A forcing axiom deciding the generalized Souslin Hypothesis
aa r X i v : . [ m a t h . L O ] S e p A FORCING AXIOM DECIDING THE GENERALIZED SOUSLINHYPOTHESIS
CHRIS LAMBIE-HANSON AND ASSAF RINOT
Abstract.
We derive a forcing axiom from the conjunction of square anddiamond, and present a few applications, primary among them being the exis-tence of super-Souslin trees. It follows that for every uncountable cardinal λ ,if λ ++ is not a Mahlo cardinal in G¨odel’s constructible universe, then 2 λ = λ + entails the existence of a λ + -complete λ ++ -Souslin tree. Introduction
Trees. A tree is a partially ordered set ( T, < T ) with the property that forevery x ∈ T , the downward cone x ↓ := { y ∈ T | y < T x } is well-ordered by < T .The order type of ( x ↓ , < T ) is denoted by ht( x ), and the α th -level of the tree is theset T α := { x ∈ T | ht( x ) = α } . The tree ( T, < T ) is said to be χ -complete if forevery chain C ⊆ T of size < χ , there is x ∈ T such that C ⊆ x ↓ ∪ { x } .If κ is a regular, uncountable cardinal, then a κ -Aronszajn tree is a tree of size κ having no chains or levels of size κ , and a κ -Souslin tree is a tree of size κ havingno chains or antichains of size κ . As tree levels are antichains, any κ -Souslin treeis a κ -Aronszajn tree.In 1920, Mikhail Souslin [24] asked whether every ccc, dense, complete linearordering with no endpoints is isomorphic to the real line. In [12], Kurepa showedthat a negative answer to Souslin’s question is equivalent to the existence of an ℵ -Souslin tree. Attempts to settle the question by constructing an ℵ -Souslin treeproved unsuccessful but did lead to Aronszajn’s construction of an ℵ -Aronszajntree, which is described in [12]. The question remained open until it was proven,in [26], [9], [11], and [23], that, in contrast to the existence of ℵ -Aronszajn trees,the existence of ℵ -Souslin trees is independent of the usual axioms of set theory( ZFC ).As these objects proved incredibly useful and important, a systematic study oftheir consistency and interrelation was carried out. Following standard conventions,we let TP κ stand for the nonexistence of κ -Aronszajn trees (the tree property at κ ), SH κ stand for the nonexistence of κ -Souslin trees (the Souslin Hypothesis at κ ),and CH λ stand for 2 λ = λ + . Two early results read as follows: Theorem 1.1 (Specker, [25]) . For every cardinal λ , CH λ implies the failure of TP λ ++ . Mathematics Subject Classification.
Primary 03E05; Secondary 03E35, 03E57.
Key words and phrases.
Souslin tree, square, diamond, sharply dense set, forcing axiom, SDFA.This research was partially supported by the Israel Science Foundation (grant Here, ccc is a consequence of separability, asserting that every pairwise-disjoint family of openintervals is countable. By a cardinal , we always mean an infinite cardinal.
Theorem 1.2 (Jensen, [10]) . In G¨odel’s constructible universe, L , for every regu-lar, uncountable cardinal κ , the following are equivalent: • TP κ ; • SH κ ; • κ is a weakly compact cardinal. We remind the reader that a cardinal κ is weakly compact iff it is uncountableand Ramsey’s theorem holds at the level of κ , i.e., every graph of size κ contains aclique or an anticlique of size κ .Ever since Jensen’s result, the general belief has been that the consistency of SH κ for κ of the form λ ++ requires the consistency of a weakly compact cardinal.This conjecture is supported by the following later results: Theorem 1.3 (Mitchell and Silver, [15]) . The existence of a regular cardinal λ forwhich TP λ ++ holds is equiconsistent with the existence of a weakly compact cardinal.In particular, the consistency of a weakly compact cardinal gives the consistency of ¬ CH λ together with SH λ ++ . Theorem 1.4 (Laver and Shelah, [14]) . For every cardinal λ , if there is a weaklycompact cardinal above λ , then there is a forcing extension by a λ + -directed closedforcing notion in which CH λ and SH λ ++ both hold. Theorem 1.5 (Rinot, [17]) . For every cardinal λ , if CH λ , CH λ + , and SH λ ++ allhold, then λ ++ is a weakly compact cardinal in L . Whether the hypotheses of Theorem 1.5 are consistent, relative to any largecardinal assumption, is a major open problem.In this paper, we are interested in a possible converse for Theorem 1.4. As of now,the best result in this direction gives a lower bound of an inaccessible cardinal. Theorem 1.6 (Shelah and Stanley, [18]) . For every cardinal λ , if CH λ and SH λ ++ both hold, then λ ++ is an inaccessible cardinal in L . Here, we establish the following.
Theorem A.
For every uncountable cardinal λ , if CH λ and SH λ ++ both hold, then λ ++ is a Mahlo cardinal in L . The following table provides a clear summary of all of these results.Theorem λ CH λ CH λ + SH λ ++ lower bound upper bound1.3 regular ✗ ✓ ✓ weakly compact1.5 arbitrary ✓ ✓ ✓ weakly compact1.4 arbitrary ✓ ✗ ✓ weakly compact1.6 arbitrary ✓ ✗ ✓ inaccessibleA uncountable ✓ ✗ ✓ Mahlo1.2.
Combinatorial constructions.
In order to prove Theorem A, we develop ageneral framework for carrying out combinatorial constructions. It turns out that,in this and other applications, it is often desirable to be able to construct an objectof size κ + , where κ is a regular, uncountable cardinal, using approximations to that Recall that any weakly compact cardinal admits stationarily many Mahlo cardinals below it,and any Mahlo cardinal admits stationarily many inaccessible cardinals below it.
FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 3 object of size < κ . When one attempts to carry out such a construction just usingthe axioms of
ZFC , though, one naturally runs into problems: the constructionseems to require κ + steps, but the approximations may become too large after only κ steps.The usual way to attempt to overcome this problem is to assume, in additionto ZFC , certain nice combinatorial features of κ or κ + . One such feature, whosedefinition is motivated by precisely such constructions, is the existence of a ( κ, §
4] or [5, Chapter VIII]). Velleman [30], and Shelah and Stanley[18], present frameworks for carrying out constructions of objects of size κ + using a( κ, ♦ ( κ ) to build objects of size κ + out of approximations of size < κ . Ideas from these papers were used by Foreman, Magidor, and Shelah [6] toprove, assuming the consistency of a huge cardinal, the consistency of the existenceof an ultrafilter U on ω such that | ω ω / U| = ℵ , and later by Foreman [7] to prove,again assuming the consistency of a huge cardinal, the consistency of the existenceof an ℵ -dense ideal on ℵ .In this paper, we present a framework for constructions of objects of size κ + using ♦ ( κ ) and (cid:3) Bκ , a weakening of (cid:3) κ that, unlike (cid:3) κ itself, is implied by the existenceof a ( κ, P κ , introducethe notion of a sharply dense system , and formulate a forcing axiom, SDFA( P κ ),that asserts that for every P from the class P κ and every sequence hD i | i < κ i ofsharply dense systems, there is a filter G on P that meets each D i everywhere.The last two sections of the paper are devoted to the proof of the following: Theorem B.
For every regular uncountable cardinal κ , if ♦ ( κ ) and (cid:3) Bκ both hold,then so does SDFA( P κ ) . In Section 3, we give a few simple applications of the forcing axiom SDFA( P κ ).We open by pointing out that the Cohen forcing Add( κ, κ + ) is a member of theclass P κ . Then, we show that SDFA( P κ ) entails κ <κ = κ and (cid:3) Bκ . This has threeconsequences. First, it shows that our square hypothesis in Theorem B is optimal: Theorem B’.
Suppose that κ is a regular, uncountable cardinal and ♦ ( κ ) holds.Then the following are equivalent: • (cid:3) Bκ holds; • SDFA( P κ ) holds. Second, by Shelah’s theorem [19] stating that CH λ entails ♦ ( λ + ) for every un-countable cardinal λ , it gives cases in which the diamond hypothesis is optimal, aswell: Theorem B”.
For every successor cardinal κ > ℵ , the following are equivalent: • ♦ ( κ ) and (cid:3) Bκ both hold; • SDFA( P κ ) holds. Third, it implies that SDFA( P κ ) entails the existence of a strong stationarycoding set, i.e., a stationary subset of [ κ + ] <κ on which the map x sup( x ) is CHRIS LAMBIE-HANSON AND ASSAF RINOT injective. This is of interest because the existence of such a set was previouslyobtained by Shelah and Stanley [21] from their forcing axiom S κ ( ♦ ), which isequivalent to the existence of a ( κ, κ, λ , a λ ++ -super-Souslin tree is a λ ++ -tree ( T, < T ) with a certain highly absolutecombinatorial property that ensures that ( T, < T ) has a λ ++ -Souslin subtree in any ZFC extension W of the universe V that satisfies P W ( λ ) = P V ( λ ) and ( λ ++ ) W =( λ ++ ) V . These trees were introduced in a paper by Shelah and Stanley [18], wherethe existence of super-Souslin trees provided the primary application of the forcingaxiom isolated in that paper. In particular, they proved that the existence ofa λ ++ -super-Souslin tree follows from the existence of a ( λ + , CH λ . In [30] and [21] the same hypotheses are shown to entail the existenceof a λ ++ -super-Souslin tree which is moreover λ + -complete. Here, we prove thefollowing analogous result. Theorem C.
For every cardinal λ , SDFA( P λ + ) entails the existence of a λ + -complete λ ++ -super-Souslin tree. By Theorems B and C, and the fact that for any super-Souslin tree (
T, < T ),there exists some x ∈ T such that ( x ↑ , < T ) is Souslin, we obtain: Corollary 1.
For every cardinal λ , if ♦ ( λ + ) and (cid:3) Bλ + both hold, then there is a λ + -complete λ ++ -Souslin tree. Recalling Jensen’s theorem [10] stating that if (cid:3) κ fails, then κ + is a Mahlocardinal in L , and Shelah’s theorem [19] stating that CH λ entails ♦ ( λ + ) for everyuncountable cardinal λ , we see that Theorem A follows from Corollary 1.We also obtain a corollary concerning partition relations. Recall that, for ordinals α, β , and γ , the statement α → ( β, γ ) asserts that, for every coloring c : [ α ] →{ , } , either there exists B ⊆ α of order type β which is 0-monochromatic, orthere exists C ⊆ α of order type γ which is 1-monochromatic. By a recent theoremof Raghavan and Todorcevic [16], the existence of a κ + -Souslin tree entails κ + ( κ + , log κ ( κ + ) + 2) , where log κ ( κ + ) denotes the least cardinal ν such that κ ν > κ .We thus obtain the following corollary: Corollary 2.
Suppose that λ is an uncountable cardinal. If CH λ and λ ++ → ( λ ++ , λ + + 2) both hold, then λ ++ is a Mahlo cardinal in L . Note that by a theorem of Erd˝os and Rado, CH λ entails λ ++ → ( λ ++ , λ + + 1) .1.3. Notations and conventions.
We write c.o.i. as a shorthand for “continuous,order-preserving injection.” In particular, a c.o.i. is a map π from a set of ordinalsinto the ordinals such that π is continuous, order-preserving, and injective, and,moreover, dom( π ) is closed in its supremum. Thus, when we write, for example,“ π : y → κ + is a c.o.i.,” it is implicit that y is closed in its supremum. For ordinals θ < µ , let (cid:0) µθ (cid:1) := { Im( π ) | π : θ → µ is a c.o.i. } , i.e., (cid:0) µθ (cid:1) consists of all closed copiesof θ in µ .For a set of ordinals x , otp( x ) denotes the order type of x and, for all i < otp( x ), x ( i ) denotes the unique element α of x such that otp( x ∩ α ) = i . We write ssup( x ) :=sup { α + 1 | α ∈ x } , acc( x ) := { α ∈ x | sup( x ∩ α ) = α > } , nacc( x ) := x \ acc( x ), FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 5 acc + ( x ) := { α < ssup( x ) | sup( x ∩ α ) = α > } , and cl( x ) := x ∪ acc + ( x ). Byconvention, ssup( ∅ ) = sup( ∅ ) = 0. For sets of ordinals x and y , we write x ⊑ y iff y is an end-extension of x , i.e., y ∩ ssup( x ) = x . For cardinals λ < µ , let E µλ := { α < µ | cf( α ) = λ } , let E µ<λ := { α < µ | cf( α ) < λ } , let [ µ ] <λ := { x ⊆ µ || x | < λ } , and let [ µ ] := { ( α, β ) | α < β < µ } . Also, let H µ denote the collection ofall sets of hereditary cardinality less than µ .Throughout the paper, κ stands for an arbitrary regular, uncountable cardinal.For simplicity, the reader may assume that κ = ℵ .2. The forcing axiom
We begin by introducing the class P κ of forcing notions that will be of interest. Definition 2.1. P κ consists of all triples ( P , ≤ P , Q ) such that ( P , ≤ P ) is a forcingnotion, P ∈ Q ⊆ P , and all of the following requirements hold.(1) (Realms) For all p ∈ P , there is a unique x p ∈ [ κ + ] <κ , which we refer to asthe realm of p . The map p x p is a projection from ( P , ≤ P ) to ([ κ + ] <κ , ⊇ ):(a) x P = ∅ ;(b) for all q ≤ P p , we have x q ⊇ x p ;(c) for all p ∈ P and x ∈ [ κ + ] <κ with x ⊇ x p , there is q ≤ P p with x q = x .(2) (Scope) For all y ⊆ κ + , let P y := { p ∈ P | x p ⊆ y } and Q y := Q ∩ P y .Then P ∅ = { P } and P κ ⊆ H κ .(3) (Actions of c.o.i.’s) For every y ⊆ κ + and every c.o.i. π : y → κ + , π actson P y in such a way that, for all p, q ∈ P y :(a) π.p is in P with x π.p = π “ x p , and if p ∈ Q y , then π.p is in Q ;(b) π.q ≤ P π.p iff q ≤ P p ;(c) if π is the identity map, then π.p = p ;(d) if π ′ : y ′ → κ + is a c.o.i. with Im( π ) ⊆ y ′ , then π ′ . ( π.p ) = ( π ′ ◦ π ) .p ;(e) if π ′ : y ′ → κ + is a c.o.i. with x p ⊆ y ′ , then π ↾ x p = π ′ ↾ x p impliesthat π.p = π ′ .p .(4) (Restrictions) For all p ∈ P and α < κ + , there is a unique ≤ P -leastcondition r such that x r = x p ∩ α and p ≤ P r . This condition r is referredto as p ↾ ↾ α . Moreover:(a) if p ∈ Q , then p ↾ ↾ α ∈ Q ;(b) if q ≤ P p , then q ↾ ↾ α ≤ P p ↾ ↾ α .(5) (Vertical limits) Suppose that ξ < κ and h p η | η < ξ i is a sequence ofconditions from P such that, for all η < η ′ < ξ , we have p η = p η ′ ↾ ↾ ssup( x p η ).Then there is a unique condition p ∈ P such that x p = S η<ξ x p η and, forall η < ξ , p η = p ↾ ↾ ssup( x p η ). Moreover, if p η ∈ Q for all η < ξ , then p ∈ Q .(6) (Sharpness) For all q ∈ Q , x q is closed in its supremum. Moreover, for all p ∈ P , there is q ≤ P p with x q = cl( x p ) such that q ∈ Q .(7) (Controlled closure) Suppose that ξ < κ and h q η | η < ξ i is a decreasingsequence of conditions from Q . Let x := S η<ξ x q η . Suppose that α < ssup( x ) and that r ∈ Q ssup( x ∩ α ) is a lower bound for h q η ↾ ↾ α | η < ξ i . Thenthere is q ∈ Q such that:(a) q ↾ ↾ ssup( x ∩ α ) = r ;(b) x q = cl( x r ∪ x );(c) q is a lower bound for h q η | η < ξ i . CHRIS LAMBIE-HANSON AND ASSAF RINOT (8) (Amalgamation)
For all p ∈ Q , α < ssup( x p ), and q ∈ P α with q ≤ P p ↾ ↾ α ,we have that p and q have a unique ≤ P -greatest lower bound r . Moreover,it is the case that x r = x q ∪ x p and r ↾ ↾ α = q .We now introduce the class of families of dense sets that we will be interested inmeeting. Definition 2.2 (Sharply dense set) . Suppose that ( P , ≤ P , Q ) ∈ P κ and D is anonempty subset of P . Denote x D := T { x p | p ∈ D } . We say that D is sharplydense iff for every p ∈ P , there is q ∈ D with q ≤ P p such that x q = cl( x p ∪ x D ). Definition 2.3 (Sharply dense system) . Suppose that ( P , ≤ P , Q ) ∈ P κ . We saythat D ⊆ P ( P ) is a sharply dense system iff there exists an ordinal θ D < κ suchthat D is of the form { D x | x ∈ (cid:0) κ + θ D (cid:1) } , where for all x ∈ (cid:0) κ + θ D (cid:1) : • D x is sharply dense with x D x = x ; • for every p ∈ P , and every c.o.i. π : y → κ + with x ⊆ x p ⊆ y , we have p ∈ D x iff π.p ∈ D π “ x . Definition 2.4.
Suppose that ( P , ≤ P , Q ) ∈ P κ and D is a sharply dense system.We say that a filter G on P meets D everywhere iff, for all D ∈ D , G ∩ D = ∅ .We are now ready to formulate our forcing axiom for sharply dense systems. Definition 2.5.
SDFA( P κ ) is the assertion that, for every ( P , ≤ P , Q ) ∈ P κ andevery collection {D i | i < κ } of sharply dense systems, there exists a filter G on P such that, for all i < κ , G meets D i everywhere.3. Applications
In this section we present a few applications of SDFA( P κ ). Just before that, letus point out two features of members of the class P κ . Proposition 3.1.
Suppose that ( P , ≤ P , Q ) ∈ P κ . Then:(1) ( Q , ≤ P ) is κ -closed.(2) For all x ⊆ κ + , denote D x := { q ∈ Q | x q ⊇ x } . Then, for all θ < κ , { D x | x ∈ (cid:0) κ + θ (cid:1) } is a sharply dense system.Proof. (1) Suppose that ξ < κ and ~q = h q η | η < ξ i is a decreasing sequence ofconditions from Q . Note that if x := S η<ξ x q η is empty, then P is a lower boundfor ~q , so we may assume that x is nonempty. Since P ∈ Q and P = { P } , weinfer from Clause (4) of Definition 2.1 that { q η ↾ ↾ | η < ξ } = Q = { P } . So, byClause (7) of Definition 2.1, using α := 0 and r := P , we infer that ~q admits alower bound.(2) By Clauses (3a) and (6) of Definition 2.1. (cid:3) Next, we show that the actions of c.o.i.’s behave as expected with respect to therestriction operation.
Proposition 3.2.
Suppose that ( P , ≤ P , Q ) ∈ P κ , p ∈ P , α ∈ x p , and π : y → κ + isa c.o.i. with x p ⊆ y ⊆ κ + . Then π. ( p ↾ ↾ α ) = ( π.p ) ↾ ↾ π ( α ) .Proof. Let r := π. ( p ↾ ↾ α ). Since p ≤ P p ↾ ↾ α , Clause (3b) (of Definition 2.1) impliesthat π.p ≤ P r . In addition, by Clauses (3a) and (4), and since α ∈ y , we have: x r = π “ x p ↾ ↾ α = π “( x p ∩ α ) = π “ x p ∩ π “ α = π “ x p ∩ π ( α ) = x π.p ∩ π ( α ) . FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 7
This shows that r is a candidate for being ( π.p ) ↾ ↾ π ( α ). To finish the proof, fix anarbitrary q ∈ P such that x q = x π.p ∩ π ( α ) and π.p ≤ P q . We have to verify that r ≤ P q .Let π ′ := { ( δ, ε ) | ( ε, δ ) ∈ π } , so that π ′ is a c.o.i. and π ′ ◦ π and π ◦ π ′ are theidentity maps on their respective domains. Since π.p ≤ P q and x q ⊆ Im( π ), andby Clauses (3c) and (3d), we have p = π ′ . ( π.p ) ≤ P π ′ .q . Moreover, x π ′ .q = x p ∩ α ,so, by Clause (4), p ↾ ↾ α ≤ P π ′ .q . Now another application of Clauses (3b) and (3c)yields r ≤ P q . (cid:3) A warm-up example.
Let us point out that P := Add( κ, κ + ) belongs tothe class P κ . Specifically, p ∈ P iff p is a function from a subset of κ + × κ to 2and | p | < κ . Let p ≤ P q iff p ⊇ q . Let x p := { β ∈ κ + | ∃ η [( β, η ) ∈ dom( p )] } .Let Q := { p ∈ P | x p = cl( x p ) } . Whenever π is a c.o.i. from a subset of κ + to κ + and p ∈ P dom( π ) , we let π.p := { (( π ( β ) , η ) , i ) | (( β, η ) , i ) ∈ p } . We also let p ↾ ↾ α := { (( β, η ) , i ) ∈ p | β < α } . The reader is now encouraged to verify that, withthis definition, ( P , ≤ P , Q ) ∈ P κ .3.2. Cardinal arithmetic.
In this subsection, we identify a simple member of P κ and use it to prove that SDFA( P κ ) implies κ <κ = κ . Definition 3.3. P consists of all pairs p = ( x, f ) such that:(1) x ∈ [ κ + ] <κ ;(2) f is a function such that:(a) | f | < κ ;(b) dom( f ) ⊆ x × κ ;(c) for all ( β, η ) ∈ dom( f ), we have f ( β, η ) ⊆ β ∩ x .The coordinates of a condition p ∈ P will often be identified as x p and f p ,respectively. Definition 3.4.
For all p, q ∈ P , we let q ≤ P p iff x q ⊇ x p and f q ⊇ f p . Definition 3.5. Q := { p ∈ P | x p = cl( x p ) } . Definition 3.6.
Suppose that π is a c.o.i. from a subset of κ + to κ + . For each p ∈ P dom( π ) , we let π.p be the condition ( x, f ) such that:(1) x = π “ x p ;(2) f = { (( π ( β ) , η ) , π “ z ) | (( β, η ) , z ) ∈ f p } . Definition 3.7.
Suppose that p ∈ P and α < κ + . Then we define p ↾ ↾ α to be thecondition ( x, f ) such that:(1) x = x p ∩ α ;(2) f = { (( β, η ) , z ) ∈ f p | β < α } .It is readily verified that, with these definitions, ( P , ≤ P , Q ) is a member of P κ . Theorem 3.8.
Suppose κ <κ > κ . Then ( P , ≤ P , Q ) witnesses that SDFA( P κ ) fails.Proof. We commence with a simple observation.
Claim 3.8.1.
There exists a cardinal λ < κ for which | (cid:0) λλ (cid:1) | > κ .Proof. Since κ is regular, we have κ <κ = P λ<κ λ λ . So, since κ <κ ≥ κ + and κ + isregular, we may fix a cardinal λ < κ such that λ λ ≥ κ + . For every A ⊆ λ , let C A := acc( λ ) ∪ { α + 1 | α ∈ A } . CHRIS LAMBIE-HANSON AND ASSAF RINOT
Then A C A is an injection from P ( λ ) to (cid:0) λλ (cid:1) , and we are done. (cid:3) Fix a cardinal λ < κ such that | (cid:0) λλ (cid:1) | > κ . For each x ∈ (cid:0) κ + λ +1 (cid:1) , let D x be the setof all conditions ( x p , f p ) ∈ P such that: • x ⊆ x p ; • there is η < κ with (max( x ) , η ) ∈ dom( f p ) such that f p (max( x ) , η ) = x ∩ max( x ) . Evidently, D := { D x | x ∈ (cid:0) κ + λ +1 (cid:1) } is a sharply dense system.Towards a contradiction, suppose that SDFA( P κ ) holds. In particular, thereexists a filter G on P that meets D everywhere. Let f := S p ∈ G f p , so that f is afunction from a (possibly proper) subset of κ + × κ to P ( κ + ). Put Λ := { f ( λ, η ) |∃ η < κ [( λ, η ) ∈ dom( f )] } . Clearly, | Λ | ≤ κ . Finally, let C ∈ (cid:0) λλ (cid:1) be arbitrary. Since C ∪ { λ } ∈ (cid:0) κ + λ +1 (cid:1) , we have G ∩ D C ∪{ λ } = ∅ , and hence C ∈ Λ. It follows that (cid:0) λλ (cid:1) ⊆ Λ, contradicting the fact that | (cid:0) λλ (cid:1) | > κ ≥ | Λ | . (cid:3) Corollary 3.9.
SDFA( P κ ) entails κ <κ = κ . (cid:3) Baumgartner’s square.
In unpublished work, Baumgartner introduced theprinciple (cid:3) Bκ , which is a natural weakening of Jensen’s (cid:3) κ principle. Definition 3.10. A (cid:3) Bκ -sequence is a sequence h C β | β ∈ Γ i such that:(1) E κ + κ ⊆ Γ ⊆ acc( κ + );(2) for all β ∈ Γ, C β is club in β and otp( C β ) ≤ κ ;(3) for all β ∈ Γ and all α ∈ acc( C β ), we have α ∈ Γ and C α = C β ∩ α .The principle (cid:3) Bκ asserts the existence of a (cid:3) Bκ -sequence. Some basic facts about (cid:3) Bκ can be found in [30], where it goes by the name“weak (cid:3) κ .” In particular, it is shown in [30] that (cid:3) Bκ follows from the existence ofa ( κ, Theorem 3.11.
Suppose that
SDFA( P κ ) holds. Then so does (cid:3) Bκ . The rest of this subsection is devoted to proving Theorem 3.11. We must firstidentify a relevant member of P κ , which will be a slight modification of the posetused to add (cid:3) Bκ in [30, § Definition 3.12. P consists of all pairs p = ( x, f ) such that:(1) x ∈ [ κ + ] <κ ;(2) f is a function from x to P ( x ) such that for all β ∈ x :(a) f ( β ) is a closed subset of β ; (b) for all α ∈ acc( f ( β )), we have f ( α ) = f ( β ) ∩ α .The coordinates of a condition p ∈ P will often be identified as x p and f p ,respectively. Definition 3.13.
For all p, q ∈ P , we let q ≤ P p iff: • x p ⊆ x q ; • for all β ∈ x p , we have f p ( β ) ⊑ f q ( β ); Note that (cid:3) Bκ is equivalent to the principle (cid:3) κ ( κ + , ⊑ κ ) from [1, § We say that c is a closed subset of β iff c ⊆ β and for every α < β , c ∩ α = ∅ = ⇒ sup( c ∩ α ) ∈ c . FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 9 • for all β ∈ x p , if sup( x p ∩ β ) = β , then f q ( β ) = f p ( β ). Definition 3.14. Q is the set of all conditions p ∈ P such that:(1) x p = cl( x p );(2) for all β ∈ nacc( x p ) \ { min( x p ) } , we have max( f p ( β )) = max( x p ∩ β ).In order to show that ( P , ≤ P , Q ) ∈ P κ , we must define the actions of c.o.i.’s on P and a restriction operation. Definition 3.15.
Suppose that π is a c.o.i. from a subset of κ + to κ + . For each p ∈ P dom( π ) , we define π.p to be the condition ( x, f ) ∈ P such that:(1) x = π “ x p ;(2) for all α ∈ x p , we have f ( π ( α )) = π “ f p ( α ). Definition 3.16.
Suppose that p ∈ P and α < κ + . Then p ↾ ↾ α is the condition( x, f ) ∈ P such that x = x p ∩ α and f = f p ↾ x .Naturally, for each p ∈ P , we let x p denote the realm of p . With these definitions,it is immediate that ( P , ≤ P , Q ) satisfies Clauses (1)–(5) of Definition 2.1. We nowverify Clauses (6)–(8), in order. Lemma 3.17.
Suppose that p ∈ P . Then there is q ∈ Q with q ≤ P p such that x q = cl( x p ) .Proof. Set x q := cl( x p ), so that nacc( x q ) = nacc( x p ) and acc( x q ) ⊇ acc( x p ). Next,define f q : x q → P ( x q ) by stipulating: f q ( α ) := f p ( α ) ∪ { max( x q ∩ α ) } if α ∈ nacc( x q ) \ { min( x q ) } ; ∅ if α ∈ x q \ x p ; f p ( α ) otherwise . It is clear that q := ( x q , f q ) is as desired. (cid:3) Lemma 3.18.
Suppose that ξ < κ and h q η | η < ξ i is a decreasing sequenceof conditions from Q . Let x := S η<ξ x q η , and suppose that α < ssup( x ) and r ∈ Q ssup( x ∩ α ) is a lower bound for h q η ↾ ↾ α | η < ξ i . Then there is q ∈ Q such that:(1) q ↾ ↾ ssup( x ∩ α ) = r ;(2) x q = cl( x r ∪ x ) ;(3) q is a lower bound for h q η | η < ξ i .Proof. We will construct a condition q = ( x q , f q ) as desired. We are required to let x q := cl( x r ∪ x ) and to ensure that f q ↾ x r := f r . As x r ⊑ x q , it remains to determine f q ↾ ( x q \ ssup( x ∩ α )). We will define f q ( β ) by recursion on β ∈ ( x q \ ssup( x ∩ α )),maintaining the hypothesis that ( x q ∩ ( β + 1) , f q ↾ ( β + 1)) is an element of Q and alower bound for h q η ↾ ↾ ( β +1) | η < ξ i . For notational ease, if β ∈ nacc( x q ) \{ min( x q ) } ,then let β − := max( x q ∩ β ). ◮ If β ∈ acc( x ), then fix η β < ξ such that β ∈ x q ηβ , and let f q ( β ) := S η ∈ [ η β ,ξ ) f q η ( β ). There are two possibilities to consider here. If there is η ∗ ∈ [ η β , ξ )such that sup( x q η ∗ ∩ β ) = β , then it follows from Definition 3.13 that f q ( β ) = f q η ∗ ( β ).If, on the other hand, there is no such η ∗ , then the fact that each x q η is closedin its supremum implies that, for all η ∈ [ η β , ξ ), we have β ∈ nacc( x q η ) and hence max( f q η ( β )) = max( x q η ∩ β ). Since β ∈ acc( x ), it then follows that f q ( β ) is clubin β . ◮ If β ∈ acc + ( x ) \ x , then let γ := min( x \ ( β + 1)). There is η β < ξ suchthat, for all η ∈ [ η β , ξ ), we have γ ∈ x q η and x q η ∩ β = ∅ . For all such η , let δ η := max( x q η ∩ β ). It follows that sup { δ η | η ∈ [ η β , ξ ) } = β and, for all η ∈ [ η β , ξ ),we have max( f q η ( γ )) = δ η . We can therefore let f q ( β ) := S η ∈ [ η β ,ξ ) f q η ( γ ). ◮ If β ∈ nacc( x ) and β − / ∈ x , then, by the construction in the previous case, wehave f q ( β − ) = S η<ξ f q η ( β ). We can therefore let f q ( β ) := f q ( β − ) ∪ { β − } . ◮ If β ∈ nacc( x ) and β − ∈ x , then there is η β < ξ such that { β, β − } ⊆ x q ηβ .But then, for all η ∈ [ η β , ξ ), we have f q η ( β ) = f q ηβ ( β ) and max( f q η ( β )) = β − . Wecan therefore let f q ( β ) := f q ηβ ( β ).It is easily verified that q , constructed in this manner, is as desired. (cid:3) Lemma 3.19.
Suppose that p ∈ Q , α < ssup( x p ) , q ∈ P α , and q ≤ P p ↾ ↾ α . Then p and q have a ≤ P -greatest lower bound, r . Moreover, we have x r = x p ∪ x q and r ↾ ↾ α = q .Proof. Let x r := x p ∪ x q , so that x r ∩ α = x q . Define f r : x r → P ( x r ) by stipulating: f r ( β ) := ( f q ( β ) if β < α ; f p ( β ) otherwise . To see that r := ( x r , f r ) is a condition, we fix arbitrary β ∈ x r and γ ∈ acc( f r ( β )), and verify that f r ( γ ) = f r ( β ) ∩ γ . To avoid trivialities, suppose that β ≥ α > γ . Since f r ( β ) = f p ( β ) ⊆ x p , we have sup( x p ∩ γ ) = γ , so, since q ≤ P p ↾ ↾ α ,we infer that f q ( γ ) = f p ( γ ) = f p ( β ) ∩ γ , i.e., f r ( γ ) = f r ( β ) ∩ γ .It is now readily checked that r has the desired properties. (cid:3) It follows that ( P , ≤ P , Q ) ∈ P κ . For each x ∈ (cid:0) κ + (cid:1) , let D x := { p ∈ Q | x p ⊇ x } .By Proposition 3.1(2), D := { D x | x ∈ (cid:0) κ + (cid:1) } is a sharply dense system, so wecan apply SDFA( P κ ) to obtain a filter G on P that meets D everywhere. For all β ∈ E κ + κ , let C β := S { f p ( β ) | p ∈ G, β ∈ x p } . Note that for all p ∈ G and β ∈ x p ,we have | f p ( β ) | ≤ | x p | < κ . Claim 3.20.
Suppose that β, γ ∈ E κ + κ . Then:(1) C β is club in β and otp( C β ) = κ ;(2) For all α ∈ acc( C β ) ∩ acc( C γ ) , we have C β ∩ α = C γ ∩ α .Proof. (1) By the definition of P and the fact that G is a filter, it follows that C β is a subset of β , closed in its supremum, such that every proper initial segment of C β has size < κ . It thus suffices to verify that C β is unbounded in β . To this end,fix α < β . Since G meets D everywhere, we can find p ∈ G ∩ D { α,β,β +1 } . Sincecf( β ) = κ and | x p | < κ , we have β ∈ nacc( x p ). Therefore, since p ∈ Q , we havemax( f p ( β )) = max( x p ∩ β ) ≥ α , so C β ∩ [ α, β ) = ∅ .(2) Given α ∈ acc( C β ) ∩ acc( C γ ), we fix p ∈ G ∩ D { α,β,γ } . As in the previous case,we have max( f p ( β )) ≥ α and max( f p ( γ )) ≥ α . Consequently, C β ∩ α = f p ( β ) ∩ α and C γ ∩ α = f p ( γ ) ∩ α . By the definition of P , it then follows that C β ∩ α = f p ( α ) = C γ ∩ α . (cid:3) Let Γ := E κ + κ ∪ S { acc( C β ) | β ∈ E κ + κ } . For each α ∈ Γ \ E κ + κ , find β ∈ E κ + κ such that α ∈ acc( C β ), and let C α := C β ∩ α . By the preceding Claim, this is FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 11 independent of the choice of β . It follows that h C α | α ∈ Γ i is a (cid:3) Bκ -sequence, thuscompleting the proof of Theorem 3.11.3.4. Strong stationary coding sets.
In [21], Shelah and Stanley derive a sta-tionary coding set from the existence of a ( κ, ♦ . Specifically,they obtain a stationary subset S of [ κ + ] <κ on which the map x sup( x ) is one-to-one. By Theorem 3.11 and the next proposition, this also follows from the forcingaxiom SDFA( P κ ). Proposition 3.21 (folklore) . If (cid:3) Bκ holds, then there exists a stationary subset of [ κ + ] <κ on which the map x sup( x ) is one-to-one.Proof. Let h C β | β ∈ Γ i be a (cid:3) Bκ -sequence. Enlarge it to a sequence ~C = h C β | β < κ i by letting, for all limit β ∈ κ \ Γ, C β be an arbitrary club in β of order typecf( β ), and letting C β +1 := { β } for all β < κ .Let ρ ~C : [ κ + ] → κ denote the associated maximal weight function from [27, § β < κ + , let ρ β : β → κ denote the fiber map ρ ~C ( · , β ). Note that: • for all β < κ + , ρ β [ C β ] = otp( C β ); • for all β < κ + , ρ β is ( < κ )-to-1; • for all β ∈ Γ and α ∈ acc( C β ), we have ρ α ⊆ ρ β .In particular, for every β ∈ E κ + <κ , we have that x β := ( ρ β ) − [otp( C β )]is a cofinal subset of β of size < κ . Thus, we are left with proving the following. Claim 3.22. { x β | β ∈ E κ + <κ } is stationary in [ κ + ] <κ .Proof. Given a function f : [ κ + ] <ω → κ + , let us fix some γ ∈ E κ + κ such that f “[ γ ] <ω ⊆ γ . Define g : κ → κ by letting, for all ε < κ , g ( ε ) := sup( ρ γ “ f “[ ρ − γ [ ε ]] <ω ) . Fix ǫ ∈ acc( κ ) such that g [ ǫ ] ⊆ ǫ . Put β := C γ ( ǫ ), so that otp( C β ) = ǫ and ρ β ⊆ ρ γ . To see that f “[ x β ] <ω ⊆ x β , let { α i | i < n } ∈ [ x β ] <ω be arbitrary. Since x β = ( ρ β ) − [ ǫ ] and ρ β ⊆ ρ γ , we have { ρ γ ( α i ) | i < n } = { ρ β ( α i ) | i < n } ∈ [ ǫ ] <ω . Fix a large enough ε < ǫ such that { α i | i < n } ∈ [ ρ − γ [ ε ]] <ω . Since g ( ε ) < ǫ , wethen have f ( { α i | i < n } ) ∈ x β . (cid:3) This completes the proof. (cid:3)
Note that, by [8, § GCH -freeversion of ♦ . For more information on stationary coding sets, see [31].4. Super-Souslin trees
Throughout this section, λ denotes an arbitrary cardinal.The notion of a λ ++ -super-Souslin tree was isolated by Shelah in response towork by Laver on trees with ascent paths. Ascent paths provide obstacles to atree being special; super-Souslin trees are designed to present a similar obstaclethat entails the existence not only of a non-special tree but of a Souslin one. InSubsection 4.1, we provide, as a means of helping to motivate and provide context for the definition of super-Souslin trees, some remarks on the connection betweenthese notions. In Subsection 4.2, we provide a proof of Theorem C: Theorem C.
Suppose that
SDFA( P λ + ) holds. Then there exists a λ + -complete λ ++ -super-Souslin tree. Introduction to super-Souslin trees.
A tree (
T, < T ) is said to be a κ -tree if for every α < κ , T α is a nonempty set of size < κ and T κ = ∅ . The tree is saidto be splitting if every node in the tree admits at least two immediate successors.It is said to be normal if, for all α < β < κ and all u ∈ T α , there is v ∈ T β suchthat u < T v . It is said to be Hausdorff if for all limit α < κ and all u, v ∈ T α , theequality u ↓ = v ↓ implies u = v . For convenience, we will not require that a treebe Hausdorff. Note, however, that any splitting (resp. normal) tree ( T, < T ) caneasily be turned into a splitting (resp. normal) Hausdorff tree ( T ′ , < T ′ ) by shiftingall levels T α to be T ′ α +1 and, for limit α < κ , letting T ′ α consist of unique limits ofall branches through S β<α T β that are continued in T α . Definition 4.1.
Let θ be an arbitrary cardinal. For each α < κ , let T θα denote thecollection of all injections a : θ → T α . Let T θ denote S α<κ T θα .An element of T θ will be referred to as a θ -level sequence from T (or, simply, a level sequence from T ). For a, b ∈ T θ , we abuse notation and write a < T b iff, forall i < θ , a ( i ) < T b ( i ). Likewise, a ≤ T b iff, for all i < θ , a ( i ) ≤ T b ( i ). Definition 4.2. [ T θ ] := { ( a, b ) ∈ T θ × T θ | a < T b } . Definition 4.3 (Shelah, [18]) . A λ ++ -super-Souslin tree is a normal, splitting λ ++ -tree ( T, < T ) for which there exists a function F : [ T λ ] → λ + satisfying thefollowing condition: for all a, b, c ∈ T λ with a < T b, c , if F ( a, b ) = F ( a, c ), thenthere is i < λ such that b ( i ) and c ( i ) are < T -comparable. Fact 4.4 (Shelah, [18]) . Suppose ( T, < T ) is a λ ++ -super-Souslin tree. If W is anouter model of V with the same P ( λ ) and λ ++ , then, in W , there exists some x ∈ T such that ( x ↑ , < T ) is a λ ++ -Souslin tree. The next lemma shows that the two-dimensional function F witnessing that atree ( T, < T ) is λ ++ -super-Souslin cannot be replaced by a one-dimensional function. Lemma 4.5.
Suppose that ( T, < T ) is a normal, splitting κ -tree, and θ, µ are cardi-nals < κ (e.g., κ = λ ++ , µ = λ + , and θ = λ .) There exists no function F : T θ → µ such that, for every a, b ∈ T θ , if F ( a ) = F ( b ) , then there is i < θ such that a ( i ) and b ( i ) are < T -comparable.Proof. Suppose for sake of contradiction that there is such a function F . We firstargue that ( T, < T ) is a κ -Souslin tree. Furthermore: Claim 4.5.1.
Suppose W is an outer model of V in which κ is not collapsed. Then ( T, < T ) is a κ -Souslin tree in W .Proof. Work in V . As the proof of Claim A.7.1 of [3] makes clear, the fact that( T, < T ) is normal and splitting implies that for every u ∈ T , we may find some a u ∈ T θ such that u < T a u ( i ) for all i < θ . Next, let us work in W , where W is anouter model of V in which κ is not collapsed. Since ( T, < T ) is a splitting κ -tree, toshow that ( T, < T ) is κ -Souslin, it suffices to show that it has no antichains of size κ . Towards a contradiction, suppose that U := { u α | α < κ } is an antichain. While FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 13 it is possible that U ∈ W \ V , we nevertheless have { a u α | α < κ } ⊆ V . Since κ is not collapsed, we may find ordinals α < β < κ such that F ( a u α ) = F ( a u β ).Pick i < θ such that a u α ( i ) and a u β ( i ) are < T -comparable. Then u α and u β are < T -comparable. This is a contradiction. (cid:3) Now force over V with the forcing notion P := ( T, > T ) (i.e., the order of P isthe reverse of the tree order). As ( T, < T ) is a κ -Souslin tree in V , we have that P has the κ -c.c. and does not collapse κ . Therefore, the preceding claim impliesthat ( T, < T ) is a κ -Souslin tree in V P , contradicting the fact that P adds a cofinalbranch through ( T, < T ). (cid:3) The next lemma shows that the range of the function F witnessing that a tree( T, < T ) is λ ++ -super-Souslin cannot be smaller than λ + . In particular, there isno straightforward generalization of the notion of super-Souslin tree to inaccessiblecardinals. Lemma 4.6.
Suppose that ( T, < T ) is a normal, splitting κ -tree, and θ, µ are car-dinals < κ . If µ + < κ , then there exists no function F : [ T θ ] → µ such that, forall a, b, c ∈ T θ with a < T b, c , if F ( a, b ) = F ( a, c ) , then there is i < θ such that b ( i ) and c ( i ) are < T -comparable.Proof. Suppose that F is a counterexample. Fix an arbitrary a ∈ T θ . As the proofof Claim A.7.1 of [3] makes clear, the fact that ( T, < T ) is normal and splittingimplies that there exists some large enough β < κ and an injection b : µ + × θ → T β such that for all η < µ + and all i < θ , a ( i ) < T b ( η, i ). For each η < µ + , define b η : θ → T β by stipulating b η ( i ) := b ( η, i ). Now, find η < ζ < µ + such that F ( a, b η ) = F ( a, b ζ ). Then there must exist some i < θ such that b η ( i ) and b ζ ( i ) are < T -comparable, contradicting the fact that b η ( i ) and b ζ ( i ) are two distinct elementsof T β . (cid:3) Now, we move on to deal with the notion of an ascent path.
Definition 4.7 (Laver) . Suppose that θ is a cardinal < κ and F is a familysatisfying θ ∈ F ⊆ P ( θ ). An F -ascent path through a κ -tree ( T, < T ) is a sequence ~f = h f α | α < κ i such that for all α < β < κ :(1) f α is a function from θ to T α ;(2) { i < θ | f α ( i ) < T f β ( i ) } ∈ F . Definition 4.8.
For every cardinal θ , write F fin θ := { Z ⊆ θ | | θ \ Z | < ω } , F bd θ := { Z ⊆ θ | sup( θ \ Z ) < θ } , and F θ := P ( θ ) \ {∅} .By [22], if ( T, < T ) is a special λ + -tree that admits an F bd θ -ascent path, thencf( θ ) = cf( λ ). By [28], if λ is regular and ( T, < T ) is a special λ + -tree that admitsan F θ -ascent path, then θ = λ . A construction of a special λ + -tree with an F bdcf( λ ) -ascent path may be found in [13]. Constructions of κ -Souslin trees with F fin θ -ascentpaths may be found in [3]. Proposition 4.9 (folklore) . Any λ ++ -super-Souslin tree ( T, < T ) admits an F λ -ascent path.Proof. Suppose (
T, < T ) is a λ ++ -super-Souslin tree with a witnessing map F :[ T λ ] → λ + . Fix an arbitrary a ∈ T λ . Let ǫ be such that a ∈ T λǫ . By normality of( T, < T ), for each β ∈ λ ++ \ ǫ , we may fix a β ∈ T λβ with a ≤ T a β . Pick a cofinal subset B ⊆ λ ++ \ ǫ on which the map β F ( a, a β ) is constant. Then h a β | β ∈ B i induces an F θ -ascent path ~f = h f α | α < κ i , as follows. For every α < λ ++ , let β ( α ) := min( B \ α ), and define f α : λ → T α by letting f α ( i ) be the unique elementof T α which is ≤ T a β ( α ) ( i ). (cid:3) Aiming for an F bd λ -ascent path, one may want to strengthen Definition 4.3 toassert that for all a, b, c ∈ T λ with a < T b, c , if F ( a, b ) = F ( a, c ), then I ( b, c ) := { i < λ | b ( i ) and c ( i ) are < T -comparable } is in F bd λ . However, this is impossible: Lemma 4.10.
Suppose ( T, < T ) is a normal, splitting λ ++ -tree, F : [ T λ ] → λ + ,and F is a proper filter on λ . Then there are ( a, b ) , ( a, c ) ∈ [ T λ ] with F ( a, b ) = F ( a, c ) such that I ( b, c ) / ∈ F .Proof. Towards a contradiction, suppose that for all ( a, b ) , ( a, c ) ∈ [ T λ ] with F ( a, b ) = F ( a, c ), we have I ( b, c ) ∈ F . For all a ∈ T λ and η < λ + , let U a := { b ∈ T λ | a ≤ T b } and U ηa := { b ∈ U a | F ( a, b ) = η } . Now, fix some a ∈ T λ arbi-trarily, and, for every η < λ + , let U η := { b ∈ U a | U b ∩ U ηa = ∅} be the downwardclosure of U ηa within U a . Claim 4.10.1.
Suppose that η < λ + and b, c ∈ U η . Then I ( b, c ) ∈ F .Proof. Pick b ′ ∈ U b ∩ U ηa and c ′ ∈ U c ∩ U ηa . Since F ( a, b ′ ) = η = F ( a, c ′ ), and byassumption, we have that I ( b ′ , c ′ ) ∈ F .Let β, β ′ , γ, γ ′ be such that b ∈ T λβ , b ′ ∈ T λβ ′ , c ∈ T λγ , and c ′ ∈ T λγ ′ . Withoutloss of generality, β ′ ≤ γ ′ . Now, there are two relevant configurations of the otherordinals to consider. Case 1: β ≤ β ′ < γ . In this case, for all i ∈ I ( b ′ , c ′ ), we have b ( i ) ≤ T b ′ ( i ) and b ′ ( i ) , c ( i ) ≤ T c ′ ( i ), so b ( i ) and c ( i ) are < T -comparable. Case 2: β, γ ≤ β ′ . In this case, for all i ∈ I ( b ′ , c ′ ), we have b ( i ) , c ( i ) ≤ T b ′ ( i )and again, b ( i ) and c ( i ) are < T -comparable. (cid:3) For any two distinct ordinals η, ζ below λ + , let δ η,ζ denote the least ordinal δ below λ ++ such that there are b ∈ U η ∩ T λδ and c ∈ U ζ ∩ T λδ for which I ( b, c ) = ∅ ,if such an ordinal exists; otherwise, leave δ η,ζ undefined. Claim 4.10.2.
Suppose δ η,ζ is defined. Then U η ∩ U ζ ⊆ S { T λβ | β < δ η,ζ } .Proof. Towards a contradiction, suppose that d ∈ U η ∩ U ζ ∩ T λβ for some β ≥ δ η,ζ .Since U η and U ζ are downward closed, we may simply assume that β = δ η,ζ .Using the fact that β = δ η,ζ , fix b ∈ U η ∩ T λβ and c ∈ U ζ ∩ T λβ such that I ( b, c ) = ∅ .By Claim 4.10.1, since b, d ∈ U η and c, d ∈ U ζ , we have that I ( b, d ) and I ( c, d ) arein F . In particular, I ( b, d ) ∩ I ( c, d ) = ∅ , contradicting the fact that I ( b, c ) = ∅ . (cid:3) As λ + < λ ++ , let β < λ ++ be large enough so that, if η, ζ are two distinctordinals below λ + and δ η,ζ is defined, then δ η,ζ < β . By increasing β if necessary, wemay assume that U a ∩ T λβ = ∅ . Fix d ∈ U a ∩ T λβ . By the fact that ( T, < T ) is splitting,for each i < λ we may fix e ( i ) = e ( i ), both in T β +1 , with d ( i ) < T e ( i ) , e ( i ).Let η := F ( a, e ) and ζ := F ( a, e ). Clearly, I ( e , e ) = ∅ , so that δ η,ζ is defined.So, by our choice of β , we have δ η,ζ < β . However, since d < T e , e , we have d ∈ U η ∩ U ζ ∩ T λβ , contradicting Claim 4.10.2. (cid:3) FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 15
Proof of Theorem C.
The rest of this section is devoted to proving Theo-rem C. We will define a poset ( P , ≤ P , Q ) ∈ P λ + and a collection {D i | i < λ + } ofsharply dense systems such that any filter that meets each D i everywhere gives riseto a λ + -complete λ ++ -super-Souslin tree. We intend to construct a tree ( T, < T )with underlying set λ ++ × λ + , such that, furthermore, T α = { α } × λ + for all α < λ ++ . We start by defining P . Definition 4.11. P consists of all quintuples p = ( x, < , t, < , f ) satisfying thefollowing requirements.(1) x ∈ [ λ ++ ] <λ + .(2) < is a partial ordering on x such that for all β ∈ x , we have thatpred p ( β ) := { α ∈ x | α < β } is a closed subset of β which is well-ordered by < .(3) t ∈ [ x × λ + ] <λ + . In a slight abuse of notation, and anticipating the genericobject, for all α ∈ x , we let t α denote t ∩ ( { α } × λ + ), and we let t λα denotethe set of injective functions from λ to t α . For each a in t λ := S α ∈ x t λα , welet Lev( a ) denote the unique ordinal α such that a ∈ t λα .(4) < is a tree order on t such that, for all β ∈ x and all v ∈ t β , lettingpred p ( v ) := { u ∈ t | u < v } , we have that { α ∈ x | pred p ( v ) ∩ t α = ∅} = pred p ( β ) . Let [ t λ ] := { ( a, b ) | a, b ∈ t λ , a < b } , where for a, b ∈ t λ , we write a < b iff a ( i ) < b ( i ) for all i < λ .(5) f is a partial function from [ t λ ] to [ λ + ] <λ + \ {∅} , and | f | ≤ λ .(6) Suppose that ( a, b ) , ( a, c ) ∈ dom( f ). If f ( a, b ) ∩ f ( a, c ) = ∅ and Lev( b ) ≤ Lev( c ), then |{ i < λ | b ( i ) ≤ c ( i ) }| = λ .(7) For all ( a, c ) ∈ dom( f ) and all b ∈ t λ such that a < b < c , we have( a, b ) ∈ dom( f ) and f ( a, b ) ⊇ f ( a, c ).The coordinates of a condition p ∈ P will often be identified as x p , < p , t p , < p ,and f p , respectively. Definition 4.12.
For all p, q ∈ P , we let q ≤ P p iff: • x q ⊇ x p ; • < q ⊇ < p ; • t q ⊇ t p ; • < q ⊇ < p ; • dom( f q ) ⊇ dom( f p ); • for all ( a, b ) ∈ dom( f p ), we have f q ( a, b ) ⊇ f p ( a, b ). Definition 4.13. Q is the set of all conditions p ∈ P such that:(1) x p = cl( x p );(2) < p is the usual ordinal ordering on x p .We now show that ( P , ≤ P , Q ) is in P λ + . For p ∈ P , x p is the realm of p . We nextdescribe how c.o.i.’s act on P . In order to make it easier to refer to and manipulatelevel sequences in our conditions, we introduce the following notation. Notation 4.14.
By SDFA( P λ + ) and Corollary 3.9, CH λ holds, and we can let h σ δ | δ < λ + i injectively enumerate λ λ + . For all α < λ ++ and δ < λ + , let a α,δ : λ → { α } × λ + be defined by stipulating a α,δ ( i ) := ( α, σ δ ( i )). Note that everylevel sequence in our desired tree ( T, < T ) will be of the form a α,δ for a unique pair( α, δ ) ∈ λ ++ × λ + . Definition 4.15.
Suppose that π is a c.o.i. from a subset of λ ++ to λ ++ . For each p ∈ P dom( π ) , we define π.p to be the condition ( x, < , t, < , f ) ∈ P such that:(1) x = π “ x p ;(2) < = { ( π ( α ) , π ( β )) | ( α, β ) ∈ < p } ;(3) t = { ( π ( α ) , η ) | ( α, η ) ∈ t p } ;(4) < = { (( π ( α ) , η ) , ( π ( β ) , ζ )) | (( α, η ) , ( β, ζ )) ∈ < p } ;(5) f = { (( a π ( α ) ,δ , a π ( β ) ,ǫ ) , z ) | (( a α,δ , a β,ǫ ) , z ) ∈ f p } .Finally, we describe the restriction operation. Definition 4.16.
Suppose that p ∈ P and α < λ ++ . Then p ↾ ↾ α is the condition( x, < , t, < , f ) such that: • x = x p ∩ α ; • < = < p ∩ x ; • t = t p ∩ ( α × λ + ); • < = < p ∩ t ; • f = { (( a, b ) , z ) ∈ f p | ( a, b ) ∈ [ t λ ] } .With these definitions, it follows easily that ( P , ≤ P , Q ) satisfies Clauses (1)–(5)of Definition 2.1. We now verify Clauses (6)–(8), in order. Lemma 4.17.
Suppose p ∈ P . Then there is q ∈ Q with q ≤ P p such that x q =cl( x p ) .Proof. We need to define q = ( x q , < q , t q , < q , f p ). Of course, we let x q := cl( x p )and let < q be the usual ordinal ordering on x q . Thus, the main task is in findingsuitable t q , < q and f q . Our strategy is to define the first two and then derive f q byminimally extending f p so as to satisfy Clause (7) of Definition 4.11. To be precise,once t q and < q are determined, we will letdom( f q ) := dom( f p ) ∪ { ( a, b ) ∈ [ t λq ] | ∃ c ∈ t λp (cid:0) a < q b < q c and ( a, c ) ∈ dom( f p ) (cid:1) } and, for all ( a, b ) ∈ dom( f q ), we will let f q ( a, b ) := f p ( a, b ) ∪ [ { f p ( a, c ) | ( a, c ) ∈ dom( f p ) and a < q b < q c } . We now turn to defining t q and < q to ensure that Clauses (4) and (6) of Defini-tion 4.11 hold. By our intended definition of f q and < q , these clauses dictate that,for all β ∈ x q and α ∈ x q ∩ ( β + 1):(4 ′ ) for all v ∈ ( t q ) β , there is some u ∈ ( t q ) α with u ≤ q v ;(6 ′ ) for all ( a, b ) , ( a, c ) ∈ dom( f p ) with f p ( a, b ) ∩ f p ( a, c ) = ∅ , if b ∈ ( t p ) λα and c ∈ ( t p ) λβ , then |{ i < λ | b ( i ) ≤ q c ( i ) }| = λ .In order to satisfy Clause (4 ′ ), it is possible that we will have to add new nodesto t q , i.e., that t q \ t p = ∅ . However, we will do so in such a way that each elementof t q \ t p will be a < q -predecessor of an element of t p . Consequently, to define t q and < q , it suffices to specify pred q ( v ) for all v ∈ t p .Now, by recursion on β ∈ x p , we define pred q ( v ) for all v ∈ ( t p ) β in a waythat ensures that Clauses (4 ′ ) and (6 ′ ) hold for all α ∈ x q ∩ ( β + 1). Suppose FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 17 that β ∈ x p and, for all α ∈ x p ∩ β , we have specified pred q ( u ) for every u ∈ ( t p ) α . Let t <β denote the underlying set of the tree we have defined thus far, i.e., S α ∈ x p ∩ β S u ∈ ( t p ) α (pred q ( u ) ∪ { u } ).If pred p ( β ) = ∅ and v ∈ ( t p ) β , then let B be a maximal branch through t <β .It might be the case that B is bounded below β , i.e., there is γ ∈ x q ∩ β with B ∩ ( { γ } × λ + ) = ∅ . If this is the case, then, for each such γ , add a new elementfrom { γ } × λ + to t q and require that these new elements, together with B , forma branch whose levels are unbounded in x q ∩ β . Let this unbounded branch bedenoted by B ∗ , and set pred q ( v ) := B ∗ .If pred p ( β ) is unbounded in β , then, for all v ∈ ( t p ) β , we are obliged to letpred q ( v ) be precisely S u ∈ pred p ( v ) (pred q ( u ) ∪ { u } ).It remains to consider the case in which pred p ( β ) is nonempty and bounded in β . Put β ′ := sup(pred p ( β )). Since pred p ( β ) is a closed, nonempty subset of β , wehave β ′ ∈ x p . If there is no γ ∈ x p with β ′ < γ < β , then, for all v ∈ ( t p ) β , we areagain obliged to let pred q ( v ) := S u ∈ pred p ( v ) (pred q ( u ) ∪ { u } ). Thus, from now on,suppose that x p ∩ ( β ′ , β ) = ∅ .Let h ( a ℓ , b ℓ , c ℓ ) | ℓ < λ i enumerate all triples ( a, b, c ) such that: • ( a, b ) , ( a, c ) ∈ dom( f p ); • f p ( a, b ) ∩ f p ( a, c ) = ∅ ; • c ∈ ( t p ) λβ and there is α ∈ x p ∩ ( β ′ , β ) such that b ∈ ( t p ) λα .Moreover, assume that each such triple is enumerated as ( a ℓ , b ℓ , c ℓ ) for λ -many ℓ < λ . (If there are no such triples, then simply define pred q ( v ) arbitrarily for each v ∈ ( t p ) β subject to the constraint pred q ( v ) ⊇ pred p ( v ).)Now, by recursion on ℓ < λ , we choose nodes v ℓ ∈ ( t p ) β and specify pred q ( v ℓ ).Suppose that ℓ < λ and we have chosen v ℓ ′ and pred q ( v ℓ ′ ) for all ℓ ′ < ℓ . Considerthe triple ( a ℓ , b ℓ , c ℓ ).Suppose first that a ℓ ∈ ( t p ) β ′ . We have that, for all i < λ , a ℓ ( i ) < p b ℓ ( i ) , c ℓ ( i ).In particular, since β ′ = max(pred p ( β )), we have, for all i < λ , pred q ( b ℓ ( i )) ⊇ pred p ( b ℓ ( i )) ⊇ pred p ( c ℓ ( i )). Choose i < λ such that c ℓ ( i ) / ∈ { v ℓ ′ | ℓ ′ < ℓ } , set v ℓ := c ℓ ( i ), and let B be a maximal branch through t <β with b ℓ ( i ) ∈ B . As in thecase in which pred p ( β ) = ∅ , extend B , by adding nodes if necessary, to a branch B ∗ whose levels are unbounded in x q ∩ β , and set pred q ( v ℓ ) := B ∗ .Suppose next that a ℓ ∈ ( t p ) <β ′ . Let c ′ ∈ ( t p ) λβ ′ be the unique level sequencesuch that a ℓ < p c ′ < p c ℓ . Since p ∈ P , we have ( a ℓ , c ′ ) ∈ dom( f p ) and f p ( a ℓ , c ′ ) ⊇ f p ( a ℓ , c ℓ ). In particular, f p ( a ℓ , c ′ ) ∩ f p ( a ℓ , b ℓ ) = ∅ , so, by our inductive hypothesis,we know that, for λ -many i < λ , we have c ′ ( i ) < q b ℓ ( i ). Choose such an i with c ℓ ( i )
6∈ { v ℓ ′ | ℓ ′ < ℓ } and let v ℓ := c ℓ ( i ). As in the previous case, by adding nodesif necessary, fix a branch B ∗ whose levels are unbounded in x q ∩ β with b ℓ ( i ) ∈ B ∗ ,and set pred q ( v ℓ ) := B ∗ .At the end of this process, if there are nodes in ( t p ) β \ { v ℓ | ℓ < λ } , then assigntheir < q -predecessors arbitrarily. We must verify that we have maintained theinductive hypothesis. To this end, fix ( a, b, c ) such that: • ( a, b ) , ( a, c ) ∈ dom( f p ); • f p ( a, b ) ∩ f p ( a, c ) = ∅ ; • c ∈ ( t p ) λβ and there is α ∈ x p ∩ β such that b ∈ ( t p ) λα . Suppose first that α ≤ β ′ . This implies that a ∈ ( t p ) λ<β ′ . Therefore, we can let c ′ ∈ ( t p ) λβ ′ be the unique level sequence such that a < p c ′ < p c . Then f p ( a, c ′ ) ⊇ f p ( a, c ), so, by the inductive hypothesis applied at β ′ , we have that, for λ -many i < λ , b ( i ) ≤ q c ′ ( i ) ≤ q c ( i ), so we are done.Next, suppose β ′ < α < β . In this case, for λ -many ℓ < λ , we have ( a, b, c ) =( a ℓ , b ℓ , c ℓ ). For each such ℓ , at stage ℓ of the construction, we chose a distinct i < λ and ensured that b ℓ ( i ) < q c ℓ ( i ), so, for λ -many i < λ , we have b ( i ) < q c ( i ), asdesired. (cid:3) Lemma 4.18.
Suppose that ξ < λ + and h q η | η < ξ i is a decreasing sequence from Q . Let x := S η<ξ x q η . Suppose that α < ssup( x ) and that r ∈ Q ssup( x ∩ α ) is a lowerbound for h q η ↾ ↾ α | η < ξ i . Then there is q ∈ Q such that: • q is a lower bound for h q η | η < ξ i ; • q ↾ ↾ ssup( x ∩ α ) = r ; • x q = cl( x r ∪ x ) .Proof. x q and < q are determined by the requirements of the Lemma. We nowspecify t q , < q , and f q . We must let q ↾ ↾ ssup( x ∩ α ) = r , so we only deal with theparts of t q , < q , and f q related to levels at ssup( x ∩ α ) or higher.Fix β ∈ x q \ ssup( x ∩ α ). If β ∈ x , then let ( t q ) β := S η<ξ ( t q η ) β . If β x , thenlet γ := min( x \ β ), and let ( t q ) β := { ( β, ζ ) | ( γ, ζ ) ∈ S η<ξ ( t q η ) γ } .We define < q by specifying pred q ( v ) for all v ∈ t q . This is already done forall v ∈ ( t q ) < ssup( x ∩ α ) . We take care of the v ∈ ( t q ) ≥ ssup( x ∩ α ) by recursion on the β ∈ x q such that v ∈ ( t q ) β . Thus, suppose β ∈ x q \ ssup( x ∩ α ) and we have definedpred q ( u ) for all u ∈ ( t q ) <β .Suppose first that β x , and let γ := min( x \ β ). If v = ( β, ζ ) ∈ ( t q ) β ,then let v ′ := ( γ, ζ ) ∈ ( t q ) γ , and let pred q ( v ) be the < q -downward closure of S η<ξ pred q η ( v ′ ).Suppose next that β ∈ x and β ′ := sup( x q ∩ β ) x . If v = ( β, ζ ) ∈ ( t q ) β , thenlet v ′ := ( β ′ , ζ ) ∈ ( t q ) β ′ , and let pred q ( v ) := { v ′ } ∪ pred q ( v ′ ).Finally, suppose that β ∈ x and sup( x q ∩ β ) ∈ x . Then, for all v ∈ ( t q ) β , letpred q ( v ) be the < q -downward closure of S η<ξ pred q η ( v ).To finish, we define f q . Suppose that β ∈ x q \ (ssup( x ∩ α ) ∪ x ) and b ∈ ( t q ) λβ .Let γ β := min( x \ β ), and let b ′ ∈ ( t q ) λγ β be given by letting b ′ ( i ) be the unique( γ β , ζ ) such that b ( i ) = ( β, ζ ). Note that b < q b ′ . We setdom( f q ) := dom( f r ) ∪ [ η<ξ dom( f q η ) ∪ ( a, b ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ β ∈ x q \ (ssup( x ∩ α ) ∪ x ) b ∈ ( t q ) λβ and ( a, b ′ ) ∈ [ η<ξ dom( f q η ) . If ( a, b ) ∈ dom( f r ), then we set f q ( a, b ) := f r ( a, b ). If ( a, b ) ∈ S η<ξ dom( f q η ) \ dom( f r ), then we let f q ( a, b ) := S η<ξ f q η ( a, b ). If ( a, b ) is such that b ∈ ( t p ) λβ forsome β ∈ x p \ (ssup( x ∩ α ) ∪ x ) and ( a, b ′ ) ∈ S η<ξ dom( f q η ), then let f q ( a, b ) = S η<ξ f q η ( a, b ′ ) = f q ( a, b ′ ). It is easily verified that q is as desired. (cid:3) FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 19
Lemma 4.19.
Suppose p ∈ Q , α < ssup( x p ) , and q ≤ p ↾ ↾ α with q ∈ P α . Then thereis r ∈ P that is a greatest lower bound for p and q . Moreover, we have x r = x p ∪ x q and r ↾ ↾ α = q .Proof. We construct such an r by doing as little as possible while still satisfyingDefinition 4.11 and extending both p and q . Let x r := x p ∪ x q , and require that r ↾ ↾ α = q . Suppose that β ∈ x p \ α . Let pred r ( β ) := pred p ( β ) ∪ S γ ∈ pred p ( β ) ∩ α pred q ( γ ).Let t r := t p ∪ t q . If v ∈ t p \ t q , then let pred r ( v ) := pred p ( v ) ∪ S u ∈ pred p ( v ) ∩ t q pred q ( u ).Finally, let dom( f r ) := dom( f p ) ∪ dom( f q ). If ( a, b ) ∈ dom( f q ), then let f r ( a, b ) := f q ( a, b ). If ( a, b ) ∈ dom( f p ) \ dom( f q ), then let f r ( a, b ) := f p ( a, b ).The only clauses of Definition 4.11 that are non-trivial to check are (6) and(7). Let us first deal with Clause (6). To this end, fix a, b, c ∈ t λr such that( a, b ) , ( a, c ) ∈ dom( f r ) and f r ( a, b ) ∩ f r ( a, c ) = ∅ . If we have either ( a, b ) , ( a, c ) ∈ dom( f q ) or ( a, b ) , ( a, c ) ∈ dom( f p ) \ dom( f q ), then the conclusion of Clause (6)follows from the fact that p, q ∈ P . Thus, we may assume without loss of generalitythat ( a, b ) ∈ dom( f q ) and ( a, c ) ∈ dom( f p ) \ dom( f q ). Let β, γ ∈ x r be such that b ∈ ( t r ) λβ and c ∈ ( t r ) λγ . By assumption, we have β < α ≤ γ .If β r γ , then there is nothing to check. Thus, assume that β ≤ r γ . By thedefinition of ≤ r , it follows that there is β ′ ∈ ( x p ∩ α ) such that β ≤ q β ′ and β ′ ≤ p γ .Let c ′ ∈ ( t p ) λβ ′ be the unique level sequence such that a < p c ′ < p c . Since p ∈ P , itfollows that ( a, c ′ ) ∈ dom( f p ) and f p ( a, c ′ ) ⊇ f p ( a, c ). Since q ≤ p ↾ ↾ α , we must have( a, c ′ ) ∈ dom( f q ) and f q ( a, c ′ ) ⊇ f p ( a, c ). Thus, we have f q ( a, c ′ ) ∩ f q ( a, b ) = ∅ .Since q ∈ P and β ≤ q β ′ , we have that, for λ -many i < λ , b ( i ) ≤ q c ′ ( i ). But then,for all such i < λ , we also have b ( i ) ≤ r c ( i ), as required.Finally, we check Clause (7). Suppose that ( a, c ) ∈ dom( f r ) and b ∈ t λr is suchthat a < r b < r c . If ( a, c ) ∈ dom( f q ), then the conclusion follows from the factthat q ∈ P . Thus suppose that ( a, c ) ∈ dom( f p ) \ dom( f q ). Let β ∈ x r be such that b ∈ ( t r ) λβ , and let γ ∈ x p be such that c ∈ ( t p ) λγ . If β ∈ x p , then we have a < p b < p c ,and the conclusion follows from the fact that p ∈ P . Thus, assume that β ∈ x q \ x p .Then there is β ′ ∈ x p ∩ α such that β ≤ q β ′ and β ′ ≤ p γ . Let c ′ ∈ ( t p ) λβ ′ be theunique level sequence such that a < p c ′ < p c . Since p ∈ P , we have ( a, c ′ ) ∈ dom( f p )and f p ( a, c ′ ) ⊇ f p ( a, c ). Since q ≤ P p ↾ ↾ α , we have f q ( a, c ′ ) ⊇ f p ( a, c ). Finally, since q ∈ P and a < q b < q c ′ , we have ( a, b ) ∈ dom( f q ) and f q ( a, b ) ⊇ f q ( a, c ′ ). Thus,( a, b ) ∈ dom( f r ) and f r ( a, b ) ⊇ f r ( a, c ), as required. (cid:3) It now follows that ( P , ≤ P , Q ) is in P λ + . We are thus left with isolating therelevant sharply dense systems. The following are all straightforward. Lemma 4.20 (Normal and splitting) . Suppose η < λ + . For every α < β < λ ++ ,let D nsη, { α,β } be the set of all conditions p ∈ Q such that: • { α, β } ⊆ x p ; • ( α, η ) , ( β, η ) ∈ t p ; • ( α, η ) has at least two < p -successors in ( t p ) β .Then D nsη := { D nsη,x | x ∈ (cid:0) λ ++ (cid:1) } is a sharply dense system. (cid:3) Lemma 4.21 (Complete) . Suppose that µ < λ + is a regular cardinal and g : µ → λ + . For every x ∈ (cid:0) λ ++ µ +1 (cid:1) , let D com g,x be the set of all conditions p ∈ Q such that: • x ⊆ x p ; • for all i < µ , we have ( x ( i ) , g ( i )) ∈ t p ; • if { ( x ( i ) , g ( i )) | i < µ } forms a < p -chain, then it has a < p -upper bound in ( t p ) x ( µ ) .Then D com g := { D com g,x | x ∈ (cid:0) λ ++ µ +1 (cid:1) } is a sharply dense system. (cid:3) Lemma 4.22 (Super-Souslin) . Suppose δ, ǫ < λ + . For all α < β < λ ++ , let E δ,ǫ, { α,β } be the set of all conditions p ∈ Q such that: • { α, β } ⊆ x p ; • a α,δ , a β,ǫ ∈ t λp ; • if a α,δ < p a β,ǫ , then ( a α,δ , a β,ǫ ) ∈ dom( f p ) .Then E δ,ǫ := { E δ,ǫ,x | x ∈ (cid:0) λ ++ (cid:1) } is a sharply dense system. (cid:3) By SDFA( P λ + ), we can find a filter G on P such that: • for every η < λ + , G meets D nsη everywhere; • for every regular cardinal µ < λ + and every function g : µ → λ + , G meets D com g everywhere; • for all δ, ǫ < λ + , G meets E δ,ǫ everywhere.Now define a tree ( T, < T ) as follows. Let T := λ ++ × λ + . Let ( α, η ) < T ( β, ξ ) iffthere is p ∈ G such that ( α, η ) , ( β, ξ ) ∈ t p and ( α, η ) < p ( β, ξ ). The fact that G meets D nsη everywhere for all η < λ + ensures that ( T, < T ) is a normal, splittingtree and T α = { α } × λ + for all α < λ ++ . The fact that G meets D com g everywherefor all regular µ ≤ λ and g : µ → λ + ensures that ( T, < T ) is λ + -complete.Finally, we define a function F : [ T λ ] → λ + witnessing that ( T, < T ) is a super-Souslin tree. Fix α < β < λ ++ and δ, ǫ < λ + such that a α,δ < T a β,ǫ . Find p ∈ G ∩ E δ,ǫ, { α,β } . Since p ∈ Q and a α,δ < T a β,ǫ , it follows that a α,δ < p a β,ǫ .Therefore, ( a α,δ , a β,ǫ ) ∈ dom( f p ). Let F ( a α,δ , a β,ǫ ) be an arbitrary element of f p ( a α,δ , a β,ǫ ).To verify that F is as sought, fix a, b, c ∈ T λ such that ( a, b ) , ( a, c ) ∈ [ T λ ] and F ( a, b ) = F ( a, c ). Without loss of generality, suppose there are β ≤ γ < λ ++ suchthat b ∈ T λβ , and c ∈ T λγ . Find p b ∈ G such that ( a, b ) ∈ dom( f p b ) and F ( a, b ) ∈ f p b ( a, b ). Similarly, find p c ∈ G such that ( a, c ) ∈ dom( f p c ) and F ( a, c ) ∈ f p c ( a, c ).Find q ∈ G ∩ Q with q ≤ P p b , p c . Then ( a, b ) , ( a, c ) ∈ dom( f q ), F ( a, b ) ∈ f q ( a, b ),and F ( a, c ) ∈ f q ( a, c ). In particular, f q ( a, b ) ∩ f q ( a, c ) = ∅ . Since q ∈ Q it followsthat there are λ -many i < λ such that b ( i ) ≤ q c ( i ). But then, for all such i < λ , wehave b ( i ) ≤ T c ( i ). Thus, F witnesses that ( T, < T ) is a λ ++ -super-Souslin tree, soour proof of Theorem C is now complete.5. Square and diamond
In this section, we use (cid:3) Bκ and ♦ ( κ ) to construct combinatorial objects that willhelp us prove Theorem B in Section 6.5.1. Enlarged direct limit.
In this short subsection, we introduce an “enlargeddirect limit” operator. This operator motivates our application of (cid:3) Bκ that will becarried out in the next subsection. Recall that by Corollary 3.9, SDFA( P λ + ) implies | <λ + λ + | = λ + . FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 21
Definition 5.1.
For a linearly ordered set ( Y, ⊳ ) and a subset Z ⊆ Y , we definedouble Z ( Y, ⊳ ) as a linearly ordered set whose underlying set is ( Z ×{ } ) ⊎ ( Y ×{ } ),ordered lexicographically by letting ( y, i ) ⊳ l ( y ′ , i ′ ) iff one of the following holds: • y ⊳ y ′ ; • y = y ′ and ( i, i ′ ) = (0 , Y, ⊳ ) we have in mind is a direct limit of a systemof well-ordered sets, and the choice of the subset Z ⊆ Y (to be doubled) will bedefined momentarily. The following is obvious. Lemma 5.2.
For all Z ⊆ Y , if Y is well-ordered by ⊳ , then, double Z ( Y, ⊳ ) iswell-ordered by ⊳ l . (cid:3) We start with a system of well-ordered sets. Specifically, suppose that ~θ = h θ η | η < ξ i and ~π = h π η,η ′ | η < η ′ < ξ i are such that: • ξ is a limit ordinal; • ~θ is a non-decreasing sequence of ordinals; • for all η < η ′ < ξ , π η,η ′ : θ η → θ η ′ is a c.o.i.; • for all η < η ′ < η ′′ < ξ , we have π η,η ′′ = π η ′ ,η ′′ ◦ π η,η ′ .As is well-known, the direct limit of the system ( ~θ, ~π ) is defined as follows: • Put X := { ( η, γ ) | η < ξ, γ < θ η } . • For ( η, γ ) , ( η ′ , γ ′ ) ∈ X with η < η ′ , let ( η, γ ) ∼ ( η ′ , γ ′ ) iff π η,η ′ ( γ ) = γ ′ . • Let Y consists of all equivalence classes [( η, γ )] for ( η, γ ) ∈ X . • Order Y by letting [( η , γ )] ⊳ [( η , γ )] iff there exists some η ≥ max { η , η } and γ ′ < γ ′ such that ( η , γ ) ∼ ( η, γ ′ ) and ( η , γ ) ∼ ( η, γ ′ ). • For each η < ξ , define a map π η : θ η → Y by stipulating π η ( γ ) := [( η, γ )]. Definition 5.3 (Direct limit) . lim( ~θ, ~π ) stands for ( Y, ⊳ , h π η | η < ξ i ).Next, we let Z be the set of equivalence classes in Y such that, for every repre-sentative ( η, γ ) from the equivalence class, π η ↾ γ is bounded below π η ( γ ), i.e., Z := { z ∈ Y | ∀ ( η, γ ) ∈ z ∃ y ∈ Y ∀ β < γ [ π η ( β ) ⊳ y ⊳ π η ( γ )] } . Let W := double Z ( Y, ⊳ ), and let ̟ denote the map from Y to its canonical copyinside W , i.e., ̟ ( y ) = ( y, Definition 5.4 (Enlarged direct limit) . lim ∗ ( ~θ, ~π ) stands for ( W, ⊳ l , h π ∗ η | η < ξ i ),where π ∗ η := ̟ ◦ π η for each η < ξ .Finally, by Lemma 5.2, in the special case that ( Y, ⊳ ) is well-ordered, we knowthat ( W, ⊳ l ) is well-ordered. In this case, we put θ := otp( W, ⊳ l ), and let π ∗ : W → θ be the collapse map. Then, we define: Definition 5.5 (Ordinal enlarged direct limit) . lim + ( ~θ, ~π ) stands for ( θ, ∈ , h π + η | η < ξ i ), where π + η := π ∗ ◦ π ∗ η for all η < ξ .5.2. Square.
Fix a (cid:3) Bκ -sequence, h C β | β ∈ Γ i . Enlarge the preceding to a se-quence ~C = h C β | β < κ i by letting, for all limit β ∈ κ \ Γ, C β be an arbitrary clubin β of order type cf( β ), and letting C β +1 := { , β } for all β < κ . In particular, forevery β ∈ E κω \ Γ, we have acc( C β ) = ∅ . Thus, without loss of generality, we mayassume that E κω ⊆ Γ. For convenience, assume also that 0 ∈ C β for all nonzero β < κ . We now turn to constructing a matrix ~B = h B βη | β < κ + , η < κ i such that S η<κ B βη = β + 1 for all β < κ + . From this matrix, for each β < κ + , we shallderive the following additional objects: ◦ we shall let η β denote the least η < κ such that B βη = ∅ ; ◦ for each ξ ∈ acc( κ \ η β ), we write B β<ξ := S η<ξ B βη ; ◦ for each η < κ , we shall set θ βη := otp( B βη ) and let π βη : B βη → θ βη denotethe unique order-preserving bijection; ◦ for each η < ξ < κ , π βη,ξ : θ βη → θ βξ will denote the order-preserving injectionindicating how B βη “sits inside” B βξ , i.e., π βη,ξ := π βξ ◦ ( π βη ) − .We shall also derive a “distance function” d : [ κ + ] → κ by letting for all α < β < κ + : d ( α, β ) := min { η < κ | α ∈ B βη } . Lemma 5.6.
There exists a matrix ~B = h B βη | β < κ + , η < κ i such that, for each β < κ + , the following hold:(1) h B βη | η < κ i is a ⊆ -increasing sequence of closed sets, each of size < κ ,that converges to β + 1 , and β ∈ B βη β ;(2) for all η < κ and α ∈ B βη , we have B αη = B βη ∩ ( α + 1) and π αη = π βη ↾ ( α + 1) ;(3) for all η < κ , if cf( β ) = κ , then max( B βη ∩ β ) = C β ( ωη ) ;(4) if β ∈ Γ ∩ E κ + <κ , then η β = otp(acc( C β )) and acc( C β ) ⊆ B βη β ;(5) for all ξ ∈ acc( κ \ η β ) , all of the following hold:(a) B βξ is the ordinal closure of B β<ξ ;(b) for every α ∈ B βξ \ B β<ξ , letting γ := min( B βξ \ ( α + 1)) , we have cf( γ ) = κ and α = C γ ( ωξ ) ;(c) cf( β ) = κ iff ssup( π βη,ξ “ π βη ( β )) < π βξ ( β ) for all η ∈ [ η β , ξ ) .Proof. The construction is by recursion on β < κ + . Case 0: β = 0 . Set B βη := { } for all η < κ . It is trivial to see that Clauses(1)–(5) all hold. Case 1: β = α + 1 . For all η < η α , let B βη := ∅ , and for all η ∈ [ η α , κ ), let B βη := { β } ∪ B αη . It is trivial to see that Clauses (1)–(5) all hold. Case 2: β ∈ acc( κ ) and sup(acc( C β )) < β . In particular, a := C β \ sup(acc( C β )) is a cofinal subset of β of order type ω . Note that, sincecf( β ) = ω , we have β ∈ Γ. Put η β := otp(acc( C β )) and η ∗ := max { η β , sup( d “[ a ] ) } .Now, for all η < κ , define B βη as follows: ◮ If η < η β , then let B βη := ∅ . Clauses (2)–(5) are trivially satisfied. ◮ If η β ≤ η ≤ η ∗ , then let α ∗ := min( a ) and put B βη := { β } ∪ B α ∗ η . Since α ∗ ∈ acc( C β ) ∪ { } and β ∈ Γ, we have C α ∗ ⊑ C β , which ensuresClause (4). As for Clause (5c), for all η < ξ in [ η β , η ∗ ], we havessup( π βη,ξ “ π βη ( β )) = π βη,ξ ( π βη ( α ∗ ) + 1) = π βξ ( α ∗ ) + 1 = π βξ ( β ) . The other clauses are easily seen to be satisfied. ◮ Otherwise, let B βη := { β } ∪ S α ∈ a B αη . Since η > η ∗ , for every pair ofordinals α < α ′ from a , we have α ∈ B α ′ η , so that B αη = B α ′ η ∩ ( α + 1).It follows that h B αη | α ∈ a i is an ⊑ -increasing sequence of closed sets. FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 23
In particular, B βη ∩ β is a club in β , which takes care of Clause (5c).So, all clauses are satisfied. Case 3: cf( β ) < κ and sup(acc( C β )) = β . Put η β := sup( d “[acc( C β )] ),and, for all η < κ , define B βη as follows: ◮ If η < η β , then let B βη := ∅ . Clauses (2)–(5) are trivially satisfied. ◮ If η ≥ η β , then let B βη := { β }∪ S α ∈ acc( C β ) B αη . Since η ≥ sup( d “[acc( C β )] ),we have that h B αη | α ∈ acc( C β ) i is an ⊑ -increasing sequence of closedsets. So B βη ∩ β is a club in β , and all clauses except Clause (4) areeasily seen to be satisfied. Now, if β ∈ Γ, then, since Clause (4) holdsfor all α ∈ acc( C β ), we have η β = otp(acc( C β )), so that Clause (4)holds for β , as well. Case 4: β ∈ E κ + κ . For all η < κ , let α η := C β ( ωη ) and B βη := { β } ∪ B α η η , sothat Clause (3) is satisfied.Since cf( β ) = κ , we have β ∈ Γ. Hence, for all η < ξ < κ , we have α ξ ∈ Γ, so that α η ∈ B α ξ ξ by Clause (4). It follows that h B βη ∩ β | η < κ i is ⊆ -increasing. In particular, Clauses (1) and (2) are satisfied. It also followsthat, for all ξ ∈ acc( κ ) and α ∈ B βξ ∩ β , we have B β<ξ ∩ ( α + 1) = B α<ξ , sothat Clauses (5a) and (5b) are satisfied.Finally, to verify Clause (5c), fix an arbitrary ξ ∈ acc( κ ) and η < ξ . ByClauses (2) and (3), we have B βη = { β } ∪ B α η η and B βξ = { β } ∪ B α ξ ξ , so that π βη ( β ) = π βη ( α η ) + 1 and π βξ ( β ) = π βξ ( α ξ ) + 1. Therefore, we have π βη,ξ “ π βη ( β ) ⊆ π βη,ξ ( π βη ( α η )) + 1 = π βξ ( α η ) + 1 < π βξ ( α ξ ) < π βξ ( β ) . (cid:3) The next lemma assumes familiarity with the previous subsection.
Lemma 5.7.
Suppose β < κ + and ξ ∈ acc( κ \ η β ) . Write ~θ := h θ βη | η < ξ i and ~π := h π βη,η ′ | η < η ′ < ξ i . Then lim + ( ~θ, ~β ) is defined and, letting ( θ, ∈ , h π + η | η < ξ i ) := lim + ( ~θ, ~π ) , we have θ = θ βξ and h π + η | η < ξ i = h π βη,ξ | η < ξ i .Proof. Let h Y, ⊳ , h π η | η < ξ i ) := lim(( ~θ, ~π )). For every class y ∈ Y and represen-tatives ( η, γ ) , ( η ′ , γ ′ ) ∈ y with η < η ′ , we have π βη,η ′ ( γ ) = γ ′ , i.e., ( π βη ′ ) − ( γ ′ ) =( π βη ) − ( γ ). Therefore, for each y ∈ Y , we may let α y := ( π βη ) − ( γ ) for an arbitrarychoice of ( η, γ ) ∈ y . Note that, for all y, y ′ in Y , we have y ⊳ y ′ iff α y < α y ′ .Therefore, the order type of ( Y, ⊳ ) is precisely otp( B β<ξ ). In particular, lim( ~θ, ~π ) iswell-ordered, so lim + ( ~θ, ~π ) is defined. Write ( θ, ∈ , h π + η | η < ξ i ) for lim + ( ~θ, ~π ).Let Z be the set of equivalence classes in Y such that, for every representative( η, γ ) from the class, we have that π η ↾ γ is bounded below π η ( γ ). By Clause (5c)of Lemma 5.6, we know that Z = { z ∈ Y | cf( α z ) = κ } .For all α ∈ B β<ξ ∩ E κ + κ , we have C α ( ωξ ) ∈ B βξ \ B β<ξ and, moreover, α =min( B βξ \ ( C α ( ωξ ) + 1)). Also, by Clauses (5a) and (5b) of Lemma 5.6, we knowthat B βξ = B β<ξ ∪ { C α ( ωξ ) | α ∈ B β<ξ ∩ E κ + κ } . Now, for all z ∈ Z , the additionof ( z,
0) when passing from Y to W := double Z ( Y, ⊳ ) corresponds precisely to theaddition of C α z ( ωξ ) when passing from B β<ξ to B βξ . It follows that otp( W, ⊳ l ) =otp( B βξ ) = θ βξ . That is, θ = θ βξ . Letting π ∗ : W → θ be the collapse map, we have that, for all z ∈ Y , π ∗ ( z,
1) = π βξ ( α z ), so, for all η < ξ and γ < θ βη , we have π + η ( γ ) = π ∗ ([( η, γ )] ,
1) = π βξ (( π βη ) − ( γ )) = π βη,ξ ( γ ) , so h π + η | η < ξ i = h π βη,ξ | η < ξ i . (cid:3) Diamond.
Our next goal is to prove the following.
Lemma 5.8.
Suppose that ♦ ( κ ) holds and that ( P , ≤ P , Q ) ∈ P κ . Then there arearrays h ϑ ξη | η ≤ ξ < κ i , h ̟ ξη,η ′ | η < η ′ ≤ ξ < κ i , and h q ξη | η < ξ < κ i satisfying the following: For every β < κ + and every decreasing sequence h p η | η < κ i ∈ Q η<κ P B βη , there are stationarily many ξ < κ such that: • h ϑ ξη | η ≤ ξ i = h θ βη | η ≤ ξ i ; • h ̟ ξη,η ′ | η < η ′ ≤ ξ i = h π βη,η ′ | η < η ′ ≤ ξ i ; • h q ξη | η < ξ i = h π βη .p η | η < ξ i . The rest of this subsection will be devoted to proving Lemma 5.8. To avoidthe use of codings, we shall make use of the following equivalent version of ♦ ( κ )(see [2]). Definition 5.9. ♦ − ( H κ ) asserts the existence of a sequence h A ξ | ξ < κ i such that,for every A ⊆ H κ and p ∈ H κ + , there exists an elementary submodel M ≺ H κ + ,with p ∈ M , such that κ M := M ∩ κ is an ordinal < κ and A ∩ M = A κ M .Fix a ♦ − ( H κ )-sequence, h A ξ | ξ < κ i . Definition 5.10.
We say that ξ < κ is good if ξ ∈ acc( κ ) and A ξ = { ( ϑ ξη , q ξη , ̟ ξη,η ′ , η, η ′ ) | η < η ′ < ξ } , where, for all η < η ′ < η ′′ < ξ , we have • ϑ ξη ≤ ϑ ξη ′ < κ ; • q ξη ∈ P ϑ ξη ; • ̟ ξη,η ′ : ϑ ξη → ϑ ξη ′ is a c.o.i., and q ξη ′ ≤ P ̟ ξη,η ′ .q ξη ; • ̟ ξη,η ′′ = ̟ ξη ′ ,η ′′ ◦ ̟ ξη,η ′ ; • lim( h ϑ ξη | η < ξ i , h ̟ ξη,η ′ | η < η ′ < ξ i ) is well-ordered. ◮ If ξ < κ is good, then h ϑ ξη | η < ξ i , h ̟ ξη,η ′ | η < η ′ < ξ i , and h q ξη | η < ξ i arealready defined, and we let:( ϑ ξξ , ∈ , h ̟ ξη,ξ | η < ξ i ) := lim + ( h ϑ ξη | η < ξ i , h ̟ ξη,η ′ | η < η ′ < ξ i ) . ◮ If ξ < κ is not good, then let h ϑ ξη | η ≤ ξ i , h ̟ ξη,η ′ | η < η ′ ≤ ξ i , and h q ξη | η < ξ i be arbitrary.We claim that the arrays thus defined satisfy the conclusion of Lemma 5.8. Toverify this, fix β < κ + , a decreasing sequence h p η | η < κ i ∈ Q η<κ P B βη , and a club D in κ . Put A := { ( θ βη , π βη .p η , π βη,η ′ , η, η ′ ) | η < η ′ < κ } . Since A ⊆ H κ and D ∈ H κ + , we can let p := { A, D } and fix an elementary submodel M ≺ H κ + with p ∈ M such that ξ := M ∩ κ is in κ and A ∩ M = A ξ . By the FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 25 fact that D ∈ M and the elementarity of M , we have ξ ∈ D . Since M ∩ κ = ξ and A ∈ M , and by the elementarity of M , we have A ξ = { ( θ βη , π βη .p η , π βη,η ′ , η, η ′ ) | η < η ′ < ξ } . In particular, ξ is good. By Lemma 5.7, we have ϑ ξξ = θ βξ and, for all η < ξ , ̟ ξη,ξ = π βη,ξ . Therefore, ξ ∈ D satisfies the three bullet points in the statement ofLemma 5.8. Since D was arbitrary, this completes the proof of the lemma.6. Proof of Theorem B
This section is devoted to the proof of Theorem B, which forms the main resultof this paper.
Theorem B.
Suppose that (cid:3) Bκ and ♦ ( κ ) both hold. Then so does SDFA( P κ ) . Setup.
Fix an arbitrary ( P , ≤ P , Q ) ∈ P κ along with a collection {D i | i < κ } of sharply dense systems. For each i < κ , write D i = { D i,x | x ∈ (cid:0) κ + θ D i (cid:1) } .Let ~B be given by Lemma 5.6, and let h ϑ ξη | η ≤ ξ < κ i , h ̟ ξη,η ′ | η < η ′ ≤ ξ < κ i ,and h q ξη | η < ξ < κ i be given by Lemma 5.8 applied to ( P , ≤ P , Q ). Definition 6.1.
Let X denote the set of ξ ∈ acc( κ ) such that: • ξ is good, in the sense of Definition 5.10; • h ̟ ξη,ξ .q ξη | η < ξ i admits a lower bound in P ϑ ξξ .Let ⊳ κ be some well-ordering of H κ . Using κ <κ = κ (which follows from ♦ ( κ )),enumerate all elements of S i<κ { i } × κ × (cid:0) κθ D i (cid:1) as a sequence h ( i η , j η , z η ) | η < κ i . Lemma 6.2.
There is a sequence of conditions h s ξ | ξ ∈ X i ∈ Q ξ ∈ X Q ϑ ξξ , suchthat, for all ξ ∈ X : • s ξ is a lower bound for h ̟ ξη,ξ .q ξη | η < ξ i ; • for all η < ξ , if j η < ξ and z η ⊆ ϑ ξj η , then there is q ∈ D i η ,̟ ξjη,ξ “ z η suchthat s ξ ≤ P q .Proof. Let ξ ∈ X be arbitrary. We first define a sequence h s η | η ≤ ξ i ∈ Q η ≤ ξ Q ϑ ξξ by recursion on η : ◮ For η = 0, use Clauses (1c) and (6) of Definition 2.1 and the fact that ξ ∈ X to find s ∈ Q such that x s = ϑ ξξ and s is a lower bound for h ̟ ξη,ξ .q ξη | η < ξ i . ◮ For η < ξ , with j η < ξ and z η ⊆ ϑ ξj η , use the fact that D i η is a sharply densesystem and that ̟ ξj η ,ξ “ z η ⊆ ϑ ξξ to find s η, ∗ ∈ D i η ,̟ ξjη,ξ “ z η such that s η, ∗ ≤ P s η and x s η, ∗ = ϑ ξξ . Then, use Clause (6) of Definition 2.1 to find s η +1 ∈ Q such that s η +1 ≤ P s η, ∗ and x s η +1 = ϑ ξξ . ◮ For η < ξ with j η ≥ ξ or z η ϑ ξj η , simply let s η +1 := s η . ◮ For η ∈ acc( ξ + 1), assuming that h s ζ | ζ < η i has already been defined, useClause (7) of Definition 2.1 to let s η be a lower bound for h s ζ | ζ < η i in Q with x s η = ϑ ξξ .Having constructed h s η | η ≤ ξ i , it is clear that s ξ := s ξ is as sought. (cid:3) Fix a sequence h s ξ | ξ ∈ X i as in the preceding lemma. We will construct amatrix of conditions h p βη | β < κ + , η < κ i satisfying:(i) for all β < κ + and η < κ , we have p βη ∈ P B βη ;(ii) for all β < κ + , h p βη | η < κ i is ≤ P -decreasing;(iii) for all β < κ + , η < κ , and α ∈ B βη , we have p βη ↾ ↾ ( α + 1) = p αη ;(iv) for all β < κ + , all i < κ , and all x ∈ (cid:0) β +1 θ D i (cid:1) , there is ξ < κ and q ∈ D i,x suchthat p βξ ≤ P q ;(v) for all β ∈ E κ + κ and all ξ ∈ acc( κ ), the sequence h π βξ .p βη | η ≤ ξ i depends onlyon the value of C β ( ωξ ).Note that if we are successful, then, letting G be the upward closure of { p βη | β < κ + , η < κ } , it follows from Clauses (i)–(iv) that G is a filter on P that, foreach η < κ , meets D η everywhere. Of course, the sequence h s ξ | ξ ∈ X i , which wasderived from ♦ , will be a key to ensuring Clause (iv).6.2. Hypotheses.
The construction of h p βη | β < κ + , η < κ i will be by recursionon η < κ and, for fixed η , by recursion on β < κ + . We will maintain requirements(i)–(iii) and (v) as recursion hypotheses. In order to ensure that the constructionwill be successful, we need to carry along some further hypotheses. Suppose that β < κ + , ξ ∈ acc( κ ), and h p αη | α < κ + , η < ξ i has been constructed. Definition 6.3.
We say that the pair ( β, ξ ) is active if ξ ∈ X , θ βξ ≤ ϑ ξξ , and one ofthe following holds: • ξ > η β and, for all η < ξ , s ξ ≤ P π βξ .p βη ; or, • ξ = η β and there is γ ∈ E κ + κ such that β ∈ acc( C γ ) and ( γ, ξ ) is active.In our construction, we will require that, for all active ( β, ξ ), we have π βξ .p βξ = s ξ ↾ ↾ θ βξ . In particular, if ( β, ξ ) is active, then p βξ ∈ Q and x p βξ = B βξ . Moreover, forall β < κ + , we will arrange that, if ξ < κ is least such that β ∈ x p βξ , then either( β, ξ ) is active or ( ξ = η β and p βξ ∈ Q ). Lemma 6.4.
Suppose that β < κ + , ξ ∈ X , and ( β, ξ ) is active. Then ( α, ξ ) isactive for all α ∈ B βξ .Proof. Let α ∈ B βξ be arbitrary. As B βξ ∩ ( α +1) = B αξ , we have θ αξ < θ βξ ≤ ϑ ξξ = x s ξ . ◮ If ξ > η β and α ∈ B β<ξ , then ξ > η α and, for all sufficiently large η < ξ , wehave p αη = p βη ↾ ↾ ( α + 1). By Clause (2) of Lemma 5.6, then, s ξ ≤ P π βη .p αη = π αη .p αη ,so ( α, ξ ) is active. ◮ If ξ > η β and α ∈ B βξ \ B β<ξ , then let γ := min( B βξ \ ( α + 1)). By Clause (5b)of Lemma 5.6, we know that cf( γ ) = κ and α = C γ ( ωξ ). It follows that α ∈ Γand hence, by Clause (4) of Lemma 5.6, we have ξ = η α . Since | B βξ | < κ , and byClause (5a) of Lemma 5.6, we know that γ ∈ B β<ξ , so, by the previous paragraph,( γ, ξ ) is active. Hence, by Definition 6.3, ( α, ξ ) is active as well. ◮ If ξ = η β and γ ∈ E κ + κ is such that β ∈ acc( C γ ) and ( γ, ξ ) is active, thenby Clauses (2) and (3) of Lemma 5.6, B γξ ∩ ( β + 1) = B βξ , so α ∈ B γξ . Moreover, ξ > η γ = 0, so, by the previous cases, we again conclude that ( α, ξ ) is active. (cid:3) FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 27
Our final recursion hypotheses concern non-active pairs ( β, ξ ).First, suppose that ( β, ξ ) is not active and ξ = η β . If ξ ∈ acc( κ ) and there is γ ∈ E κ + κ such that β ∈ acc( C γ ) and sup { η < ξ | ( γ, η ) is active } = ξ , then we willrequire that p βξ ∈ Q and x p βξ = B βξ .Next, suppose that ( β, ξ ) is not active and ξ > η β . Let η ∗ := max { sup { η < ξ | ( β, η ) is active } , η β } . ◮ If η ∗ = ξ , then we will require that p βξ ∈ Q and x p βξ = B βξ . ◮ If η ∗ < ξ and β ∈ x p βη ∗ , then we will have p βη ∗ ∈ Q and will require that p βξ is the ≤ P -greatest condition q such that q ≤ P p βη ∗ and, for all α ∈ B βξ ∩ β , q ↾ ↾ ( α + 1) = p αξ .6.3. The construction.
We now turn to the actual construction. Suppose that β < κ + , ξ < κ , and we have already constructed h p αη | α < κ + , η < ξ i and h p αξ | α < β i . We now construct p βξ . There are a number of cases to consider. In all cases,unless explicitly verified, it will be trivial to check that the recursion hypothesesare maintained. Case 0: ξ < η β . Let p βη := P . Case 1: ξ = η β . There are now a few subcases to consider.
Subcase 1a: ( β, ξ ) is active. In particular, x s ξ = ϑ ξξ ≥ θ βξ . Let p βξ bethe unique condition q such that x q = B βξ and π βξ .q = s ξ ↾ ↾ θ βξ , i.e., p βξ = ( π βξ ) − . ( s ξ ↾ ↾ θ βξ ). Note that, for all α ∈ B βξ , Lemma 6.4 impliesthat ( α, ξ ) is active. We therefore have π βξ .p αξ = π αξ .p αξ = s ξ ↾ ↾ θ βξ , so p βξ ↾ ↾ ( α + 1) = p αξ and requirement (iii) is satisfied. Subcase 1b: ( β, ξ ) is not active and there is γ ∈ E κ + κ such that β ∈ acc( C γ ) and sup { η < ξ | ( γ, η ) is active } = ξ . Fix such a γ .Note that B γ<ξ ∩ β is unbounded in B βξ ∩ β and, for all α ∈ B γ<ξ ∩ β , sup { η < ξ | ( α, η ) is active } = ξ . Therefore, by our recursionhypotheses, for all α ∈ B γ<ξ , we know that p αξ ∈ Q and x p αξ = B αξ . ByClause (5) of Definition 2.1, there is a unique condition q ∈ Q suchthat x q = B βξ ∩ β and, for all α ∈ B βξ , we have q ↾ ↾ ( α + 1) = p αξ . ByClause (7) of Definition 2.1, there is a lower bound p for h p γη | η < ξ i such that: • p ∈ Q ; • x p = B γξ = B βξ ∪ { γ } ; • p ↾ ↾ β = q .Fix such a lower bound p with a ⊳ κ -minimal possible value for π γξ .p ,and let p βξ := p ↾ ↾ ( β +1). Note that, by requirement (v), the constructionin this Subcase is independent of our choice of γ . Subcase 1c: Otherwise.
Let p βξ be the unique condition, given byClause (5) of Definition 2.1, such that x p βξ = S { x p αξ | α ∈ ( B βξ ∩ β ) } and, for all α ∈ ( B βξ ∩ β ), we have p βξ ↾ ↾ ( α + 1) = p αξ . Case 2: ξ > η β . There are again a few subcases to consider.
Subcase 2a: ( β, ξ ) is active. Let p βξ be the unique condition q suchthat x q = B βξ and π βξ .q = s ξ ↾ ↾ θ βξ . By Definition 6.3, we have that s ξ ≤ P π βξ .p βη for all η < ξ , which implies that p βξ ≤ P p βη for all η < ξ ,so requirement (i) holds. Subcase 2b: ( β, ξ ) is not active, sup { η < ξ | ( β, η ) is active } = ξ ,and β E κ + κ . In this Subcase, we have that ξ ∈ acc( κ ) and B β<ξ ∩ β isunbounded in B βξ ∩ β . Since, for all α ∈ B β<ξ , we know that sup { η < ξ | ( α, η ) is active } = ξ , it follows as in Subcase 1b that there is a uniquecondition q ∈ Q such that x q = B βξ ∩ β and, for all α ∈ B βξ , we have q ↾ ↾ ( α + 1) = p αξ . By Clause (7) of Definition 2.1, there is p ∈ Q suchthat: • p is a lower bound for h p βη | η < ξ i ; • p ∈ Q ; • x p = B βξ ; • p ↾ ↾ β = q .Let p βξ be such a p . Subcase 2c: ( β, ξ ) is not active, sup { η < ξ | ( β, η ) is active } = ξ ,and β ∈ E κ + κ . Let α := C β ( ωξ ), so that B βξ = B αξ ∪ { β } . Whendefining p αξ , we were in Subcase 1b. In that Subcase, we considered a γ ∈ E κ + κ such that α ∈ acc( C γ ), produced a condition p with x p = B γξ ,and let p αξ := p ↾ ↾ ( α + 1). Let π : B γξ → B βξ be the unique order-preserving bijection, and let p βξ = π.p . Since, by requirement (v), wehave h π βξ .p βη | η < ξ i = h π γξ .p γη | η < ξ i , and since π ↾ B αξ is the identity,the recursion hypotheses are all easily verified. Subcase 2d: ( β, ξ ) is not active and there is no η < ξ such that β ∈ x p βη . Let p βξ be the unique condition q such that x q = S { x p αξ | α ∈ ( B βξ ∩ β ) } and, for all α ∈ ( B βξ ∩ β ), we have q ↾ ↾ ( α + 1) = p αξ . Subcase 2e: Otherwise.
Let η ∗ := max { sup { η < ξ | ( β, η ) is active } , η β } . Since we are not in any of the previous Subcases, it must be the casethat η ∗ < η , p βη ∗ ∈ Q , and x p βη ∗ = B βη ∗ . For all η ∈ ( η ∗ , ξ ], let q η be theunique condition, given by Clause (5) of Definition 2.1, such that x q η = S { x p αη | α ∈ ( B βη ∩ β ) } and, for all α ∈ ( B βη ∩ β ), we have q ↾ ↾ ( α +1) = p αη .By the recursion hypotheses, we know that, for all η ∈ ( η ∗ , ξ ), p βη isthe ≤ P -greatest lower bound of p βη ∗ and q η , as given by Clause (8) ofDefinition 2.1. Therefore, if we let p βξ be the ≤ P -greatest lower boundof p βη ∗ and q ξ , which again exists by Clause (8) of Definition 2.1, it willfollow that p βξ ≤ P p βη for all η < ξ , so requirement (i) holds. The otherrequirements are easily verified.This completes the construction. We have maintained requirements (i)–(iii) and(v) throughout. We now verify requirement (iv). To this end, fix β < κ + , i < κ ,and x ∈ (cid:0) β +1 θ D i (cid:1) . We will find ξ < κ and q ∈ D i,x such that p βξ ≤ P q . FORCING AXIOM DECIDING THE GENERALIZED SOUSLIN HYPOTHESIS 29
Fix j < κ such that x ⊆ B βj , and fix η ∗ < κ such that ( i η ∗ , j η ∗ , z η ∗ ) = ( i, j, π βj “ x ).Find ξ ∈ acc( κ \ (max { j, η ∗ , η β } + 1)) such that: • h ϑ ξη | η ≤ ξ i = h θ βη | η ≤ ξ i ; • h ̟ ξη,η ′ | η < η ′ ≤ ξ i = h π βη,η ′ | η < η ′ ≤ ξ i ; • h q ξη | η < ξ i = h π βη .p βη | η < ξ i .The following two claims now suffice for the verification of requirement (iv). Claim 6.5. ( β, ξ ) is active.Proof. We verify the requirements in Definition 6.3. We clearly have θ βξ ≤ ϑ ξξ and ξ > η β . Moreover, for all η < ξ , we have ̟ ξη,ξ .q ξη = π βη,ξ .π βη .p βη = π βξ .p βη . Since p βξ ∈ P B βξ is a lower bound for h p βη | η < ξ i , it follows that π βξ .p βξ ∈ P ϑ ξξ is alower bound for h ̟ ξη,ξ .q ξη | η < ξ i . In particular, ξ ∈ X . It follows that s ξ is a lowerbound for h ̟ ξη,ξ .q ξη | η < ξ i = h π βξ .p βη | η < ξ i , which completes the verification. (cid:3) Claim 6.6.
There is q ∈ D i,x such that p βξ ≤ P q .Proof. Since ( β, ξ ) is active and θ βξ = ϑ ξξ , we have π βξ .p βξ = s ξ . It thus suffices tofind q ′ ∈ D i,π βξ “ x such that s ξ ≤ P q ′ .Note that η ∗ , j < ξ and π βj “ x ⊆ θ βj = ϑ ξj . Therefore, since ( i, j, π βj “ x ) =( i η ∗ , j η ∗ , z η ∗ ) and h s ξ | ξ ∈ X i satisfies the conclusion of Lemma 6.2, it followsthat there is q ′ ∈ D i,̟ ξj,ξ “ π βj “ x = D i,π βξ “ x such that s ξ ≤ P q ′ , as desired. (cid:3) Acknowledgments
In 2010, a few days after attending his talk at the , Foreman wrote to the second author that one can con-struct an ℵ -Souslin tree from the conjunction of (cid:3) ℵ and ♦ ( ℵ ), using ideas fromShelah’s “models with second order properties” papers and [7]. This work was neverpublished, and no details of the construction were provided, but this hint turnedout to be quite stimulating. We thank him for pointing us in this direction.Portions of this work were presented by the first author at the Oberseminarmathematische Logik at the University of Bonn in May 2017 and at the in Budapest in July 2017. We thank the organizers for theirhospitality.We also thank the referee for their thoughtful feedback.
References [1] Ari Meir Brodsky and Assaf Rinot. Distributive Aronszajn trees.
Fund. Math. , 2019. acceptedMarch 2018. .[2] Ari Meir Brodsky and Assaf Rinot. A microscopic approach to Souslin-tree constructions.Part I.
Ann. Pure Appl. Logic , 168(11):1949–2007, 2017.[3] Ari Meir Brodsky and Assaf Rinot. Reduced powers of Souslin trees.
Forum Math. Sigma ,5(e2):1–82, 2017.[4] Keith J. Devlin.
Aspects of constructibility . Lecture Notes in Mathematics, Vol. 354. Springer-Verlag, Berlin-New York, 1973.[5] Keith J. Devlin.
Constructibility . Perspectives in Mathematical Logic. Springer-Verlag, Berlin,1984. [6] M. Foreman, M. Magidor, and S. Shelah. Martin’s maximum, saturated ideals and nonregularultrafilters. II.
Ann. of Math. (2) , 127(3):521–545, 1988.[7] Matthew Foreman. An ℵ -dense ideal on ℵ . Israel J. Math. , 108:253–290, 1998.[8] Moti Gitik and Assaf Rinot. The failure of diamond on a reflecting stationary set.
Trans.Amer. Math. Soc. , 364(4):1771–1795, 2012.[9] Tom´aˇs Jech. Non-provability of Souslin’s hypothesis.
Comment. Math. Univ. Carolinae ,8:291–305, 1967.[10] R. Bj¨orn Jensen. The fine structure of the constructible hierarchy.
Ann. Math. Logic , 4:229–308; erratum, ibid. 4 (1972), 443, 1972. With a section by Jack Silver.[11] Ronald B Jensen. Souslin’s hypothesis is incompatible with V=L.
Notices Amer. Math. Soc ,15(6), 1968.[12] ¯Duro Kurepa. Ensembles ordonn´es et ramifi´es.
Publications de l’Institut Math´ematiqueBeograd , 4:1–138, 1935.[13] Chris Lambie-Hanson. Aronszajn trees, square principles, and stationary reflection.
MLQMath. Log. Q. , 63(3-4):265–281, 2017.[14] Richard Laver and Saharon Shelah. The ℵ -Souslin hypothesis. Trans. Amer. Math. Soc. ,264(2):411–417, 1981.[15] William Mitchell. Aronszajn trees and the independence of the transfer property.
Ann. Math.Logic , 5:21–46, 1972/73.[16] Dilip Raghavan and Stevo Todorcevic. Suslin trees, the bounding number, and partitionrelations. to appear in
Israel J. Math. , 2018.[17] Assaf Rinot. Higher Souslin trees and the GCH, revisited.
Adv. Math. , 311(C):510–531, 2017.[18] S. Shelah and L. Stanley. S -forcing. I. A “black-box” theorem for morasses, with applicationsto super-Souslin trees. Israel J. Math. , 43(3):185–224, 1982.[19] Saharon Shelah. Diamonds.
Proc. Amer. Math. Soc. , 138(6):2151–2161, 2010.[20] Saharon Shelah, Claude Laflamme, and Bradd Hart. Models with second order properties.V. A general principle.
Ann. Pure Appl. Logic , 64(2):169–194, 1993.[21] Saharon Shelah and Lee Stanley. S -forcing. IIa. Adding diamonds and more applications:coding sets, Arhangelskii’s problem and L [ Q <ω , Q ]. Israel Journal of Mathematics , 56:1–65, 1986.[22] Saharon Shelah and Lee Stanley. Weakly compact cardinals and nonspecial Aronszajn trees.
Proc. Amer. Math. Soc. , 104(3):887–897, 1988.[23] R. M. Solovay and S. Tennenbaum. Iterated Cohen extensions and Souslin’s problem.
Ann.of Math. (2) , 94:201–245, 1971.[24] Mikhail Yakovlevich Souslin. Probl`eme 3.
Fundamenta Mathematicae , 1(1):223, 1920.[25] E. Specker. Sur un probl`eme de Sikorski.
Colloquium Math. , 2:9–12, 1949.[26] S. Tennenbaum. Souslin’s problem.
Proc. Nat. Acad. Sci. U.S.A. , 59:60–63, 1968.[27] Stevo Todorcevic.
Walks on ordinals and their characteristics , volume 263 of
Progress inMathematics . Birkh¨auser Verlag, Basel, 2007.[28] Stevo Todorcevic and Victor Torres Perez. Conjectures of Rado and Chang and special Aron-szajn trees.
MLQ Math. Log. Q. , 58(4-5):342–347, 2012.[29] Dan Velleman. Souslin trees constructed from morasses. In
Axiomatic set theory (Boulder,Colo., 1983) , volume 31 of
Contemp. Math. , pages 219–241. Amer. Math. Soc., Providence,RI, 1984.[30] Daniel J. Velleman. Morasses, diamond, and forcing.
Ann. Math. Logic , 23(2-3):199–281(1983), 1982.[31] William S. Zwicker. P k λ combinatorics. I. Stationary coding sets rationalize the club filter. In Axiomatic set theory (Boulder, Colo., 1983) , volume 31 of
Contemp. Math. , pages 243–259.Amer. Math. Soc., Providence, RI, 1984.
Department of Mathematics, Bar-Ilan University, ramat-gan 5290002, Israel.
URL : http://math.biu.ac.il/~lambiec Department of Mathematics, Bar-Ilan University, ramat-gan 5290002, Israel.
URL ::