Controlling cardinal characteristics without adding reals
Martin Goldstern, Jakob Kellner, Diego A. Mejía, Saharon Shelah
aa r X i v : . [ m a t h . L O ] M a y CONTROLLING CARDINAL CHARACTERISTICSWITHOUT ADDING REALS
MARTIN GOLDSTERN, JAKOB KELLNER, DIEGO A. MEJ´IA, AND SAHARON SHELAH
Abstract.
We investigate the behavior of cardinal characteristics of the realsunder extensions that do not add new <κ -sequences (for some regular κ ).As an application, we show that consistently the following cardinal char-acteristics can be different: The (“independent”) characteristics in Cicho´n’sdiagram, plus ℵ < m < p < h < add( N ). (So we get thirteen different values,including ℵ and continuum). Introduction
In this work we investigate how to preserve and how to change certain cardinalcharacteristics of the continuum in NNR extensions, i.e., extensions that do not addreals; or more generally that do not add <κ -sequences of ordinals for some regular κ . It is known that the “Blass-uniform” characteristics (see Definition 2.1) tendto keep their values in such extensions (cf. Mildenberger’s [Mil98, Prop. 2.1]), andwe give some explicit results in that direction. Other cardinal characteristics tendto keep a value θ only if θ < κ . We will use this effect to combine various forcingnotions (most of them already known) to get models with many simultaneouslydifferent “classical” characteristics.In particular, we look at the entries of Cicho´n’s diagram, which we call Ci-cho´n-characteristics (see Figure 1, we assume that the reader is familiar with thisdiagram), and the following characteristics:
Definition 1.1.
Let P be a class of posets.(1) m ( P ) denotes the minimal cardinal where Martin’s axiom for the posetsin P fails. More explicitly, it is the minimal κ such that, for some poset Q ∈ P , there is a collection D of size κ of dense subsets of Q such thatthere is no filter in Q intersecting all the members of D .(2) m := m (ccc).(3) Write a ⊆ ∗ b iff a r b is finite. Say that a ∈ [ ω ] ℵ is a pseudo-intersection of F ⊆ [ ω ] ω if a ⊆ ∗ b for all b ∈ F .(4) The pseudo-intersection number p is the smallest size of a filter base of afree filter on ω that has no pseudo-intersection in [ ω ] ℵ . Date : 2020-05-02.2010
Mathematics Subject Classification. cov( N ) / / non( M ) / / / / cof( N ) / / ℵ b / / O O d O O ℵ / / add( N ) / / O O / / O O cov( M ) / / O O non( N ) O O Figure 1.
Cicho´n’s diagram with the two “dependent” valuesremoved, which are add( M ) = min( b , cov( M )) and cof( M ) =max(non( M ) , d ). An arrow x → y means that ZFC proves x ≤ y .cov( N ) (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂❂❂❂❂ non( M ) (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂❂❂❂ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂❂❂❂❂ cof( N ) / / ℵ b O O d ℵ / / m / / p / / h / / add( N ) O O cov( M ) O O non( N ) O O Figure 2.
The model we construct in this paper; here x → y means that x < y . Any number of the < signs can be replaced by= as desired.(5) The tower number t is the smallest order type of a ⊆ ∗ -decreasing sequencein [ ω ] ℵ without pseudo-intersection.(6) The distributivity number h is the smallest size of a collection of densesubsets of ([ ω ] ℵ , ⊆ ∗ ) whose intersection is empty.(7) A family D ⊆ [ ω ] ℵ is groupwise dense if(i) a ⊆ ∗ b and b ∈ D implies a ∈ D , and(ii) whenever ( I n : n < ω ) is an interval partition of ω , there is some a ∈ [ ω ] ℵ such that S n ∈ a I n ∈ D .The groupwise density number g is the smallest size of a collection of group-wise dense sets whose intersection is empty.The known ZFC provable relations between these cardinals are(1.2) m ≤ p = t ≤ h ≤ g , m ≤ add( N ) , t ≤ add( M ) , h ≤ b , g ≤ d . Also, with the exception of m and d , all the cardinals in (1.2) are known to be regular(and uncountable), 2 < t = c and g ≤ cof( c ). For details see e.g. Blass [Bla10], butfor p = t see [MS16] with Malliaris. Recently [GKS19] constructed, assuming four strongly compact cardinals, a ZFCmodel where the ten (non-dependent) Cicho´n-characteristics are pairwise different.This orders the characteristics as shown in Figure 2. In [GKMS20] we give aconstruction that does not require large cardinals.To continue with this line of work, we ask whether other classical cardinal charac-teristics of the continuum can be included and forced to be pairwise different. Our However, only the trivial inequality p ≤ t is used in this text. ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 3 main result is that we can additionally force that ℵ < m < p < h = g < add( N ),thus yielding a model where 13 classical cardinal characteristics are pairwise differ-ent.We now give an outline of this paper: S. 2, p. 4: Preliminaries.
We review some aspects of the Cicho´n’s Maximumconstruction (the construction from [GKMS20] that gives 10 different values inCicho´n’s diagram). In particular, we mention Blass-uniform characteristics and theLCU and COB properties.
S. 3, p. 7: NNR extensions.
We define some classes of cardinal characteris-tics and show how they are affected (or unaffected) by extensions that do not addnew <κ -sequences for some regular κ ; in particular: under <κ -distributive forc-ing extensions; and when intersecting the poset with some <κ -closed elementarysubmodel. S. 4, p. 11: m . Using classical methods of Barnett and Todorˇcevi´c [Tod86, Tod89,Bar92], we modify the Cicho´n’s Maximum construction to additionally force m = λ m for any given regular value λ m between ℵ and add( N ).In addition to m , we can control the Knaster-numbers m ( k -Knaster) as well. Butthis does not give a larger number of simultaneously different characteristics (as allKnaster numbers bigger than ℵ have the same value, which is also the value of m (precaliber)). We give models for all possible constellations (at least for regular λ ): All Knaster numbers (and m (precaliber)) can be ℵ . And there can be a k ≥ m ( ℓ -Knaster) = ℵ for all 1 ≤ ℓ < k and m ( ℓ -Knaster) = λ for ℓ ≥ k .(For notational convenience, we identify 1-Knaster with ccc.) S. 5, p. 15: m (precaliber). We deal with a case that was left open in theprevious section: We construct a model where all Knaster numbers are ℵ , and theprecaliber number is some regular λ > ℵ . S. 6, p. 18: h . Given a poset P , we show how to obtain a complete subposet P ′ of P forcing smaller values to g and c , while preserving certain other values for cardinalcharacteristics already forced by P . This method allows us to get p = h = g . S. 7, p. 20: p . Based on a result with Dow [DS], we show that the product of a ξ -cc poset P with the poset ξ <ξ may add a tower of length ξ , while preserving thecardinal h above ξ and the values for the Cicho´n-characteristics that were alreadyforced by P .This allows us to prove the main theorem, thirteen pairwise different character-istics. S. 8, p. 21: Extensions.
We remark on alternative initial forcings (i.e., forcingsfor the left hand side of Cicho´n’s diagram) and an alternative order.
Notation.
When we are investigating a characteristic x and plan to force a specificvalue to it, we will usually call this value λ x . Let us stress that calling a cardinal λ x is not an implicit assumption that P (cid:13) x = λ x for the P under investigation; itis just an (implicit) declaration of intent. Acknowledgements.
We would like to thank Teruyuki Yorioka for pointing outthe reference [Bar92], which is cited in Section 4.
MARTIN GOLDSTERN, JAKOB KELLNER, DIEGO A. MEJ´IA, AND SAHARON SHELAH Preliminaries
We mention some of the required definitions and constructions from [GKS19]and [GKMS20]. We will not give all required proofs and not even the completeconstruction, as it is rather involved. We will have to assume that the reader eitherknows this construction, or is willing to accept it as a blackbox.2.1.
LCU and COB, the initial forcing P pre for the left side.Definition 2.1. A Blass-uniform cardinal characteristic is a characteristic of theform d R := min {| D | : D ⊆ ω ω and ( ∀ x ∈ ω ω ) ( ∃ y ∈ D ) xRy } for some Borel R .Such characteristics have been studied systematically since at least the 1980s bymany authors, including Fremlin [Fre84], Blass [Bla93, Bla10] and Vojt´aˇs [Voj93].Note that its dual cardinal b R := min {| F | : F ⊆ ω ω and ( ∀ y ∈ ω ω ) ( ∃ x ∈ F ) ¬ xRy } is also Blass-uniform because b R = d R ⊥ where xR ⊥ y iff ¬ ( yRx ). Remark.
All Blass-uniform characteristics in this paper, and many others, such asthose in Blass’ survey [Bla10] or those in [GS93], are in fact of the form b R or d R forsome Σ relation R which is invariant under finite modifications of its arguments.When we restrict to such relations, there is no ambiguity as to which Blass-uniformcardinal characteristics are of the form b R and which are of the form d R . It wasshown by Blass [Bla93] that for such relations R we must have b R ≤ non( M ) and d R ≥ cov( M ), thus b R is always on the left side of Cicho´n’s diagram, and d R is onthe right side. Remark 2.2.
It can be more practical to consider more generally relations on X × Y for some Polish spaces X , Y other than ω ω , in particular as many examplesof Blass-uniform cardinals are naturally defined in such spaces.To cover such cases, one can either modify the definition, or use a Borel isomor-phisms to translate the relation to ω ω .The Cicho´n-characteristics are all Blass-uniform, defined by natural relations.Accordingly, they come in pairs ( b R , d R ) for the according Borel relation R :(add( N ) , cof( N )), (cov( N ) , non( N )), (add( M ) , cof( M )), (non( M ) , cov( M )), and( b , d ). (The last pair, for example, is defined by eventual domination ≤ ∗ .)Another example for a Blass-uniform pair is ( s , r ) = ( b R , d R ) where s is splittingnumber and r the reaping number and R is the relation on [ ω ] ℵ that states xRy iff “ x does not split y ”.We will often have a situation where ( b R , d R ) = ( λ, µ ) is “strongly witnessed”,as follows: We could just as well assume that R is analytic or co-analytic. More specifically, for all resultsin this paper, it is enough to assume that R is absolute between the extensions we consider; inour case between extensions that do not add new reals. So even projective relations would beOK. However, all concrete relations that we will actually use are Borel, even of very low rank.Regarding “on ω ω ”, see Remark 2.2. The relations R used to define the following characteristics are “natural”, but not entirely“canonical”. For example, a different choice of a natural relation R such that b R = s leads to adifferent dual d R = r σ . See [Bla10, Example 4.6]. ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 5
Definition 2.3.
Fix a Borel relation R , λ a regular cardinal and µ an arbitrarycardinal. We define two properties: Linearly cofinally unbounded:
LCU R ( λ ) means: There is a family ¯ f = ( f α : α < λ ) of reals such that:(2.4) ( ∀ g ∈ ω ω ) ( ∃ α ∈ λ ) ( ∀ β ∈ λ \ α ) ¬ f β Rg.
Cone of bounds:
COB R ( λ, µ ) means: There is a <λ -directed partial order E on µ , and a family ¯ g = ( g s : s ∈ µ ) of reals such that(2.5) ( ∀ f ∈ ω ω ) ( ∃ s ∈ µ ) ( ∀ t D s ) f Rg t . Fact 2.6.
LCU R ( λ ) implies b R ≤ λ ≤ d R . COB R ( λ, µ ) implies b R ≥ λ and d R ≤ µ . Remark 2.7.
COB R ( λ, µ ) clearly implies COB R ( λ ′ , µ ) whenever λ ′ ≤ λ . Theproperty COB R (2 , µ ), the weakest of these notions, just says that there is a witnessfor d R ≤ µ , or in other words: there is an R -dominating family of size µ .Also, COB R ( λ, µ ) implies COB R ( λ, µ ′ ) whenever µ ′ ≥ µ . And note that, fornontrivial R , COB R ( λ, µ ) implies λ ≤ µ .Informally, we call the objects ¯ f in the definition of LCU and ( E , ¯ g ) for COB“strong witnesses”, and say that the corresponding cardinal inequalities (or equal-ities) are “strongly witnessed”.In [GKS19] (building on [GMS16]) the following is shown: Lemma 2.8.
Assume GCH and ℵ < ν < ν < ν < ν < θ ∞ are all successorsof regular cardinals. Then there is a ccc countable support iteration P pre of length θ ∞ + θ ∞ forcing that ℵ < add( N ) = ν < cov( N ) = ν < b = ν < non( M ) = ν < c = θ ∞ . Moreover, all the equalities are strongly witnessed; all iterands in P are ( σ, k ) -linked(see Definition 4.1) for all k ; and in the first θ ∞ many steps we add Cohen reals. In this work, we will modify this construction P pre to get similar iterations P thatallow us to add additional characteristics. We claim that these modifications willnot change the fact that the characteristics in Lemma 2.8 are strongly witnessed.A reader who doesn’t know the proof of Lemma 2.8 will hopefully trust us on this;for the others we give the (simple) argument: • We get the required COB properties simply by bookkeeping, when forcingwith “partial random”, or “partial eventually different”, etc., forcings. Thiswill not change when we add additional iterands (as long as, cofinally often,we choose the iterands as in the original construction). • Fix a (left hand) Cicho´n-characteristic x other than b . We get the strongwitness LCU R ( ν ) (for R a relation connected to x and ν the according ν i )because all the iterands are “( ν, R )-good”.Any forcing of size <ν is automatically good, so adding small iterandswill not be a problem. In [BCM18] (and in other related work), a family with LCU R ( λ ) is said to be strongly λ - R -unbounded of size λ , while a family with COB R ( λ, µ ) is said to be strongly λ - R -dominating ofsize µ . I.e., every subset of µ of cardinality <λ has a E -upper bound Formally: D ⊆ ω ω is R -dominating iff ( ∀ x ∈ ω ω ) ( ∃ y ∈ D ) xRy . MARTIN GOLDSTERN, JAKOB KELLNER, DIEGO A. MEJ´IA, AND SAHARON SHELAH
Also, σ -centered forcings are always good for the characteristics add( N )and cov( N ). • For b , it is more cumbersome to prove LCU R ( ν ), but at least it is clearthat adding additional iterands of size <ν will not interfere with the proof.So we can summarize: Claim 2.9.
We can add to P pre arbitrary iterands that all are • either of size <ν , • or σ -centered and of size <ν ,and still force strong witnesses for the Cicho´n-characteristics of Lemma 2.8. (Of course these new iterands have to be added in a way so that we still use theold iterands unboundedly often; we cannot just add new iterands at the end.) Remark.
Instead of the construction of [GKS19], one can use alternative construc-tions that require weaker assumptions, cf. Section 8.3.2.2.
The Cicho´n’s Maximum construction.
As before, we will not require ordescribe the construction in detail, but only present the basic structure and certainproperties.The following is the main Theorem (3.1) of [GKMS20]. As we will use theassumptions of the theorem repeatedly, we make them explicit:
Assumption 2.10.
Assume GCH, and that ℵ ≤ κ ≤ λ add( N ) ≤ λ cov( N ) ≤ λ b ≤ λ non( M ) ≤≤ λ cov( M ) ≤ λ d ≤ λ non( N ) ≤ λ cof( N ) ≤ λ ∞ are regular cardinals, with the possible exception of λ ∞ , for which we only require λ <κ ∞ = λ ∞ . Theorem 2.11.
Under these assumptions, there is a ccc poset P fin forcing strongwitnesses for ℵ ≤ add( N ) = λ add( N ) ≤ cov( N ) = λ cov( N ) ≤ b = λ b ≤ non( M ) = λ non( M ) ≤ cov( M ) = λ cov( M ) ≤ d = λ d ≤ non( N ) = λ non( N ) ≤ cof( N ) = λ cof( N ) ≤ c = λ ∞ . Note that κ does not make much sense in this theorem, as you can just set κ = ℵ (resulting in the weakest requirement λ ℵ ∞ = λ ∞ ). Indeed this is what isdone in [GKMS20] (where κ is not mentioned at all); but mentioning κ explicitlyhere will be useful in Lemma 2.12 below.The construction in [GKMS20] is as follows:(A) Pick a sequence of successors of regular cardinals (strictly) above λ ∞ : ξ < ν < ξ < ν < ξ < ν < ξ < ν < θ ∞ , (B) Start with any initial κ -cc poset P pre for the “left hand side”, which forces“strong witnesses” foradd( N ) = ν < cov( N ) = ν < b = ν < non( M ) = ν < c = θ ∞ (So we can use the forcing of Lemma 2.8, or any modification satisfyingClaim 2.9.)The proof in [GKMS20] can then be formulated as the following: ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 7
Lemma 2.12.
Under Assumption 2.10, and given a forcing P pre as in (A) and (B),there is a <κ -closed elementary submodel N ∗ of H ( χ ) such that P fin := P pre ∩ N ∗ witnesses Theorem 2.11. (As usual, χ is a sufficiently large, regular cardinal.)2.3. History.
We briefly remark on the history of the result of this section.A (by now) classical series of results by various authors [Bar84, BJS93, CKP85,JS90, Kam89, Kra83, Mil81, Mil84, RS83] (summarized by Bartoszy´nski and Ju-dah [BJ95]) shows that any assignments of {ℵ , ℵ } to the Cicho´n-characteristicsthat satisfy the well known ZFC restrictions is consistent. This leaves the ques-tions how to show that many values can be simultaneously different. The “lefthand side” part was done in [GMS16] and uses eventually different forcing E toensure non( M ) ≥ λ non( M ) and ultrafilter-limits of E to show that b remains small.It relies heavily on the notion of goodness, introduced in [JS90] (with Judah) andby Brendle [Bre91], and summarized in e.g. [GMS16] or [CM19] (with Cardona).Based on this construction, [GKS19] uses Boolean ultrapowers to get simulta-neously different values for all (independent) Cicho´n-characteristics, modulo fourstrongly compact cardinals.For this, the construction for the left hand side first has to be modified to get accc forcing starting with a ground model satisfying GCH.Then Boolean ultrapowers are applied to separate the cardinals on the right side.[KTT18] (with Tˇanasie and Tonti) gives an introduction to the Boolean ultrapowerconstruction. Such Boolean ultrapowers are applied four times, once for each pairof cardinals on the right side that are separated.For this it is required that there is a strongly compact cardinal between twovalues corresponding to adjacent cardinals characteristics on the left side, so thecardinals on this side are necessarily very far apart. [BCM18] improves the lefthand side construction of [GMS16] to include cov( M ) < d = non( N ) = c . Thisis achieved by using matrix iterations of partial Frechet-linked posets (the latterconcept is originally from [Mej19]). Then the same method of Boolean ultrapowersas before can be applied, in the same way, to force different values for all Cicho´n-characteristics, modulo three strongly compact cardinals.Finally, in [GKMS20] we can get the result without assuming large cardinals;this is the construction we use in this paper.3. Cardinal characteristics in extensions without new <κ -sequences Let us consider <κ -distributive forcing extensions for some regular κ . (In par-ticular these extensions are NNR, i.e., do not add new reals.) For such extensions,we can also preserve strong witnesses in some cases: Lemma 3.1.
Assume that Q is θ -cc and <κ -distributive for κ regular uncountable,and let λ be a regular cardinal and R a Borel relation.(1) If LCU R ( λ ) , then Q (cid:13) LCU R (cof( λ )) .So if additionally λ ≤ κ or θ ≤ λ , then Q (cid:13) LCU R ( λ ) .(2) If COB R ( λ, µ ) and either λ ≤ κ or θ ≤ λ , then Q (cid:13) COB R ( λ, | µ | ) .So for any λ , COB R ( λ, µ ) implies Q (cid:13) COB R (min( | λ | , κ ) , | µ | ) . [GKMS20] uses the case κ = ℵ , so we get only a countably closed N ∗ . But the the proofthere works for any uncountable regular κ , with only the trivial change: We let N be a <κ -closedmodel of size λ ∞ , and note that then N ∗ is <κ -closed as well. MARTIN GOLDSTERN, JAKOB KELLNER, DIEGO A. MEJ´IA, AND SAHARON SHELAH
Proof.
For (1) it is enough to assume that Q does not add reals: Take a strongwitness for LCU R ( λ ). This object still satisfies (2.4) in the Q -extension (as thereare no new reals), but the index set will generally not be regular any more; we canjust take a cofinal subset of order type cof( λ ) which will still satisfy (2.4).Similarly, a strong witness for COB R ( λ, µ ) still satisfies (2.5) in the Q extension.However, the index set is generally not <λ -directed any more, unless we eitherassume λ ≤ κ (as in that case there are no new small subsets of the partial order)or Q is λ -cc (as then every small set in the extension is covered by a small set fromthe ground model). (cid:3) If P forces strong witnesses, then any complete subforcing that includes namesfor all witnesses also forces strong witnesses: Lemma 3.2.
Assume that R is a Borel relation, P ′ is a complete subforcing of P , λ regular and µ is a cardinal, both preserved in the P -extension.(a) If P (cid:13) LCU R ( λ ) witnessed by some ˙¯ f , and ˙¯ f is actually a P ′ -name, then P ′ (cid:13) LCU R ( λ ) .(b) If P (cid:13) COB R ( λ, µ ) witnessed by some ( ˙ E , ˙¯ g ) , and ( ˙ E , ˙¯ g ) is actually a P ′ -name, then P ′ (cid:13) COB R ( λ, | µ | ) .Proof. Let V be the P -extension and V the intermediate P ′ -extension. For LCU:(2.4) holds in V , V ⊆ V and ( f i ) i<λ ∈ V , and R is absolute between V and V ,so (2.4) holds in V . The argument for COB is similar. (cid:3) We now define three properties of cardinal characteristics (more general thanBlass-uniform) that have implications for their behaviour in extensions withoutnew <κ -sequences. We call these properties e.g. t -like to refer to the “typical”representative t . But note that this is very superficial: There is no deep connectionor similarity to t for all t -like characteristics, it is just that t is a well known examplefor this property, and “ t -like” seems easier to memorize than other names we cameup with. Definition 3.3.
Let x be a cardinal characteristic.(1) x is t -like , if it has the following form: There is a formula ψ ( x ) (possiblywith, e.g., real parameters) absolute between universe extensions that donot add reals, such that x is the smallest cardinality λ of a set A of realssuch that ψ ( A ).All Blass-uniform characteristics are t -like; other examples are t , u , a and i .(2) x is called h -like , if it satisfies the same, but with A being a family of setsof reals (instead of just a set of reals).Note that t -like implies h -like, as we can include “the family of sets ofreals is a family of singletons” in ψ . Examples are h and g .(3) x is called m -like , if it has the following form: There is a formula ϕ (possiblywith, e.g., real parameters) such that x is the smallest cardinality λ suchthat H ( ≤ λ ) (cid:15) ϕ . Concretely, if M ⊆ M are transitive (possibly class) models of a fixed, large fragment ofZFC, with the same reals, then ψ is absolute between M and M . ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 9
Any infinite t -like characteristic is m -like: If ψ witnesses t -like, then wecan use ϕ = ( ∃ A ) [ ψ ( A )&( ∀ a ∈ A ) a is a real] to get m -like (since H ( ≤ λ )contains all reals). Examples are m , m (Knaster), etc.(Actually, we do not know anything about t -like characteristics in general, apartfrom the fact that they are both m -like and h -like.) Lemma 3.4.
Let V ⊆ V be models (possibly classes) of set theory (or a sufficientfragment), V transitive and V is either transitive or an elementary submodel of H V ( χ ) for some large enough regular χ , such that V ∩ ω ω = V ∩ ω ω .(a) If x is h -like, then V (cid:15) x = λ implies V (cid:15) x ≤ | λ | .In addition, whenever κ is uncountable regular in V and V <κ ∩ V ⊆ V :(b) If x is m -like, then V (cid:15) x ≥ κ iff V (cid:15) x ≥ κ .(c) If x is m -like and λ < κ , then V (cid:15) x = λ iff V (cid:15) x = λ .(d) If x is t -like and λ = κ , then V (cid:15) x = λ implies V (cid:15) x = λ .Proof. First note that (d) follows by (a) and (b) because any t -like characteristicis both m -like and h -like.Assume V is transitive. For (a), if ψ witnesses that x is h -like, A ∈ V and V satisfies ψ ( A ), then the same holds in V . For (b) and (c), note that H V ( ≤ µ ) = H V ( ≤ µ ) for all µ < κ (easily shown by ∈ -induction).The case V = N (cid:22) H V ( χ ) is similar. Note that H V ( χ ) is a transitive subsetof V , so (a) follows by the previous case. For (b) and (c), work inside V . Notethat κ ⊆ N (by induction). Whenever µ < κ , µ is regular iff N | =“ µ regular”, and H ( ≤ µ ) ⊆ N . So N | =“ H ( ≤ µ ) | = φ ” iff H ( ≤ µ ) | = φ .Alternatively, the case V (cid:22) H V ( χ ) is a consequence of the first case. Workin V . Let π : V → ¯ V be the transitive collapse of V . Note that π ( x ) = x forany x ∈ ω ω ∩ V , so ω ω ∩ ¯ V = ω ω ∩ V = ω ω . To see (a), V (cid:15) x = λ implies¯ V (cid:15) x = π ( λ ), so x ≤ | π ( λ ) | ≤ | λ | by the transitive case.Now assume V <κ ⊆ V (still inside V ), so we also have ¯ V <κ ⊆ ¯ V . To see (b), V | = x ≥ κ iff ¯ V | = x ≥ π ( κ ) = κ , iff V | = x ≥ κ by the transitive case. Property(c) follows similarly by using π ( λ ) = λ (when λ < κ ). (cid:3) We apply this to three situations: Boolean ultrapowers (which we will not applyin this paper), extensions by distributive forcings, and complete subforcings:
Corollary 3.5.
Assume that κ is uncountable regular, P (cid:13) x = λ , and(i) either Q is a P -name for a <κ -distributive forcing, and we set P + := P ∗ Q and j ( λ ) := λ ;(ii) or P is ν -cc for some ν < κ , j : V → M is a complete embedding into atransitive <κ -closed model M , cr( j ) ≥ κ , and we set P + := j ( P ) ,(iii) or P is κ -cc, M (cid:22) H ( χ ) is <κ -closed, and we set P + := P ∩ M and j ( λ ) := | λ ∩ M | . (So P + is a complete subposet of P ; and if λ ≤ κ then j ( λ ) = λ .)Then we get:(a) If x is m -like and λ ≥ κ , then P + (cid:13) x ≥ κ .(b) If x is m -like and λ < κ , then P + (cid:13) x = λ .(c) If x is h -like then P + (cid:13) x ≤ | j ( λ ) | . Concretely, m can be characterized as the smallest λ such that there is in H ( ≤ λ ) a ccc forcing Q and afamily ¯ D of dense subsets of Q such that “there is no filter F ⊆ Q meeting all D i ” holds. for (i): P + (cid:13) x ≤ | λ | ;for (ii): P + (cid:13) x ≤ | j ( λ ) | ;for (iii): P + (cid:13) x ≤ | λ ∩ M | .(d) So if x is t -like and λ = κ , then for (i) and (iii) we get P + (cid:15) x = κ .Proof. Case (i).
Follows directly from Lemma 3.4.
Case (ii).
Since M is <κ -closed and P is ν -cc, P (or rather: the isomorphicimage j ′′ P ) is a complete subforcing of j ( P ). Let G be a j ( P )-generic filter over V . As j ( P ) is in M (and M is transitive), G is generic over M as well. Then V := M [ G ] is <κ closed in V := V [ G ].First note that V and V have the same <κ -sequences of ordinals. Let ˙¯ x =( ˙ x i ) i ∈ µ be a sequence of j ( P )-names for members of M with µ < κ . Each ˙ x i isdetermined by an antichain, which has size <ν and therefore is in M , so each ˙ x i isin M . Hence ˙¯ x is in M .By elementaricity, P (cid:13) x = λ implies M | =“ j ( P ) (cid:13) x = j ( λ )”. So V | = x = j ( λ ),and we can apply Lemma 3.4: In the case that x is m -like, if λ ≥ κ , then j ( λ ) ≥ j ( κ ) ≥ κ , so V | = x ≥ κ ; If λ < κ , then j ( λ ) = λ , so V | = x = λ ; if x is h -like, then V | = x ≤ | j ( λ ) | . Case (iii).
Let π : M → ¯ M be the transitive collapse. Set ¯ P := π ( P ) ∈ ¯ M .Note that π ( κ ) = κ and that ¯ M is <κ -closed. Also, any condition in P is M -generic since, for any antichain A in P , A ∈ M iff A ⊆ M (by <κ -closedness).Let G + be P + -generic over V . We can extend G + to a P -generic G over V (as P + is a complete subforcing of P ), and we get G + = G ∩ P + = G ∩ M . Now work in V [ G ]. Note that M [ G ] is an elementary submodel of H V [ G ] ( χ ) (and obviously nottransitive), and that the transitive collapse π : M [ G ] → V extends π (as there areno new elements of V in M [ G ]). We claim that V = ¯ M [ ¯ G + ] where ¯ G + := π ′′ G + (which is ¯ P -generic over ¯ M , also ¯ G + = π ( G )), and that ¯ τ [ ¯ G + ] = π ( τ [ G ]) for any P -name τ ∈ M , where ¯ τ := π ( τ ). So in particular, V is a subset of V := V [ G + ](the P + -generic extension of V ) because π and M (and therefore ¯ M ) are elementsof V , so G + (and therefore ¯ G + ) are elements of V [ G + ]. In fact, ¯ G + is ¯ P -generic over V because ¯ M is <κ -closed and ¯ P is κ -cc, moreover, V = V [ ¯ G + ] (this is reflectedby the fact that, in V , π ↾ P + is an isomorphism between P + and ¯ P ).We claim:( ∗ ) V is an NNR extension of V , moreover V is <κ -closed in V .To show this, work in V . We argue with ¯ P . Let τ be a ¯ P -name of an element of V = ¯ M [ ¯ G + ]. So we can find a maximal antichain A in ¯ P and, for each a ∈ A ,a ¯ P -name σ a in ¯ M such that a (cid:13) ¯ P τ = σ a . Since | A | < κ and ¯ P ⊆ ¯ M and ¯ M is <κ -closed, A , as well as the function a σ a , are in ¯ M . Mixing the names σ a along A to a name σ ∈ ¯ M , we get ¯ M (cid:15) a (cid:13) ¯ P σ a = σ for all a ∈ A , which implies V (cid:15) a (cid:13) ¯ P σ a = σ because the forcing relation of atomic formulas is absolute. So¯ P (cid:13) τ = σ .Now fix a ¯ P name ~τ = ( τ α ) α<µ of a sequence of elements of V , with µ < κ .Again we use closure of ¯ M and get a sequence ( σ α ) α<µ in ¯ M such that ¯ P forcesthat τ α = σ α [ ¯ G + ], and so the evaluation of the sequence ~τ is in ¯ M [ ¯ G + ] = V . Thisproves ( ∗ ).Now assume that x is either h -like or m -like, and P (cid:13) x = λ . By elementaricity,this holds in M , so ¯ M (cid:15) ¯ P (cid:13) x = π ( λ ). Now let ¯ G + be ¯ P -generic over V , V := This can be proved by induction on the rank of τ , and uses that M [ G ] (cid:22) H V [ G ] ( χ ). ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 11 (cid:15) (cid:15) o o (cid:15) (cid:15) · · · o o (cid:15) (cid:15) σ -centered (cid:15) (cid:15) o o ccc 2-Knaster o o o o · · · o o precaliber o o Figure 3.
Some classes of ccc forcings¯ M [ ¯ G + ] and V := V [ ¯ G + ], so V | = x = π ( λ ). If x is h -like then, by Lemma 3.4(a), V | = x ≤ | π ( λ ) | = | λ ∩ M | ; if x is m -like and λ < κ , then V | = x = λ and so thesame is satisfied in V by Lemma 3.4(c); otherwise, if λ ≥ κ then V | = x = π ( λ ) ≥ π ( κ ) = κ , so V | = x ≥ κ by Lemma 3.4(b).In any of the cases above, (d) is a direct consequence of (a) and (c). (cid:3) Dealing with m We show how to deal with m . It is easy to check that the Cicho´n’s Maximumconstruction from [GKS19] forces m = ℵ , and can easily be modified to force m = add( N ) (by forcing with all small ccc forcings during the iteration). With abit more work it is also possible to get ℵ < m < add( N ).Let us start by recalling the definitions of some well-known classes of ccc forcings: Definition 4.1.
Let λ be an infinite cardinal, k ≥ Q be a poset.(1) Q is ( λ, k ) -Knaster if, for every A ∈ [ Q ] λ , there is a B ∈ [ A ] λ which is k -linked (i.e., every c ∈ [ B ] k has a lower bound in Q ). We write k -Knaster for ( ℵ , k )-Knaster; Knaster means 2-Knaster; ( λ, -Knaster denotes λ -ccand 1 -Knaster denotes ccc. (2) Q has precaliber λ if, for every A ∈ [ Q ] λ , there is a B ∈ [ A ] λ which iscentered, i.e., every finite subset of B has a lower bound in Q . We sometimesshorten “precaliber ℵ ” to “precaliber”.(3) Q is ( σ, k ) -linked if there is a function π : Q → ω such that π − ( { n } ) is k -linked for each n .(4) Q is σ -centered if there is a function π : Q → ω such that each π − ( { n } ) iscentered.The implications between these notions (for λ = ℵ ) are listed in Figure 3. Toeach class C of forcing notions, we can define the Martin’s Axiom number m ( C )in the usual way (recall Definition 1.1). An implication C ← C in the diagramcorresponds to a ZFC inequality m ( C ) ≤ m ( C ). Recall that m ( σ -centered) = p = t . Also recall that, in the old constructions, all iterands were ( σ, k )-linked for all k . Lemma 4.2. (1) If there is a Suslin tree, then m = ℵ .(2) After adding a Cohen real c over V , in V [ c ] there is a Suslin tree.(3) Any Knaster poset preserves Suslin trees.(4) The result of any finite support iteration of ( λ, k ) -Knaster posets ( λ un-countable regular and k ≥ ) is again ( λ, k ) -Knaster.(5) In particular, when k ≥ , if P is a f.s. iteration of forcings such that alliterands are either ( σ, k ) -linked or smaller than λ , then P is ( λ, k ) -Knaster. This is just an abuse of notation that turns out to be convenient for stating our results. (6) Let C be any of the forcing classes of Figure 3, and assume m ( C ) = λ > ℵ .(Or just assume that C is a class of ccc forcings closed under Q Q <ω , thefinite support product of countably many copies of Q , and under ( Q, p ) q : q ≤ p } for p ∈ Q .)If Q ∈ C , then every subset A of Q of size <λ is “ σ -centered in Q ” (i.e.,there is a function π : A → ω such that every finite π -homogeneous subsetof A has a common lower bound in Q ).So in particular, for all µ < λ of uncountable cofinality, Q has precaliber µ and is ( µ, ℓ ) -Knaster for all ℓ ≥ .(7) m > ℵ implies m = m (precaliber) . m ( k -Knaster) > ℵ implies m ( k -Knaster) = m (precaliber) .Proof. (1): Clear. (2): See [She84, Tod89] or Velleman [Vel84]. (3): Recall thatthe product of a Knaster poset with a ccc poset is still ccc. Hence, if P is Knasterand T is a Suslin tree, then P × T = P ∗ ˇ T is ccc, i.e., T remains Suslin in the P -extension.(4): Well-known, see e.g. Kunen [Kun11, Lemma V.4.10] for ( ℵ , σ, k )-linked implies ( µ, k )-Knaster (for all uncountable regular µ ),and since every forcing of size <µ is ( µ, k )-Knaster (for any k ).(6): First note that it is well known that MA ℵ (ccc) implies that every cccforcing is Knaster, and hence that the class C of ccc forcings is closed under Q Q <ω . (For the other classes C in Figure 3, the closure is immediate.)So let C be a closed class, m ( C ) = λ > ℵ , Q ∈ C and A ∈ [ Q ] <λ . Given a filter G in Q <ω and q ∈ Q , set c ( q ) = n iff n is minimal such that there is a ¯ p ∈ G with p ( n ) = q . Note that for all q , the set D q = { p ∈ Q <ω : ( ∃ n ∈ ω ) q = p ( n ) } is dense, and that c ( q ) is defined whenever G intersects D q . Pick a filter G meetingall D q for q ∈ A . This defines c : A → ω such that c ( a ) = c ( a ) = · · · = c ( a ℓ − ) = n implies that all a i appear in G ( n ) and thus they are compatible in Q . Hence, A isthe union of countably many centered (in Q ) subsets of Q .(7): Follows as a corollary. (cid:3) This shows that it is not possible to simultaneously separate more than twoKnaster numbers. More specifically: ZFC proves that there is a 1 ≤ k ∗ ≤ ω and, if k ∗ < ω , a λ > ℵ , such that for all 1 ≤ ℓ < ω (4.3) m ( ℓ -Knaster) = ( ℵ if ℓ < k ∗ λ otherwise . (Recall that m (1-Knaster) = m (ccc) by our definition.)In this section, we will show how these constellations can be realized togetherwith the previous values for the Cicho´n-characteristics.In the case k ∗ < ω , we know that m (precaliber) = λ as well. We briefly commentthat m (precaliber) = ℵ (in connection with the Cicho´n-values) is possible too. Inthe next section, we will deal with the remaining case: k ∗ = ω , i.e., all Knasternumbers are ℵ , while m (precaliber) > ℵ . See, e.g., Jech [Jec03, 16.21] (and the historical remarks, where the result is attributed to(independently) Kunen, Rowbottom and Solovay), or [BJ95, 1.4.14] or Galvin [Gal80, Pg. 34].
ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 13
The central observation is the following, see [Tod86, Tod89] and [Bar92, Sect. 3].
Lemma 4.4.
Let k ∈ ω , k ≥ and λ be uncountable regular. Let C be the finitesupport iteration of λ many copies of Cohen forcing. Assume that C forces that P is ( λ, k + 1) -Knaster. Then C ∗ P forces m ( k -Knaster) ≤ λ .The same holds for k = 1 and λ = ℵ . For k = 1 this trivially follows from Lemma 4.2: The first Cohen forcing adds aSuslin tree, which is preserved by the rest of the Cohen posets composed with P .So we get m = ℵ . The proof for k > Remark 4.5.
Adding the Cohen reals first is just for notational convenience. Thesame holds, e.g., in a f.s. iteration where we add Cohen reals on a subset of theindex set of order type λ ; and we assume that the (limit of the) whole iteration is( λ, k + 1)-Knaster. Lemma 4.6.
Under the assumption of Lemma 4.4, for k ≥ : We interpret eachCohen real η α ( α ∈ λ ) as an element of ( k + 1) ω . C ∗ P forces: For all X ∈ [ λ ] λ , ( ⋆ ) ( ∃ ν ∈ ( k + 1) <ω ) ( ∃ α , . . . , α k ∈ X ) ( ∀ ≤ i ≤ k ) ν ⌢ i ⊳ η α i Proof.
Let p ∗ ∈ C ∗ P force that X ∈ [ λ ] λ . By our assumption, first note that p ∗ ↾ λ forces that there is some X ′ ∈ [ λ ] λ and a k + 1-linked set { r α : α ∈ X ′ } ofconditions in P below p ∗ ( λ ) such that r α (cid:13) P α ∈ X for any α ∈ X ′ .Since X ′ is a C -name, there is some Y ∈ [ λ ] λ and, for each α ∈ Y , some p α ≤ p ∗ ↾ λ in C forcing α ∈ X ′ . We can assume that α ∈ dom( p α ) and, by thinning out Y ,that dom( p α ) forms a ∆-system with heart a below each α ∈ Y , h p α ↾ a : α ∈ Y i isconstant, and that p α ( α ) is always the same Cohen condition ν ∈ ( k + 1) <ω .For each α ∈ Y let q α ∈ C ∗ P such that q α ↾ λ = p α and q α ( λ ) = r α . It isclear that h q α : α ∈ Y i is k + 1-linked and that q α (cid:13) α ∈ X . Pick α , . . . , α k ∈ Y and q ≤ q α , . . . , q α k . We can assume that q ↾ λ is just the union of the q α i ↾ λ . Inparticular, we can extend q ( α i ) = ν to ν ⌢ i , satisfying ( ⋆ ) after all. This proves theclaim. (cid:3) Lemma 4.7.
Under the assumption of Lemma 4.4, for k ≥ : In V C define R K,k to be the set of finite partial functions p : u → ω , u ⊆ λ finite, such that ( ⋆ ) failsfor all p -homogeneous X ⊆ u . Then P forces the following:(a) There is no filter on R K,k meeting all dense D α ( α ∈ λ ), where we set D α = { p : α ∈ dom( u ) } .(b) R K,k is k -Knaster. Note that this proves Lemma 4.4, as R K,k is a witness.
Proof.
Clearly each D α is dense (as we can just use a hitherto unused color). If G is a filter meeting all D α , then G defines a total function p ∗ : λ → ω , and there issome n ∈ ω such that X := p ∗− ( { n } ) has size λ . So ( ⋆ ) holds for X , witnessedby some α , . . . , α k . Now pick some q ∈ G such that all α i are in the domain of q .Then q contradicts the definition of R K,k . R K,k is k -Knaster: Given ( r α : u α → ω ) α ∈ ω , we thin out so that u α formsa ∆-system of sets of the same size and such that each r α has the same “type”,independent of α , where the type contains the following information: The color Say that X ⊆ u is p -homogeneous if p ↾ X is a constant function. assigned to the n -the element of u α ; the (minimal, say) h such that all η β ↾ h aredistinct for β ∈ u α , and η β ↾ h + 1.We claim that the union of k many such r α is still in R K,k : Assume towards acontradiction that there is a S i For each of the following items (1) to (3), and ℵ ≤ λ ≤ ν regular, P pre can be modified to some forcing P ′ which still strongly witnesses the Cicho´n-characteristics, and additionally satisfies:(1) Each iterand in P ′ is ( σ, ℓ ) -linked for all ℓ ≥ ; and P ′ forces ℵ = m = m (precaliber) ≤ p = b . (2) Fix k ≥ . Each iterand in P ′ is k + 1 -Knaster, and additionally either ( σ, ℓ ) -linked for all ℓ or of size less than λ ; and P ′ forces ℵ = m = m ( k -Knaster) < m ( k + 1-Knaster) = m (precaliber) = λ ≤ p = b . (3) Each iterand in P ′ is either ( σ, ℓ ) -linked for all ℓ , or ccc of size less than λ ; and P ′ forces m = m (precaliber) = λ ≤ p = b . Proof. An argument like in [Bre91] works. We first modify P pre as follows:We construct an iteration P with the same index set δ as P pre ; we partition δ into two cofinal sets δ = S old ∪ S new of the same size. For α ∈ S old we define Q α as we defined Q ∗ α for P pre . For α ∈ S new , pick (by suitable book-keeping) a small(less than ν , the value for b ) σ -centered forcing Q α .As cof( δ ) ≥ λ b , we get that P forces p ≥ ν . Also, P still adds strong witnessesfor the Cicho´n-characteristics, according to Claim 2.9: All new iterands are smallerthan ν and σ -centered.Note that all iterands are still ( σ, k )-linked for all k (as the new ones are even σ -centered).To deal with ℓ -Knaster, recall that the first λ ∞ iterands are Cohen forcings; andwe call these Cohen reals η α ( α ∈ λ ∞ ). Given ℓ , we can (and will) interpret theCohen real η α as an element of ( ℓ + 1) ω .(1) Recall from [Bar92, Sect. 2] that, after a Cohen real, there is a precaliber ω poset Q ∗ such that no σ -linked poset adds a filter intersecting certain ℵ -many densesubsets of Q ∗ . Therefore, the P we just constructed forces m (precaliber) = ℵ . To be more precise, after one Cohen real there is a sequence ¯ r = h r α : ω → α ∈ ω limit i such that, for any ladder system ¯ c from the ground model, the pair (¯ c, ¯ r ), as a ladder systemcoloring, cannot be uniformized in any stationary subset of ω . Furthermore, this property is ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 15 (2) Just as with the modification from P pre to P , we now further modify P to force(by some bookkeeping) with all small (smaller than λ ) k + 1-Knaster forcings. Sothe resulting iteration obviously forces m ( k + 1-Knaster) ≥ λ .Note that now all iterands are either smaller than λ ≤ ν or σ -linked (so wecan again use Claim 2.9); and additionally all iterands are k + 1-Knaster. So P is both ( ℵ , k + 1)-Knaster and ( λ, ℓ )-Knaster for any ℓ . Again by Lemma 4.4, P forces both m ( k -Knaster) = ℵ and m ( ℓ -Knaster) ≤ λ for any ℓ , (which implies m ( k + 1-Knaster) = λ ).(3) This is very similar, but this time we use all small ccc forcings (not just the k + 1-Knaster ones). This obviously results in m ≥ λ ; and the same argument asabove shows that still m ( ℓ -Knaster) ≤ λ for all ℓ . (cid:3) This, together with Corollary 3.5 and Lemma 2.12 gives us 11 characteristics.However, we postpone this collorally until a time we can also add h = g = p = κ inLemma 6.4. 5. Dealing with the precaliber number Recall the possible constellations for the Knaster numbers and the definition of k ∗ given in (4.3). Note that if k ∗ < ω , then m (precaliber) = λ as well.In this section, we construct models for all Knaster numbers being ℵ and m (precaliber) = λ for some given regular ℵ < λ ≤ add( N ) (and the “old” valuesfor the Cicho´n-characteristics, as in the previous section). Definition 5.1. Let λ > ℵ be regular. A condition p ∈ P cal = P cal ,λ consists of(i) finite sets u p , F p ⊆ λ ,(ii) a function c p : [ u p ] → α ∈ F p , a function d p,α : P ( u p ∩ α ) → ω satisfying( ⋆ ) if α ∈ F p and s , s are 1-homogeneous (w.r.t. c p ) subsets of u p ∩ α with d p,α ( s ) = d p,α ( s ), then s ∪ s is 1-homogeneous.The order is defined by q ≤ p iff u p ⊆ u q , F p ⊆ F q , c p ⊆ c q and d p,α ⊆ d q,α for any α ∈ F p . Lemma 5.2. P cal has precaliber ω (and in fact precaliber µ for any regular un-countable µ ) and forces the following:(1) The generic functions c : [ λ ] → { , } and d α : [ α ] < ℵ → ω for α < λ aretotally defined.(2) Whenever ( s i ) i ∈ I is a family of finite, 1-homogeneous (w.r.t. c ) subsetsof α , and d α ( s i ) = d α ( s j ) for i, j ∈ I , then S i ∈ I s i is 1-homogeneous.(3) If A ⊆ [ λ ] < ℵ is a family of size λ of pairwise disjoint sets, then there aretwo sets u = v in A such that c ( ξ, η ) = 0 for any ξ ∈ u and η ∈ v .(4) Whenever u ∈ [ λ ] < ℵ , the set { η < λ : ∀ ξ ∈ u ( c ( ξ, η ) = 1) } is unboundedin λ .Proof. For any α < λ , the set of conditions p ∈ P cal such that α ∈ F p is dense. Starting with p such that α / ∈ F p , we set u q = u p , F q = F p ∪ { α } , and we picknew and unique values for all d q,α ( s ) for s ⊆ u q ∩ α = u p ∩ α , as well as new and preserved after any σ -linked poset. Also recall from [DS78] (with Devlin) that m (precaliber) > ℵ implies that any ladder system coloring can be uniformized. Say that s ⊆ u p ∩ α is 1 -homogeneous w.r.t. c p if c p ( ξ ) = 1 for any ξ ∈ s . unique values for all d q,β ( s ) for s ⊆ u q ∩ β with α ∈ s . We have to show that q ∈ P cal , i.e., that it satisfies ( ⋆ ): Whenever s , s satisfy the assumptions of ( ⋆ ),then α / ∈ s i (for i = 1 , ⋆ ) holds for p . (1) and (4) For any ξ < λ , the set of q ∈ P cal such that ξ ∈ u q is dense. Starting with p with ξ / ∈ u p , we set u q = u p ∪ { ξ } and F q = F p . Again, pick new(and different) values for all d q,α ( s ) with ξ ∈ s , and we can set c ( x, ξ ) to whateverwe want. The same argument as above shows that q ∈ P cal . In particular we canset all c ( x, ξ ) = 1, which shows that P cal forces (4). (2) follows from ( ⋆ ) for I = { , } , and this trivially implies the case for arbitrary I . (For x , x ∈ S i ∈ I s i , pick i , i ∈ I such that x ∈ s i and x ∈ s i ; then apply( ⋆ ) to { i , i } .) Amalgamation. Let p ∈ P and u ⊆ u p and F ⊆ F p (let us call u, F the “heart”).Then we define the type of p (with respect to the heart) as the following structure:Let i be the order-preserving bijection (Mostowski’s collapse) of u p ∪ F p to some N ∈ ω , which also translates the partial functions c p and d p and the subsets u p and F p . Then the type is the induced structure on N . Between any two conditionswith same type there is a natural isomorphism.Assume p , p , . . . p n − are in P cal . We set u i := u p i , and we do the same for F , c , and d . Assume u i and F i form ∆-systems with hearts u and F , and that p i havethe same type for i ∈ n with respect to u, F . Then all c i and all d i agree on thecommon domain. Then we define an “amalgamation” q = q ( p , p , . . . , p n − ) as follows: u q := S i ∈ n u i , F q := S i ∈ n F i , d q extends all d i and has a unique new value for each newelement in its domain, c q extends all c i ; and yet undefined c q ( x, y ) are set to 0 if x, y > max( F ) (and 1 otherwise).To see that q ∈ P cal , assume that α ∈ F q and s , s are as in ( ⋆ ). This impliesthat d q,α ( s k ) for both k = 1 , by one of the p i (for i ∈ n ),otherwise we would have picked a new value.If they are both defined by the same p i , we can use ( ⋆ ) for p i . So assumeotherwise, for notational simplicity assume that s i is defined by p i ; and let x i ∈ s i .We have to show c q ( x , x ) = 1. Note that α ∈ F ∩ F = F . If x or x are notin u , then we have set c q ( x , x ) to 1 (as x i < α ∈ F ), so we are done. So assume x , x ∈ u . The natural isomorphism between p and p maps s onto some s ′ ⊆ u ,and we get that s ′ is 1-homogeneous and that d ,α ( s ) = d ,α ( s ) = d ,α ( s ′ ). Sowe use that p satisfies ( ⋆ ) to get that c ( a, b ) = 1 for all a ∈ s and b ∈ s ′ . As theisomorphism does not move x , we can use a = x and b = x . Precaliber. P cal has precaliber µ for any uncountable regular µ . Let { p ξ : ξ < µ } be a set of conditions in P cal . For ξ < µ denote u ξ := u p ξ , F ξ := F p ξ and so on. We can assume that the u ξ ’s and F ξ ’s form ∆-systems withroots u and F respectively, and that the type of p ξ does not depend on ξ . Thenany finite subset of these conditions is compatible, witnessed by its amalgamation. (3) Let p ∈ P cal and assume that p forces that ˙ A ⊆ [ λ ] < ℵ is a family of size λ ofpairwise disjoint sets. We can find, in the ground model, a family A ′ ⊆ [ λ ] < ℵ ofsize λ and conditions p v ≤ p for v ∈ A ′ such that v ⊆ u p v , and p v forces v ∈ ˙ A . I.e., for α < β in u , c i ( α, β ) = c j ( α, β ), and for α ∈ F and s ⊆ u , d i,α ( s ) = d j,α ( s ). By which we mean α ∈ F i and s k ⊆ u i for both k = 1 , ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 17 We again thin out to a ∆-system as above; this time we can additionally assumethat the heart of the F v is below the non-heart parts of all u v , i.e., that max( F ) isbelow u v \ u for all v .Pick any two p v , p ′ v in this ∆-system, and let q be the amalgamation definedabove. Then q witnesses that p v , p w are compatible, which implies v ∩ w = 0, i.e., v, w are outside the heart; which by construction of q implies that c q is constantlyzero on v × w (as their elements are above max( F )). (cid:3) The poset P cal ,λ adds generic functions c and d α . We now use them to define aprecaliber ω poset Q cal witnessing m (precaliber) ≤ λ : Lemma 5.3. In V P cal , define the poset Q cal := { u ∈ [ λ ] < ℵ : u is 1-homogeneous } ,ordered by ⊇ (By 1-homogeneous, we mean 1-homogeneous with respect to c .) Thenthe following is satisfied (in V P cal ):(1) Q cal is an increasing union of length λ of centered sets (so in particular ithas precaliber ℵ ).(2) For α < λ , the set D α := { u ∈ Q cal : u * α } is open dense. So Q cal adds acofinal generic 1-homogeneous subset of λ .(3) There is no 1-homogeneous set of size λ (in V P cal ). In other words, thereis no filter meeting all D α .Proof. For (1) set Q α cal = Q cal ∩ [ α ] < ℵ . Then d α : Q α cal → ω is a centering function,according to Lemma 5.2(2). Precaliber ℵ is a consequence of λ cal > ℵ .Property (2) is a direct consequence of Lemma 5.2(4), and (3) follows fromLemma 5.2(3). (cid:3) This shows that P cal ,λ (cid:13) m (precaliber) ≤ λ . We now show that this is preservedin further Knaster extensions. Lemma 5.4. In V P cal , assume that P ′ is a ccc λ -Knaster poset. Then, in V P cal ∗ P ′ , m (precaliber) ≤ λ .Proof. We claim that in V P cal ∗ P ′ , Q cal still has precaliber ℵ , and there is no filtermeeting each open dense subset D α ⊆ Q cal for α < λ .Precaliber follows from Lemma 5.3(1). So we have to show that λ has no 1-homogeneous set (w.r.t. c ) of size λ in V P cal ∗ P ′ .Work in V P cal and assume that ˙ A is a P ′ -name and p ∈ P ′ forces that ˙ A is in [ λ ] λ .By recursion, find A ′ ∈ [ λ ] λ and p ζ ≤ p for each ζ ∈ A ′ such that p ζ (cid:13) ζ ∈ ˙ A . Since P ′ is λ -Knaster, we may assume that { p ζ : ζ ∈ A ′ } is linked. By Lemma 5.2(3),there are ζ = ζ ′ in A ′ such that c ( ζ, ζ ′ ) = 0. So there is a condition q strongerthat both p ζ and p ζ ′ forcing that ζ, ζ ′ ∈ ˙ A and c ( ζ, ζ ′ ) = 0, i.e., that ˙ A is not1-homogeneous. (cid:3) We can now add another case to Lemma 4.8: Lemma 5.5. For ℵ ≤ λ ≤ ν regular, P pre can be modified to some forcing P ′ which still strongly witnesses the Cicho´n-characteristics, and additionally satisfies:For all k ∈ ω , m ( k -Knaster) = ℵ ; m (precaliber) = λ ; and p = b .Proof. The case λ = ℵ was already dealt with in the previous section, so we assume λ > ℵ .We modify P pre as follows: We start with the forcing P cal ,λ . From then on, use(by bookkeeping) all precaliber forcings of size <λ , all σ -centered ones of size <ν , the value for b (and in between we use all the iterands required for the originalconstruction). So each new iterand either has precaliber ℵ and is of size <λ , oris ( σ, k )-linked for any k ≥ 2. Therefore, the limits are k + 1-Knaster (for any k ).Accordingly, the limit forces that each k -Knaster number is ℵ .Also, each iterand is either of size <λ or σ -linked; so the limit is λ -Knaster, andby Lemma 5.4 it forces that the precaliber number is ≤ λ ; our bookkeeping gives ≥ λ . And, as before, we get p ≥ ν by bookkeeping. (cid:3) Dealing with h The following is a very useful tool to deal with g . Lemma 6.1 (Blass [Bla89, Thm. 2], see also Brendle [Bre10, Lem. 1.17]) . Let ν be an uncountable regular cardinal and let ( V α ) α ≤ ν be an increasing sequence oftransitive models of ZFC such that(i) ω ω ∩ ( V α +1 r V α ) = ∅ ,(ii) ( ω ω ∩ V α ) α<ν ∈ V ν , and(iii) ω ω ∩ V ν = S α<ν ω ω ∩ V α .Then, in V ν , g ≤ ν . This result gives an alternative proof of the well-known: Corollary 6.2. g ≤ cof( c ) . Proof. Put ν := cof( c ) and let ( µ α ) α<ν be a cofinal increasing sequence in c formedby limit ordinals. By recursion, we can find an increasing sequence ( V α ) α<ν oftransitive models of (a large enough fragment of) ZFC such that (i) of Lemma 6.1is satisfied, µ α ∈ V α , | V α | = | µ α | and S α<ν ω ω ∩ V α = ω ω . Set V ν := V , soLemma 6.1 applies, i.e., g ≤ ν = cof( c ). (cid:3) The following lemma is our main tool to modify the values of g and c via acomplete subposet of some forcing, while preserving m -like and Blass-uniform valuesfrom the original poset. This is a direct consequence of Lemmas 3.2 and 6.1 andCorollary 3.5. As we are only interested in finitely many characteristics, the indexsets I , I , J and K will be finite when we apply the lemma. Lemma 6.3. Assume the following:(1) ℵ ≤ κ ≤ ν ≤ µ , where κ and ν are regular and µ = µ <κ ≥ ν ,(2) P is a κ -cc poset forcing c > µ .(3) For some Borel relations R i ( i ∈ I ) on ω ω and some regular λ i ≤ µ : P forces LCU R i ( λ i ) (4) For some Borel relations R i ( i ∈ I ) on ω ω , λ i ≤ µ regular and a cardinal ϑ i ≤ µ : P forces COB R i ( λ i , ϑ i ) .(5) For some m -like characteristics y j ( j ∈ J ) and λ j < κ : P (cid:13) y j = λ j .(6) For some m -like characteristics y ′ k ( k ∈ K ): P (cid:13) y ′ k ≥ κ .(7) | I ∪ I ∪ J ∪ K | ≤ µ .Then there is a complete subforcing P ′ of P of size µ forcing(a) y j = λ j , y ′ k ≥ κ , LCU R i ( λ i ) and COB R i ′ ( λ i ′ , ϑ i ′ ) for all i ∈ I , i ′ ∈ I , j ∈ J and k ∈ K ;(b) c = µ and g ≤ ν . A more elementary proof can be found in [Bla10, Thm.8.6, Cor. 8.7] ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 19 Proof. Construct an increasing sequence of elementary submodels ( M α : α < ν )of some ( H ( χ ) , ∈ ) for some sufficiently large χ , where each M α is <κ -closed withcardinality µ , in a way that M := M ν = S α<ν M α satisfies:(i) µ ∪ { µ } ⊆ M ,(ii) I ∪ I ∪ J ∪ K ⊆ M ,(iii) M contains all the definitions of the characteristics we use,(iv) M contains all the P -names of witnesses of each LCU R i ( λ i ) ( i ∈ I ),(v) for each i ∈ I and some chosen name ( ˙ E i , ˙¯ g i ) of a witness of COB R i ( λ i , ϑ i ):for all ( s, t ) ∈ ϑ i × ϑ i , ˙ g is ∈ M and the maximal antichain deciding “ s ˙ E i t ”belongs to M ,(vi) M α +1 contains P -names of reals that are forced not to be in the P ∩ M α -extension (this is because P forces c > µ ).Note that M is also a <κ -closed elementary submodel of H ( χ ) of size µ , and that P α := P ∩ M α (for α ≤ ν ) is a complete subposet of P . Put P ′ := P ν .According to Corollary 3.5, in the P ′ -extension, each m -like characteristic below κ is preserved (as in the P -extension) and for the others “ y ′ k ≥ κ ” is preserved; andaccording to Lemma 3.2 the LCU and COB statements are preserved as well. Thisshows (a).It is clear that P α is a complete subposet of P β for every α < β ≤ ν , and that P ′ is the direct limit of the P α . Therefore, if V ′ denotes the P ′ -extension and V α denotes the P α -intermediate extensions, then ω ω ∩ V α +1 r V α = ∅ (by (vi)) and ω ω ∩ V ′ ⊆ S α<ν V α . Hence, by Lemma 6.1, V ′ | = g ≤ ν . Clearly, V ′ | = c = µ . (cid:3) We are now ready to add h = g = p to our characteristics: Lemma 6.4. For ℵ ≤ λ m ≤ κ ≤ ν regular, P pre can be modified to some forc-ing P ′ which still strongly witnesses the Cicho´n-characteristics, and additionallysatisfies: m = λ m ≤ h = g = p = κ In addition to m = λ m we can get m = m (precaliber), which is case (3) ofLemma 4.8; and instead of m = λ m we can alternatively force case (1) or (2) ofLemma 4.8, or the situation of Lemma 5.5. Proof. We start with the (appropriate) P from Lemma 4.8 (or from Lemma 5.5);but for the “inflated” continuum θ + ∞ instead of θ ∞ .We then apply Lemma 6.3 for µ := θ ∞ , and ν := κ . This gives a subforcing P ′ which still forces: • Strong witnesses for all the Cicho´n-characteristics;as they fall under Lemma 6.3(3,4). • p ≥ κ ; an instance of Lemma 6.3(6) as P forces p = ν ≥ κ . • g ≤ ν ; according to Lemma 6.3(b).As ZFC proves p ≤ h ≤ g and ν = κ , this implies p = h = g = κ . • If λ m < κ , we get m = λ m < κ as instance of Lemma 6.3(5). • If λ m = κ , we get m ≥ κ by Lemma 6.3(6);but as m ≤ p this also implies m = λ m . • Alternatively: The same argument for m (precaliber) and/or m ( k -Knaster)instead of / in addition to m ; as required by the desired case of Lemma 4.8or 5.5. (cid:3) We can now get twelve different characteristics: Corollary 6.5. Under Assumption 2.10, and for ℵ ≤ λ m ≤ κ regular, we can geta ccc poset P ′′ which forces, in addition to Theorem 2.11, m = λ m ≤ h = g = p = κ (The comment after Lemma 6.4 regarding various Martins axiom numbers ap-plies here as well.) Proof. The resulting P ′ we just constructed still satisfies the requirements forLemma 2.12, so we apply this lemma and get P ′′ := P ′ ∩ N ∗ (for a <κ -closed N ∗ ) which forces the desired values to all Cicho´n-characteristics. Additionally P ′′ forces: • p ≥ κ , by Corollary 3.5(iii)(a), as P ′ forces p = κ . • g ≤ κ , by Corollary 3.5(iii)(c), as P ′ forces g = κ . • p = h = g = κ , as ZFC proves p ≤ g . • In case λ m < κ : m = λ m by Corollary 3.5(iii)(b). • In case λ m = κ : m ≥ κ by Corollary 3.5(iii)(a), which again implies m = λ m , as ZFC proves m ≤ p = κ . (cid:3) Products, dealing with p We start reviewing a basic result in forcing theory. Lemma 7.1 (Easton’s lemma) . Let ξ be an uncountable cardinal, P a ξ -cc posetand let Q be a <ξ -closed poset. Then P forces that Q is <ξ -distributive.Proof. See e.g. [Jec03, Lemma 15.19]. Note that there the lemma is proved forsuccessor cardinals only, but literally the same proof works for any regular cardinal;for singular cardinals ξ note that <ξ -closed implies <ξ + -closed so we even get <ξ + -distributive. (cid:3) Lemma 7.2. Assume ξ <ξ = ξ , P is ξ -cc, and set Q = ξ <ξ (ordered by extension).Then P forces that Q V preserves all cardinals and cofinalities. Assume P (cid:13) x = λ (in particular that λ is a cardinal), and let R be a Borel relation.(a) If x is m -like: λ < ξ implies P × Q (cid:13) x = λ ; λ ≥ ξ implies P × Q (cid:13) x ≥ ξ .(b) If x is h -like: P × Q (cid:13) x ≤ λ .(c) P (cid:13) LCU R ( λ ) implies P × Q (cid:13) LCU R ( λ ) .(d) P (cid:13) COB R ( λ, µ ) implies P × Q (cid:13) COB R ( λ, µ ) .Proof. We call the P + -extension V ′′ and the intermediate P -extension V ′ .In V ′ , all V -cardinals ≥ ξ are still cardinals, and Q is a <ξ -distributive forcing(due to Easton’s lemma). So we can apply Lemma 3.1 and Corollary 3.5. (cid:3) The following is shown in [DS]: Lemma 7.3. Assume that ξ = ξ <ξ and P is a ξ -cc poset that forces ξ ≤ p . In the P -extension V ′ , let Q = ( ξ <ξ ) V . Then,(a) P × Q = P ∗ Q forces p = ξ (b) If in addition P forces ξ ≤ p = h = κ then P × Q forces h = κ . ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 21 Proof. Work in the P -extension V ′ . Q preserves cardinals and cofinalities, and itforces p ≥ ξ by Lemma 7.2.There is an embedding F from h Q, ( i into h [ ω ] ℵ , ) ∗ i preserving the order andincompatibility (using the fact that ξ ≤ p = t and that every infinite set can besplit into ξ many almost disjoint sets). Now, Q adds a new sequence z ∈ ξ ξ \ V ′ and forces that ˙ T = { F ( z ↾ α ) : α < ξ } is a tower (hence t ≤ ξ ). If this were notthe case, some condition in Q would force that ˙ T has a pseudo-intersection a , butactually a ∈ V ′ and it determines uniquely a branch in ξ ξ , and this branch wouldbe in fact z , i.e., z ∈ V ′ , a contradiction. So we have shown P × Q (cid:13) t = ξ .For (b): We already know that Q (cid:13) h ≤ κ . To show that h does not decrease,again work in V ′ . Note that h [ ω ] ℵ , ⊆ ∗ i is <κ -closed (as t = κ ). We claim that Q forces that h [ ω ] ℵ , ⊆ ∗ i is <κ -distributive, (which implies Q (cid:13) h ≥ κ ).If κ = ξ then h [ ω ] ℵ , ⊆ ∗ i is still <ξ -closed because Q is <ξ -distributive; soassume ξ < κ . Then Q is κ -cc (because | Q | = ξ ), so h [ ω ] ℵ , ⊆ ∗ i is forced to be <κ -distributive by Easton’s Lemma (recall that Q does not add new reals). (cid:3) We are now ready to formulate the main theorem, the consistency of 13 differentvalues (see Figure 2): Theorem 7.4. Assume GCH, and that ℵ ≤ λ m ≤ ξ ≤ κ ≤ λ add( N ) ≤ λ cov( N ) ≤ λ b ≤ λ non( M ) ≤≤ λ cov( M ) ≤ λ d ≤ λ non( N ) ≤ λ cof( N ) ≤ λ ∞ are regular cardinals, with the possible exception of λ ∞ , for which we only require λ <κ ∞ = λ ∞ . Then we can force that ℵ ≤ λ m ≤ p = ξ ≤ h = g = κ ≤ add( N ) = λ add( N ) ≤ cov( N ) = λ cov( N ) ≤ b = λ b ≤ non( M ) = λ non( M ) ≤ cov( M ) = λ cov( M ) ≤ d = λ d ≤ non( N ) = λ non( N ) ≤ cof( N ) = λ cof( N ) ≤ ℵ = λ ∞ and we can additionally chose any one of the following: • m = m (precaliber) = λ m . • For a fixed ≤ k < ω , m ( k -Knaster) = ℵ and m ( k + 1-Knaster) = λ m . • m ( k -Knaster) = ℵ for all k < ω , and m (precaliber) = λ m .Proof. Start with the appropriate forcing P ′′ of Corollary 6.5. Then P ′′ × ξ <ξ forces: • Strong witnesses to all Cicho´n-characteristics; by Lemma 7.2(c,d). • p = ξ and h = κ ; by Lemma 7.3. • g ≤ κ by Lemma 7.2(b) as P ′′ forces g = κ and g is h -like. This implies g = κ , as ZFC proves h ≤ g . • The desired values to the Martin axiom numbers; by Lemma 7.2(a) (andby the fact that m ≤ p , in case λ m = ξ ). (cid:3) Alternatives The methods of this paper can be used for other initial forcings on the left handside and for the Boolean ultrapower method instead of the method of intersectionswith elementary submodels. Also, it allows us to compose many forcing notionswith collapses while preserving cardinal characteristics. cov( N ) / / non( M ) (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂❂❂❂ / / cof( N ) / / ℵ b f f ◆◆◆◆◆◆◆◆◆ d ℵ / / m / / p / / h / / add( N ) / / cov( M ) / / non( N ) f f ◆◆◆◆◆◆◆◆ Figure 4. An alternative order that we get when we start withthe initial forcing from [KST19]. (Any → can be interpreted aseither < or = as desired.)All these topics are described in more detail in [GKMS]; in the following we justgive an overview.8.1. Another order. In [KST19], another ordering of Cicho´n’s maximum is shownto be consistent (using large cardinals), namely the ordering shown in Figure 4.The initial (left hand side) forcing is based on ideas from [She00], and in partic-ular the notion of finite additive measure (FAM) limit introduced there for randomforcing. In addition, a creature forcing Q similar to the one defined in [HS] (withHorowitz) is introduced, which forces non( M ) ≥ λ non( M ) and which has FAM-limits similar to random forcing (which is required to keep b small).In [GKMS20], we show that we can remove the large cardinal assumptions forthis ordering as well (using the same method).It is straightforward to check that the method in this paper allows us to add m , p , h to this ordering as well; so we get Theorem 7.4 with both ( b and cov( N )) and( d and non( N )) exchanged. In particular, we get (see Figure 4): Theorem 8.1. Consistently, ℵ < m < p < h < add( N ) < b < cov( N ) < non( M ) << cov( M ) < non( N ) < d < cof( N ) < ℵ . Boolean ultrapowers. As mentioned in Subsection 2.3, the original Cicho´nMaximum construction [GKS19] uses four strongly compact cardinals: First, theleft side of Cicho´n’s diagram is separated with P pre of 2.8, where we assume thatthere are compacts between each of ℵ < ν < ν < ν < ν . Then four Booleanultrapowers are applied to this poset (one for each compact cardinal) to constructa forcing P ∗ that separates, in addition, the right hand side, while preserving theleft side values already forced by P pre .In view of Corollary 3.5(ii), we can use the methods of Sections 4–7 to force, inaddition, m < p < h < add( N ).In contrast with Theorem 7.4, we can now force not only the continuum to besingular, but also cov( M ). The reason is that the poset for the left side can forcecov( M ) = c singular, and the value of cov( M ) is not changed after Booleanultrapowers (and the other methods). The same applies to the alternate orderfrom [KST19] as well. This is not explicitly mentioned in [GKS19]. ONTROLLING CARDINAL CHARACTERISTICS WITHOUT ADDING REALS 23 Alternative left hand side forcings. According to subsection 2.3, [BCM18]provides an alternative proof of Cicho´n’s maximum, using three strongly compactcardinals. As in [GKS19], this results from applying Boolean ultrapowers to a cccposet that separates the left side, but the new initial forcing additionally givescov( M ) < d = non( N ) = c , where this value of d can be singular. The methodsof this work also apply, and we can obtain a consistency result as in Theorem 7.4,but there d and c are forced to be singular.8.4. Reducing gaps with collapsing forcing. 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E-mail address : [email protected] URL : http://dmg.tuwien.ac.at/kellner/ Creative Science Course (Mathematics), Faculty of Science, Shizuoka University,Ohya 836, Suruga-ku, Shizuoka-shi, Japan 422-8529. E-mail address : [email protected] URL : Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The HebrewUniversity of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathematics,Rutgers University, New Brunswick, NJ 08854, USA. E-mail address : [email protected] URL ::