ⊆ κ is a stationary set consisting of ordinals of countable cofinality, then theusual forcing to shoot a club through A with countable conditions is subcomplete.The present version of the lemma implies this, under the additional assumptionthat κ > ω .More importantly, the original lemma assumes that κ > ω , while I need κ > ω .It was first observed by Sean Cox that there is a step in Jensen’s proof that seemsto only go through if κ > ω , and I thank him sincerely for pointing this out tome. The assumption is actually needed for the lemma, as I observe in the notefollowing the statement of the lemma, and it is also needed for the original lemma,as was originally noticed by Hiroshi Sakai (as communicated by Corey Switzer).The assumption is used in Claim (5) of the proof.On the other hand, while Jensen’s lemma was missing a needed assumption, theits proof made an assumption that is not needed, namely that A belongs to therange of the embedding σ . It will play a role later on, in Lemma 3.29, that this isunnecessary. In fact, the present version of the lemma does not mention A .In the article containing the original lemma, Jensen works with a variant of sub-completeness [33, Def. on p. 114] in which the “suprema condition” (3) of Definition2.21 is replaced with a “hull condition” which implies the suprema condition. Ac-cordingly, the proof of his lemma establishes a potentially stronger property than(b) and (e). But the proof of the present version of the lemma establishes thatcondition as well, as I point out in Remark 3.6, after the proof.Finally, Jensen’s lemma does not mention clause (c), and this clause will also beimportant in the aforementioned application.I will carry the proof out in considerable detail, because it is a subtle argumentin which it is easy to overlook problems (and this has happened in the past, as Iexpained). Lemma 3.5. Let κ > ω be a regular cardinal, θ > κ regular, N a transitivemodel of ZFC − (in a finite language) with a definable well-ordering of its universeand H θ ⊆ N , σ : ¯ N ≺ N , where ¯ N is countable and full and κ ∈ ran( σ ) . Let η ∈ κ ∩ ran( σ ) be such that η ω < κ . Let ¯ κ, ¯ η be the preimages of κ, η under σ ,respectively. Let h ¯ λ i | i < n i be regular cardinals in ¯ N , each greater than ¯ κ . Let ¯ a be some element of ¯ N .Then there is an ω -club of ordinals κ < κ (i.e., the set of such κ is unboundedin κ and closed under limits of countable cofinality) for which there is an embedding σ ′ : ¯ N ≺ N with the following properties: (a) Letting p = { ¯ a, ¯ κ, ¯ η, ¯ λ , . . . , ¯ λ n − } , we have that σ ↾ p = σ ′ ↾ p . (b) For i < n , sup σ “¯ λ i = sup σ ′ “¯ λ i . (c) σ ↾ ¯ η = σ ′ ↾ ¯ η . (d) sup σ ′ “¯ κ = κ . (e) sup σ “(On ∩ ¯ N ) = sup σ ′ “(On ∩ ¯ N ) .Note: (1) Instead of a single ¯ a , one can choose finitely many members of ¯ N , say¯ a , . . . , ¯ a n − , and find a σ ′ as in the lemma that moves each of these el-ements the same way σ does, because one can apply the lemma to thesequence h ¯ a , . . . , ¯ a n − i .(2) The assumption that κ > ω is necessary, because if σ ′ : ¯ N ≺ N , then σ ′ ↾ ((2 ω ) ¯ N + 1) = σ ↾ ((2 ω ) ¯ N + 1). Clearly, σ ((2 ω ) ¯ N ) = 2 ω = σ ′ ((2 ω ) ¯ N ).And if f is the N -least bijection from P ( ω ) to 2 ω , and ¯ f is the ¯ N -leastbijection from ( P ( ω )) ¯ N to (2 ω ) ¯ N , then σ ( ¯ f ) = f = σ ′ ( ¯ f ). It follows thatfor any ordinal ξ < (2 ω ) ¯ N , letting x = ¯ f − ( ξ ), then σ ( ξ ) = σ ( ¯ f ( x )) = σ ( ¯ f )( σ ( x )) = σ ′ ( ¯ f )( σ ′ ( x )) = σ ′ ( ¯ f ( x )) = σ ′ ( ξ ). Proof. Let me define a = σ (¯ a ), λ i = σ (¯ λ i ), for i < n , and η = σ (¯ η ).Since N has a definable well-order of its universe, for every subset X of theuniverse of N , there is a minimal (with respect to inclusion) subset Y of the universeof N that contains X such that N | Y ≺ N . I denote this Y by Hull N ( X ), and I willuse this notation for other models that have a definable well-order as well. Clearly,the set C = { α < κ | Hull N ( α ∪ ran( σ )) ∩ κ = α } is club in κ . I claim that whenever κ ∈ C has countable cofinality, then there is a σ ′ as described, proving the lemma.For the purpose of the proof, let me define that for transitive models ¯ M and M of ZFC − , an embedding j : ¯ M ≺ M is cofinal if for every x ∈ M there is a y ∈ ¯ M such that x ∈ j ( y ). If ¯ δ is a cardinal in ¯ M , then j : ¯ M ≺ M is ¯ δ -cofinal if for every x ∈ M there is a y ∈ ¯ M of ¯ M -cardinality less than ¯ δ with x ∈ j ( y ). I will use thefollowing facts several times throughout the proof.(A) If j : ¯ M ≺ M , ¯ δ is a cardinal in ¯ M , δ = j (¯ δ ) and M = Hull M (ran( j ) ∪ δ ),then j is (¯ δ + ) ¯ M -cofinal.(B) If j : ¯ M ≺ M is ¯ δ -cofinal, then j is continuous at every ¯ λ ∈ ¯ M withcf ¯ M (¯ λ ) ≥ ¯ δ , that is, j (¯ λ ) = sup j “¯ λ . Proof of (A) & (B).. To see (A), let x ∈ M be given. By assumption, x is definablein M from some ordinal α < δ and some elements a , . . . , a n − of ran( j ). Let’s say x is the unique z such that M | = ϕ ( z, α, ~a ). Let ¯ a , . . . , ¯ a n − be the preimages of a , . . . , a n − under j , and consider the function ¯ f : ¯ δ −→ ¯ M defined in ¯ M by letting¯ f (¯ α ) be the unique z such that ϕ ( z, ¯ α, ~ ¯ a ) holds, if such a z exists, and let ¯ f (¯ α ) = 0otherwise. Clearly, if we set y = ran( ¯ f ), then x ∈ j ( y ), and y has ¯ M -cardinality atmost ¯ δ .To see (B), fix a ¯ λ as stated. To show that j (¯ λ ) = sup j “¯ λ , note that the righthand side is obviously less than or equal to the left hand side. For the converse,suppose α < j (¯ λ ). By ¯ δ -cofinality, let y ∈ ¯ N have ¯ N -cardinality less than ¯ δ , with α ∈ j ( y ). We may assume that y ⊆ ¯ λ (by intersecting it with ¯ λ if necessary). Butthen y is bounded in ¯ λ , since the ¯ N -cofinality of ¯ λ is at least ¯ δ , say y is boundedby ξ < ¯ λ . Then α ∈ j ( y ) ⊆ j ( ξ ) < sup j “¯ λ . (cid:3) Now, let me fix κ ∈ C with cf( κ ) = ω , let N be the transitive collapse ofHull N ( κ ∪ ran( σ )), and let k be the inverse of the collapse. We have: k : N ≺ N, crit( k ) = κ , k ( κ ) = κ. Note that since η, ω ∈ ran( σ ) ∩ κ , it follows that κ > η, ω . ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 17 ¯ N ¯ η ¯ κ ¯ λ i N ω ηκ κλ i N ηλ i, k σσ Figure 1. The first interpolation.Since ran( σ ) ⊆ ran( k ), there is anelementary embedding σ : ¯ N ≺ N defined by σ = k − ◦ σ . Then we have • σ : ¯ N ≺ N , • σ (¯ κ ) = κ , • k ◦ σ = σ .Let me define a = σ (¯ a ), η = σ (¯ η )and λ i, = σ (¯ λ i ), for i < n .By (A), it follows that(1) σ : ¯ N ≺ N is a (¯ κ + ) ¯ N -cofinalembedding.Figure 1 on the right summarizes thesituation so far. The circles indicatethe cofinal image, that is, the function α sup f “ α . An arrow with superim-posed circles indicates that the functionat hand is continuous at this point, thatis, that the point is mapped to its cofi-nal image by the function.Next, I will define an intermediatemodel in between ¯ N and N . The con-struction is similar to forming an ex-tender ultrapower of ¯ N by an extenderderived from σ ↾ ¯ κ .More precisely, N is the Mostowskicollapse of the set H = { σ ( ¯ f )( α ) | ∃ β < ¯ κ ¯ f : β −→ ¯ N, ¯ f ∈ ¯ N , α < σ ( β ) } and, letting k be the inverse of the collapsing isomorphism, σ = k − ◦ σ . It isnot hard to check that H ≺ N , so that this makes sense. Then: σ : ¯ N ≺ N , k : N ≺ N , σ = k ◦ σ . Let me set a = σ (¯ a ), η = σ (¯ η ), κ = σ (¯ κ ) and λ i, = σ (¯ λ i ), for i < n . Thesituation is illustrated in Figure 2It is easy to see from the definition of N that(2) σ : ¯ N ≺ N is ¯ κ -cofinal.Namely, given y ∈ N , it follows that k ( y ) is of the form σ ( ¯ f )( α ), for somefunction ¯ f : β −→ ¯ N in ¯ N , where β < ¯ κ and α < σ ( β ). Thus, letting x = ran( ¯ f ),we have that x has cardinality less than ¯ κ in ¯ N , and k ( y ) ∈ σ ( x ) = k ( σ ( x )),and so, pulling back via k − , we have that y ∈ σ ( x ).By (B), we have:(3) For any ¯ λ ∈ ¯ N of ¯ N -cofinality at least ¯ κ , it follows that sup σ “¯ λ = σ (¯ λ ).It is also clear that(4) k ↾ κ = id. As a result, σ ↾ ¯ κ = σ ↾ ¯ κ . This is because κ ⊆ H : suppose α < κ . Then by (3), α < σ (¯ κ ) = sup σ “¯ κ .So let β < ¯ κ be such that α < σ ( β ) ≤ σ ( β ). Then α = σ (id ↾ β )( α ) ∈ H .¯ N ¯ η ¯ κ ¯ λ i N ω ηκ κ κλ i N λ i, k σ N λ i, σ σ k Figure 2. The second interpolation.The next step in the proof is crucial.It is where the assumption that κ > ω is used, and it is the verification of thenext claim that is missing in Jensen’s[33, Lemma 6.3].(5) ¯ N ∈ N .To see this, let us view ¯ N as a sub-set of ω temporarily; it can easily becoded that way. Note that by elemen-tarity, P ( ω ) ∈ N , and also P ( ω ) ∈ N .The cardinality of P ( ω ) is 2 ω , whichis less than κ , by assumption. Thus,since κ ∈ C , 2 ω < κ as well. Sincecrit( k ) = κ , it follows that (2 ω ) N =2 ω . By elementarity, there is in N a bijection g : 2 ω −→ P ( ω ). Sincecrit( k ) = κ > ω , it follows that k ( g ) = g , and hence that g is actu-ally a bijection between 2 ω and P ( ω ).Thus, P ( ω ) ⊆ N , since 2 ω ⊆ κ ⊆ N .Moreover, P ( ω ) ⊆ H . This is becauseif we let ¯ c = (2 ω ) ¯ N , then ¯ c < ¯ κ , and so,letting ¯ g : ¯ c −→ P ( ω ) ¯ N be a bijection,it follows that σ (¯ g ) : σ (¯ c ) −→ P ( ω )is a bijection and for every ξ < σ (¯ c ), σ (¯ g )( ξ ) ∈ H . Every subset of ω isof this form. Thus, ¯ N ∈ H , and so,¯ N = k − ( ¯ N ) ∈ N .Notice that clause (c) of the lemma can be equivalently expressed as σ “¯ η = σ ′ “¯ η .This is why the following point will be relevant.(6) σ “¯ η = σ “¯ η ∈ N .The reasoning is much like the argument for the previous claim. First, though,since k ↾ κ = id, it follows that σ “¯ η = σ “¯ η , as ¯ η < ¯ κ , and it also follows that η = η = η , since κ ≤ k ( κ ) = κ and k ↾ κ = id.Clearly, η < κ , and since η ω < κ , it also follows that η ω < κ , as κ ∈ C . Itfollows as before that [ η ] ω ⊆ Hull N (ran( σ ) ∪ κ ), and since crit( k ) = κ , it followsthat [ η ] ω ⊆ N . Further, since in ¯ N , ¯ η ω < ¯ κ , it follows as before that [ η ] ω ⊆ H .Hence, k − “[ η ] ω = [ η ] ω ⊆ N . In particular, σ “¯ η ∈ N .Now let α ( N ) be the least β > L β ( N ) is admissible, and let N +1 = L α ( N ) ( N ). By Claims (5) and (6), ¯ N and σ “¯ η are elements of N , whichallows us to define the ZFC − -theory L on N +1 that has an extra constant symbol˙ σ and the following additional axioms: • ˙ σ : ¯ N ≺ N ¯ κ -cofinally. ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 19 • ˙ σ (¯ κ, ¯ a, ¯ η, ¯ λ , . . . , ¯ λ n − ) = κ , a , η , λ , , . . . , λ n − , . • ˙ σ “¯ η = σ “¯ η .It is crucial here that ¯ N , σ “¯ η ∈ N , so that the first and third item of the theorymake sense.Clearly, this theory is consistent, since h H κ , ∈ , σ i is a model, for example.Since σ = k ◦ σ : ¯ N ≺ N is cofinal, by Claim (1), it follows that k : N ≺ N is cofinal as well, and hence, we know by the Transfer Lemma 3.4, which isapplicable since ¯ N is full, that the theory L =“ k ( L )” on N +0 = L α ( N ) ( N )is also consistent. In more detail, L is the ZFC − -theory on N +0 with the extraconstant symbol ˙ σ and the additional axioms • ˙ σ : ¯ N ≺ N ¯ κ -cofinally. • ˙ σ (¯ κ, ¯ a, ¯ η, ¯ λ , . . . , ¯ λ n − ) = κ , a , η , λ , , . . . , λ n − , . • ˙ σ “¯ η = σ “¯ η .Notice here that k ( ¯ N ) = ¯ N , since by elementarity, N sees that ¯ N is coded bya real, and that real is not moved by k . And k ( σ “¯ η ) = σ “¯ η = σ “¯ η since k ↾ κ = id and κ > σ (¯ η ) = η .In the last step of the proof, I would like to use Barwise completeness, ideally tofind an elementary embedding as described in L . But Barwise completeness onlyapplies to countable theories, so the idea is to put all the relevant information insidea sufficiently rich model, and take a countable elementary substructure. Thus, let M = h H κ , ∈ , N , κ , σ , a , η , ~λ i . Let π : ˜ M ≺ M be such that ˜ M is countableand transitive. Note that ¯ N is definable from σ and N , and so are ¯ κ, ¯ η and ~ ¯ λ ,and so, all of these objects are in the range of π . Note further that π − ( ¯ N ) = ¯ N ,since ¯ N is coded by a real number in M , and moreover, π ↾ ¯ N = id. I will write ˜ x for π − ( x ) when x ∈ ran( π ). So ˜ σ : ¯ N ≺ ˜ N , and it follows that σ = π ◦ ˜ σ , becausefor x ∈ ¯ N , σ ( x ) = π (˜ σ )( x ) = π (˜ σ )( π ( x )) = π (˜ σ ( x )).Clearly, L is definable in M , and hence in the range of π . Its preimage is˜ L = π − ( L ). ˜ L is then a consistent language on the structure ˜ N + = π − ( N +0 ).Since this structure is countable (in V), it has a model, say A , whose well-foundedpart may be chosen to be transitive. Let ˜ σ ′ = ˙ σ A . Note that we do not know that˜ σ ′ ∈ ˜ M . Set σ ′ = k ◦ π ◦ ˜ σ ′ . I claim that σ ′ has the desired properties. To see this, it will be useful to write outwhat the axioms in ˜ L express: • the basic axioms and ZFC − . • ˙ σ : ¯ N ≺ ˜ N ¯ κ -cofinally. • ˙ σ (¯ κ, ¯ a, ¯ η, ¯ λ , . . . , ¯ λ n − ) = ˜ κ , ˜ a , ˜ η , ˜ λ , , . . . , ˜ λ n − , . • ˙ σ “¯ η = ˜ σ “¯ η .Since A is a model of this theory, we have that • ˜ σ ′ : ¯ N ≺ ˜ N is ¯ κ -cofinal, • ˜ σ ′ (¯ κ, ¯ a, ¯ η, ¯ λ , . . . , ¯ λ n − ) = ˜ κ , ˜ a , ˜ η , ˜ λ , , . . . , ˜ λ n − , , and • (˜ σ ′ )“¯ η = ˜ σ “¯ η .Composing with π , and writing ˆ σ = π ◦ ˜ σ ′ , this translates to: • ˆ σ : ¯ N ≺ N , • ˆ σ (¯ κ, ¯ a, ¯ η, ¯ λ , . . . , ¯ λ n − ) = κ , a , η , λ , , . . . , λ n − , , and • ˆ σ “¯ η = σ “¯ η . Remembering that k ↾ η = id, composing with k results in: • σ ′ : ¯ N ≺ N , • σ ′ (¯ κ, ¯ a, ¯ η, ¯ λ , . . . , ¯ λ n − ) = κ, a, η, λ , . . . , λ n − , and • ( σ ′ )“¯ η = σ “¯ η .In particular, clauses (a) and (c) of the lemma are satisfied. For the remainingclauses, it will be useful to analyze ˆ σ in more detail. The fact that ˜ σ ′ : ¯ N ≺ ˜ N is¯ κ -cofinal gives some more information about this embedding. It is this argumentin which I will make use of the fact that κ has countable cofinality.(7) ˆ σ : ¯ N ≺ N is ¯ κ -cofinal.To see this, let a ∈ N . Since σ : ¯ N ≺ N is ¯ κ + -cofinal, there is a b ∈ ¯ N of¯ N -cardinality ¯ κ with a ∈ σ ( b ). Let f : ¯ κ −→ b be surjective, f ∈ ¯ N , and let ξ < κ be such that a = σ ( f )( ξ ). Since M sees that κ has countable cofinality,˜ M sees that ˜ κ has countable cofinality, and it follows that π “˜ κ is unbounded in κ . So let β < ˜ κ be such that ξ < π ( β ). Let b ′ = ˜ σ ( f )“ β , so that a ∈ π ( β ′ ), since π ( b ′ ) = σ ( f )“ π ( β ) ∋ σ ( f )( ξ ) = a . Since ˜ σ ′ : ¯ N ≺ ˜ N is ¯ κ -cofinal, there is a c ∈ ¯ N of ¯ N -cardinality less than ¯ κ and such that b ′ ∈ ˜ σ ′ ( c ). Since the ˜ N -cardinality of b ′ is less than ˜ κ , we may assume that every element of c has size less than ¯ κ in¯ N , by shrinking c if necessary. Now, since ¯ κ is regular in ¯ N , it follows that S c has¯ N -cardinality less than ¯ κ , and a ∈ π ( b ′ ) ⊆ π (˜ σ ′ ( S c )) = ˆ σ ( S c ).(8) If cf ¯ N (¯ λ ) ≥ ¯ κ , then ˆ σ (¯ λ ) = sup ˆ σ “¯ λ , and if cf ¯ N (¯ λ ) > ¯ κ , then σ (¯ λ ) =sup σ “¯ λ .This follows from the previous claim, (1) and (B).It is now obvious that clause (d) holds, that is, that sup σ ′ “¯ κ = κ . This isbecause sup σ ′ “¯ κ = sup k “ˆ σ “¯ κ = sup k “ˆ σ (¯ κ ) = sup k “ κ = κ .The next claim shows that σ ′ satisfies clause (b) of the lemma.(9) For i < n , sup σ “¯ λ i = sup ˆ σ “¯ λ i and sup σ ′ “¯ λ i = sup σ “¯ λ i .The first part follows from the previous claim, becausesup σ “¯ λ i = σ (¯ λ i ) = λ i, = ˆ σ (¯ λ i ) = sup ˆ σ “¯ λ i . The second part follows from the first, since σ ′ = k ◦ ˆ σ and σ = k ◦ σ .Finally, let me check that clause (e) is satisfied, that is, that sup σ “(On ∩ ¯ N ) =sup σ ′ “(On ∩ ¯ N ). This is easy to see: Both σ and ˆ σ cofinal, and hence,sup σ “(On ∩ ¯ N ) = sup k “ sup σ “(On ∩ ¯ N ) = sup k “(On ∩ N )= sup k “ sup ˆ σ “(On ∩ ¯ N ) = sup σ ′ “(On ∩ ¯ N ) . (cid:3) I will not use the following remark, but I would like to state and prove it anyway,since in some of Jensen’s writings, he defines subcompleteness by requiring that,in the notation of Definition 2.21, the embedding σ ′ satisfy the “hull condition”that Hull N (ran( σ ) ∪ δ ) = Hull N (ran( σ ′ ) ∪ δ ), where δ = δ ( P ) is the density of theforcing in question, instead of the “suprema condition,” that is, condition (3) ofthat Definition. Remark . In the notation of the previous lemma, the embedding σ ′ can beguaranteed to have the property thatHull N (ran( σ ) ∪ κ ) = Hull N (ran( σ ′ ) ∪ κ ) . ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 21 Proof. The embedding constructed in the proof has this property. To see this, letme freely use notation from the proof. By construction, N = Hull N (ran( σ ) ∪ κ ),and so, it suffices to show that Hull N (ran(ˆ σ ) ∪ κ ) = N as well, since σ / σ ′ resultfrom composing σ /ˆ σ with k . But I showed that ˆ σ : ¯ N ≺ N is ¯ κ -cofinal, whichimmediately implies this. (cid:3) It is maybe worth mentioning that, still in the notation of Lemma 3.5, if σ ′ has the property stated in the previous remark, and actually it is enough thatHull N (ran( σ ) ∪ κ ) = Hull N (ran( σ ′ ) ∪ κ ), which is a weaker condition, then for any¯ λ > ¯ κ of ¯ N -cofinality greater than ¯ κ , if σ (¯ λ ) = σ ′ (¯ λ ), then sup σ “¯ λ = sup σ ′ “¯ λ . Sothis condition is a strong form of clause (b) of the lemma. For a proof, see [16, Fact1.6].3.2. Friedman’s problem, the failure of square, and SCH . In Section 2, Irelativized the strong reflection principle to a forcing class Γ by saying that everyΓ-projective stationary set contains a continuous elementary chain of length ω . Iwould now like to derive some consequences of this Γ-fragment of SRP , and thereason why these consequences arise is that certain sets are Γ-projective station-ary. In order to be able to keep track of the sets that are responsible for theseconsequences, it will be useful to name them. Definition 3.7. For an uncountable regular cardinal κ , let S lifting ( κ ) = { lifting ( A, [ H κ ] ω ) ∩ C | A ⊆ S κω is stationary in κ and C ⊆ [ H κ ] ω is club } . Given a collection S of stationary subsets of H θ , the S -fragment of the strongreflection principle , SRP ( S ), asserts that if S ∈ S , then there is a continuous ∈ -chain of length ω through S .So S lifting ( κ ) consists of all the liftings of stationary subsets of S κω to [ H κ ] ω , andtheir intersections with clubs; see Definition 2.4. The reason why I isolated theclass S lifting is that SRP ( S lifting ( κ )) has some interesting consequences hinted at inthe title of the present subsection, and follows from Γ- SRP , for the classes Γ ofinterest. The following fact has been known for a long time. Fact 3.8 (Feng & Jech [11, Example 2.2]) . Let κ > ω be a regular cardinal, andlet A ⊆ S κω be stationary. Then the set S = { X ∈ [ H κ ] ω | sup( X ∩ κ ) ∈ A } = lifting ( A, [ H κ ] ω ) is projective stationary. By Observation 2.6, this can be restated by saying that for regular κ > ω , everyset in S lifting ( κ ) is projective stationary. With Lemma 3.5 at my disposal, I am nowready to prove the corresponding fact about spread out sets. But I do need that κ > ω . Lemma 3.9. Let κ > ω be regular. Then S lifting ( κ ) consists of spread out sets.Proof. By Observation 2.29, it suffices to show that if B ⊆ S κω is stationary, then S = { X ∈ [ H κ ] ω | sup( X ∩ κ ) ∈ B } is spread out. To this end, let X ≺ N = L Aτ , σ : ¯ N −→ X the inverse of the collapse of X , S ∈ X , ¯ N countable and full, H θ ⊆ N ,where θ is sufficiently large. Let a = σ (¯ a ) be fixed, and assume that κ = σ (¯ κ ) ∈ ran( σ ). By Lemma 3.5, there is an ω -club of ordinals κ less than κ such that thereis a σ ′ : ¯ N ≺ N with σ ′ (¯ a ) = a , σ ′ (¯ κ ) = κ and sup σ ′ “¯ κ = κ . Since B ⊆ S κω is stationary, there is such a κ ∈ B . Let σ ′ be the corresponding embedding, and set Y = ran( σ ′ ). Since Y ∩ κ = σ ′ “¯ κ , we have that sup( Y ∩ κ ) = sup σ ′ “¯ κ = κ ∈ B ,so Y ∩ H κ ∈ S . Thus, since π = σ ′ ◦ σ − : X −→ Y is an isomorphism fixing a , S is spread out. (cid:3) Clauses (b) and (e) of Lemma 3.5 can be used to show that in the situation ofthe previous lemma, S lifting ( κ ) actually consists of fully spread out sets.The following theorem explains the import of S lifting ( κ ). Of course, only the lastpart mentioning ∞ - SC - SRP is new. For the definition of FP κ , see Definition 2.1. Theorem 3.10. Let κ > ω be regular.(1) SRP ( S lifting ( κ )) implies FP κ .(2) SRP ( κ ) implies SRP ( S lifting ( κ )) and hence FP κ .(3) If κ > ω , then ∞ - SC - SRP ( κ ) implies SRP ( S lifting ( κ )) and hence FP κ .Proof. For (1), let A ⊆ S κω be stationary. Then the set S = { X ∈ [ H κ ] ω | sup( X ∩ κ ) ∈ A } is in S lifting ( κ ). By SRP ( S lifting ( κ )), let h M i | i < ω i be a continuous elementarychain through S . Define f : ω −→ A by f ( i ) = sup M i ∩ κ . Then f is a normalfunction, verifying the instance of FP κ given by A .Now (2) follows from Fact 3.8 and (1), and (3) from Lemma 3.9 and (1). (cid:3) In item (3) of the previous theorem, ∞ - SC - SRP ( κ ) can be replaced with SC - SRP ( κ ), since the relevant sets are fully spread out, as pointed out before.It is well-known that for a cardinal κ , FP κ + implies the failure of Jensen’s prin-ciple (cid:3) κ , so as a consequence of the previous theorem, one obtains: Corollary 3.11. ∞ - SC - SRP implies that for every cardinal κ ≥ ω , (cid:3) κ fails. Again, SC - SRP is sufficient here. The following terminology expands on [17]. Definition 3.12. Let κ be an uncountable regular cardinal.The strong Friedman Property at κ , denoted SFP κ , says that for any partition h D i | i < ω i of ω into stationary sets, and for any sequence h S i | i < ω i of sta-tionary subsets of S κω , there is a normal function f : ω −→ S i<ω S i such that forevery i < ω , f “ D i ⊆ S i .This is a somewhat technical concept, but I will extract from it something morenatural. The following expands on [26, Def. 8.17]. Definition 3.13. Let κ be a cardinal of uncountable cofinality, and let S ⊆ κ bestationary. Then an ordinal δ < κ is a reflection point of S if δ has uncountablecofinality and S ∩ δ is stationary in δ . It is an exact reflection point of S if S ∩ δ contains a club in δ .If ~S = h S i | i < λ i is a sequence of stationary subsets of κ , then an ordinal δ < κ is a simultaneous reflection point of ~S if for every i < λ , δ is a reflection point of S i . It is an exact simultaneous reflection point of ~S if it is a simultaneous reflectionpoint of ~S and δ ∩ ( S i<λ S i ) contains a club in δ .The trace of S , denoted Tr ( S ) is the set of reflection points of S , and similarly,the trace of ~S , denoted Tr ( ~S ), is the set of simultaneous reflection points of ~S .Similarly, the exact trace of S , denoted eTr ( S ), is the set of exact reflection pointsof S , and eTr ( ~S ) is the set of exact reflection points of ~S . ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 23 Clearly, FP κ implies not only that every stationary subset A of S κω has a reflectionpoint, but that it has an exact reflection point. SFP κ has a similar effect on ω -sequences of stationary subsets of S κω , as the following observation shows - notethat it obviously implies (a) if S = S κω , and hence each of the equivalent conditionsstated. In fact, the observation shows that (a) is maybe a more natural version of SFP κ . Observation 3.14. Let κ > ω be regular and fix a stationary subset S of κ . Thefollowing are equivalent: (a) Whenever ~S = h S i | i < ω i is a sequence stationary subsets of S , there area partition h D i | i < ω i of ω into stationary sets and a normal function f : ω −→ κ such that for all i < ω , f “ D i ⊆ S i . (b) Whenever ~S = h S i | i < ω i is a sequence of stationary subsets of S , then eTr ( ~S ) = ∅ . (c) Whenever ~S = h S i | i < ω i is a sequence of stationary subsets of S , then eTr ( ~S ) is stationary.Proof. (a) = ⇒ (c): Let ~S be as in (c), and let C ⊆ κ be club. Then let ~D , f beas in (a) with respect to the sequence h S i ∩ C | i < ω i . Letting δ = sup ran( f ), itfollows that δ ∈ C ∩ eTr ( ~S ).(c) = ⇒ (b): trivial.(b) = ⇒ (a): Let ~S = h S i | i < ω i be a sequence of stationary subsets of S κω . Let ~S ′ = h S ′ i | i < ω i be a refinement of ~S into a sequence of pairwise disjoint stationarysets. Such a sequence ~S ′ exists because κ > ω , so that the nonstationary idealon κ is “nowhere ω -saturated”, see Baumgartner-Hajnal-M´at´e [5, Lemma 2.1] fordetails. By (b), let δ be an exact reflection point of ~S ′ . Let C ⊆ ( δ ∩ S i<ω S i )be club, and let f : ω −→ C be the monotone enumeration of C . For i < ω , let D i = f − “ S ′ i . Then h D i | i < ω i is a partition of ω into stationary sets such thatfor every i < ω , f “ D i ⊆ S ′ i ⊆ S i . (cid:3) The following fact is usually stated assuming some variation of SFP κ in place ofmy assumption, but it can be filtered through the “exact” reflection property of theprevious observation. See Foreman-Magidor-Shelah [14], or Jech [26, p. 686, proofof Theorem 37.13]. Note that if δ is an exact reflection point of a sequence ~S , then δ is a reflection point of each S i , but if T is a stationary subset of κ that’s disjointfrom S ~S , then δ is not a reflection point of T , and this is why I refer to it as an exact reflection point. This property is used in the proof of the following fact. Fact 3.15. Let κ > ω be a regular cardinal, and let S be a stationary subset of κ such that any ω -sequence of stationary subsets of S has an exact simultaneousreflection point. Then κ ω = κ .Proof. Fix a sequence h S i | i < κ i of pairwise disjoint stationary subsets of S . If x ∈ [ κ ] ω and δ is an exact simultaneous reflection point for ~S ↾ x , then x = R δ = { i < κ | δ is a reflection point of S i } . Thus, [ κ ] ω ⊆ { R δ | δ < κ } . (cid:3) The following is due to Feng & Jech [11, Example 2.3]. Lemma 3.16. Let κ > ω be regular. Let ~D = h D i | i < ω i be a partition of ω into stationary sets which is maximal in the sense that for every stationary T ⊆ ω , there is an i < ω such that D i ∩ T is stationary. Let ~S = h S i | i < ω i be a sequenceof stationary subsets of S κω . Then the set S = { X ∈ [ H κ ] ω | ∀ i < ω ( X ∩ ω ∈ D i → sup( X ∩ κ ) ∈ S i ) } . is projective stationary. The status of the maximality assumption on ~D here is interesting. First, let menote: Remark . Maximal partitions ~D of ω into stationary sets as in the previ-ous lemma are easy to construct: start with an arbitrary partition ~D ′ of ω intostationary sets. It can be identified with the function f ′ : ω −→ ω defined by i ∈ D ′ f ′ ( i ) . Modify f ′ to the regressive function f : ω −→ ω defined by f ( i ) = f ′ ( i )if f ′ ( i ) < i , and f ( i ) = 0 otherwise. The corresponding partition h D i | i < ω i with D i = { α | f ( α ) = i } is then as wished: each D i is stationary, because it contains D ′ i \ ( i + 1), and if T ⊆ ω is stationary, then f ↾ T is constant on a stationary subsetof T , say with value i , so T ∩ D i is stationary.Following is the version of Feng & Jech’s Lemma 3.16, with “spread out” inplace of “projective stationary.” I have to strengthen the assumption that κ > ω to κ > ω , but I can drop the maximality assumption on the partition ~D . Lemma 3.18. Let κ > ω be regular. Let ~D = h D i | i < ω i be a partition of ω into stationary sets and ~S = h S i | i < ω i a sequence of stationary subsets of S κω .Let S = { X ∈ [ H κ ] ω | ∀ i < ω ( X ∩ ω ∈ D i → sup( X ∩ κ ) ∈ S i ) } . Then S is spread out.Proof. Let θ be sufficiently large, H θ ⊆ L Aτ = N | = ZFC − , X ≺ N countable andfull with S ∈ X , and fix a ∈ X . Let i < ω be such that δ = X ∩ ω ∈ D i . We cannow use Lemma 3.5 as in the proof of Lemma 3.9, showing that there are a Y ≺ N and an isomorphism π ′ : X −→ Y fixing a such that κ = sup( Y ∩ κ ) ∈ S i . Since thisisomorphism has to fix the countable ordinals, it follows that Y ∩ ω = X ∩ ω ∈ S i ,and hence that Y ∩ H κ ∈ S . (cid:3) Again, using properties (b) and (e) of Lemma 3.5, one obtains that in the pre-vious lemma, S is fully spread out.Since spread out sets are also projective stationary (see Observation 2.28), theprevious lemma shows that if κ > ω , then it is not necessary to assume ~D ismaximal in Lemma 3.16. This seems to be new, and I am not sure how one wouldprove this without using something along the lines of Lemma 3.5. Theorem 3.19. Let κ > ω be regular. Then ∞ - SC - SRP ( κ ) implies SFP κ .Proof. Let ~D , ~S be as in Definition 3.12. By Lemma 3.18, the set S = { X ∈ [ H κ ] ω | ∀ i < ω ( X ∩ ω ∈ D i → sup( X ∩ κ ) ∈ S i ) } is ∞ - SC -projective stationary. By ∞ - SC - SRP , let h M i | i < ω i be a continuous ∈ -chain through S . Let C = { α < ω | M α ∩ ω = α } . Clearly, C is closed andunbounded in ω . Define f : C −→ κ by f ( i ) = sup( M i ∩ κ ). Then f is strictlyincreasing and continuous in the sense that if α is a countable limit point of C ,then f ( α ) = sup β ∈ C ∩ α f ( β ). Moreover, for j ∈ C , if j ∈ D i , then f ( j ) ∈ S i , since ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 25 j = M j ∩ ω ∈ D i and hence sup( M j ∩ κ ) ∈ S i , as M j ∈ S . So f is almost likethe function postulated to exist by SFP κ , except that it is only defined on C , aclub subset of ω , rather than on all of ω . This form of SFP κ is enough for theapplications, so let me just sketch how to obtain the full version from this.All that needs to be done is fill in the gaps. Let h ζ i | i < ω i be the monotoneenumeration of C . Fixing i < ω , consider the forcing notion P consisting of allfunctions h : ( ζ i , α ] −→ κ , continuous on their domains, such that (1), ζ i < α < ω ,(2), for all ξ ∈ dom( h ), if j is such that ξ ∈ D j , then h ( ξ ) ∈ S j , and (3), if ζ i + 1 ∈ dom( h ), then h ( ζ i + 1) > f ( ζ i ). This forcing is stationary set preserving,and for every α ∈ ( ζ i , ω ), the set D α of conditions in P whose domain contains α isdense in P . The latter property is what I need here, and an argument establishingit can be found in [26, p. 686]. Now let G i be M ζ i +1 -generic for P (we may assumethat each element of the chain is an elementary submodel of h H κ , ∈ , ~D, ~S i , so that P belongs to every model in the chain), and let g i = S G i . It then follows thatdom( g i ) = [ ζ i + 1 , ζ i +1 ), g i ( ζ i + 1) > f ( ζ i ), sup g i “ ζ i +1 = sup( M i +1 ∩ κ ) = f ( ζ i +1 ),and for all α ∈ ( ζ i , ζ i +1 ), if α ∈ D j , then g i ( α ) ∈ S j . Hence, if we define f ′ = f ∪ S i<ω g i , then f ′ is as desired. (cid:3) Again, SC - SRP ( κ ) is sufficient in this theorem. It has been well-known for a longtime that if κ > ω is regular, then SRP ( κ ) implies the version of SFP κ in which itis assumed that the partition ~D used (see Definition 3.12) is maximal in the sense ofLemma 3.16. The previous theorem shows that already the subcomplete fragmentof SRP ( κ ) implies the full SFP κ principle, provided that κ > ω . In this corollary, SC - SRP is enough. Corollary 3.20. ∞ - SC - SRP implies that for regular κ > ω , κ ω = κ .Proof. This is by the previous theorem, Observation 3.14 and Fact 3.15. (cid:3) Results of [10] can be used to derive further consequences of Theorem 3.19 interms of the failure of weak square principles. I will not go into the details here,but I would like to keep track of the fragment of SRP responsible for the latestconsequences mentioned. First, one could ask: Question 3.21. Assume κ > ω is regular. Does SRP ( S lifting ( κ )) imply SFP κ ?On the positive side, one can list the sets used to derive SFP κ . Definition 3.22. Given an uncountable regular cardinal κ , a partition ~D of ω into stationary sets and an ω -sequence ~S of stationary subsets of S κω , call the pair h ~D, ~S i a κ -correspondence, and define the lifting of such a correspondence to any X with κ ⊆ X by lifting ( h ~D, ~S i , [ X ] ω ) = { x ∈ [ X ] ω | ∀ i < ω ( x ∩ ω ∈ D i −→ sup( x ∩ κ ) ∈ S i ) } and then define the class of liftings of correspondences by letting S corr ( κ ) = { lifting ( h ~D, ~S i , [ H κ ] ω ) ∩ C | κ > ω , h ~D, ~S i is a κ -correspondenceand C ⊆ [ H κ ] ω is club } . Clearly then, for κ > ω , S corr ( κ ) consists of spread out sets, by Lemma 3.18,and SRP ( S corr ( κ )) implies SFP κ , by the proof of Theorem 3.19. In fact, fixing onepartition ~D as in Definition 3.22 would suffice. Corollary 3.23. ∞ - SC - SRP implies SCH . Actually, SRP ( S corr ( κ )) , for all κ > ω ,suffices.Proof. We have to show that if λ is a singular cardinal with 2 cf( λ ) < λ , then λ cf( λ ) = λ + . By Silver’s Theorem (see [26, Theorem 8.13]), it suffices to prove thisin the case that λ has countable cofinality. In this case, we have that λ > ω . Since SRP ( S ∞ - SC ) holds, it follows from Corollary 3.20 that ( λ + ) ω = λ + . Thus we have λ + ≤ λ cf( λ ) = λ ω ≤ ( λ + ) ω = λ + as wished. (cid:3) Of course, SC - SRP is sufficient in this corollary.3.3. Mutual stationarity. The ideas of the previous subsection can be carrieda little further. Let me recall the notion of mutual stationarity, introduced byForeman & Magidor [13]. Definition 3.24. Let K be a collection of regular cardinals with supremum δ , andlet ~S = h S κ | κ ∈ K i be a sequence such that for every κ ∈ K , S κ is a subset of κ .Then ~S is mutually stationary if for every algebra A on δ , there is an N ≺ A suchthat for all κ ∈ N ∩ K , sup( N ∩ κ ) ∈ S κ .It is easy to see that if ~S is mutually stationary, then for all κ ∈ K , S κ is astationary subset of κ . The following beautiful and fundamental fact on mutualstationarity was proved in the article in which the concept was introduced and givesa condition under which the converse is also true. Fact 3.25 (Foreman & Magidor [13, Thm. 7]) . Let K be a set of uncountable regularcardinals, and let ~S = h S κ | κ ∈ K i be a sequence such that for every κ ∈ K , S κ isa stationary subset of S κω . Then, ~S is mutually stationary: for any algebra A on sup K , there is a countable N ≺ A such that for all κ ∈ N ∩ K , sup( N ∩ κ ) ∈ S κ . It seems as though the following connection has not been made before: Corollary 3.26. Let K be a set of regular cardinals with min( K ) > ω , and let ~S = h S κ | κ ∈ K i be a sequence such that for every κ ∈ K , S κ ⊆ S κω is stationaryin κ . Let δ = sup K . Then the set S = { M ∈ [ H δ ] ω | ∀ κ ∈ M ∩ K sup( M ∩ κ ) ∈ S κ } is projective stationary.Proof. This is because if A ⊆ ω is stationary, and if we let K ′ = K ∪ { ω } and ~S ′ = ~S ∪ {h ω , A i} , then we can apply Fact 3.25 to these objects, showing that theset S A = { M ∈ S | M ∩ ω ∈ A } is a stationary subset of [ H δ ] ω . (cid:3) This can be readily improved as follows by adding in a “correspondence” as be-fore, but this time between a partition of ω into stationary sets and a correspondinglist of ω -sequences of stationary sets. ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 27 Corollary 3.27. Let K be a set of regular cardinals with min( K ) > ω , and let ~S = h S κ,i | κ ∈ K, i < ω i be a sequence such that for every κ ∈ K and every i < ω , S κ,i ⊆ S κω is stationary in κ . Let δ = sup K . Let h D i | i < ω i be amaximal partition of ω into stationary sets (in the sense of Lemma 3.16.) Thenthe set S = { M ∈ [ H δ ] ω | ∀ κ ∈ M ∩ K ∀ i < ω ( M ∩ ω ∈ D i = ⇒ sup( M ∩ κ ) ∈ S κ,i ) } is projective stationary.Proof. Let B ⊆ ω be stationary, and let i < ω be such that D i ∩ B is stationary.Let K ′ = K ∪ { ω } , and consider the sequence h S ′ κ | κ ∈ K ′ i defined by letting S ′ κ = S κ,i for κ ∈ K and S ′ ω = D i ∩ B . By Fact 3.25, the set ¯ S = { M ∈ [ H δ ] ω | ∀ κ ∈ M ∩ K ′ sup( M ∩ κ ) ∈ S ′ κ } is stationary. But ω ∈ M , for a club C of M , and ¯ S ∩ C ⊆ S B = { M ∈ S | M ∩ ω ∈ B } , showing that S B is stationary. (cid:3) This connection to mutual stationarity gives rise to a somewhat “diagonal” re-flection principle for sequences of stationary sets of ordinals which follows from SRP , in contrast to others that I will discuss in Section 4. It is a kind of simultane-ous reflection principle for sequences of stationary sets that live on different regularcardinals. To motivate it, recall that Observation 3.14 shows that a weak version of SFP κ that follows from SRP for regular κ > ω implies that whenever { S i | i < ω } is a collection of stationary subsets of S κω , then the set of exact simultaneous re-flection points of this collection, is stationary in κ . The following theorem saysthat if we are given such collections of stationary sets, living on different regularcardinals, then the sequence consisting of the sets of the exact reflection points ofthese different collections is mutually stationary. Theorem 3.28. Assume SRP . Let K be a set of regular cardinals with min( K ) >ω . Let ~S = h S κ,i | κ ∈ K, i < ω i be such that for every κ ∈ K and i < ω , S κ,i is a subset of S κω stationary in κ . For κ ∈ K , let ~S κ = h S κ,i | i < ω i . Then thesequence ~T = h eTr ( ~S κ ) | κ ∈ K i is mutually stationary.Proof. Let δ = (sup( K )) + . Fix a partition h A i | i < ω i of ω into stationary setswhich is a maximal antichain. Let S be the set of countable M ≺ H δ such that if M ∩ ω ∈ A i , then for all κ ∈ M ∩ K , sup( M ∩ κ ) ∈ S κ,i . By Corollary 3.27, thisset is projective stationary. By SRP , let h M α | α < ω i be a continuous elementarychain through S . Let M = S α<ω M α . Then M ≺ H δ . We claim that M verifiesthat ~T is mutually stationary. To see this, suppose κ ∈ M ∩ K . We have to showthat for every i < ω , s κ = sup( M ∩ κ ) ∈ eTr ( ~S κ ). That is, we have to show thatfor every i < ω , s κ is a reflection point of S κ,i , and that S i<ω S κ,i ∩ s κ containsa club.For the first part, fix any countable ordinal i . To see that S κ,i ∩ s κ is stationary,let D ⊆ s κ be club. We have to show that S κ,i ∩ D = ∅ . Let β < ω be leastsuch that κ ∈ M β , and define, for j ∈ [ β, ω ), ξ j = sup( M j ∩ κ ). Then ~ξ is strictlyincreasing, continuous, and cofinal in s κ . That is, C = { ξ j | β ≤ j < ω } is club in s κ . Since cf( s κ ) = ω , C ∩ D is club in s κ , and hence, ¯ D = { j ∈ [ β, ω ) | ξ j ∈ D } is club in ω . Similarly, the sequence h M j ∩ ω | j < ω i is strictly increasing andcontinuous, and so the set E = { j < ω | j = M j ∩ ω } is club in ω . Now since T i is stationary in ω , we can pick α ∈ T i ∩ ¯ D ∩ E . Then ξ α ∈ D , since α ∈ ¯ D ,and α = M α ∩ ω ∈ T i . Since M α ∈ S and α ≥ β , that is, κ ∈ M α , it follows that ξ α = sup( M α ∩ κ ) ∈ S κ,i . Thus, ξ α ∈ S κ,i ∩ D , as wished.For the second part, note that the club C defined in the previous paragraph iscontained in S i<ω S κ,i (cid:3) I will show in the following that the same conclusion can be drawn from ∞ - SC - SRP under the additional assumption that CH holds. Essentially, this amountsto showing a version of Corollary 3.26 with “spread out” in place of “projectivestationary”. To this end, I will use the following strengthening of Lemma 3.5. Theargument will be a construction that proceeds in ω many steps, each of which willbe an application of Lemma 3.5. Lemma 3.29. Let K be a set of regular cardinals such that min( K ) > ω and suchthat whenever κ < λ , κ, λ ∈ K , then κ ω < λ . Let ~S = h S κ | κ ∈ K i be such that forevery κ ∈ K , S κ ⊆ S κω is stationary in κ . Let θ > sup( K ) be regular, N a transitivemodel of ZFC − that has a definable well-order, with H θ ⊆ N . Let σ : ¯ N ≺ N , where ¯ N is countable and full and K ∈ ran( σ ) . Let ¯ K be the preimage of K , and let ¯ a besome element of ¯ N .Then there is an embedding σ ′ : ¯ N ≺ N with the following properties: (a) σ (¯ a ) = σ ′ (¯ a ) and σ ′ ( ¯ K ) = K , (b) for every ¯ κ ∈ ¯ K , sup σ ′ “¯ κ ∈ S σ ′ (¯ κ ) .Proof. We may assume that K is infinite. Let h ¯ κ n | n < ω i enumerate ¯ K . Let usalso fix an enumeration h ¯ a n | n < ω i of ¯ N . Also, letting ρ = sup( K ), let us fix, forevery α ∈ S ρω , an increasing and cofinal function f α : ω −→ α .We will construct sequences h σ ′ n | n < ω i , h κ n | n < ω i , h ˜ κ n | n < ω i and h β nm,ℓ | m ≤ ℓ ≤ n < ω i by simultaneous recursion on n , satisfying the following properties,for every n < ω :(i) σ ′ n : ¯ N ≺ N .(ii) Let ˜ κ n = sup σ ′ n “¯ κ n and κ n = σ ′ n (¯ κ n ). Then ˜ κ n ∈ S κ n .(iii) σ ′ n ↾ { ¯ K, ¯ a } = σ ↾ { ¯ K, ¯ a } .(iv) For m ≤ n , σ ′ n (¯ a m ) = σ ′ n +1 (¯ a m ).(v) For m < n , σ ′ m (¯ κ m ) = σ ′ n (¯ κ m ) and sup σ ′ m “¯ κ m = sup σ ′ n “¯ κ m .(vi) For m ≤ ℓ ≤ n , let β nm,ℓ < ¯ κ m be the least ordinal β such that σ ′ n ( β ) >f ˜ κ m ( ℓ ). Then σ ′ n ( β nm,ℓ ) = σ ′ n +1 ( β nm,ℓ ). Moreover, for all k < n and all m ≤ ℓ ≤ k , σ ′ n +1 ( β km,ℓ ) = σ ′ n ( β km,ℓ ).To get started, let κ = σ (¯ κ ) and apply Lemma 3.5 to κ and S κ . We don’tneed the full strength of the lemma in step 0 of the construction, but just conditions(a) and (d) (so we can let η = 0). This gives us a σ ′ : ¯ N ≺ N that moves ¯ K , ¯ a , ¯ κ the same way σ does, and such that letting ˜ κ = sup σ ′ “¯ κ , we have that ˜ κ ∈ S κ ,since there is an ω -club in κ of possibilities for ˜ κ and S κ is a stationary subsetof κ consisting of ordinals of countable cofinality. Thus, conditions (i)-(iii) aresatisfied at n = 0, and the remaining conditions are vacuous at n = 0. Define β , as in condition (vi). This is possible because σ ′ “¯ κ is cofinal in ˜ κ .Now let us assume that h σ ′ m | m ≤ n i have been constructed, and h κ m | m ≤ n i , h ˜ κ m | m ≤ n i and h β km,ℓ | m ≤ ℓ ≤ k ≤ n i have been defined accordingly, so thatall of the conditions are satisfied at each m ≤ n . Let κ n +1 = σ ′ n (¯ κ n +1 ). Let η = max( { } ∪ ( { κ m | m ≤ n } ∩ κ n +1 )). Since κ n +1 ∈ K , η < κ n +1 and either ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 29 η = 0 or η ∈ K , it follows from our assumption on K that η ω < κ n +1 . Moreover, η ∈ ran( σ ′ n ), so we can let ¯ η be the preimage of η under σ ′ n . Let ¯ λ , . . . , ¯ λ r enumeratethe finite set { ¯ κ m | m ≤ n ∧ ¯ κ m > ¯ κ n +1 } . Let p = { ¯ K, ¯ a } ∪ { ¯ a m | m ≤ n } ∪ { ¯ κ m | m ≤ n } ∪ { β km,ℓ | m ≤ ℓ ≤ k ≤ n } Now apply Lemma 3.5 to σ ′ n : ¯ N ≺ N , S κ n +1 , κ n +1 , η , ~ ¯ λ and p . This gives us anembedding σ ′ n +1 : ¯ N ≺ N such that σ ′ n +1 ↾ p = σ ′ n ↾ p , sup σ ′ n +1 “¯ λ i = sup σ ′ n “¯ λ i for i < r , σ ′ n +1 ↾ ¯ η = σ ′ n ↾ ¯ η and, letting ˜ κ n +1 = sup σ ′ n +1 “¯ κ n and κ n +1 = σ ′ n +1 (¯ κ n +1 ),we have that ˜ κ n +1 ∈ S κ n +1 .Let us check that the conditions are satisfied at n + 1. This is immediate forconditions (i), (ii), (iii) and (iv). To check condition (v), let m < n + 1. Wehave to show that σ ′ m (¯ κ m ) = σ ′ n +1 (¯ κ m ) and sup σ ′ m “¯ κ m = sup σ ′ n +1 “¯ κ m . Sinceinductively, the conditions are satisfied at n , we know that σ ′ m (¯ κ m ) = σ ′ n (¯ κ m ) andsup σ ′ m “¯ κ m = sup σ ′ n “¯ κ m . Hence, it suffices to show that σ ′ n (¯ κ m ) = σ ′ n +1 (¯ κ m ) andsup σ ′ n “¯ κ m = sup σ ′ n +1 “¯ κ m . The first part of this is clear, because we put ¯ κ m into p , and σ ′ n ↾ p = σ ′ n +1 ↾ p . For the second part, we consider two cases. If ¯ κ m < ¯ κ n +1 ,then ¯ κ m ≤ ¯ η , and σ ′ n +1 ↾ ¯ η = σ ′ n ↾ ¯ η , so in particular, sup σ ′ n “¯ κ m = sup σ ′ n +1 “¯ κ m . If¯ κ m > ¯ κ n +1 , then ¯ κ m = ¯ λ i , for some i < r , and hence, we have the desired equalitybecause sup σ ′ n +1 “¯ λ i = sup σ ′ n “¯ λ i . Finally, regarding Condition (vi), observe thatsince σ “ n +1 ¯ κ n +1 is cofinal in ˜ κ n +1 , β n +1 m,ℓ is well-defined for m ≤ ℓ ≤ n + 1. Theremainder of this condition is again clear because σ ′ n +1 ↾ p = σ ′ n ↾ p and the relevantordinals are in p .This finishes the recursive construction h σ ′ n | n < ω i , h κ n | n < ω i , h ˜ κ n | n < ω i and h β nm,ℓ | m ≤ ℓ ≤ n < ω i .Observe that for every ¯ x ∈ ¯ N , h σ ′ n (¯ x ) | n < ω i is eventually constant, so we candefine σ ′ : ¯ N −→ N by letting σ ′ (¯ x ) be the eventual value of this sequence, in otherwords, σ ′ (¯ a n ) = σ ′ n (¯ a n )for every n < ω . We claim that σ ′ is the desired embedding.First, note that σ ′ : ¯ N ≺ N , because if ϕ ( ~x ) is a formula in the language of N with j < ω free variables and ¯ a n , . . . , ¯ a n j − are parameters in ¯ N , then if we choose n ≥ max { n , . . . , n j − } , we have that σ ′ ( ~ ¯ a ) = σ ′ n ( ~ ¯ a ), and so¯ N | = ϕ ( ~ ¯ a ) ⇐⇒ N | = ϕ ( σ ′ n ( ~ ¯ a )) ⇐⇒ N | = ϕ ( σ ′ ( ~ ¯ a ))since σ ′ n is an elementary embedding. Obviously, we have that σ ′ (¯ κ n ) = κ n , forevery n < ω , and σ ′ (¯ a ) = σ (¯ a ). Thus, condition (a) is satisfied. Let us checkthe remaining condition (b). Let ¯ κ ∈ ¯ K , say ¯ κ = ¯ κ n . We have to show thatsup σ ′ “¯ κ n ∈ S σ ′ (¯ κ n ) . Since σ ′ (¯ κ n ) = κ n and ˜ κ n ∈ S κ n , it will suffice to show thatsup σ ′ “¯ κ n = ˜ κ n . Clearly, σ ′ “¯ κ n ⊆ ˜ κ n , since for every ξ < ¯ κ n , there is a k < ω with k ≥ n such that σ ′ ( ξ ) = σ ′ k ( ξ ), but by condition (v), sup σ ′ k “¯ κ n = sup σ ′ n “¯ κ n = ˜ κ n ,so σ ′ ( ξ ) < ˜ κ n . Thus, σ ′ “¯ κ n ≤ ˜ κ n . For the other inequality, we show that σ ′ “¯ κ n isunbounded in ˜ κ n . Since ran( f ˜ κ n ) is unbounded in ˜ κ n , it suffices to show that forevery ℓ < ω , there is a β < ¯ κ n such that σ ′ ( β ) > f ˜ κ n ( ℓ ). We may clearly assumethat ℓ ≥ n . Let k < ω , k ≥ max { n, ℓ } . Consider β = β kn,ℓ . By definition, β < ¯ κ n and σ ′ k ( β ) > f ˜ κ n ( ℓ ). Moreover, by the same condition, we have that σ ′ k ′ ( β ) = σ ′ k ( β )whenever k < k ′ < ω . Hence, σ ′ ( β ) = σ ′ k ( β ) > f ˜ κ n ( ℓ ), as claimed. (cid:3) We thus obtain the following version of Foreman & Magidor’s mutual stationarityFact 3.25, or rather, the equivalent Corollary 3.26. Corollary 3.30. Let K be a set of regular cardinals such that min( K ) > ω andsuch that whenever κ < λ , κ, λ ∈ K , then κ ω < λ . Let ~S = h S κ | κ ∈ K i be suchthat for every κ ∈ K , S κ ⊆ S κω is stationary in κ . Let ρ ≥ sup( K ) . Then the set S = { M ∈ [ H ρ ] ω | ∀ κ ∈ M ∩ K sup( M ∩ κ ) ∈ S κ ) } is spread out. Instead of providing a proof of this, let me build in a correspondence as before,and prove the following more general statement. This is the version of Corollary3.27. Corollary 3.31. Let K be a set of regular cardinals such that min( K ) > ω andsuch that whenever κ < λ , κ, λ ∈ K , then κ ω < λ . Let ~S = h S κ,i | κ ∈ K, i < ω i besuch that for every κ ∈ K and every i < ω , S κ,i ⊆ S κω is stationary in κ . Let h D i | i < ω i be a partition of ω into stationary sets, and let ρ ≥ sup( K ) be regular.Then the set S = { M ∈ [ H ρ ] ω | ∀ κ ∈ M ∩ K ∀ i < ω ( M ∩ ω ∈ D i = ⇒ sup( M ∩ κ ) ∈ S κ,i ) } is spread out.Proof. This is an immediate application of Lemma 3.29. Namely, to show that S is spread out, let X ≺ L Aθ as usual, and let σ : ¯ N −→ X be the inverse ofthe Mostowski collapse of X , where we assume that ¯ N is full. As usual, we mayassume that X contains the parameters we care about; in this case we choose K . Let¯ K = σ − ( K ). Fix some a ∈ X , and let ¯ a = σ − ( a ). Let δ = X ∩ ω = ω ¯ N , and let i < ω be such that δ ∈ D i . Applying Lemma 3.29 to the sequence h S κ,i | κ ∈ K i ,there is a σ ′ : ¯ N ≺ N with σ ′ ( ¯ K ) = K , and σ ′ (¯ a ) = σ (¯ a ), such that for every¯ κ ∈ ¯ K , sup σ ′ “¯ κ ∈ S σ ′ (¯ κ ) ,i . Now let Y = ran( σ ′ ). Then π = σ ′ ◦ σ − : X −→ Y isan isomorphism fixing σ (¯ a ), and Y ∩ H ρ ∈ S : since neither σ nor σ ′ move countableordinals, we have that Y ∩ ω = X ∩ ω = δ . Now let κ ∈ Y ∩ K . Let ¯ κ = σ ′− ( κ ).Since σ ′ ( ¯ K ) = K , we have that ¯ κ ∈ ¯ K . Let ˜ κ = sup σ ′ “¯ κ . Then σ ′ “¯ κ = Y ∩ κ , andso, ˜ κ = sup( Y ∩ κ ) ∈ S κ,i , as wished. (cid:3) It is easy to see that the construction in the proof of Lemma 3.29 can be modifiedso as to obtain the version of the previous corollary in which “spread out” is replacedwith “fully spread out.” The corollary can be made to be closer to Corollary3.27 by adding the assumption of ∞ - SC - SRP , because then the cardinal arithmeticrequirements on K are automatically satisfied: Corollary 3.32. Assume ∞ - SC - SRP . Let K be a set of regular cardinals such that min( K ) > ω . Let ~S = h S κ,i | κ ∈ K, i < ω i be such that for every κ ∈ K andevery i < ω , S κ,i ⊆ S κω is stationary in κ . Let h D i | i < ω i be a partition of ω into stationary sets, and let ρ ≥ sup( K ) be regular. Then the set S = { M ∈ [ H ρ ] ω | ∀ κ ∈ M ∩ K ∀ i < ω ( M ∩ ω ∈ D i = ⇒ sup( M ∩ κ ) ∈ S κ,i ) } is spread out.Proof. The point is that under ∞ - SC - SRP , we have that for any regular cardinal κ > ω , κ ω = κ , by Theorem 3.19 and Fact 3.15. Thus, Corollary 3.31 applies,completing the proof. (cid:3) ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 31 Thus, under ∞ - SC - SRP + CH , an even stronger version of Corollary 3.27 holdswith “spread out,” and even “fully spread out,” in place of “projective stationary,”since it does not assume the partition of ω into stationary sets to be maximal.Here is the promised version of Theorem 3.28 for ∞ - SC - SRP , and even SC - SRP : Theorem 3.33. Assume ∞ - SC - SRP . Let K be a set of regular cardinals with min( K ) > ω . Then the conclusions of Theorem 3.28 hold: let ~S = h S κ,i | κ ∈ K, i < ω i be such that for every κ ∈ K and i < ω , S κ,i is a subset of S κω stationary in κ . For κ ∈ K , let ~S κ = h S κ,i | i < ω i . Then the sequence ~T = h eTr ( ~S κ ) | κ ∈ K i is mutually stationary.Thus, under the additional assumption of CH , we get the full conclusion of The-orem 3.28 from ∞ - SC - SRP .Proof. The point is that under ∞ - SC - SRP , we have that for any regular cardinal κ > ω , κ ω = κ , by Theorem 3.19 and Fact 3.15. Thus, the assumptions on K inCorollary 3.31 are satisfied. The theorem now follows by an argument exactly asin the proof of Theorem 3.28. (cid:3) The following result of Jensen [30] fits in here well. I recast it as a statementabout mutual stationarity. Theorem 3.34 (Jensen) . Assume that SCFA holds, and GCH holds below λ , anuncountable cardinal. Let K ⊆ λ be a set of regular cardinals greater than ω , andlet f : K −→ { ω, ω } . Then the sequence h S κf ( κ ) | κ ∈ K i is mutually stationary. The proof of this theorem uses the subcompleteness of an intricate forcing no-tion, developed in [29], that changes the cofinality of some regular cardinals to becountable, while preserving that others have uncountable cofinality. Clearly, thisforcing is not countably distributive. It seems unlikely that in this theorem, SCFA can be replaced with SC - SRP , as the forcing notions of the form P S are countablydistributive. This question is a fitting segue into the next section, in which I willexplore stationary reflection principles that do not follow from fragments of SRP ,and consequences of SCFA that don’t follow from SC - SRP , thus separating theseassumptions. 4. Limitations and separations I will now explore limitations on the extent to which the subcomplete fragmentof SRP implies certain principles of stationary reflection, and I will develop someresults going in the direction of separating the subcomplete fragment of SRP from SCFA . In the first subsection, I will focus on achieving such results in a generalsetting, but the results strongly suggest that the addition of CH should be made.Consequently, the last two subsections deal with this scenario.4.1. The general setting. Here, I will explore a framework for obtaining limitingresults, following the approach of Larson [35], originally in the setting of the full SRP . The idea is to start in a model of set theory V in which an indestructibleversion of SRP holds, namely, SRP holds in V and in any forcing extension of Vobtained by ω -directed closed forcing. This indestructible form of SRP followsfrom MM . To show that SRP does not imply a statement ϕ , one then forces with a poset P to add a counterexample to ϕ . Let’s say the forcing extension is V[ G ].One then argues that there is in V[ G ] a further forcing T = ˙ T G such that P ∗ ˙ T is ω -distributive. Letting H be V[ G ]-generic for Q , one has then that V[ G ][ H ]satisfies SRP . This is then used to argue that V[ G ] also satisfies SRP . The crucialstep here is to show that if S is projective stationary in V[ G ], then this remainstrue in V[ G ][ H ]. The main technical problem is thus to prove the preservation ofprojective stationarity. The property ϕ in this approach is usually a statement about stationary reflec-tion, and so, the forcing P usually adds a sequence of stationary sets that does notreflect in a certain way. The forcing T is usually a forcing that destroys this coun-terexample to ϕ by destroying the stationarity of some of the sets in the sequence,that is, by shooting a club through the complement of some of these sets. Given astationary and costationary subset A of some regular uncountable cardinal κ , theforcing used is called T A (the forcing to “kill” A ), and consists of closed boundedsubsets of κ disjoint from A , and ordered by end-extension. Larson [35, Lemma4.5] has analyzed when this forcing notion preserves generalized stationarity; see[22, Lemma 4.3] for a proof of this exact formulation. Lemma 4.1. Let γ > ω be regular, X ⊇ γ a set, A ⊆ γ stationary and S ⊆ [ X ] ω also stationary, such that γ \ A is unbounded in γ and T A is countably distributive(this is the case, for example, if A ⊆ S γω and S γω \ A is stationary). Then thefollowing are equivalent: (1) S \ lifting ( A, [ X ] ω ) is stationary. (2) T A preserves the stationarity of S . (3) There is a condition p ∈ T A that forces that ˇ S is stationary. This leads to a characterization of when T A preserves projective stationarity asfollows. Lemma 4.2. Let γ > ω be regular, X ⊇ γ a set, A ⊆ γ stationary and S ⊆ [ X ] ω projective stationary, such that γ \ A is unbounded in γ and T A is countablydistributive. Then the following are equivalent: (1) S \ lifting ( A, [ X ] ω ) is projective stationary. (2) T A preserves the projective stationarity of S . (3) There is a condition p ∈ T A that forces that ˇ S is projective stationary.Proof. (1) = ⇒ (2): Suppose G ⊆ T A is generic and S is not projective sta-tionary in V[ G ]. Then there is some stationary B ⊆ ω such that in V[ G ], S ∩ lifting ( B, [ X ] ω ) is not stationary. Since T A is countably distributive, B ∈ V isstationary in V. Since in V, S \ lifting ( A, [ X ] ω ) is projective stationary, we know that( S \ lifting ( A, [ X ] ω )) ∩ lifting ( B, [ X ] ω ) is stationary. Note that this intersection is thesame as ( S ∩ lifting ( B, [ X ] ω )) \ lifting ( A, [ X ] ω ). So by Lemma 4.1, S ∩ lifting ( B, [ X ] ω )is stationary in V[ G ], a contradiction.(2) = ⇒ (3) is trivial.(3) = ⇒ (1): Let p ∈ T A force that ˇ S is projective stationary. Then, for everystationary B ⊆ ω , p forces that S ∩ lifting ( B, [ X ] ω ) is stationary. By Lemma 4.1,this implies that ( S ∩ lifting ( B, [ X ] ω )) \ lifting ( A, [ X ] ω ) is stationary. That is, forevery stationary B ⊆ ω , ( S \ lifting ( A, [ X ] ω ) ∩ lifting ( B, [ X ] ω ) is stationary, whichmeans that S \ lifting ( A, [ X ] ω ) is projective stationary, as wished. (cid:3) ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 33 To formulate some sample applications of these methods, let me recall the fol-lowing principles of stationary reflection. The simulatenous reflection principles arewell-known, see [9]. The first diagonal version was originally introduced in Fuchs[20], and its variations come from Fuchs & Lambie-Hanson [22]. Definition 4.3. Let λ be a regular cardinal, let S ⊆ λ be stationary, and let κ < λ .The simultaneous reflection principle Refl ( κ, S ) says that whenever ~S = h S i | i < κ i is a sequence of stationary subsets of S , then ~S has a simultaneous reflectionpoint (see Def. 3.13).The diagonal reflection principle DSR ( <κ, S ) says that whenever h S α,i | α <λ, i < j α i is a sequence of stationary subsets of S , where j α < κ for every α < λ ,then there are an ordinal γ < λ of uncountable cofinality and a club F ⊆ γ suchthat for every α ∈ F and every i < j α , S α,i ∩ γ is stationary in γ .The version of the principle in which j α ≤ κ is denoted DSR ( κ, S ).If F is only required to be unbounded, then the resulting principle is called uDSR ( <κ, S ), and if it is required to be stationary, then it is denoted sDSR ( <κ, S ).These principles can be viewed as different ways of generalizing the followingreflection principle, due to Larson [35]. Definition 4.4. The principle OSR ω (for “ordinal stationary reflection”) statesthat whenever ~S = h S i | i < ω i is a sequence of stationary subsets of S ω ω , there isan ordinal ρ < ω of uncountable cofinality which is a simultaneous reflection pointof ~S ↾ ρ .The following fact shows that the principles introduced above indeed generalize OSR ω . Fact 4.5 (special case of [22, Lemma 2.4]) . The following are equivalent: (1) uDSR ( ω , S ω ω )(2) DSR ( ω , S ω ω )(3) OSR ω The result of Larson [35, Thm. 4.7] most relevant here is that if Martin’s Max-imum is consistent, then it is consistent that SRP holds but OSR ω fails. On theother hand, MM implies OSR ω . Hence, OSR ω is a consequence of MM which isnot captured by SRP , thus separating MM from SRP . The following lemma showsthat, under the appropriate assumptions, the ∞ -subcomplete fragment of SRP doesnot even imply Refl (1 , S ω ω ), giving a much stronger failure of reflection. Lemma 4.6. Assume that SRP holds and continues to hold after ω -directed closedforcing. Then there is an ω -strategically closed forcing of size ω in whose exten-sions the following hold:(a) ∞ - SC - SRP ,(b) ω = ω ,(c) there is a nonreflecting stationary subset of S ω ω .Proof. Let P be the forcing to add a nonreflecting stationary subset of S ω ω by closedinitial segments. Since SRP holds, we have that ω ω = ω , and hence, the cardinalityof P is ω , and it is well-known that P is ω -strategically closed. A generic filterfor P may be identified with its union, which is a nonreflecting stationary subset of S ω ω . Let A be such a generic set. V[ A ] then has the desired properties, and only property (a) needs to be verified. To see this, let me prove a general claim, whichmay be useful in other contexts as well.(1) Suppose ω < λ ≤ ω , λ regular, and B is a stationary subset of S λω . Let κ ≥ λ be regular, and let S ⊆ [ H κ ] ω be spread out. Then S ∩ lifting ( B, [ H κ ] ω )is projective stationary. Proof of (1). Let D ⊆ ω be stationary, and let f : ( H κ ) <ω −→ H κ . We have tofind an x ∈ S such that S ∩ ω ∈ D , x is closed under f and sup( x ∩ λ ) ∈ B . To thisend, let θ witness that S is spread out. Let H θ ⊆ N = L Iτ , where S, θ, f ∈ N and τ is a cardinal in L [ I ]. Let τ ′ = ( τ + ) L [ I ] , and let N ′ = L Iτ ′ . Let x ′ be countablewith N ′ | x ′ ≺ N and S, θ, f, λ ∈ x ′ . Moreover, let x ′ ∩ ω ∈ D and sup( x ′ ∩ λ ) ∈ B .This is possible by Fact 3.25. Let x = x ′ ∩ L τ [ I ]. Then x is full, and hence, since S is spread out, there is a y such that N | x is isomorphic to N | y via an isomorphism π which fixes f , and such that ¯ y = y ∩ H κ ∈ S . But since λ ≤ ω , π is theidentity on λ (see the paragraph before Observation 2.32), so that ¯ y is as wished:¯ y ∩ ω = x ∩ ω ∈ D , sup(¯ y ∩ λ ) = sup( x ∩ λ ) ∈ B , ¯ y ∈ S , and since f = π ( f ) ∈ y ,¯ y is closed under f . (cid:3) Now, working in V[ A ], A is a nonreflecting stationary subset of S ω ω . It followsthat B = S ω ω \ A is stationary. Hence, by Claim (1), S ∩ lifting ( B, [ H κ ] ω ) = S \ lifting ( A, [ H κ ] ω ) is projective stationary. By Lemma 4.2, it follows that theforcing T A preserves the projective stationarity of S .Now it is well-known that P ∗ ˙ T ˙ A is forcing equivalent to an ω -directed closedforcing of size ω , so that if we let H be generic for T A over V[ G ], then SRP holdsin V[ G ][ H ], where S is projective stationary. Working in V[ G ][ H ], let h y α | α < ω i be a continuous ∈ -chain through S ↑ [ H V[ G ][ H ] κ ] ω . Then letting x α = y α ∩ H V[ G ] κ ,it follows that ~x is a continuous ∈ -chain through S , and ~x ∈ V[ G ], since T A is ω -distributive in V[ G ]. (cid:3) Note that it was crucial in the proof of (1) that λ ≤ ω , and the case λ = ω iswhat was used in the remainder of the proof. Thus, we are in the funny situationthat if we want ∞ - SC - SRP to be similar to SRP , in that it should yield consequenceslike Refl ( ω , S ω ω ), for example, then we should add the assumption that CH holds,which contradicts SRP . In fact, already Observation 2.32 was a clear indicationthat one should add CH to ∞ - SC - SRP in order to make it stronger.I also have to point out that the previous lemma does not separate ∞ - SC - SRP from ∞ - SCFA . In fact, Hiroshi Sakai observed that methods of Sean Cox can beused to show that ∞ - SCFA + 2 ω = ω is consistent with the failure of Refl (1 , S ω ω )as well, and more, for example, ∞ - SCFA , if consistent, is consistent with (cid:3) ω . Thedetails of the argument, and variations of it, have been worked out by Corey Switzer(forthcoming).Thus, Lemma 4.6 provides a fairly strong limitation of ∞ - SC - SRP , but fails to separate it from ∞ - SCFA . However, it is possible to obtain a separating result asfollows. The first fact I need is that for a regular cardinal κ , DSR ( ω , S κω ) followsfrom Martin’s Maximum if κ > ω , and also from SCFA , for κ > ω , see [20].Second, the principle does not follow from SRP . In fact: Theorem 4.7 (Fuchs & Lambie-Hanson [22, Thm. 4.4]) . Let κ > ω be regular,and suppose that SRP holds and continues to hold in any forcing extension obtained ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 35 by κ -directed closed forcing. Then there is a κ -strategically closed forcing notionthat produces forcing extensions in which (1) SRP continues to hold, but (2) uDSR (1 , S κω ) fails. Thus, a model as in the previous theorem will satisfy ∞ - SC - SRP (since it evensatisfies the full SRP ), but not SCFA , since κ is regular and greater than ω =2 ω in that model, so if SCFA held, then this would imply DSR ( ω , S κω ) by theabovementioned fact, contradicting that even uDSR (1 , S κω ) fails. Thus: Fact 4.8. Assuming the consistency of MM , ∞ - SC - SRP does not imply SCFA . While this is in a sense a satisfying separation result, I view the fact that ∞ - SC - SRP ( ω ) is trivial if 2 ω ≥ ω , and that ∞ - SC - SRP , and even ∞ - SCFA , do not imply Refl (1 , S ω ω ) if CH fails, as strong indications that these axioms should be augmentedwith the assumption of CH . But in the model that witnesses the separation factjust stated, we have that 2 ω = ω , so I have not separated ∞ - SC - SRP + CH from ∞ - SCFA . Since these are the axioms of the main interest, I will focus on themfor the remainder of this article. The goal result would be that the answer to thefollowing question is no: Question 4.9. Does ∞ - SC - SRP + CH imply uDSR (1 , S κω ), for any regular κ > ω ,or any of the diagonal reflection principles?Note that answering this question in the negative would strengthen the conclu-sion of Theorem 4.7. The reason why I am hopeful that this might be the case isthat the assumption of CH might replace the assumption that the nonstationaryideal in ω -dense in the following. Corollary 4.10 (Fuchs & Lambie-Hanson [22, Cor. 4.7]) . Suppose that SRP holdsand continues to hold in any forcing extension obtained by an ω -directed closedforcing notion. Assume furthermore that the density of the nonstationary ideal on ω is ω . Then there is an ω -strategically closed forcing notion which producesforcing extensions where (1) SRP continues to hold, but (2) uDSR (1 , S ω ω ) fails. However, there are obstacles to obtaining versions of Theorem 4.7 or the corollaryjust mentioned for ∞ - SC - SRP in the context of CH . The main problem is to find aversion of the Preservation Lemma 4.2 for spread out sets rather than for projectivestationarity.4.2. The CH setting: a less canonical separation. I will now move on toa partial separation result which may be cause for optimism that the answer toQuestion 4.9 is negative. It concerns the fragment of ∞ - SC - SRP that says that SRP holds for all the spread out sets shown to be spread out in this article. Theseare the ones from Subsection 3.3. Since this fragment of SRP does not correspondto a forcing class, I don’t consider it to be canonical, hence the title of the presentsubsection. Definition 4.11. Let ρ be regular, and let K ⊆ ρ be a set of regular cardinals suchthat min( K ) > ω . Let ~S = h S κ,i | κ ∈ K, i < ω i be such that for every κ ∈ K and every i < ω , S κ,i ⊆ S κω is stationary in κ . Let h D i | i < ω i be a partition of ω into stationary sets, and let ρ ≥ sup( K ) be regular. Let’s call h ~D, ~S i a K -correspondence, and define lifting ( ~D, ~S, [ H ρ ] ω ) = { x ∈ [ H ρ ] ω | ∀ κ ∈ x ∩ K ∀ i < ω ( x ∩ ω ∈ D i = ⇒ sup( x ∩ κ ) ∈ S κ,i ) } . Let S + ( ρ ) = { lifting ( h ~D, ~S i , [ H ρ ] ω ) ∩ C | h ~D, ~S i is a K -correspondencefor some K ⊆ ρ , and C ⊆ [ H ρ ] ω is club } . Let’s write SRP ( S + ) for the statement that SRP ( S + ( ρ )) holds for every regular ρ > ω . Observation 4.12. It follows from the results of the previous subsections that: (1) ∞ - SC - SRP + CH implies SRP ( S + ) . (2) If SRP ( S + ) + CH holds, then for all regular κ > ω , κ ω = κ , and hence,every set in S + ( ρ ) , for ρ > ω , is spread out. The assumptions of the following theorem are implied by SCFA + CH ; see Fuchs& Rinot [23]. Theorem 4.13. Assume that SRP ( S + ) holds in V and in any forcing extensionobtained by a λ -directed closed forcing, and that CH is true. Then there is a λ -strategically closed forcing in whose forcing extensions (1) SRP ( S + ) holds, yet (2) uDSR (1 , S ω ω ) fails.Remark . I recommend reading the proof with Question 4.9 in mind. Theproblem is the preservation of an arbitrary spread out set without knowing that itbelongs to some S + ( ρ ). Proof. The forcing notion in question is P , the forcing to add a counterexampleto uDSR (1 , S ω ω ) used in the proof of [22, Theorem 3.10], where it is shown that P is ω -strategically closed (see claim 3.11 in the proof). Let G be generic for P . In V[ G ], there is a sequence ~A = h A α | α < ω i which witnesses the failure of uDSR (1 , S λω ). That is, for every α < ω of uncountable cofinality, the set { ξ <α | A ξ ∩ α is stationary in α } is bounded in α . Thus, I will be done if I can showthat V[ G ] satisfies SRP ( S + ).To this end, let S ⊆ [ H λ ] ω ∈ S + ( ρ ), for some regular ρ > ω . Let ~D , ~S , C witness this, that is, ~D = h D i | i < ω i is a partition of ω into stationary sets, ~S = h S κ,i | κ ∈ K, i < ω i , where K ⊆ ρ is a set of regular cardinals greater than ω , C ⊆ [ H ρ ] ω is club and S = lifting ( ~D, ~S, [ H ρ ] ω ) ∩ C .For β < λ , let A ≥ β = S β ≤ α<ω A α , and let T β be the forcing to shoot a clubthrough the complement of A ≥ β . By Claim 3.13 of the proof of [22, Theorem 3.9],the forcing P ∗ ˙ T ≥ β has a dense subset that’s ω -directed closed and has size ω inV. It is used here that ω ω = ω , which follows from SRP ( S + ) + CH . Clearly then, P ∗ ˙ T ≥ β preserves cofinalities.Let me assume for a second that ω ∈ K . By Claim 3.13 of the aforementionedproof, for every i < ω , there is a β i < ω such that S ω ,i \ A ≥ β i is stationary. Let β = sup i<ω β i . Then β < ω , and for every i < ω , S ω ,i \ A ≥ β is stationary, which ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 37 means that forcing with T β preserves the stationarity of each S ω ,i , see [22, Lemma3.7.(2)].If ω / ∈ K , then let β = 0.Now let H be T β -generic over V[ G ]. Since P ∗ ˙ T β has a dense subset that’s ω -directed closed, it follows from our indestructible SRP ( S + ) assumption in V that SRP ( S + ) holds in V[ G ][ H ]. Since P ∗ ˙ T β has an ω -directed closed subset, it addsno sequences of ground model sets of length less than ω . In V[ G ][ H ], it is stillthe case that C is a club subset of [ H V[ G ] ρ ] ω , and it is still the case that h ~D, ~S i isa correspondence: ~D obviously is still a partition of ω into stationary sets. Since P ∗ T ≥ β preserves cofinalities, K is still a set of regular cardinals greater than ω .Now, for κ ∈ K and i < ω , S κ,i is still a stationary subset of S κω - in the case where κ = ω , this is by our choice of β , and if κ > ω , then T β preserves the stationarityof S κ,i because it is κ -c.c.; it has size ω < κ .Let C ∗ = C ↑ [ H V[ G ][ H ] ρ ] ω - so C ∗ contains a club C ′ . In V[ G ][ H ], let S ∗ = lifting ( ~D, ~S, [ H V[ G ][ H ] ρ ] ω ) ∩ C ′ . Then S ∗ ∈ S + ( ρ ) in V[ G ][ H ], so by SRP ( S + ) inV[ G ][ H ], there is a continuous ∈ -chain h x ′ i | i < ω i through S ∗ . But then, for every i < ω , x i = x ′ i ∩ H V ρ ∈ S . This is because x ′ i ∈ C ′ ⊆ C ↑ [ H V[ G ][ H ] ρ ] ω , so that x i ∈ C ,and because x ′ i ∈ lifting ( ~D, ~S, [ H V[ G ][ H ] ρ ] ω ), if we let j = x ′ i ∩ ω , then j = x i ∩ ω ,and further, if κ ∈ K ∩ x i , then κ ∈ K ∩ x ′ i , so that sup x i ∩ κ = sup x ′ i ∩ κ ∈ S κ,j .Now the sequence h x i | i < ω i already exists in V[ G ], since T β is ω -distributive inV[ G ].Thus, V[ G ] satisfies SRP ( S + ) but not uDSR (1 , S λω ), as desired. (cid:3) In particular, it is not the case that SFP λ + CH implies uDSR (1 , S λω ), for regular λ > ω . This is because SRP ( S + ) does imply SFP λ , and because of the previoustheorem.4.3. The CH setting: a canonical separation at ω . In the present subsec-tion, I work with subcompleteness rather than ∞ -subcompleteness. The goal isto separate, to some degree, the subcomplete forcing axiom from the subcompletefragment of SRP in the presence of the continuum hypothesis. The idea is to takea consequence of SCFA that is obtained using a subcomplete forcing that is notcountably distributive (as suggested by the discussion after Theorem 3.34), and toargue that it does not follow from the subcomplete fragment of SRP . The forc-ing in question is going to be Namba forcing, changing a regular cardinal to havecountable cofinality. The first ingredient I will need is an iteration theorem for thesubclass of subcomplete forcing notions consisting of those that preserve uncount-able cofinalities. Thus, this class excludes Namba forcing. Definition 4.15. A forcing notion P is uncountable cofinality preserving if forevery ordinal ε of uncountable cofinality, (cid:13) P “ ε has uncountable cofinality.”In the proof of the iteration theorem, I will use the Boolean algebraic approachto forcing iterations and to revised countable support, as described in [33]. Anextensive account of the method is the manuscript [38], which also contains a proofof the following fact, originally due to Baumgartner, see [38, Theorem 3.13]: Fact 4.16. Let h B i | i < λ i be an iteration such that for every α < λ , B α is <λ -c.c.,and such that the set of α < λ such that B α is the direct limit of ~ B ↾ α is stationary.Then the direct limit of ~ B is <λ -c.c. I will use the following iteration theorem in the separation result I am workingtowards, but it may be of independent interest as well. Theorem 4.17. Let h B i | i ≤ δ i be an RCS iteration of complete Boolean algebrassuch that for all i + 1 ≤ α , the following hold: (1) B i = B i +1 , (2) (cid:13) B i (ˇ B i +1 / ˙ G B i is subcomplete and uncountable cofinality preserving ) , (3) (cid:13) B i +1 ( δ (ˇ B i ) has cardinality at most ω ) .Then for every i ≤ δ , B i is subcomplete and uncountable cofinality preserving.Proof. I will have to use some basic facts about iterated forcing with completeBoolean algebras and revised countable support. I will state these facts as I needthem. The basic setup is that h B i | i ≤ δ i is a tower of complete Boolean algebras,with projection functions h j,i : B j −→ B i for i ≤ j ≤ δ defined by h j,i ( b ) = V B i { a ∈ B i | b ≤ B j a } . Since for i ≤ j ≤ j ≤ δ , h j ,i ↾ B j = h j ,i , I’ll just write h i for h δ,i ,so that for all i ≤ j ≤ δ , h j,i = h i ↾ B j .I claim that:for every h < δ , if G h is B h -generic over V, then in V[ G h ], forevery i ∈ [ h, δ ], B i /G h is subcomplete and uncountable cofinalitypreserving.Jensen’s proofs show that in this situation, B i /G h is subcomplete (see [34, § δ be a minimal such that there is an iteration of length δ that formsa counterexample.Since the successor case is trivial, let me focus on the case that δ is a limit ordinal.Let ε be an ordinal of uncountable cofinality in V[ G h ]. Note that by minimality of δ , this is equivalent to saying that ε has uncountable cofinality in V. I will showthat in V[ G h ], B δ /G h still forces that ˇ ε has uncountable cofinality. Case 1: there is an i < δ such that cf V[ G h ] ( ε ) ≤ δ ( B i ).Then by (3), in V[ G h ], (cid:13) B i +1 /G h cf(ˇ ε ) ≤ ω . But by minimality of δ , we alsoknow that (cid:13) B i +1 cf(ˇ ε ) ≥ ω . Thus, B i +1 /G h forces over V[ G h ] that the cofinalityof ε is ω . But the tail of the iteration is subcomplete, and hence it preserves ω .It follows that the cofinality of ε stays uncountable in V[ G h ] B δ /G h . Case 2: case 1 fails. So for all i < δ , cf V[ G h ] ( ε ) > δ ( B i ).Let κ = cf V[ G h ] ( ε ). Note that for i < δ , B i is δ ( B i ) + -c.c., so since κ > δ ( B i ), itfollows that κ is a regular uncountable cardinal greater than δ ( B i ) in V B i . We mayalso assume that κ > ω , for if κ = ω , then it will still have uncountable cofinalityin V[ G h ] B δ /G h , since B δ /G h is subcomplete in V[ G h ] and hence preserves ω , as incase 1. Case 2.1: there is an i < δ such that cf V[ G h ] ( δ ) ≤ δ ( B i ).Then, since B i +1 collapses δ ( B i ) to ω , it follows that in V[ G h ] B i +1 /G h , δ hascofinality at most ω . For notational simplicity, let me pretend that h = 0, so thatI can write V B i +1 in place of V[ G h ] B i +1 /G h , for example. So from what I said above, κ is a regular uncountable cardinal in what we call V now, and it is greater thaneach δ ( B j ). So It suffices to show that B δ preserves the fact that κ has uncountablecofinality. I’m using here that the tail of the iteration in V[ G h ] is a revised countablesupport iteration.Towards a contradiction, let ˙ f be a B δ -name, and let a ∈ B δ be a conditionsuch that a forces that ˙ f is a function from ω to κ whose range is unbounded in ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 39 κ . I will find a condition extending a that forces that the range of ˙ f is boundedin κ , a contradiction.Note that if cf( δ ) = ω , then for every i < δ , (cid:13) B i cf(ˇ δ ) = ˇ ω , as B i preserves ω .Thus, B δ is the direct limit of ~ B ↾ δ , that is, S i<δ B i is dense in B δ in this case. Letme denote this dense set by X .If, on the other hand, cf( δ ) = ω , then since ~ B is an RCS iteration, the set { V i<δ t i | h t i | i < δ i is a thread in ~ B ↾ δ } is dense in B δ – here, ~t is a thread in ~ B ↾ δ if for all i ≤ j < δ , 0 = t i = h i ( t j ). In case cf( δ ) = ω , let X be that dense subsetof B δ . For more background on threads in RCS iterations, I refer the reader to [33,p. 124].Let π : ω −→ δ be cofinal, with π (0) = 0.Let N = L Aτ with H θ ∪ κ + 1 ⊆ N , where θ verifies the subcompleteness ofeach B i , for i ≤ δ and for each B j /G i whenever i < j < δ and G i is B i -generic.Let S = h κ, ε, δ, ~ B , ˙ f , a , π i . Let σ : ¯ N ≺ N , where ¯ N is transitive, countableand full, and S ∈ ran( σ ). Let σ ( ¯ S ) = S , and let ¯ S = h ¯ κ, ¯ ε, ¯ δ, ~ ¯ B , ˙¯ f, ¯ a , ¯ π i . Let˜ κ = sup σ “ κ < κ , and let us also fix an enumeration ¯ N = { e n | n < ω } . Let¯ G ⊆ ¯ B ¯ δ be generic over ¯ N , with ¯ a ∈ ¯ G .Let h ν n | n < ω i be a sequence of ordinals ν n < ω ¯ N such that if we let ¯ γ n =¯ π ( ν n ), it follows that h ¯ γ n | n < ω i is cofinal in ¯ δ , and such that ν = 0, so that¯ γ = 0. Hence, letting γ n = σ (¯ γ n ), we have that sup n<ω γ n = sup ran( σ ∩ δ ) = ˜ δ .Moreover, whenever σ ′ : ¯ N ≺ N is such that σ ′ (¯ π ) = π , it follows that for every n < ω , σ ′ (¯ γ n ) = γ n = π ( ν n ), since σ ′ (¯ γ n ) = σ ′ (¯ π ( ν n ) = σ ′ (¯ π )( ν n ) = π ( ν n ).By induction on n < ω , construct sequences h ˙ σ n | n < ω i , h c n | n < ω i , with c n ∈ B γ n , ˙ σ n ∈ V B γn , such that for every n < ω , c n forces the following statementswith respect to B γ n :(1) ˙ σ n : ˇ¯ N ≺ ˇ N ,(2) ˙ σ n ( ˇ¯ S ) = ˇ S , and for all k < n , ˙ σ n (ˇ e k ) = ˙ σ k (ˇ e k ),(3) ˙ σ n “ ˇ¯ G ∩ ˇ¯ B ¯ γ n ⊆ ˙ G B γn ,(4) sup ˙ σ n “ˇ¯ κ = ˇ˜ κ ,(5) c n − = h γ n − ( c n ) (for n > n = 0, we set c = 1l, ˙ σ = ˇ σ .Now suppose h ˙ σ n | n ≤ n i , h c m | m ≤ n i have been constructed, so that the aboveconditions are satisfied so far. Let G γ n be B γ n -generic with c n ∈ G γ n , and workin V[ G n ] temporarily. Let σ n = ˙ σ G γn n . Then σ n extends to σ ∗ n : ¯ N [ ¯ G ¯ γ n ] ≺ N [ G γ n ]such that σ ∗ n ( ¯ G ¯ γ n ) = G n . Since θ verifies the subcompleteness of B ′ = B γ n +1 /G γ n ,and since ¯ N [ ¯ G ¯ γ n ] is full, there is a condition c ′ ∈ ( B ′ ) + such that whenever G ′ is B ′ -generic over V[ G n ] with c ′ ∈ G ′ , there is a σ ′ : ¯ N [ ¯ G ¯ γ n ] ≺ N [ G n ] such that σ ′ ( ¯ G ¯ γ n ) = G γ n , σ ′ ( ¯ S ) = S and σ ′ ( e k ) = σ ∗ n ( e k ) = σ n ( e k ) for k ≤ n , ( σ ′ )“ ¯ G ∩ ¯ B ¯ G ⊆ G ′ , where ¯ G ′ = ¯ G ¯ γ n +1 / ¯ G ¯ γ n , and sup( σ ′ )“¯ κ = ˜ κ = sup( σ ∗ “)¯ κ . The point is that κ > δ ( B γ n +1 ), so that the suprema condition can be employed in order to ensurethis last point.Since the situation described in the previous paragraph arises whenever G γ n isgeneric with c n ∈ G γ n , it is forced by c n , and there are B γ n -names ˙ c ′ , ˙ σ n +1 forthe condition c ′ and the restriction of the embedding σ ′ to ¯ N . We may choose thename ˙ c ′ in such a way that (cid:13) B γn ˙ c ′ ∈ ˇ B γ n +1 / ˙ G B γn and c n = J ˙ c ′ = 0 K B γn . Namely,given the original ˙ c ′ such that c n forces that ˙ c ′ ∈ (ˇ B γ n +1 / ˙ G B γn ) + and all the other statements listed above, there are two cases: if c n = 1l B γn , then since c n ≤ J ˙ c ′ = 0 K ,it already follows that c n = J ˙ c ′ = 0 K . If c n < B γn , then let ˙ e ∈ V B γn be a namesuch that (cid:13) B γn ˙ e = 0 ˇ B γn +1 / ˙ G B γn , and mix the names ˙ c ′ and ˙ e to get a name ˙ d ′ such that c n (cid:13) B γn ˙ d ′ = ˙ c ′ and ¬ c n (cid:13) B γn ˙ d ′ = ˙ e . Then ˙ d ′ is as desired. Clearly, (cid:13) B γn ˙ d ′ ∈ ˇ B γ n +1 / ˙ G B γn . Since c n (cid:13) B γn ˙ d ′ = ˙ c ′ , it follows that c n ≤ J ˙ d ′ = 0 K , andsince ¬ c n (cid:13) B γn ˙ d ′ = ˙ e , it follows that ¬ c n ≤ J ˙ d ′ = 0 K = ¬ J ˙ d ′ = 0 K , so J ˙ d ′ = 0 K ≤ c n .So we could replace ˙ c ′ with ˙ d ′ .So let us assume that ˙ c ′ already has this property, that is, c n = J ˙ c ′ = 0 K . Then,by [33, § 0, Fact 4], there is a unique c n +1 ∈ B γ n +1 such that (cid:13) B γn ˇ c n +1 / ˙ G B γn = ˙ c ′ ,and it follows by [33, § 0, Fact 3] that h ( c n +1 ) = J ˇ c n +1 / ˙ G B γn = 0 K B γn = J ˙ c ′ = 0 K B γn = c n as wished.This finishes the construction of h ˙ σ n | n < ω i and h c n | n < ω i .Now, the sequence h c n | n < ω i is a thread, and so, c = V n<ω c n ∈ B + δ . Let G be B δ -generic with c ∈ G . In V[ G ], let σ = [ n<ω ˙ σ G ∩ B in n ↾ { e , . . . , e n } . Jensen’s arguments then show that σ : ¯ N ≺ N , and σ “ ¯ G ⊆ G . For the latter, wecan argue as in [33, p. 141, proof of (d)]: clearly, σ “ ¯ G ∩ ¯ B ¯ γ n ⊆ G , for every n < ω ,since if ¯ a ∈ ¯ G ∩ ¯ B ¯ γ n , then for some m ≥ n , σ ( a ) = σ m ( a ) ∈ G ∩ B γ m ⊆ G (letting a = e k , this is true whenever m ≥ max( n, k )). If cf( δ ) = ω , then this impliesdirectly that σ “ ¯ G ⊆ G , because then, in ¯ N , ¯ B ¯ δ is the direct limit of ~ B ↾ ¯ δ , so S i< ¯ δ ¯ B i is dense in ¯ B δ . So if ¯ a ∈ ¯ G , me may assume that ¯ a ∈ ¯ G ∩ B ¯ γ n , for some n < ω , sothat a ∈ G ∩ B γ n .If cf( δ ) = ω , that is, in ¯ N , cf(¯ δ ) = ω , then B ¯ δ is the inverse limit of h ¯ B i | i < ¯ δ i . Hence, letting ¯ a ∈ ¯ G , we may assume that ¯ a = V i< ¯ δ ¯ a i , where h ¯ a i | i < ¯ δ i is a thread in h ¯ B i | i < ¯ δ i , since the set of such conditions is dense in B +¯ δ . Let σ ( ~ ¯ a ) = ~a = h a i | i < δ i . Then ~a is a thread in h B i | i < δ i . Moreover, for each n < ω , σ (¯ a ¯ γ n ) = a γ n ∈ G . Thus, σ (¯ a ) = V n<ω a γ n ∈ G , by the completeness of G (since h a γ n | n < ω i ∈ V, as ~a ∈ N ⊆ V and h γ n | n < ω i ∈ V.)Thus, σ lifts to an elementary embedding σ ∗ : ¯ N [ ¯ G ] ≺ N [ G ]. Note that ¯ a ∈ ¯ G ,and so, a = σ ∗ (¯ a ) ∈ G . Moreover, σ “¯ κ ⊆ ˜ κ , because if ξ < ¯ κ , then for some n < ω , ξ = e n , and so, σ ( ξ ) = σ ( e n ) = σ n ( e n ) = σ n ( ξ ) < ˜ κ , since σ n “¯ κ ⊆ ˜ κ . But then, itfollows that ran( ˙ f G ) ⊆ ˜ κ < κ , because ran( ˙ f G ) = σ “ran( ˙¯ f ¯ G ) ⊆ σ “¯ κ ⊆ ˜ κ < κ . Thiscontradicts the fact that a ∈ G and a forces that the range of ˙ f is unbounded in κ . Case 2.2: for all i < δ , cf( δ ) > δ ( B i ).We may also assume that cf( δ ) > ω , for otherwise, cf( δ ) ≤ ω and the argumentof case 2.1 goes through.It follows as in [33, p. 143, claim (2)] that for i < δ , | i | ≤ δ ( B i ). But then, δ must be regular, for otherwise, if i = cf( δ ) < δ , it would follow that cf( δ ) = i ≤ δ ( B i ) < cf( δ ). That fact should read: “Let A ⊆ B , and let (cid:13) A ˙ b ∈ ˇ B / ˙ G A , where ˙ b ∈ V A . There is a unique b ∈ B such that (cid:13) A ˙ b = ˇ b/ ˙ G A .” That’s what the proof given there shows. ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 41 Thus, δ is a regular cardinal, and δ ≥ ω . Hence, S δω , the set of ordinals less than δ with cofinality ω , is stationary in δ . For γ ∈ S δω , since B γ , being subcomplete,preserves ω , it follows that for every i < γ , (cid:13) B i cf(ˇ γ ) > ω . Thus, B γ is the directlimit of ~ B ↾ γ . Moreover, since for i < δ , δ ( B i ) < δ = cf( δ ), it follows that B i is δ ( B i ) + -c.c., and hence δ -c.c.It follows by Fact 4.16 that the direct limit of ~ B ↾ δ is δ -c.c.Again, since for all i < δ , B i is δ -c.c., B i forces that the cofinality of δ is un-countable, so that B δ is the direct limit of ~ B ↾ δ , which is δ -c.c., as we have just seen.So B δ is δ -c.c.Let G be B δ -generic over V, and suppose that in V[ G ], κ has countable cofinality.Since B δ is δ -c.c., it must be that κ < δ . But letting f : ω −→ κ be cofinal, f ∈ V[ G ],it then follows that f ∈ V[ G ∩ B i ], for some i < δ . This contradicts the minimalityof δ and completes the proof. (cid:3) Looking back, it turns out that the theorem shows iterability with countablesupport. Corollary 4.18. Let h B i | i ≤ δ i be a countable support iteration of complete Booleanalgebras such that for all i + 1 ≤ α , the following hold: (1) B i = B i +1 , (2) (cid:13) B i (ˇ B i +1 / ˙ G B i is subcomplete and uncountable cofinality preserving ) , (3) (cid:13) B i +1 ( δ (ˇ B i ) has cardinality at most ω ) .Then for every i ≤ δ , B i is subcomplete and uncountable cofinality preserving.Proof. It is easy to see by induction on i ≤ δ that B i is subcomplete and uncountablecofinality preserving, and that if i is a limit ordinal, then B i is (isomorphic to) thercs limit of ~ B ↾ i . The successor case of the induction is trivial, so let i ≤ δ bea limit ordinal. If we take B ′ i to be the revised countable support limit of ~ B ↾ i ,then the resulting iteration ~ B ↾ i ⌢ B ′ i is an rcs iteration, because inductively, ~ B ↾ i is.Hence, by the theorem, B ′ i is subcomplete and uncountable cofinality preserving.But moreover, by the proof of the theorem, whenever h ≤ k < i and G h is B h -generic, it follows that B k /G h is uncountable cofinality preserving in V[ G h ]. Itfollows that the rcs limit of ~ B ↾ i is the same as the countable support limit, andhence that B i = B ′ i (modulo isomorphism). So B i is subcomplete and uncountablecofinality preserving. (cid:3) The second set of ingredients I need is centered around bounded forcing axiomsand their consistency strengths. These axioms were introduced in [25] as follows,albeit with a different notation. Definition 4.19. Let Γ be a class of forcings, and let κ, λ be cardinals. Then BFA (Γ , ≤ κ, ≤ λ ) is the statement that if P is a forcing in Γ, B is its complete Booleanalgebra, and A is a collection of at most κ many maximal antichains in B , each ofwhich has size at most λ , then there is a A -generic filter in B , that is, a filter thatintersects each antichain in A . When κ = ω , then I usually don’t mention κ , that is, BFA (Γ , ≤ λ ) is short for BFA (Γ , ≤ ω , ≤ λ ). And when κ = λ = ω , then the resultingprinciple is abbreviated to BFA (Γ). If Γ is the class of subcomplete forcings, thenI write BSCFA for BFA ( SC ) and BSCFA ( ≤ λ ) for BFA ( SC , ≤ λ ). Similarly, if Γ is theclass of proper forcing, then BPFA denotes BFA (Γ), and similarly for BPFA ( ≤ λ ). Bounded forcing axioms can be expressed as generic absoluteness principles asfollows. Theorem 4.20 (Bagaria [2, Thm. 5]) . Let κ be a cardinal of uncountable cofinality,and let P be a poset. Then BFA ( { P } , ≤ κ, ≤ κ ) is equivalent to Σ ( H κ + ) -absolutenessfor P . The latter means that whenever g is P -generic, ϕ ( x ) is a Σ -formula and a ∈ H κ + , then V | = ϕ ( a ) iff V[ g ] | = ϕ ( a ) . For any class Γ of forcings, the principles BFA ( ≤ κ ) give closer and closer ap-proximations to FA (Γ), as κ increases; in fact, FA (Γ) is BFA (Γ , ≤∞ ), or, for all κ , BFA ( ≤ κ ). The following characterization of these axioms is easily seen to beequivalent to the one given in [7, Thm. 1.3], see also [3]. Fact 4.21. BFA ( { Q } , ≤ κ ) is equivalent to the following statement: if M = h| M | , ∈ , h R i | i < ω ii is a transitive model for the language of set theory with ω manypredicate symbols h ˙ R i | i < ω i , of size κ , and ϕ ( x ) is a Σ -formula, such that (cid:13) Q ϕ ( ˇ M ) , then there are in V a transitive ¯ M = h| ¯ M | , ∈ , h ¯ R i | i < ω ii and anelementary embedding j : ¯ M ≺ M such that ϕ ( ¯ M ) holds. Miyamoto has analyzed the strength of these principles for proper forcing andintroduced the following large cardinal concept, with slightly different terminology. Definition 4.22 ([36, Def. 1.1]) . Let κ be a regular cardinal, α an ordinal, and λ = κ + α . Then κ is H λ -reflecting, or I will say + α -reflecting, iff for every a ∈ H λ and any formula ϕ ( x ), the following holds: if there is a cardinal θ such that H θ | = ϕ ( a ), then the set of N ≺ H λ such that(1) N has size less than κ ,(2) a ∈ N ,(3) if π N : N −→ H is the Mostowski-collapse of N , then there is a cardinal¯ θ < κ such that H ¯ θ | = ϕ ( π N ( a ))is stationary in P κ ( H λ ).The concept of a reflecting cardinal was previously introduced in [25], and iscontained in this definition, as it is not hard to see that being reflecting is equiva-lent to being +0-reflecting. The +1-reflecting cardinals are also known as stronglyunfoldable cardinals, introduced independently in [39]. In the context of boundedforcing axioms, it seems to make the most sense to emphasize that they generalizereflecting cardinals, as it was shown in [25] that the consistency strength of BPFA is precisely a reflecting cardinal, and it was shown in [36, Def. 1.1] that the consis-tency strength of BPFA ( ≤ ω ) is a +1-reflecting cardinal. I showed that the sameconsistency strength results hold for BSCFA and BSCFA ( ≤ ω ) as well, in [17]. Itwill be important for the upcoming argument that +1-reflecting cardinals may existin L ; in fact, I showed in [17] that if BSCFA ( ≤ ω ) holds, then the cardinal ω V2 is+1-reflecting in L .Given this background, it is easy to observe that SC - SRP ( ω ) + ¬ CH does notimply BSCFA ( ≤ ω ), since the consistency strength of SC - SRP ( ω ) + ¬ CH is equalto that of ZFC (in fact, ¬ CH implies SC - SRP ( ω ) by Observation 2.32), while theconsistency strength of BSCFA ( ≤ ω ) is a +1-reflecting cardinal. I have now assem-bled the main tools needed to show that such a separation can also be arrangedwhen CH holds. This will be achieved by constructing a model of CH + SC - SRP ( ω )in which the consequence of BSCFA ( ≤ ω ) stated in the following lemma fails (in anextreme way). ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 43 Lemma 4.23. Assume BSCFA ( ≤ ω ) . Then the set { α < ω | cf( α ) = ω and α is regular in L } is stationary in ω .Proof. If 0 exists, then, letting I be the class of Silver indiscernibles, I ∩ ω is clubin ω and consists of L -regular cardinals. Therefore, I ∩ S ω ω is a stationary subsetof the set in question.So let me assume that 0 does not exist. The following argument traces back toTodorˇcevi´c (unpublished), but see [1, Lemma 2.4]. A variant of the argument wasused in [17, Lemma 4.11].Let κ = ω , and let γ be some singular strong limit cardinal, and let θ = γ + = ( γ + ) L , by Jensen’s covering lemma. Let E ⊆ κ be some club subset.Let X ⊆ H ω have cardinality ω , ω ⊆ X , and H ω | X ≺ H ω . Let M = h X, ∈ , E, , . . . , ξ, . . . , ω i ξ<ω . So the universe of M has size ω .Let h C ξ | ξ is a singular ordinal in L i be the canonical global (cid:3) sequence for L ofJensen [28]. It is Σ -definable in L and has the properties that for every L -singularordinal ξ , the order type of C ξ is less than ξ , and if ζ is a limit point of C ξ , then ζ is singular in L and C ζ = C ξ ∩ ζ .Let B = { ξ < θ | κ < ξ < θ and cf( ξ ) = ω } . Note that by covering, every ξ ∈ B is singular in L , since a countable cofinal subset of ξ in V can be covered by a setin L of cardinality at most ω , so that its order type will be less than κ , and henceless than ξ . So C ξ is defined for every ξ ∈ B , and since the function ξ otp( C ξ ) isregressive, there is a stationary subset A of B on which this function is constant.Let g ⊆ κ be an ω -sequence cofinal in κ , added by Namba forcing, which issubcomplete. Since Namba forcing certainly has cardinality less than θ , A remainsstationary in V[ g ]. Working in V[ g ] now, since A consists of ordinals of cofinality ω and is stationary in a regular cardinal greater than 2 ω , the forcing P A , whichadds a subset F of A that’s closed and unbounded in θ and has oder type ω , byforcing with closed initial segments, is subcomplete – this follows from Lemma 3.5,see also [33, p. 134ff., Lemma 6.3], where the assumption that θ > ω is omitted.Let h be generic over V[ g ][ F ] for Col( ω , M ). Clearly, the composition of Nambaforcing, P A and Col( ω , M ) is subcomplete.In V[ g ][ F ][ h ], the Σ -statement Φ( M ) saying “there are an ordinal θ ′ > On ∩ M ,sets g ′ and F ′ , and a function h ′ such that h ′ is a surjection from ω M onto theuniverse of M , F ′ is a club in θ ′ of order type ω M such that for all ξ, ζ ∈ F ′ ,otp( C ξ ) = otp( C ζ ), and g ′ is a cofinal subset of On ∩ M of order type ω , and in L θ ′ , On ∩ M is a regular cardinal greater than ω M ” holds, as witnessed by θ , g , F and h . It is important here that the definition of the canonical global (cid:3) sequenceis Σ . This does not depend on the particular choice of the generics g , F and h ,which means that it is forced by the trivial condition in the composition of thesesubcomplete forcings that Φ( ˇ M ) holds. So according to the characterization of BSCFA ( ≤ ω ) given by Fact 4.21, there are a transitive ¯ M = h| ¯ M | , ∈ , ¯ E, h ξ | ξ < ω ii such that Φ( ¯ M ) holds, and an elementary embedding j : ¯ M ≺ M . It follows that j is the identity. This is because we have constants for the countable ordinals, sothat ω ¯ M = ω M = ω , and since M believes that the transitive closure of any sethas cardinality at most ω .Let ¯ κ = On ¯ M , and let ¯ θ , ¯ g , ¯ F , ¯ h witness that Φ( ¯ M ) holds. Then ¯ h : ω −→ | ¯ M | is onto, so ¯ κ < κ . Moreover, since ¯ M ∈ H ω , ¯ θ may be chosen to be less than ω . Note that ¯ κ ∈ E . This is because ¯ E = ¯ κ ∩ E , and by elementarity, ¯ E is unboundedin ¯ κ , so since E is closed in κ , it follows that ¯ κ ∈ E . Moreover, ¯ κ has countablecofinality, since ¯ g is a cofinal subset of ¯ κ of order type ω . I claim that ¯ κ hasuncountable cofinality in L , thus completing the proof.The key point is that ¯ θ is a regular cardinal in L . To see this, assume that ¯ θ issingular in L . Then C ¯ θ is defined. Note that cf(¯ θ ) = ω , since otp( ¯ F ) = ω . So,letting C ′ ¯ θ be the set of limit points of C ¯ θ , C ′ ¯ θ ∩ ¯ F is club in ¯ θ . Now take ξ < ζ ,both in C ′ ¯ θ ∩ ¯ F . Then, since ξ, ζ ∈ ¯ F , C ξ and C ζ have the same order type, butsince both ξ and ζ are limit points of C ¯ θ , C ξ = C ¯ θ ∩ ξ , which is a proper initialsegment of C ζ = C ¯ θ ∩ ζ , a contradiction.So, since by Φ( ¯ M ), ¯ κ is a regular cardinal in L ¯ θ , it follows that ¯ κ is an uncount-able regular cardinal in L , as wished. (cid:3) Lemma 4.24. Let Γ be a forcing class. (1) 2 ω = ω + BFA (Γ , ≤ ω ) implies Γ - SRP ( ω ) . (2) Let Γ be the class of all subcomplete, uncountable cofinality preserving forc-ing notions. Then BFA (Γ , ≤ ω ) implies SC - SRP ( ω ) .Proof. To prove (1), first note that H ω has cardinality ω . To see that ∆- SRP ( ω )holds, let S ⊆ [ H ω ] ω be ∆-projective stationary. I will use the characterizationof BFA (∆ , ≤ ω ) given by Fact 4.21. Thus, let M = h H ω , ∈ , S, , , . . . , ξ, . . . i ξ<ω .Now the forcing P S is in Γ, since S is ∆-projective stationary, and the size of theuniverse of M is ω , so that Fact 4.21 is applicable. Namely, the Σ -statementexpressing that there is a continuous ω M -chain of models through ˙ S M is forced by P S . So, by the fact, there are a transitive model ¯ M of the same language as M ,and an elementary embedding j : ¯ M ≺ M such that the same Σ -statement is trueof ¯ M . As in the proof of Lemma 4.23, j is the identity.To see (2), let Γ be the class of all subcomplete, uncountable cofinality preservingforcing notions, and suppose that BFA (Γ , ≤ ω ) holds. If CH fails, then SC - SRP ( ω )holds trivially, by Observation 2.32. So let me assume CH . It then follow from BFA (Γ , ≤ ω ) that SFP ω holds (see Definition 3.12). To see this, let ~D = h D i | i < ω i be a partition of ω into stationary sets, and let h S i | i < ω i be a sequenceof stationary subsets of S ω ω . By (the remark after) Lemma 3.18, the set S = { a ∈ [ H ω ] ω | ∀ i < ω a ∩ ω ∈ D i −→ sup( a ∩ ω ) ∈ S i } is fully spread out. This means that the forcing P S is subcomplete, and by Fact2.9, it is countably distributive and hence in Γ. Now let N = h H ω , ∈ , ~D, ~S, , , ~ξ i ,viewed as a model of the language which has a predicate symbol for each D i , S i and i , for i < ω . Let ω ⊆ X be such that M = N | X ≺ N and such that X has cardinality ω . Let ~M be a continuous ∈ -chain through S , added by P S , andlet’s assume that each M i is an elementary submodel of h H κ , ∈ , ~D, ~S i . Using theargument of Theorem 3.19, it follows that in V[ ~M ], there is a normal function f : ω −→ ω V2 , cofinal in On ∩ M , such that for every i < ω , f “ D i ⊆ S i . Thiscan be expressed as a Σ -statement about M . By BFA (Γ , ≤ ω ), there are in V atransitive model ¯ M and an elementary embedding j : ¯ M ≺ m such that this Σ -statement is true of ¯ M . Because of the availability of the constant symbols for thecountable ordinals, it follows that ω ⊆ ran( j ), and hence j ↾ ω = id. And since ω is the largest cardinal in M , the same is true in ¯ M , and hence, letting ¯ κ = On ∩ ¯ M ,it follows that j ↾ ¯ κ is the identity. In fact, since the transitive closure of every set ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 45 in ¯ M has size at most ω , it follows that j is the identity. In any case, for i < ω ,we have that ˙ S ¯ Mi = S i ∩ ¯ κ , and ˙ D ¯ Mi = D i , for i < ω . Now let f witness that theabove statement is true of ¯ M . Then f : ω −→ ¯ κ is cofinal, normal, and for all i < ω , f “ D i ⊆ S i . This shows that SFP ω holds.By Observation 3.14 and Fact 3.15, SFP ω implies that ω ω = 2 ω = ω . Hence,by (1), we see that Γ- SRP ( ω ) holds. But Γ- SRP ( ω ) implies SC - SRP ( ω ) (andby the way, the converse is also true, because Γ ⊆ SC ), because if S ⊆ [ H κ ] ω isfully spread out, then P S is subcomplete, and P S is always countably distributive,hence uncountable cofinality preserving. Thus, P S is in Γ, making S Γ-projectivestationary. Therefore, by Γ- SRP ( ω ), there is a continuous ∈ -chain through S . (cid:3) Now I’m ready to put the pieces together and construct the model in which SC - SRP ( ω ) holds but BSCFA ( ≤ ω ) fails. Theorem 4.25. Let Γ be the class of subcomplete, uncountable cofinality preservingforcing notions. If ZFC is consistent with BFA (Γ , ≤ ω ) , then ZFC is consistent withthe conjunction of the following statements: (1) CH , (2) BFA (Γ , ≤ ω ) , (3) SC - SRP ( ω ) , (4) L is correct about uncountable cofinalities, that is, for every ordinal α , if cf L ( α ) > ω , then cf( α ) > ω , (5) ¬ BSCFA ( ≤ ω ) .Proof. The construction starts in a model of BFA (Γ , ≤ ω ). The argument of [17,Lemma 3.10] then shows that κ = ω is +1-reflecting in L . Indeed, going throughthe proof shows that only forcing notions in Γ are used. So let us work in L of thatmodel, where κ is +1-reflecting. By [17, Lemma 4.9], κ is remarkably ≤ κ -reflectingin L (I do not want to go in the details here and explain what this means, but ratheruse the results of [17] as a black box as much as possible). Now the argument of[17, Lemma 4.13] shows that in L , there is a κ -c.c. forcing notion P such that if G is generic for P , then in L [ G ], a principle called wBFA (Γ , ≤ κ ) holds (I don’t wantto define here what this principle says, since it will turn out to be equivalent to BFA (Γ , ≤ κ ) in the present situation). L [ G ] is the desired model. I will show thatit satisfies (1)-(5).The forcing P is of the form P ∗ ˙ P , where P is Woodin’s fast function forcingat κ and ˙ P is a P -name for an iteration of forcings in Γ as in Theorem 4.17. Theforcing P is κ -c.c. and (much more than) countably closed. It follows that thecomposition P = P ∗ ˙ P is in Γ, and hence, it follows that in L [ G ], L is correctabout uncountable cofinalities, that is, (4) holds in L [ G ]. This implies, by Lemma4.23, that BSCFA ( ≤ ω ) fails, since otherwise, stationarily many ordinals below ω would have to be regular in L yet of countable cofinality in L [ G ] (but there is nota single ordinal like that). So (5) holds in L [ G ].Let G = G ∗ G , where G is P -generic over L and G is P = ˙ P G -genericover L [ G ]. By [17, Lemma 4.13], κ is still +1-reflecting in L [ G ], and in particularinaccessible.Working in L [ G ] temporarily, let me analyze the iteration giving rise to P . Itis an iteration of length κ such that each initial segment of the iteration is in V κ (in the sense of L [ G ]). Due to the intermediate collapses in the iteration, it followsthat κ = ω L [ G ]2 . Thus, in L [ G ], we have wBFA (Γ , ≤ ω ), which, by [17, Obs. 4.7], is equivalent to BFA (Γ , ≤ ω ). Thus, we have (2). By part (2) of Lemma 4.24, thisimplies SC - SRP ( ω ), so that (3) is satisfied. The collapses in the iteration will force CH , and once CH is true, it remains true, since no reals are added, so we have (1),completing the proof. (cid:3) In the follow-up article [8], joint with Sean Cox, we will introduce a diagonalversion of SRP , which strengthens SRP , and we will consider the canonical frag-ments of this principle. The principle is designed in such a way that it capturesthose MM -consequences on diagonal reflection that SRP fails to capture. Thus,the subcomplete fragment of the diagonal SRP implies that for regular κ > ω , DSR ( ω , S κω ) holds, and the full principle (that is, the stationary set preservingfragment) implies this for regular κ > ω . Thus, neither Larson’s result separating SRP from MM , nor Theorem 4.7 serve to separate the subcomplete fragment ofthe diagonal SRP from SCFA . However, the previous result, Theorem 4.25, doesprovide such a separation at the level ω because, using the terminology used inits statement, in the model constructed, we have BFA (Γ , ≤ ω ) + CH + 2 ω = ω ,and this implies the subcomplete fragment of the diagonal SRP . Since BSCFA ( ≤ ω )fails in the model, this shows that the subcomplete fragment of the diagonal SRP at ω does not imply BSCFA ( ≤ ω ).5. Questions I will list questions by the related section in this article.Section 2: in subsection 2.2, the present formulation of the Γ-fragment of SRP is given, postulating that if the natural forcing P S to add a continuous ∈ -chainthrough a stationary set S is in Γ, then such a sequence exists. A potentiallystronger formulation would ask that if there is any forcing in Γ that adds sucha sequence, then such a sequence should exist. The question is whether this canbe a stronger principle. More broadly, can there be forcings in Γ that add such asequence when P S is not in Γ? Subsection 2.3 introduced the subcomplete fragmentof SRP , and an early observation was that SC - SRP ( κ ) holds trivially if κ ≤ ω . Itis then natural to ask whether SC - SRP is consistent with 2 ω > ω . The same canbe asked about SCFA instead of SC - SRP .Section 3: there is a lot of room for questions here. Many consequences of SRP obviously don’t follow from its subcomplete fragment, but many others might.For example, the weak reflection principle WRP , or the strong Chang conjecture,would be candidates. One may ask the same questions about the full forcing axiom SCFA . In fact, these principles follow from SCFA + , so assuming the consistency ofa supercompact cardinal, they are consistent with SCFA + , together with ♦ , say. Itwould also be interesting to explore consequences of Theorem 3.33 on the mutualstationarity of sequences of sets of exact simultaneous reflection points. Anotherquestion is whether MM implies the full principle SFP ω in which it is not assumedthat the sequence ~D is a maximal partition of ω into stationary sets (see Definition3.12. If not, then this would be a consequence of the subcomplete fragment of SRP with CH that does not follow from MM .Section 4: the first main question for this section concerns subsections 4.1 and4.2, and asks whether the combination of CH with the subcomplete fragment of SRP implies OSR ω , or uDSR ( λ, S κω ), for any regular κ ≥ ω and any λ with 1 ≤ λ ≤ ω . A negative answer would separate SC - SRP + CH from SCFA , in fact, itwould even separate SC - SRP + CH from the subcomplete fragment of the diagonal ANONICAL FRAGMENTS OF THE STRONG REFLECTION PRINCIPLE 47 strong reflection principle mentioned at the end of Section 4.3, to be introduced in[8]. The underlying question here is how to guarantee the preservation of spreadout sets. Regarding subsection 4.3, there is a fundamental problem concerning thedifference between ∞ -subcompleteness and subcompleteness: can a forcing be foundthat is ∞ -subcomplete but not subcomplete? Does the Iteration Theorem 4.17 gothrough for ∞ -subcomplete forcing? Can the separation result be modified to showthat ∞ - SC - SRP ( ω ) + CH does not imply ∞ - SCFA ( ω )? Finally, is there a globalversion of the result, separating SCFA from the combination of the subcompletefragment of SRP with CH ? This would most likely also separate the combination ofthe subcomplete fragment of the diagonal strong reflection principle with CH from SCFA .Questions abound. References [1] D. Asper´o. A maximal bounded forcing axiom. Journal of Symbolic Logic , 67(1):130–142,2002.[2] J. Bagaria. Bounded forcing axioms as principles of generic absoluteness. Archive for Math-ematical Logic , 39:393–401, 2000.[3] J. Bagaria, V. Gitman, and R. Schindler. Remarkable cardinals, structural reflection, and theweak proper forcing axiom. Archive for Mathematical Logic , 56(1):1–20, 2017.[4] J. Barwise. Admissible Sets and Structures . Springer, Berlin, 1975.[5] J. E. Baumgartner, A. Hajnal, and A. M´at´e. Weak saturation properties of ideals. In A. Ha-jnal, R. Rado, and V. T. S´os, editors, Infinite and finite sets, vol. I , pages 137–158. North-Holland, Amsterdam, 1973.[6] M. Bekkali. Topics in Set Theory. Lebesgue Measurability, Large Cardinals, Forcing Axioms,Rho Functions . Springer, 1991. ISBN 978-3-540-47422-7.[7] B. Claverie and R. Schindler. Woodin’s axiom ( ∗ ), bounded forcing axioms, and precipitousideals on ω . Journal of Symbolic Logic , 77(2):475–498, 2012.[8] S. Cox and G. Fuchs. The diagonal strong reflection principle and its fragments. in prepara-tion .[9] J. Cummings, M. Foreman, and M. Magidor. Squares, scales and stationary reflection. Journalof Mathematical Logic , 01(01):35–98, 2001.[10] J. Cummings and M. Magidor. Martin’s Maximum and weak square. Proceedings of theAmerican Mathematical Society , 139(9):3339–3348, 2011.[11] Q. Feng and T. Jech. Projective stationary sets and a strong reflection principle. Journal ofthe London Mathematical Society , 58(2):271–283, 1998.[12] Q. Feng, T. Jech, and J. Zapletal. On the structure of stationary sets. Science in China SeriesA: Mathematics , (50):615–627, 2007.[13] M. Foreman and M. Magidor. Mutually stationary sequences of sets and the non-saturationof the non-stationary ideal on on P κ ( λ ). Acta Mathematica , 186:271–300, 2001.[14] M. Foreman, M. Magidor, and S. Shelah. Martin’s maximum, saturated ideals, and non-regular ultrafilters. Part I. Annals of Mathematics , 127(1):1–47, 1988.[15] H. Friedman. On closed sets of ordinals. Proceedings of the American Mathematical Society ,43(1):393–401, 1974.[16] G. Fuchs. Closure properties of parametric subcompleteness. Archive for Mathematical Logic ,57(7-8):829–852, 2018.[17] G. Fuchs. Hierarchies of forcing axioms, the continuum hypothesis and square principles. Journal of Symbolic Logic , 83(1):256–282, 2018.[18] G. Fuchs. Hierarchies of (virtual) resurrection axioms. Journal of Symbolic Logic , 83(1):283–325, 2018.[19] G. Fuchs. Subcomplete forcing principles and definable well-orders. Mathematical Logic Quar-terly , 64(6):487–504, 2018.[20] G. Fuchs. Diagonal reflections on squares. Archive for Mathematical Logic , 58(1):1–26, 2019.[21] G. Fuchs. Aronszajn tree preservation and bounded forcing axioms. Submitted , 2020. Preprintat arXiv:2001.03105 [math.LO]. [22] G. Fuchs and C. Lambie-Hanson. Separating diagonal stationary reflection principles. Ac-cepted for publication in the Journal of Symbolic L , 2020. Preprint at arXiv:2002.12862[math.LO].[23] G. Fuchs and A. Rinot. Weak square and stationary reflection. Acta Mathematica Hungarica ,155(2):393–405, 2018.[24] G. Fuchs and C. Switzer. Iteration theorems for subversions of forcing classes. In preparation ,2019.[25] M. Goldstern and S. Shelah. The bounded proper forcing axiom. Journal of Symbolic Logic ,60(1):58–73, 1995.[26] T. Jech. Set Theory: The Third Millenium Edition, Revised and Expanded . Springer Mono-graphs in Mathematics. Springer, Berlin, Heidelberg, 2003.[27] T. Jech. Stationary sets. In M. Foreman, A. Kanamori, and M. Magidor, editors, Handbookof Set Theory , volume 1, pages 93–128. Springer, 2009.[28] R. B. Jensen. The fine structure of the constructible hierarchy. Annals of Mathematical Logic Handwritten notes , 2010.[33] R. B. Jensen. Subcomplete forcing and L -forcing. In C. Chong, Q. Feng, T. A. Slaman, W. H.Woodin, and Y. Yang, editors, E-recursion, forcing and C ∗ -algebras , volume 27 of LectureNotes Series, Institute for Mathematical Sciences, National University of Singapore Journal of Symbolic Logic , 65(1):247–258, 2000.[36] T. Miyamoto. A note on weak segments of PFA. In C. Chong, Q. Feng, D. Ding, Q. Huang,and M. Yasugi, editors, Proceedings of the Sixth Asian Logic Conference , pages 175–197,1998.[37] S. Todorˇcevi´c. Notes on Forcing Axioms . World Scientific, 2014.[38] M. Viale, G. Audrito, and S. Steila. A boolean algebraic approach to semiproper iterations.2014. Preprint: arXiv:1402.1714 [math.LO].[39] A. Villaveces. Chains of end elementary extensions of models of set theory. Journal of Sym-bolic Logic , 63(3):1116–1136, 1998. The College of Staten Island (CUNY), 2800 Victory Blvd., Staten Island, NY 10314The Graduate Center (CUNY), 365 5th Avenue, New York, NY 10016 E-mail address : [email protected] URL ::