Laver Trees in the Generalized Baire Space
Yurii Khomskii, Marlene Koelbing, Giorgio Laguzzi, Wolfgang Wohofsky
LLaver Trees in the Generalized Baire Space
Yurii Khomskii ∗ , Marlene Koelbing † ,Giorgio Laguzzi ‡ , Wolfgang Wohofsky § September 7, 2020
Abstract
We prove that any suitable generalization of Laver forcing to the space κ κ , for uncountable regular κ , necessarily adds a Cohen κ -real. We alsostudy a dichotomy and an ideal naturally related to generalized Laverforcing. Using this dichotomy, we prove the following stronger result: if κ <κ = κ , then every <κ -distributive tree forcing on κ κ adding a dominat-ing κ -real which is the image of the generic under a continuous functionin the ground model, adds a Cohen κ -real. This is a contribution to thestudy of generalized Baire spaces and answers a question from [1]. Mathematics Subject Classification (2010). [03E40, 03E17, 03E05]
In set theory of the reals, a basic question is whether a forcing adds
Cohenreals or dominating reals . It is well-known that Cohen forcing adds Cohen butnot dominating reals while Laver forcing does the opposite. In the languageof cardinal characteristics of the continuum, this means that an appropriateiteration of Cohen forcing starting from CH yields a model where b < cov( M ), ∗ Amsterdam University College, Postbus 94160, 1090 GD Amsterdam, The Netherlandsand Universi¨at Hamburg, Bundesstraße 55, 20146 Hamburg, Germany. [email protected] . Sup-ported by the European Unions Horizon 2020 research and innovation programme under theMarie Sk(cid:32)lodowska-Curie grant agreement No 706219 (REGPROP) † Kurt G¨odel Research Center (KGRC), Universit¨at Wien, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria. [email protected] .Supported by the ¨OAW Doc fellowship. ‡ Albert-Ludwigs-Universit¨at Freiburg, Ernst-Zermelo Str. 1, 79104 Freiburg im Breisgau,Germany. [email protected] § Kurt G¨odel Research Center (KGRC), Universit¨at Wien, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria. [email protected]
Supported by the German Research Foundation(DFG) under Grant SP 683/4-1, and the Austrian Science Fund (FWF) under InternationalProject number: I 4039This project was partially supported by the Isaac Newton Institute for Mathematical Sciencesin the programme Mathematical, Foundational and Computational Aspects of the HigherInfinite (HIF) funded by EPSRC grant EP/K032208/1. a r X i v : . [ m a t h . L O ] S e p hile an appropriate iteration of Laver forcing starting from CH yields a modelwhere cov( M ) < b .In recent years, the study of generalized Baire spaces has caught the atten-tion of an increasing number of set theorists. For a regular, uncountable cardinal κ one considers elements of κ κ or 2 κ as “ κ -reals” and looks at the correspond-ing space with the bounded topology. The generalized Cantor space is definedanalogously using 2 κ and 2 <κ .It is straightforward to generalize the above notions from the classical tothe generalized Baire spaces. Thus, we have the concepts dominating κ -real and the cardinal characteristic b κ (see Definition 2.1). Likewise, we can define M κ as the ideal of κ -meager sets, i.e., those obtained by κ -unions of nowheredense, giving rise to the cardinal characteristic cov( M κ ) defined in the usualway. κ -Cohen forcing is the partial order of basic open sets ordered by inclusion.It is not hard to see that κ -Cohen forcing does not add dominating κ -reals, soan appropriate iteration of κ -Cohen forcing, starting from a model of GCH , yieldsa model in which b κ < cov( M κ ), mirroring the classical situation. A naturalmethod for the converse direction, i.e., proving the consistency of cov( M κ ) < b κ ,would be to iterate a forcing which adds dominating κ -reals but not Cohen κ -reals. The authors of [1, p. 36] asked whether a forcing with such a propertyexisted, and in particular, whether some generalization of Laver forcing had thisproperty.In this paper, we show that any generalization of Laver forcing necessarilyadds a Cohen κ -real (Theorem 3.5). If we assume κ <κ = κ , then this holds foran even wider class of trees (Theorem 3.7). Later, we use a dichotomy result andsimilar techniques to show that if κ <κ = κ and P is any <κ -distributive forcingwhose conditions are limit-closed trees on κ <κ , and which adds a dominating κ -real obtained as the image of the generic under a continuous function in theground model, then P necessarily adds a Cohen κ -real (Theorem 5.10). It isan open question whether there exists some other <κ -distributive and/or <κ -closed forcing which adds dominating κ -reals but not Cohen κ -reals (Question5.1). We should note that a model for cov( M κ ) < b κ was recently constructed byShelah (private communication). However, Shelah’s method was to start froma model of cov( M κ ) = b κ = 2 κ > κ + and add a witness to cov( M κ ) = κ + bya short forcing iteration. It is therefore still open whether an alternative proofexists by using a forcing iteration starting from a model of GCH which addsdominating κ -reals and no Cohen κ -reals.When working in generalized Baire spaces, a common assumption is κ <κ = κ ,which is sufficient to prove many pleasant properties of generalized Baire spaces,e.g., that the topology has a base of size κ . Nevertheless, our first main theorem In an earlier version of this paper, we claimed that every <κ -closed forcing adding dom-inating κ -reals adds Cohen κ -reals, but the proof contained a gap, so, to our knowledge, thequestion is still open. <κ -distributive tree forcings. We work in the setting where κ is an uncountable, regular cardinal, and considerthe generalized Baire space κ κ with the bounded topology generated by basicopen sets of the form [ σ ] := { x ∈ κ κ : σ ⊆ x } for σ ∈ κ <κ . The generalizedCantor space κ is defined analogously.We refer the reader to [3] for a good introduction to generalized Baire spaces,and to [8] for an overview of the current state of the field and a list of openproblems. Definition 2.1.
Let f, g ∈ κ κ . We say that g dominates f , notation f ≤ ∗ g ,iff ∃ α ∀ α > α ( f ( α ) ≤ g ( α )). The generalized bounding number b κ is definedas the least size of a family F ⊆ κ κ such that for all g ∈ κ κ there is f ∈ F suchthat f (cid:54)≤ ∗ g . If M is a model of set theory, then d is a dominating κ -real over M if d dominates every f ∈ κ κ ∩ M .A tree in κ <κ is a subset closed under initial segments. If T is a tree, weuse [ T ] to denote the set of branches (of length κ ) through T , that is [ T ] := { x ∈ κ κ : ∀ α < κ ( x (cid:22) α ∈ T ) } . The same holds for trees in 2 <κ . For σ ∈ T we use the notation T ↑ σ := { τ ∈ T : σ ⊆ τ ∨ τ ⊆ σ } . A tree T ⊆ κ <κ iscalled limit-closed if for any limit ordinal λ < κ and any ⊆ -increasing sequence (cid:104) σ α : α < λ (cid:105) from T , the limit of the sequence σ := (cid:83) α<λ σ α is itself an elementof T . We call a set C superclosed if C = [ T ] for a limit-closed tree T .Every closed subset of κ κ is the set of branches through a tree but notnecessarily a limit-closed tree, so one could say that being superclosed is atopologically stronger property than being closed. We will also need to considersets of branches of length shorter than κ . For any limit ordinal λ < κ we usethe notation [ T ] λ := { σ ∈ κ λ : ∀ α < λ ( σ (cid:22) α ∈ T ) } . Notice that T is limit-closediff [ T ] λ ⊆ T for all limit ordinals λ < κ . Definition 2.2. A Laver tree is a tree T ⊆ ω <ω with the property that forevery σ ∈ T extending stem( T ), | Succ T ( σ ) | = ω . Laver forcing L is the partialorder of Laver trees ordered by inclusion. Other terminology used is “ <κ -closed” and “sequentially closed”. Laver prop-erty , a well-known iterable property implying that no Cohen reals are added.There have been several attempts in the literature to generalize Laver forcingto κ κ . Definition 2.3. A κ -Laver tree is a tree T ⊆ κ <κ which is limit-closed andsuch that for every σ ∈ T extending stem( T ), | Succ T ( σ ) | = κ . Let L κ denotethe partial order of all κ -Laver trees ordered by inclusion.This partial order itself is not well-suited as a forcing on κ κ and has neverbeen proposed as an option. But there have been other attempts at generaliza-tions of Laver forcing, usually by putting stronger requirements on “splitting”in the tree. For example, club Laver forcing (see [4]) consists of trees satisfyingthe additional condition “Succ T ( σ ) contains a club on κ ” for all σ beyond thestem. This forcing is <κ -closed and adds a dominating κ -real, but it is easy tosee that it also adds a Cohen κ -real: if S is a stationary, co-stationary subsetof κ and ϕ : κ κ → κ is given by ϕ ( x )( α ) = 1 ⇔ x ( α ) ∈ S , then ϕ ( x gen ) is aCohen κ -real.Yet another attempt is measure-one Laver forcing , where the requirement isstrengthened to “Succ T ( σ ) ∈ U ” for some <κ -complete ultrafilter on a measur-able cardinal κ . This forcing is also <κ -closed and adds a dominating κ -real,and until now it was not known whether it adds a Cohen κ -real. Of course, onecould think of further clever requirements on Laver trees in order to ensure thatno Cohen κ -reals are added.However, by the results of this paper, none of these approaches can work. In this section we will prove our first main result. The main ingredient of ourproofs in this and subsequent sections is the following game.
Definition 3.1.
Let S ⊆ κ . The supremum game G sup ( S ) is played by twoplayers, for ω moves, as follows:I A A . . . II β β . . . where A n ⊆ κ , | A n | = κ and β n ∈ A n for all n < ω . Player II wins iffsup { β n : n < ω } ∈ S . Lemma 3.2.
Let S be a stationary subset of Cof ω ( κ ) = { α < κ : cf( α ) = ω } .Then Player I does not have a winning strategy in G sup ( S ) . It is not hard to see that such a partial order would not be <κ -closed, and in fact noteven ω -distributive. Compare this to a recent result of Mildenberger and Shelah [11] showingthat a similarly “plain” version of κ -Miller forcing collapses 2 κ to ω . roof. Let σ be a strategy for Player I in G sup ( S ). We will show that σ is not awinning strategy. Let θ be sufficiently large and let M ≺ H θ be an elementarysubmodel such that σ ∈ M , | M | < κ , and δ := sup( M ∩ κ ) ∈ S . Note that wecan always do that, because the set { sup( M ∩ κ ) : M ≺ H θ , σ ∈ M, | M | < κ } contains a club.Fix a sequence (cid:104) γ n : n < ω (cid:105) cofinal in δ , such that every γ n ∈ M (but thesequence itself is not). Inductively, Player II will construct a run of the gameaccording to strategy σ .At each step n , inductively assume A k and β k for k < n have been fixed ac-cording to the rules of the game and the strategy σ , and assume they are all in M . Let A n := σ ( A , β , . . . , A n − , β n − ). Since the finite sequence was in M and the strategy σ is in M , A n is also in M . Furthermore, since | A n | = κ , thefollowing statement is true: ∃ β > γ n ( β ∈ A n ) . This statement holds in H θ , so by elementarity, it also holds in M . Thus, thereexists β n ∈ M with β n > γ n and β n ∈ A n . This completes the construction.We have produced a sequence (cid:104) β n : n < ω (cid:105) with β n ∈ M for all n . But clearlysup n β n = sup n γ n = δ ∈ S , so Player II wins this game, proving that thestrategy was not winning for Player I. Definition 3.3. A short κ -Laver tree is a tree L ⊆ κ <ω (i.e., height ω ), suchthat for all σ ∈ L extending stem( L ) we have | Succ L ( σ ) | = κ . Corollary 3.4.
Let S ⊆ κ be a stationary subset of Cof ω ( κ ) . For every short κ -Laver tree L there exists a branch η ∈ [ L ] ω such that sup n η ( n ) ∈ S .Proof. The short κ -Laver tree L induces a strategy σ L for Player I in the supre-mum game: σ L ( A , β , . . . , A n , β n ) := Succ L (stem( L ) (cid:95) (cid:104) β , . . . , β n (cid:105) ) . Whenever (cid:104) A , β , A , β , . . . (cid:105) is a run of the game according to σ L , stem( L ) (cid:95) (cid:104) β , β , . . . (cid:105) is an element of [ L ] ω .By Lemma 3.2, there exists a run of the game in which Player I follows σ L butPlayer II wins. This yields a branch η ∈ [ L ] ω such that sup n η ( n ) ∈ S .With this, we immediately obtain our main result. Theorem 3.5 (Main Theorem 1) . Let P be any forcing whose conditions are κ -Laver trees ( i.e., P ⊆ L κ ) and which is closed under the following condition:if T ∈ P and σ ∈ T , then T ↑ σ ∈ P . Then P adds a Cohen κ -real.Proof. We will use the following notation: if T ∈ κ <κ is a tree and σ ∈ T , then T (cid:22) ω σ := { τ ∈ κ <ω : σ (cid:95) τ ∈ T } . Note that if T is a κ -Laver tree, then for every σ ∈ T extending stem( T ), T (cid:22) ω σ is a short κ -Laver tree (with empty stem).5et S ∪ S be a stationary/co-stationary partition of Cof ω ( κ ) and consider themapping ϕ : κ κ → κ defined by ϕ ( x )( α ) = 1 : ⇔ sup { x ( ω · α + n ) : n < ω } ∈ S . In other words, partition x into κ -many blocks of length ω , and map each pieceto 0 or 1 depending on whether its supremum lies in S or S . We claim that if x gen is P -generic then ϕ ( x gen ) is κ -Cohen-generic.We use ˜ ϕ : κ <κ → <κ to denote the approximations of ϕ (defined as above).Let T ∈ P be given and let D be open dense in κ -Cohen forcing. Let σ :=stem( T ), w.l.o.g. len( σ ) is a limit ordinal. Let t ∈ D extend ˜ ϕ ( σ ). Suppose˜ ϕ ( σ ) (cid:95) (cid:104) (cid:105) ⊆ t . By Corollary 3.4 there is η ∈ [ T (cid:22) ω σ ] ω such that sup n η ( n ) ∈ S .If, instead, we have ˜ ϕ ( σ ) (cid:95) (cid:104) (cid:105) ⊆ t , we can apply Corollary 3.4 and find a branch µ ∈ [ T (cid:22) ω σ ] ω such that sup n µ ( n ) ∈ S . Note that, since T is limit-closed, σ (cid:95) η resp. σ (cid:95) µ are elements of T . Now proceed analogously until reaching τ , suchthat ˜ ϕ ( τ ) = t . By assumption T ↑ τ ∈ P , and now clearly T ↑ τ (cid:13) τ ⊆ ˙ x gen andtherefore T ↑ τ (cid:13) t ⊆ ϕ ( ˙ x gen ). Thus ϕ ( x gen ) is a Cohen κ -real.Another way of looking at the above proof is as follows: the sets { η ∈ κ ω :sup n η ( n ) ∈ S } and { η ∈ κ ω : sup n η ( n ) ∈ S } form Bernstein sets with respectto short κ -Laver trees in κ <ω . Note that due to cardinality reasons, we cannotuse standard diagonalization arguments to produce such sets.If we additionally assume κ <κ = κ , we can obtain an even stronger theorem. Definition 3.6.
A tree T ⊆ κ <κ is called a pseudo- κ -Laver tree if it is limit-closed and has the following property: every σ ∈ T has an extension τ ∈ T suchthat T (cid:22) ω τ is a short κ -Laver tree. We use PL κ to denote the partial order ofpseudo- κ -Laver trees ordered by inclusion. Theorem 3.7 (Main Theorem 2) . Assume κ <κ = κ . Let P be any forcingwhose conditions are pseudo- κ -Laver trees ( i.e., P ⊆ PL κ ) and which is closedunder the following condition: if T ∈ P and σ ∈ T , then T ↑ σ ∈ P . Then P addsa Cohen κ -real.Proof. The method is similar, except that now we let { S t : t ∈ κ <κ } be apartition of Cof ω ( κ ) into κ -many disjoint stationary sets, which we index by κ <κ . This is possible due to the assumption κ <κ = κ . Define the mapping π : κ κ → κ by π ( x ) := t (cid:95) t (cid:95) t (cid:95) . . . , where for all α < κ , t α is such thatsup { x ( ω · α + n ) : n < ω } ∈ S t α . We also use ˜ π to denote the same operationbut from κ <κ to 2 <κ .Let x gen be the P -generic κ -real; we show that π ( x gen ) is κ -Cohen. Let D beopen dense in κ -Cohen forcing, and let T ∈ P . Find σ ∈ T such that T (cid:22) ω σ isa short κ -Laver tree. Let t ∈ D be such that ˜ π ( σ ) ⊆ t . Let u be such that˜ π ( σ ) (cid:95) u = t . By Corollary 3.4 there is η ∈ [ T (cid:22) ω σ ] ω such that sup n η ( n ) ∈ S u .It follows that ˜ π ( σ (cid:95) η ) = ˜ π ( σ ) (cid:95) u = t . Therefore T ↑ ( σ (cid:95) η ) (cid:13) t ⊆ π ( ˙ x gen ).6 The generalized Laver dichotomy
The supremum game and the arguments from Theorem 3.5 naturally lead usto consider a question in generalized descriptive set theory (this connection isexplained in Remark 4.6).We need the following strengthening of the concept of a dominating real ,which has been studied in the classical context in [5, 9, 2, 7].
Definition 4.1.
For f : κ <κ → κ and x ∈ κ κ , we say that x strongly dominates f if ∃ α ∀ α > α ( x ( α ) ≥ f ( x (cid:22) α )). If M is a model of set theory with the same κ <κ , then x is called strongly dominating over M if for all f : κ <κ → κ with f ∈ M , x strongly dominates f .Clearly, if x is strongly dominating, then it is also dominating. The converseis false in general, e.g., let d be dominating over M and let x be defined by x ( α ) := d ( α ) for odd α and x ( α ) := d ( α + 1) for even and limit α . Then x isdominating but not strongly dominating. However, the following is true: Lemma 4.2.
Assume κ <κ = κ . Let M be a model of set theory such that κ <κ ∩ M = κ <κ . Then, if there is a dominating κ -real over M there is also astrongly dominating κ -real over M .Proof. Let d be the dominating κ -real, and fix a bijection between κ <κ and κ in M . We can define a new dominating κ -real d ∗ : κ <κ → κ , i.e., such that forevery f : κ <κ → κ in M , f ( σ ) ≤ d ∗ ( σ ) holds for all but <κ -many σ ∈ κ <κ .Now define inductively e ( α ) := d ∗ ( e (cid:22) α ) . Then e is strongly dominating. Definition 4.3.
A collection X ⊆ κ κ is a strongly dominating family if forevery f : κ <κ → κ there exists x ∈ X which strongly dominates f . D κ denotesthe ideal of all X ⊆ κ κ which are not strongly dominating families.For κ = ω , the ideal D ω = D is the well-known non-strongly-dominatingideal , introduced in [5] and independently in [13], and studied among others in[2]. The main interest in it stems from a perfect-set-like dichotomy theorem forLaver trees. Theorem 4.4 (Goldstern et al. [5]) . If T ⊆ ω <ω is a Laver tree then [ T ] / ∈ D .Every analytic set A ⊆ ω ω is either in D or contains [ T ] for some Laver tree T .In particular, there is a dense embedding from the order of Laver trees into thealgebra of Borel subsets of ω ω modulo D . Dichotomies such as this one are common in classical descriptive set theory,the most notable example being the perfect set property and the closely related K σ -dichomoty ([6]), all of which are false for arbitrary sets of reals but true foranalytic sets. Interest in generalizing such dichotomies to the κ κ -context wasrecently spurred by a result of Schlicht [12] showing that the generalized perfect7et property for generalized projective sets is consistent, and L¨ucke-Motto Ros-Schlicht [10] showing that the generalized Hurewicz dichotomy for generalizedprojective sets is consistent. Thus, it might initially seem surprising that thegeneralized Laver dichotomy fails for closed sets, provably in ZFC. Theorem 4.5.
There is a closed subset of κ κ which is neither in D κ nor con-tains the branches of a generalized Laver tree.Proof. Let ϕ be as in the proof of Theorem 3.5. Let z be the constant 0 function(or any other fixed element of 2 κ ). We show that C := ϕ − [ { z } ] is a counterex-ample to the dichotomy. Given any T ∈ L κ , we can easily find x ∈ [ T ] suchthat ϕ ( x ) (cid:54) = z , therefore [ T ] (cid:54)⊆ C . We claim that C is strongly dominating. Let f : κ <κ → κ be given. Let T f := { σ ∈ κ <κ : ∀ β < len( σ ) ( σ ( β ) ≥ f ( σ (cid:22) β )) } . Clearly T f is a generalized Laver tree and stem( T f ) = ∅ . As in the proof ofTheorem 3.5, we can find x ∈ [ T f ] such that ϕ ( x ) = z . But then x stronglydominates f and x ∈ C , completing the argument. Remark 4.6.
The relevance of this lemma is that it explains why Theorem 3.5does not (as one might initially assume) yield a ZFC-proof of b κ ≤ cov( M κ ).Indeed, it is not hard to verify that cov( D κ ) = b κ and that if X ∈ M κ then ϕ − [ X ] does not contain a κ -Laver tree. Thus, if the dichotomy would holdfor generalized Borel (or just F σ ) sets then one could have concluded b κ =cov( D κ ) ≤ cov( M κ ).One could wonder whether there is any dichotomy for the ideal D κ , i.e.,whether there is any collection P of limit-closed trees, such that for every T ∈ P ,[ T ] / ∈ D κ , and every analytic (or at least closed) set not in D κ contains [ T ] forsome T ∈ P . In fact, this is not the case either. Lemma 4.7.
Let T ⊆ κ <κ be a tree such that [ T ] is strongly dominating. Thenthere exists s ∈ T such that T (cid:22) ω s contains a short κ -Laver tree.Proof. We use a slightly modified version of the game from [5]. Given A ⊆ κ ω let G (cid:63) ( A ) be the game defined by:I α α . . . II β β . . . where α n , β n < κ , α n ≤ β n for all n , and Player II wins iff (cid:104) β n : n < ω (cid:105) ∈ A .It is easy to see that if Player II has a winning strategy in G (cid:63) ( A ) then thereexists a short κ -Laver tree L (with empty stem) such that [ L ] ω ⊆ A . Also itis well-known and easy to see that if A is closed (in the topology on κ ω ) then G (cid:63) ( A ) is determined.Suppose, towards contradiction, that there is no s ∈ T such that T (cid:22) ω s con-tains a short κ -Laver tree. Then Player II does not have a winning strategy8n G (cid:63) ([ T (cid:22) ω s ] ω ) for any s ∈ T , and therefore Player I has a winning strat-egy, call it σ s . Define f : κ <κ → κ as follows: for every t ∈ T , let s ⊆ t be the maximal node of limit length, let u be such that t = s (cid:95) u , and define f ( t ) := σ s ( u ). Since [ T ] is strongly dominating there is x ∈ [ T ] and α suchthat x ( β ) ≥ f ( x (cid:22) β ) for all β > α . In particular, there is s ⊆ x , of limit length,such that x ( | s | + n ) ≥ f ( x (cid:22) ( | s | + n )) for all n < ω . Letting z ∈ κ ω be suchthat s (cid:95) z = x (cid:22) ( | s | + ω ), we see that z ( n ) ≥ f ( s (cid:95) z (cid:22) n ) = σ s ( z (cid:22) n ), for every n . Also z ∈ [ T (cid:22) ω s ] ω , therefore z satisfies the winning conditions for Player IIin the game G (cid:63) ([ T (cid:22) ω s ] ω ), contradicting the assumption that σ s was a winningstrategy for Player I. Corollary 4.8.
There exists a closed strongly dominating set without a super-closed strongly dominating subset.Proof.
Consider again the closed set C := ϕ − [ { z } ] from the proof of Theorem4.5. Towards contradiction suppose there is a limit-closed tree T such that[ T ] ⊆ C and [ T ] is strongly dominating. Without loss of generality, we mayassume that T is pruned , in the sense that for every s ∈ T there is a properextension t ∈ T .By Lemma 4.7 there is s ∈ T such that T (cid:22) ω s contains a short κ -Laver tree L . ByCorollary 3.4 there is η ∈ [ L ] ω such that sup n η ( n ) ∈ S , and by limit-closure, s (cid:95) η ∈ T . Moreover, since T is limit-closed and pruned, there is x ∈ [ T ] suchthat s (cid:95) η ⊆ x . But then ϕ ( x ) contains a “1” and thus is not equal to z , theconstant 0-function, contradicting the assumption.Lemma 4.7, whose proof is based on the game argument from [5], will be animportant ingredient in the following section. <κ -distributive tree forcings We would like to generalize the results from Section 3 about Laver trees to awider class of forcing notions. Recall that a forcing P is <κ -closed if for everydecreasing sequence of conditions of length <κ , there is a condition below all ofthem. A forcing P is <κ -distributive if the intersection of <κ -many open densesets is open dense. Since <κ -distributive forcings do not add new elements of κ <κ , it is a natural class to consider in the context of generalized Baire spaces(after all, forcing in the ordinary Baire space does not add new finite sequences).If a forcing is <κ -closed, then it is <κ -distributive, although the converse doesnot hold. One interesting difference between the two, in the context of gener-alized descriptive set theory, is that generalized- Π -absoluteness holds between <κ -closed forcing extensions (see [4, Lemma 2.7]), while it may fail for <κ -distributive forcing extensions. In this sense, the most natural question is thefollowing: Question 5.1.
Is it true that every <κ -distributive forcing adding a dominating κ -real adds a Cohen κ -real? Is it at least true for every <κ -closed forcing? <κ -distributive forcings whose conditions are limit-closed trees, andsuch that a dominating κ -real can be defined from the generic by a ground-modelcontinuous function. More generally, this holds whenever the interpretation tree of the dominating κ -real is limit-closed.In this section, we will always assume that κ <κ = κ. Definition 5.2.
Let P be any forcing notion, let ˙ x be a name, and let p ∈ P besuch that p (cid:13) ˙ x ∈ κ κ . Then the interpretation tree of ˙ x below p is defined by: T ˙ x,p = { σ ∈ κ <κ : ∃ q ≤ p ( q (cid:13) σ ⊆ ˙ x ) } . It is clear that T ˙ x,p is always a tree in the ground model, but in general itneed not be a limit-closed tree. Lemma 5.3.
Suppose P is a <κ -distributive forcing, and suppose p (cid:13) “ ˙ d isa strongly dominating κ -real”. Additionally, assume that for every q ≤ p , theinterpretation tree T ˙ d,q is limit-closed. Then p (cid:13) “there is a Cohen κ -real”.Proof. Let π be the function defined in Theorem 3.7. We will show that p (cid:13) “ π ( ˙ d )is κ -Cohen”. Let D be κ -Cohen dense and q ≤ p arbitrary. Claim: [ T ˙ d,q ] is a strongly dominating set. Proof.
Let f : κ <κ → κ . Since q forces that ˙ d is strongly dominating, inparticular q (cid:13) ∃ β ∀ α > β ( ˙ d ( α ) ≥ ˇ f ( ˙ d (cid:22) α )). By <κ -distributivity, there is a β and q ≤ q which decides ˙ d (cid:22) β =: σ and forces the following: ∀ α > β ( ˙ d ( α ) ≥ ˇ f ( ˙ d (cid:22) α )) . ( ∗ )Consider the interpretation tree T ˙ d,q . Let x be any branch in [ T ˙ d,q ] ⊆ [ T ˙ d,q ].To see that such a branch exists, notice that for any σ ∈ T ˙ d,q there is a condition q (cid:48) deciding σ ⊆ ˙ d , and by <κ -distributivity, we can find a stronger condition q (cid:48)(cid:48) ≤ q (cid:48) deciding τ ⊆ ˙ d for a proper extension τ of σ . Moreover, at limit nodeswe can continue since T ˙ d,q is limit-closed by assumption.Now we see that for any initial segment σ ⊆ x which is longer than σ , weknow that some q (cid:48) ≤ q forces σ ⊆ ˙ d . Since q (cid:48) also forces ( ∗ ), we must have σ ( α ) ≥ f ( σ (cid:22) α ) for all α in the domain of σ with α > β . Thus we conclude that x ( α ) ≥ f ( x (cid:22) α ) holds for every α > β . (Claim)From the Claim and Lemma 4.7, it follows that there is σ ∈ T ˙ d,q such that T ˙ d,q (cid:22) ω σ contains a short κ -Laver tree. Just as in the proof of Theorem 3.7, let t ∈ D be such that ˜ π ( σ ) ⊆ t , u such that ˜ π ( σ ) (cid:95) u = t , and find η ∈ [ T ˙ d,q (cid:22) ω σ ] ω such that sup n η ( n ) ∈ S u . Now, notice that by the assumption that T ˙ d,q islimit-closed, σ (cid:95) η ∈ T ˙ d,q , hence there is r ≤ q forcing σ (cid:95) η ⊆ ˙ d . But then r (cid:13) t = ˜ π ( σ ) (cid:95) u = ˜ π ( σ (cid:95) η ) ⊆ π ( ˙ d ) , r (cid:13) π ( ˙ d ) ∈ [ t ].Next we look at forcings P whose conditions are limit-closed trees on κ <κ . Definition 5.4.
A forcing partial order P is called a tree forcing if its conditionsare limit-closed trees T ⊆ κ <κ , and for every T ∈ P and σ ∈ T , the restriction T ↑ σ ∈ P .We need to review continuous functions on κ κ . Let us call a function h : κ <κ → κ <κ pre-continuous if:1. σ ⊆ τ ⇒ h ( σ ) ⊆ h ( τ ), and2. ∀ x ∈ κ κ , { len( h ( σ )) : σ ⊆ x } is cofinal in κ .If h is pre-continuous, let f = lim( h ) be the function defined as f ( x ) := (cid:83) { h ( σ ) : σ ⊆ x } . Just as in the classical situation, it is easy to check that if h is pre-continuous, then lim( h ) is continuous, and for every continuous f there exists apre-continuous h such that f = lim( h ).Unlike the classical situation, “being pre-continuous” is not necessarily anabsolute notion. The statement (2) above is a generalized- Π -statement, soit will be absolute between <κ -closed forcing extensions, but not necessarilybetween arbitrary <κ -distributive forcing extensions. However in our case, thiswill not present a problem. We will always talk about pre-continuous functionsin the ground model, and implicitly assume that the continuous function in theextension is well-defined at least on the generic κ -real.The main point is that for tree forcings, the interpretation trees are directlyrelated to the forcing conditions. For a tree T and a pre-continuous function h ,we will consider the tree generated by the image of T under h :tr( h (cid:48)(cid:48) T ) := { τ : ∃ σ ∈ T ( τ ⊆ h ( σ )) } . Lemma 5.5.
Let P be a <κ -distributive tree forcing, ˙ x a name for a κ -real, h apre-continuous function in the ground model with f = lim( h ) , and suppose that T ∈ P is such that T (cid:13) ˙ x = f ( ˙ x gen ) . Then T ˙ x,T = tr( h (cid:48)(cid:48) T ) .Proof. First suppose σ ∈ T . Then T ↑ σ (cid:13) σ ⊆ ˙ x gen , therefore T ↑ σ (cid:13) h ( σ ) ⊆ f ( ˙ x gen ) = ˙ x . Therefore h ( σ ) ∈ T ˙ x,T .Conversely, let τ ∈ T ˙ x,T be given. We want to find σ ∈ T such that τ ⊆ h ( σ ).By definition there is S ≤ T such that S (cid:13) τ ⊆ ˙ x . But since S (cid:13) ˙ x = f ( ˙ x gen ),we also have S (cid:13) ∃ σ ⊆ ˙ x gen ( τ ⊆ h ( σ )) . In particular, part of this assumption is that T forces that { len( h ( σ )) : σ ⊆ ˙ x gen } iscofinal in κ . Recall that even if h is pre-continuous in the ground model, the second conditionmay fail to be absolute. If P is <κ -closed, then the condition is preserved by Π -absoluteness. <κ -distributivity, there exists S (cid:48) ≤ S which decides σ , i.e., we may assumethat σ is in the ground model, τ ⊆ h ( σ ) holds, and S (cid:48) (cid:13) σ ⊆ ˙ x gen . Moreover, σ ⊆ stem( S (cid:48) ), because otherwise there would be some incompatible σ (cid:48) ∈ S (cid:48) , andwe would have S (cid:48) ↑ σ (cid:48) (cid:13) σ (cid:48) ⊆ ˙ x gen , contradicting S (cid:48) (cid:13) σ ⊆ ˙ x gen . We concludethat σ ∈ S (cid:48) ⊆ S ⊆ T and τ ⊆ h ( σ ) as desired.Taking h to be the identity, an immediate corollary is that if P is a <κ -distributive tree forcing, then the interpretation trees for the generic ˙ x gen arelimit-closed. If, in addition, the generic is strongly dominating, then by Lemma5.3 we immediately know that P adds Cohen κ -reals.For our stronger result, we want to consider pre-continuous functions h otherthan the identity. In those cases, it is not guaranteed that tr( h (cid:48)(cid:48) T ) is limit-closed, even if T was. To avoid this problem we prove two technical lemmas.The main idea is that, even if the original continuous function does not preservelimit-closure, we may change it to another one which does. Definition 5.6.
A pre-continuous function h is called limit-closure-preserving if for every limit-closed tree T , the tree tr( h (cid:48)(cid:48) T ) is also limit-closed. Lemma 5.7.
For every pre-continuous function h , there exists a pre-continuousand limit-closure-preserving function j , such that for all σ and all α (in therespective domains), we have: h ( σ )( α ) ≤ j ( σ )( α ) . Proof.
Fix a function R : κ <κ × κ <κ → κ <κ such that:1. R ( ρ, ∅ ) = ∅ for all ρ .2. If σ (cid:54) = ∅ , then • len( R ( ρ, σ )) = len( σ ) for all ρ , • σ ( α ) ≤ R ( ρ, σ )( α ) for all ρ and all α < len( σ ).3. If ρ (cid:54) = ρ (cid:48) , then for any σ, σ (cid:48) (cid:54) = ∅ , we have R ( ρ, σ )(0) (cid:54) = R ( ρ (cid:48) , σ (cid:48) )(0).In words: R takes every non-empty sequence σ and shifts it coordinate-wise toa higher sequence of the same length depending on ρ ; this happens in such away that for different ρ (cid:54) = ρ (cid:48) , the first coordinates of R ( ρ, . . . ) and R ( ρ (cid:48) , . . . ) arenever the same. It is easy to see that such a function exists since κ <κ = κ .Let h be a pre-continuous function. Define j inductively: • If j ( σ ) is defined, then for every β define j ( σ (cid:95) (cid:104) β (cid:105) ) as follows: let w besuch that h ( σ ) (cid:95) w = h ( σ (cid:95) (cid:104) β (cid:105) ) ( w = ∅ is also allowed). Then let j ( σ (cid:95) (cid:104) β (cid:105) ) := j ( σ ) (cid:95) R ( σ (cid:95) (cid:104) β (cid:105) , w ) . For σ of limit length (including σ = ∅ ), let w be such that h ( σ ) = (cid:83) σ (cid:48) ⊂ σ h ( σ (cid:48) ) (cid:95) w . Note that this is always possible because h ( σ (cid:48) ) ⊆ h ( σ )for all σ (cid:48) ⊂ σ ( w = ∅ is allowed). Then let j ( σ ) := (cid:32) (cid:91) σ (cid:48) ⊂ σ j ( σ (cid:48) ) (cid:33) (cid:95) R ( σ, w ) . We claim that j is as required.Notice that, inductively, len( j ( σ )) = len( h ( σ )) for every σ . It is also clear, byconstruction, that σ ⊆ σ (cid:48) implies j ( σ ) ⊆ j ( σ (cid:48) ). Therefore j is pre-continuous.Moreover, by construction we immediately see that h ( σ )( α ) ≤ j ( σ )( α ) holds forevery σ and α < len( σ ). It remains to prove that j is limit-closure-preserving.Let T be an arbitrary limit-closed tree, and let U := tr( j (cid:48)(cid:48) T ). Let { u i : i < λ } be an increasing sequence in U of length λ < κ . We need to show that thissequence has an extension in U . For each i , let s i ∈ T be ⊆ -minimal such that u i ⊆ j ( s i ). Claim. s i ⊆ s i (cid:48) for all i < i (cid:48) < λ . Proof.
Suppose, towards contradiction, that s i (cid:54)⊆ s i (cid:48) . First, s i (cid:48) ⊂ s i (properextension) is clearly not possible, since this would imply u i ⊆ u i (cid:48) ⊆ j ( s i (cid:48) ) ⊆ j ( s i ), and thus we would have picked s i (cid:48) instead of s i . Therefore, s i and s i (cid:48) areincompatible. Let r be maximal such that r ⊆ s i and r ⊆ s i (cid:48) .Next, notice that j ( r ) ⊂ u i : otherwise, we would have u i ⊆ j ( r ), so we wouldhave picked r instead of s i .So we also know that j ( r ) ⊂ j ( s i ) and j ( r ) ⊂ j ( s i (cid:48) ). Let r be minimal suchthat r ⊆ r ⊆ s i and j ( r ) ⊂ j ( r )and let r be minimal such that r ⊆ r ⊆ s i (cid:48) and j ( r ) ⊂ j ( r ) . Note that both r and r are proper extensions of r , see Figure 1. First weconsider r : there are two cases. • Suppose r is of successor length. Then there is r such that r = r (cid:95) (cid:104) β (cid:105) and j ( r ) = j ( r ). Also, (since j ( σ ) and h ( σ ) always have the samelength), there exists w (cid:54) = ∅ such that h ( r (cid:95) (cid:104) β (cid:105) ) = h ( r ) (cid:95) w . Then bydefinition we have: j ( r ) = j ( r ) (cid:95) R ( r , w ) = j ( r ) (cid:95) R ( r , w ) . The s i ’s do not need to be distinct; e.g., they could be all equal to a unique s , or therecould be cf( λ )-many distinct s i ’s, etc. Now suppose r is of limit length. Then j ( r ) = j ( r (cid:48) ) for all r (cid:48) with r ⊆ r (cid:48) ⊂ r , but h ( r ) ⊃ (cid:83) r (cid:48) ⊂ r h ( r (cid:48) ). So (again because j ( σ ) and h ( σ ) havethe same length) there exists w (cid:54) = ∅ such that h ( r ) = (cid:83) r (cid:48) ⊂ r h ( r (cid:48) ) (cid:95) w .By definition, we have j ( r ) = (cid:32) (cid:91) r (cid:48) ⊂ r j ( r (cid:48) ) (cid:33) (cid:95) R ( r , w ) = j ( r ) (cid:95) R ( r , w ) . Thus, in both cases we have j ( r ) = j ( r ) (cid:95) R ( r , w ) for some non-empty w .By exactly the same argument but looking at r , we see that j ( r ) = j ( r ) (cid:95) R ( r , v )for some non-empty v .But r (cid:54) = r , so by condition 3 of the definition of R , the first coordinates of R ( r , w ) and of R ( r , v ) are not the same. However, we also know j ( r ) (cid:95) R ( r , w ) ⊆ j ( s i ) while j ( r ) (cid:95) R ( r , v ) ⊆ j ( s i (cid:48) ). Together with the fact that j ( r ) ⊂ u i ⊆ j ( s i )and j ( r ) ⊂ u i ⊆ u i (cid:48) ⊆ j ( s i (cid:48) ), this gives us the desired contradiction (see Figure1). We conclude that the only option is s i ⊆ s i (cid:48) . (Claim) r s i ′ u i ′ u i j ( s ) i j ( s ) i ′ T Uj j ( r ) s i rr contradiction ✷ Figure 1: Contradiction assuming s i ⊥ s i (cid:48) So we have an increasing sequence { s i : i < λ } in T , and since T is limit-closed,there is s λ ∈ T with s i ⊆ s λ for all i . Then u i ⊆ j ( s i ) ⊆ j ( s λ ) holds for all i .This completes the proof that U is limit-closed.The point of this lemma is that if h is pre-continuous in the ground modelwith f = lim( h ) and T forces that f ( ˙ x gen ) is a dominating κ -real, then letting j be as in the lemma with g = lim( j ), we know that T also forces that g ( ˙ x gen )is a dominating κ -real.The next step is to convert the dominating into a strongly dominating real.In Lemma 4.2 we mentioned how to convert a dominating to a strongly dominat-ing real, and it is easy to see that this conversion can be coded by a continuousfunction in the ground model. The problem is, this function may again fail14o be limit-closure-preserving, so we need to use a similar method as above toconstruct such a conversion function which is, in addition, limit-preserving.Let us fix an enumeration { σ i : i < κ } of κ <κ such that σ i ⊆ σ j ⇒ i < j ,using the notation (cid:112) σ (cid:113) = i iff σ = σ i . Recall that in Lemma 4.2, the conversionwas given by e ( α ) = d ∗ ( e (cid:22) α ) = d ( (cid:112) e (cid:22) α (cid:113) ). However, we may relax the conditionto e ( α ) ≥ d ( (cid:112) e (cid:22) α (cid:113) ), and the conversion would still work. Definition 5.8.
A function γ : κ κ → κ κ is called strongly-converting , if for all x and all α : γ ( x )( α ) ≥ x ( (cid:112) γ ( x ) (cid:22) α (cid:113) ) . Lemma 5.9.
There exists a pre-continuous and limit-closure preserving func-tion k such that γ = lim( k ) is strongly-converting.Proof. Fix a function R : κ <κ × κ → κ which is injective and R ( ρ, α ) ≥ α forall ρ and all α .Define k : κ <κ → κ <κ inductively as follows: • k ( σ (cid:95) (cid:104) β (cid:105) ) := (cid:40) k ( σ ) (cid:95) (cid:104) R ( σ (cid:95) (cid:104) β (cid:105) , β ) (cid:105) if len( σ ) = (cid:112) k ( σ ) (cid:113) k ( σ ) otherwise • For σ of limit length (and σ = ∅ ), k ( σ ) := (cid:83) { k ( σ (cid:48) ) : σ (cid:48) ⊂ σ } .We claim that γ = lim( k ) is as required. Checking that k is pre-continuous iseasy. Let us check that γ is strongly-converting. By construction, for every α , γ ( x )( α ) = β (cid:48) iff there is some σ (cid:95) (cid:104) β (cid:105) ⊆ x such that1. β (cid:48) = R ( σ (cid:95) (cid:104) β (cid:105) , β )2. k ( σ ) = γ ( x ) (cid:22) α
3. len( σ ) = (cid:112) k ( σ ) (cid:113) Therefore γ ( x )( α ) = β (cid:48) ≥ β = x (len( σ )) = x ( (cid:112) k ( σ ) (cid:113) ) = x ( (cid:112) γ ( x ) (cid:22) α (cid:113) ).It remains to prove that k is limit-closure-preserving. Since this is very similarto the proof of Lemma 5.7, we will leave out some details. Let T be a limit-closed tree, U := tr( k (cid:48)(cid:48) T ), and { u i : i < λ } an increasing sequence in U . Foreach i , let s i ∈ T be minimal such that u i ⊆ k ( s i ) (in this case, we actually have u i = k ( s i ), but this is not relevant). As before, we will be done if we prove thefollowing claim: Claim. s i ⊆ s i (cid:48) for all i < i (cid:48) . Proof.
Suppose s i (cid:54)⊆ s i (cid:48) . Since s i (cid:48) ⊂ s i is impossible, we must have s i ⊥ s i (cid:48) , solet r be maximal with r ⊆ s i and r ⊆ s i (cid:48) . Again we must have k ( r ) ⊂ u i ⊆ u i (cid:48) ,hence we can find least r with r ⊆ r ⊆ s i and k ( r ) ⊂ k ( r ), and least r with r ⊆ r ⊆ s i (cid:48) and k ( r ) ⊂ k ( r ). Moreover r and r are both of successor length,say with last digit β and β , respectively. Then k ( r ) = k ( r ) (cid:95) (cid:104) R ( r , β ) (cid:105) and k ( r ) = k ( r ) (cid:95) (cid:104) R ( r , β ) (cid:105) . Since r (cid:54) = r and R is injective, we obtain acontradiction as before. (Claim)15t is clear that if γ is strongly converting and T (cid:13) “ ˙ d is dominating”, then T (cid:13) “ γ ( ˙ d ) is strongly dominating”. With this, we are ready to prove the finalresult. Theorem 5.10 (Main Theorem 3) . Assume κ <κ = κ . Suppose P is a <κ -distributive tree forcing, h a pre-continuous function in the ground model with f = lim( h ) , and assume that T (cid:13) “ f ( ˙ x gen ) is a dominating κ -real”. Then T (cid:13) “there is a Cohen κ -real”.Proof. First we apply Lemma 5.7 to obtain a pre-continuous and limit-closure-preserving function j . Then, for g = lim( j ), it follows that T (cid:13) “ g ( ˙ x gen ) is adominating κ -real”.Now let k and γ be as in Lemma 5.9. Then T (cid:13) “ γ ( g ( ˙ x gen )) is strongly domi-nating”.Let ˙ e be the name such that T (cid:13) γ ( g ( ˙ x gen )) = ˙ e . Since k and j are limit-closure-preserving, so is k ◦ j . Therefore, by Lemma 5.5, T ˙ e,T = tr(( k ◦ j ) (cid:48)(cid:48) T ) islimit-closed. Of course, the same applies for any stronger condition S ≤ T , i.e., T ˙ e,S is also limit-closed for every S ≤ T . This is all we need to apply Lemma5.3, from which it follows that T (cid:13) “there is a Cohen κ -real”.Unfortunately, none of the methods in this section seem to settle Question5.1, which the authors consider very significant in the context of forcing over κ κ : “Is it true that every <κ -distributive forcing adding a dominating κ -real adds aCohen κ -real? Is it at least true for every <κ -closed forcing?” Acknowledgments.
We would like to thank Hugh Woodin and Martin Gold-stern for useful discussion and advice.
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