More about lambda-support iterations of <lambda-complete forcing notions
aa r X i v : . [ m a t h . L O ] O c t MORE ABOUT λ –SUPPORT ITERATIONS OF ( <λ ) –COMPLETEFORCING NOTIONS ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH
Abstract.
This article continues Ros lanowski and Shelah [8, 9, 10, 11, 12]and we introduce here a new property of ( <λ )–strategically complete forcingnotions which implies that their λ –support iterations do not collapse λ + (fora strongly inaccessible cardinal λ ). Introduction
The systematic studies of iterations with uncountable supports which do notcollapse cardinals were intensified with articles Shelah [13, 14]. Those works startedthe development of a theory parallel to that of “proper forcing in CS iterations”,but the drawback there was that the corresponding properties were more like thosein the case of “not adding new reals in CS iterations of proper forcings”. If wewant to investigate cardinal characteristics associated with λ λ (in a manner it wasdone for cardinal characteristics of the continuum), we naturally are interestedin iterating forcing notions which do add new elements of λ λ . The study of λ –support iterations of such forcing notions (for an uncountable cardinal λ ) has aquite long history already. For instance, Kanamori [6] considered iterations of λ –Sacks forcing notion (similar to the forcing Q , ¯ E ; see Definition 3.7 and Remark 3.8)and he proved that under some circumstances these iterations preserve λ + . Fusionproperties of iterations of other tree–like forcing notions were used in Friedmanand Zdomskyy [4] and Friedman, Honzik and Zdomskyy [3]. In particular, theyshowed that λ –support iterations of a close relative of Q λ from Definition 3.1 donot collapse λ + . Several conditions ensuring that λ + is not collapsed in λ –supportiterations were introduced in a series of previous works Ros lanowski and Shelah[8, 9, 10, 11, 12]. Also Eisworth [2] introduced a condition of this type. Each ofthose conditions was meant to be applicable to some natural forcing notions addinga new member of λ λ without adding new elements of <λ λ . In some sense, theyexplained why the relevant forcings can be iterated (without collapsing cardinals).In the present paper we introduce semi–pure properness (Definition 2.3) andwe show that for an inaccessible cardinal λ , λ –support iterations of semi–purelyproper forcing notions are proper in the standard sense (Theorem 2.7). The casesof successor λ and/or weakly inaccessible λ will be treated in a subsequent paper[7]. Date : September 2012.1991
Mathematics Subject Classification.
Primary 03E40; Secondary:03E35.
Key words and phrases.
Forcing, iterations, not collapsing cardinals, proper.We would like to thank the referee for valuable comments and suggestions.Both authors acknowledge support from the United States-Israel Binational Science Foundation(Grant no. 2006108). This is publication 942 of the second author.
The semi–pure properness is designed to cover the forcing notion Q λ mentionedabove (and its relatives given in 3.1, 3.7), but we hope it is much more general.This property has a flavor of fuzzy properness over quasi–diamonds of [10, DefinitionA.3.6] and even more so of being reasonably merry of [11, Definition 6.3]. There isalso some similarity with pure B ∗ –boundedness of [11, Definition 2.2]. However, theexact relationships between these and other properness conditions are not clear.While there are some similarities between conditions studied so far, we are farfrom the state that was achieved for CS iterations and the concept of properness.The considered properties are (unfortunatelly) tailored to fit particular forcing no-tions and they do not provide any satisfactory general framework covering all ex-amples. The search for the “right” notion of λ –propernes is still far from beingcompleted.Basic definitions concerning strategically complete forcing notions, their itera-tions and trees of conditions are reminded in the further part of the Introduction.In the second section of the paper we prove our Iteration Theorem 2.7 and in thefollowing section we present the forcing notions to which this theorem applies. Somespecial properties of and relationships between the forcings from the third sectionare investigated in the fourth section.1.1. Notation.
Our notation is rather standard and compatible with that of clas-sical textbooks (like Jech [5]). However, in forcing we keep the older conventionthat a stronger condition is the larger one .(1) Ordinal numbers will be denoted be the lower case initial letters of the Greekalphabet ( α, β, γ, δ . . . ) and also by i, j (with possible sub- and superscripts).Cardinal numbers will be called κ, λ ; λ will be always assumed to bea regular uncountable cardinal such that λ <λ = λ ; in most instances λ is even assumed to be strongly inaccessible .Also, χ will denote a sufficiently large regular cardinal; H ( χ ) is the familyof all sets hereditarily of size less than χ . Moreover, we fix a well ordering < ∗ χ of H ( χ ).(2) We will consider several games of two players. One player will be called Generic or Complete or just
COM , and we will refer to this player as “she”.Her opponent will be called
Antigeneric or Incomplete or just
INC and willbe referred to as “he”.(3) For a forcing notion P , all P –names for objects in the extension via P will bedenoted with a tilde below (e.g., τ ˜ , X ˜ ), and G ˜ P will stand for the canonical P –name for the generic filter in P . The weakest element of P will be denotedby ∅ P (and we will always assume that there is one, and that there is noother condition equivalent to it).By “ λ –support iterations” we mean iterations in which domains of con-ditions are of size ≤ λ . However, on some occasions we will pretend thatconditions in a λ –support iteration ¯ Q = h P ζ , Q ˜ ζ : ζ < ζ ∗ i are total functionson ζ ∗ and for p ∈ lim( ¯ Q ) and α ∈ ζ ∗ \ dom( p ) we will let p ( α ) = ∅ ˜ Q ˜ α .(4) A filter on λ is a non-empty family of subsets of λ closed under supersetsand intersections and do not containing ∅ . A filter is ( < λ )–complete if itis closed under intersections of <λ members. (Note: we do allow principalfilters or even { λ } .) ORE ABOUT λ –SUPPORT ITERATIONS 3 For a filter D on λ , the family of all D –positive subsets of λ is called D + . (So A ∈ D + if and only if A ⊆ λ and A ∩ B = ∅ for all B ∈ D .) Bya normal filter on λ we mean proper uniform filter closed under diagonalintersections.(5) By a sequence we mean a function whose domain is a set of ordinals. Fortwo sequences η, ν we write ν ⊳ η whenever ν is a proper initial segmentof η , and ν E η when either ν ⊳ η or ν = η . The length of a sequence η isthe order type of its domain and it is denoted by lh( η ).(6) A tree is a ⊳ –downward closed set of sequences. A complete λ –tree is atree T ⊆ <λ λ such that every ⊳ -chain of size less than λ has an ⊳ -boundin T and for each η ∈ T there is ν ∈ T such that η ⊳ ν .Let T ⊆ <λ λ be a tree. For η ∈ T we letsucc T ( η ) = { α < λ : η ⌢ h α i ∈ T } and ( T ) η = { ν ∈ T : ν ⊳ η or η E ν } . We also let root( T ) be the shortest η ∈ T such that | succ T ( η ) | > λ ( T ) = { η ∈ λ λ : ( ∀ α < λ )( η ↾ α ∈ T ) } .1.2. Background on trees of conditions.Definition 1.1.
Let P be a forcing notion.(1) For an ordinal γ and a condition r ∈ P , let a γ ( P , r ) be the following gameof two players, Complete and
Incomplete :the game lasts at most γ moves and during a play theplayers construct a sequence h ( p i , q i ) : i < γ i of pairs ofconditions from P in such a way that( ∀ j < i < γ )( r ≤ p j ≤ q j ≤ p i )and at the stage i < γ of the game, first Incomplete chooses p i and then Complete chooses q i .Complete wins if and only if for every i < γ there are legal moves for bothplayers.(2) We say that the forcing notion P is strategically ( <γ ) –complete ( strategically ( ≤ γ ) –complete , respectively) if Complete has a winning strategy in thegame a γ ( P , r ) (in the game a γ +10 ( P , r ), respectively) for each condition r ∈ P .(3) Let a model N ≺ ( H ( χ ) , ∈ , < ∗ χ ) be such that <λ N ⊆ N , | N | = λ and P ∈ N . We say that a condition p ∈ P is ( N, P ) –generic in the standardsense (or just: ( N, P ) –generic ) if for every P –name τ ˜ ∈ N for an ordinalwe have p (cid:13) “ τ ˜ ∈ N ”.(4) P is λ –proper in the standard sense (or just: λ –proper ) if there is x ∈ H ( χ )such that for every model N ≺ ( H ( χ ) , ∈ , < ∗ χ ) satisfying <λ N ⊆ N, | N | = λ and P , x ∈ N, and every condition q ∈ N ∩ P there is an ( N, P )–generic condition p ∈ P stronger than q . Definition 1.2 (Compare [10, Def. A.1.7], see also [9, Def. 2.2]) . (1) Let γ bean ordinal, ∅ 6 = w ⊆ γ . A ( w, γ –tree is a pair T = ( T, rk) such that • rk : T −→ w ∪ { γ } , • if t ∈ T and rk( t ) = ε , then t is a sequence h ( t ) ζ : ζ ∈ w ∩ ε i , ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH • ( T, ⊳ ) is a tree with root hi and • if t ∈ T , then there is t ′ ∈ T such that t E t ′ and rk( t ′ ) = γ .(2) If, additionally, T = ( T, rk) is such that every chain in T has a ⊳ –upperbound it T , we will call it a standard ( w, γ –tree We will keep the convention that T xy is ( T xy , rk xy ).(3) Let ¯ Q = h P i , Q ˜ i : i < γ i be a λ –support iteration. A tree of conditions in ¯ Q is a system ¯ p = h p t : t ∈ T i such that • ( T, rk) is a ( w, γ –tree for some w ⊆ γ , • p t ∈ P rk( t ) for t ∈ T , and • if s, t ∈ T , s ⊳ t , then p s = p t ↾ rk( s ).If, additionally, ( T, rk) is a standard tree, then ¯ p is called a standard treeof conditions .(4) Let ¯ p , ¯ p be trees of conditions in ¯ Q , ¯ p i = h p it : t ∈ T i . We write ¯ p ≤ ¯ p whenever for each t ∈ T we have p t ≤ p t .Note that our standard trees and trees of conditions are a special case of that[10, Def. A.1.7] when α = 1.2. Semi-purity and iterations
In this section we introduce a new property of ( <λ )–complete forcing notions: semi–pure properness . Then we prove that if λ is strongly inaccessible, then λ –support iterations of semi–pure proper forcing notions are proper in the standardsense (so they preserve stationarity of relevant sets and do not collapse λ + ). Definition 2.1.
Let f : λ −→ λ + 1. A forcing notion with f –complete semi-purity is a triple ( Q , ≤ , ¯ ≤ pr ) such that ¯ ≤ pr = h≤ α pr : α < λ i and ≤ , ≤ α pr are transitive andreflexive (binary) relations on Q satisfying for each α < λ :(a) ≤ α pr ⊆ ≤ ,(b) ( Q , ≤ ) is strategically ( <λ )–complete and ( Q , ≤ α pr ) is strategically ( ≤ κ )–complete for all infinite cardinals κ < f ( α ).If ( Q , ≤ , ¯ ≤ pr ) is a forcing notion with semi-purity, then all our forcing terms (like“forces”, “name”, etc) refer to ( Q , ≤ ). The relations ≤ α pr have an auxiliary characteronly and if we want to refer to them we add “ α –purely” (so “stronger” refers to ≤ and “ α –purely stronger” refers to ≤ α pr ). Remark . Note that unlike in [11, Definition 2.1], in semi-purity we do notrequire any kind of pure decidability.
Definition 2.3.
Let f : λ −→ λ + 1 and let ( Q , ≤ , ¯ ≤ pr ) be a forcing notion with f –complete semi-purity. Suppose that D is a normal filter on λ (e.g., the clubfilter).(1) A sequence ¯ Y = h Y α : α < λ i is called an indexing sequence whenever ∅ 6 = Y α ⊆ α λ and | Y α | < λ for each α < λ .(2) For an indexing sequence ¯ Y , a system ¯ q = h q α,η : α < λ & η ∈ Y α i ⊆ Q and a condition p ∈ Q we define a game a aux¯ Y ( p, ¯ q, Q , ≤ , ¯ ≤ pr , D ) betweentwo players, COM and INC as follows. A play of a aux¯ Y ( p, ¯ q, Q , ≤ , ¯ ≤ pr , D )lasts λ steps during which the players choose successive terms of a sequence h ( r α , A α , η α , r ′ α ) : α < λ i . These terms are chosen so that ORE ABOUT λ –SUPPORT ITERATIONS 5 (a) r α , r ′ α ∈ Q , A α ∈ D , η α ∈ α λ and for α < β < λ : p = r ≤ r α ≤ r ′ α ≤ r β and A β ⊆ A α and η α ⊳ η β , (b) at a stage α of the play, first COM chooses ( r α , A α , η α ) and then INCpicks r ′ α ≥ r α .At the end, COM wins the play h ( r α , A α , η α , r ′ α ) : α < λ i if and only ifboth players had always legal moves (so the play really lasted λ steps) and( ⊙ ) if γ ∈ △ α<λ A α is limit, then η γ ∈ Y γ and q γ,η γ ≤ γ pr r γ .(3) If COM has a winning strategy in a aux¯ Y ( p, ¯ q, Q , ≤ , ¯ ≤ pr , D ) then we say that the condition p is aux-generic over ¯ q, D .(4) Let ¯ Y be an indexing sequence and p ∈ Q . A game a main¯ Y ( p, Q , ≤ , ¯ ≤ pr , D )between two players, Generic and Antigeneric, is defined as follows. A playof the game lasts λ steps during which the players construct a sequence h ¯ p α , ¯ q α : α < λ i . At stage α < λ of the play, first Generic chooses a system¯ p α = h p α,η : η ∈ Y α i of pairwise incompatible conditions from Q . ThenAntigeneric answers by picking a system ¯ q α = h q α,η : η ∈ Y α i of conditionsfrom Q satisfying p α,η ≤ α pr q α,η for all η ∈ Y α . At the end, Generic wins the play h ¯ p α , ¯ q α : α < λ i if and only if, letting¯ q = h q α,η : α < λ & η ∈ Y α i ,( ⊡ ) there is an aux-generic condition p ∗ ≥ p over ¯ q, D .(5) A forcing notion Q is f –semi-purely proper over an indexing sequence ¯ Y anda filter D if for some sequence ¯ ≤ pr of binary relations on Q , ( Q , ≤ , ¯ ≤ pr ) is aforcing with the f –complete semi-purity and for every p ∈ Q Generic has awinning strategy in a main¯ Y ( p, Q , ≤ , ¯ ≤ pr , D ). We then say that the sequence ¯ ≤ pr witnesses the semi-pure properness of Q .(6) If D is the club filter on λ , then we omit it and we write a main¯ Y ( p, Q , ≤ , ¯ ≤ pr )etc. If ≤ α pr = ≤ pr for all α < λ , then we write ≤ pr instead of ¯ ≤ pr , like in a main¯ Y ( p, Q , ≤ , ≤ pr ). If f ( α ) = λ for all α , then we write λ instead of f (inphrases like λ –complete semi–purity etc). Observation 2.4. If f, g : λ −→ λ + 1 and f ≤ g , then “ g –semi-purely proper”implies “ f –semi-purely proper”. The proof of the following proposition may be considered as an introduction tothe more complicated and general proof of Theorem 2.7 dealing with the iterations.
Proposition 2.5.
Assume that f : λ −→ λ + 1 , ω + α < f ( α ) for α < λ and D isa normal filter on λ . Let ¯ Y = h Y α : α < λ i be an indexing sequence. If a forcingnotion Q is f –semi-purely proper over ¯ Y , D , then it is λ –proper in the standardsense.Proof. Let ¯ ≤ pr be a sequence witnessing the semi-pure properness of Q . Assume N ≺ ( H ( χ ) , ∈ , < ∗ χ ) satisfies <λ N ⊆ N, | N | = λ and ( Q , ≤ , ¯ ≤ pr ) , ¯ Y , D . . . ∈ N. Let p ∈ N ∩ Q . Fix a winning strategy st ∈ N of Generic in a main¯ Y ( p, Q , ≤ , ¯ ≤ pr , D )and pick a list h τ ˜ α : α < λ i of all Q –names for ordinals from N .Consider a play of a main¯ Y ( p, Q , ≤ , ¯ ≤ pr , D ) in which Generic uses st and Anti-generic chooses his answers as follows. At stage α < λ of the play, after Generic ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH played ¯ p α = h p α,η : η ∈ Y α i , Antigeneric picks the < ∗ χ –first sequence ¯ q α = h q α,η : η ∈ Y α i such that for each η ∈ Y α :( ∗ ) η p α,η ≤ α pr q α,η ,( ∗∗ ) η if β < α and there is a condition q α –purely stronger than q α,η and forcinga value to τ ˜ β , then q α,η already forces a value to τ ˜ β .Note that since ( Q , ≤ α pr ) is strategically ( ≤| α | )–complete, there are conditions q ∈ Q satisfying ( ∗ ) η + ( ∗∗ ) η . One checks inductively that ¯ p α , ¯ q α ∈ N for all α < λ (remember st ∈ N and the choice of “the < ∗ χ –first”). The play h ¯ p α , ¯ q α : α < λ i is won by Generic, so there is a condition p ∗ ≥ p which is aux-generic over ¯ q = h q α,η : α < λ & η ∈ Y α i and D . We claim that p ∗ is ( N, Q )–generic. So supposetowards contradiction that p + ≥ p ∗ , p + (cid:13) τ ˜ β = ζ , β < λ but ζ / ∈ N . Considera play h ( r α , A α , η α , r ′ α ) : α < λ i of a aux¯ Y ( p ∗ , ¯ q, Q , ≤ , ¯ ≤ pr , D ) in which COM followsher winning strategy and INC plays: • r ′ = p + , and for α > r ′ α = r α .Let γ ∈ △ α<λ A α be a limit ordinal greater than β . Since the play was won by COM,we have η γ ∈ Y γ and q γ,η γ ≤ γ pr r γ . Since p + ≤ r γ , we know that r γ (cid:13) τ ˜ β = ζ andhence (by ( ∗∗ ) η γ ) q γ,η γ (cid:13) τ ˜ β = ζ . However, q γ,η γ ∈ N , contradicting ζ / ∈ N . (cid:3) Lemma 2.6.
Assume that λ is a regular uncountable cardinal, f : λ −→ λ + 1 and ¯ Q = h P ξ , Q ˜ ξ : ξ < γ i is a λ –support iteration such that for every ξ < λ : (cid:13) P ξ “ ( Q ˜ ξ , ≤ , ¯ ≤ pr ) is a forcing notion with f –complete semi-purity ”.Let T = ( T, rk) be a standard ( w, γ –tree, w ∈ [ γ ] <λ , and let ¯ p = h p t : t ∈ T i be atree of conditions in P γ . Suppose that α < λ and Υ is a set of P γ –names for ordinalssuch that | T | · | Υ | < f ( α ) . Then there exists a tree of conditions ¯ q = h q t : t ∈ T i such that ( ⊛ ) ¯ p ≤ ¯ q and if t ∈ T , ξ ∈ w ∩ rk( t ) , then q t ↾ ξ (cid:13) P ξ p t ( ξ ) ≤ α pr q t ( ξ ) , and ( ⊛ ) if τ ˜ ∈ Υ , t ∈ T , rk( t ) = γ and there is a condition q ∈ P γ such that • q t ≤ q , and q ↾ ξ (cid:13) P ξ q t ( ξ ) ≤ α pr q ( ξ ) for all ξ ∈ w , and • q forces a value to τ ˜ ,then q t forces a value to τ ˜ .Proof. Let κ = | T | · | Υ | < f ( α ) (and we may assume κ is infinite as otherwisearguments are trivial). Let ≤ pr w be a binary relation on P γ defined by p ≤ pr w q if and only if p ≤ P γ q and for each ξ ∈ w , q ↾ ξ (cid:13) P ξ p ( ξ ) ≤ α pr q ( ξ ).The relation ≤ pr w is extended to trees of conditions in the natural way.For ξ ∈ γ \ w let st ˜ ξ be a P ξ –name for a wining strategy of Complete in a κ +10 (cid:0) ( Q ˜ ξ , ≤ ) , ∅ ˜ Q ˜ ξ (cid:1) such that it instructs her to play ∅ ˜ Q ˜ ξ as long as Incompleteplays ∅ ˜ Q ˜ ξ . For ξ ∈ w let st ˜ ξ be a name for a similar strategy for the game a κ +10 (cid:0) ( Q ˜ ξ , ≤ α pr ) , ∅ ˜ Q ˜ ξ (cid:1) .Let h ( t i , τ ˜ i ) : i < κ i list all members of { t ∈ T : rk( t ) = γ } × Υ (with possiblerepetitions). By induction on i ≤ κ we choose trees of conditions ¯ q i = h q it : t ∈ T i and ¯ r i = h r it : t ∈ T i such that( α ) ¯ p ≤ pr w ¯ q , ¯ q i ≤ pr w ¯ r i ≤ pr w ¯ q j ≤ pr w ¯ r j for i < j ≤ κ , ORE ABOUT λ –SUPPORT ITERATIONS 7 ( β ) for each t ∈ T , j ≤ κ and ξ ∈ rk( t ) \ w , q jt ↾ ξ (cid:13) P ξ “ the sequence h ( q it ( ξ ) , r it ( ξ ) : i ≤ j i is a legal partial play of a κ +10 (cid:0) ( Q ˜ ξ , ≤ ) , ∅ ˜ Q ˜ ξ (cid:1) in which Complete follows st ˜ ξ ”,( γ ) for each t ∈ T , j ≤ κ and ξ ∈ rk( t ) ∩ w , q jt ↾ ξ (cid:13) P ξ “ the sequence h ( q it ( ξ ) , r it ( ξ ) : i ≤ j i is a legal partial play of a κ +10 (cid:0) ( Q ˜ ξ , ≤ α pr ) , ∅ ˜ Q ˜ ξ (cid:1) in which Complete follows st ˜ ξ ”,( δ ) for each i < κ , if there is a condition q ∈ P γ such that(a) q it i ≤ pr w q , and(b) q forces a value to τ ˜ i ,then already q it i forces the value to τ ˜ i .So suppose we have defined ¯ q j , ¯ r j for j < i . Stipulating ¯ r − = ¯ p , t κ = t , and τ ˜ κ = τ ˜ we ask if there is a condition q ∈ P γ such that r jt i ≤ pr w q for all j < i whichforces a value to τ ˜ i . If there are such conditions, let q it i be one of them. Otherwiselet q it i be any ≤ pr w –bound to { r jt i : j < i } (there is such a bound by ( β ) + ( γ )). Thenfor t ∈ T \ { t i } define q it so that letting s = t ∩ t i : • if ξ < rk( s ), then q it ( ξ ) = q it i ( ξ ), • if rk( s ) ≤ ξ < rk( t ), ξ / ∈ w , then q it ( ξ ) is the < ∗ χ –first P ξ –name such that q it ↾ ξ (cid:13) P ξ “ q it ( ξ ) is a ≤ –upper bound to { r jt ( ξ ) : j < i } ” , • if rk( s ) ≤ ξ < rk( t ), ξ ∈ w , then q it ( ξ ) is the < ∗ χ –first P ξ –name such that q it ↾ ξ (cid:13) P ξ “ q it ( ξ ) is a ≤ α pr –upper bound to { r jt ( ξ ) : j < i } ” . It should be clear that the above demands correctly define a tree of conditions¯ q i = h q it : t ∈ T i (note the choice of “the < ∗ χ –first names”). Finally, we choose ¯ r i sothat (the respective instances of) conditions ( β ) + ( γ ) are satisfied. To ensure weend up with a tree of conditions, at each coordinate we choose “the < ∗ χ –first namesfor the answers given by the respective strategies”.After the inductive process is completed, put ¯ q = ¯ q κ . (cid:3) Theorem 2.7.
Assume that λ is a strongly inaccessible cardinal, f : λ −→ λ + 1 and ¯ κ = h κ α : α < λ i is a sequence of infinite cardinals such that ( κ α ) | α | < f ( α ) for all α < λ , and suppose also that D is a normal filter on λ . For ξ < γ let ¯ Y ξ = h Y ξα : α < λ i be an indexing sequence such that | Y ξα | ≤ κ α . Let ¯ Q = h P ξ , Q ˜ ξ : ξ < γ i be a λ –support iteration such that (cid:13) P ξ “ Q ˜ ξ is f –semi-purely proper over ¯ Y ξ , D V P ξ ”for every ξ < γ (where D V P ξ is the normal filter on λ generated in V P ξ by D ).Then P γ = lim( ¯ Q ) is λ –proper in the standard sense.Proof. The proof is very similar to that of [11, Theorem 2.7].Abusing our notation, the names for the forcing relation and a witness for thesemi-pure properness of Q ˜ ξ will be denoted ≤ and ¯ ≤ pr = h≤ α pr : α < λ i , respectively.For each ξ < γ let st ˜ ξ be the < ∗ χ –first P ξ –name for a winning strategy of Completein a λ ( Q ˜ ξ , ∅ ˜ Q ˜ ξ ) such that it instructs Complete to play ∅ ˜ Q ˜ ξ as long as her opponentplays ∅ ˜ Q ˜ ξ . ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH
Let N ≺ ( H ( χ ) , ∈ , < ∗ χ ) be such that <λ N ⊆ N , | N | = λ and ¯ Q , D, h ¯ Y ξ , ( Q ˜ ξ , ≤ , ¯ ≤ pr ) : ξ < γ i , . . . ∈ N . Let p ∈ N ∩ P γ and let h τ ˜ α : α < λ i list all P γ –names forordinals from N . Note that if ξ ∈ γ ∩ N , then st ˜ ξ ∈ N .By induction on δ < λ we will choose( ⊗ ) δ T δ , w δ , r − δ , r δ , ¯ p δ ∗ , ¯ q δ ∗ and ¯ p ˜ δ,ξ , ¯ q ˜ δ,ξ , st ˜ ξ for ξ ∈ N ∩ γ so that the following demands are satisfied.( ∗ ) All objects listed in ( ⊗ ) δ belong to N . After stage δ < λ of the construction,these objects are known for δ and ξ ∈ w δ .( ∗ ) r − δ , r δ ∈ P γ , r − (0) = r (0) = p (0), w δ ⊆ γ , | w δ | = | δ + 1 | , w = { } , w δ ⊆ w δ +1 , and if δ is limit then w δ = S α<δ w α , and [ α<λ dom( r α ) = [ α<λ w α = N ∩ γ. ( ∗ ) For each α < δ < λ we have (cid:0) ∀ ξ ∈ w α +1 (cid:1)(cid:0) r α ( ξ ) = r δ ( ξ ) (cid:1) and p ≤ r − α ≤ r α ≤ r − δ ≤ r δ .( ∗ ) If ξ ∈ ( γ \ w δ ) ∩ N , then r δ ↾ ξ (cid:13) “ the sequence h r − α ( ξ ) , r α ( ξ ) : α ≤ δ i is a legal partial play of a λ (cid:0) Q ˜ ξ , ∅ ˜ Q ˜ ξ (cid:1) in which Complete follows st ˜ ξ ”and if ξ ∈ w δ +1 \ w δ , then st ˜ ξ ∈ N is a P ξ –name for a winning strategyof Generic in a main¯ Y ξ ( r δ ( ξ ) , Q ˜ ξ , ≤ , ¯ ≤ pr , D V P ξ ). (And st ∈ N is a winningstrategy of Generic in a main¯ Y ( p (0) , Q , ≤ , ¯ ≤ pr , D ).)( ∗ ) T δ = ( T δ , rk δ ) is a standard ( w δ , γ –tree, T δ = S α ≤ γ Q ξ ∈ w δ ∩ α Y ξδ (so T δ con-sists of all sequences ¯ t = h t ξ : ξ ∈ w δ ∩ α i where α ≤ γ and t ξ ∈ Y ξδ ).( ∗ ) ¯ p δ ∗ = h p δ ∗ ,t : t ∈ T δ i and ¯ q δ ∗ = h q δ ∗ ,t : t ∈ T δ i are standard trees of conditions,¯ p δ ∗ ≤ ¯ q δ ∗ .( ∗ ) For t ∈ T δ we have that dom( p δ ∗ ,t ) = (cid:0) dom( p ) ∪ S α<δ dom( r α ) ∪ w δ (cid:1) ∩ rk δ ( t )and for each ξ ∈ dom( p δ ∗ ,t ) \ w δ : p δ ∗ ,t ↾ ξ (cid:13) P ξ “ if the set { r α ( ξ ) : α < δ } ∪ { p ( ξ ) } has an upper bound in Q ˜ ξ , then p δ ∗ ,t ( ξ ) is such an upper bound ”.( ∗ ) For ξ ∈ N ∩ γ , ¯ p ˜ δ,ξ = h p ˜ ξδ,η : η ∈ Y ξδ i and ¯ q ˜ δ,ξ = h q ˜ ξδ,η : η ∈ Y ξδ i are P ξ –namesfor systems of conditions in Q ˜ ξ indexed by Y ξδ .( ∗ ) If ξ ∈ w β +1 \ w β , β < λ , then (cid:13) P ξ “ h ¯ p ˜ α,ξ , ¯ q ˜ α,ξ : α < λ i is a play of a main¯ Y ξ ( r β ( ξ ) , Q ˜ ξ , ≤ , ¯ ≤ pr , D V P ξ )in which Generic uses st ˜ ξ ”.( ∗ ) If t ∈ T δ , rk δ ( t ) = ξ < γ , then for each η ∈ Y ξδ q δ ∗ ,t (cid:13) P ξ “ p ˜ ξδ,η = p δ ∗ ,t ∪{h ξ,η i} ( ξ ) and q ˜ ξδ,η = q δ ∗ ,t ∪{h ξ,η i} ( ξ ) ”.( ∗ ) If t ∈ T δ , rk δ ( t ) = γ and α < δ and there is a condition q ∈ P γ such that(a) q δ ∗ ,t ≤ q , and(b) q ↾ ξ (cid:13) P ξ q δ ∗ ,t ( ξ ) ≤ δ pr q ( ξ ) for all ξ ∈ w δ and(c) q forces a value to τ ˜ α ,then already the condition q δ ∗ ,t forces the value to τ ˜ α . ORE ABOUT λ –SUPPORT ITERATIONS 9 ( ∗ ) dom( r − δ ) = dom( r δ ) = S t ∈ T δ dom( q δ ∗ ,t ) and if t ∈ T δ , ξ ∈ dom( r δ ) ∩ rk δ ( t ) \ w δ ,and q δ ∗ ,t ↾ ξ ≤ q ∈ P ξ , r δ ↾ ξ ≤ q , then q (cid:13) P ξ “ if the set { r α ( ξ ) : α < δ } ∪ { q δ ∗ ,t ( ξ ) , p ( ξ ) } has an upper bound in Q ˜ ξ , then r − δ ( ξ ) is such an upper bound ”.We start with fixing an increasing continuous sequence h w α : α < λ i of subsetsof N ∩ γ such that the demands of ( ∗ ) are satisfied. Now, by induction on δ < λ we choose the other objects. So assume that we have defined all objects listed in( ⊗ ) α for α < δ .To ensure ( ∗ ) , whenever we say “choose an X such that . . . ” we mean “choosethe < ∗ χ –first X such that . . . ”. This convention will guarantee that our choices arefrom N .If δ is a successor ordinal and ξ ∈ w δ \ w δ − , then let st ˜ ξ ∈ N be a P ξ –namefor a winning strategy of Generic in a main¯ Y ξ ( r δ − ( ξ ) , Q ˜ ξ , ≤ , ¯ ≤ pr , D V P ξ ). We also pick¯ p ˜ α,ξ , ¯ q ˜ α,ξ for α < δ so that ( ∗ ) + ( ∗ ) hold (note that we already know r δ − ( ξ ) andby ( ∗ ) it is going to be equal to r δ ( ξ )).Clause ( ∗ ) fully describes T δ . Note that, by the assumptions on ¯ Y , ¯ κ ,( ∗ ) | T δ | ≤ ( κ δ ) | δ | < f ( δ ) so also | T δ | · | δ | < f ( δ ).For each ξ ∈ w δ we choose a P ξ –name ¯ p ˜ δ,ξ such that (cid:13) P ξ “ ¯ p ˜ δ,ξ = h p ˜ ξδ,η : η ∈ Y ξδ i is given to Generic by st ˜ ξ as an answer to h ¯ p ˜ α,ξ , ¯ q ˜ α,ξ : α < δ i in the game a main¯ Y ξ ( r β ( ξ ) , Q ˜ ξ , ≤ , ¯ ≤ pr , D V P ξ ) , ”where β < δ is such that ξ ∈ w β +1 \ w β . (Note that for each ξ ∈ w δ and distinct η , η ∈ Y ξδ we have (cid:13) P ξ “ the conditions p ˜ ξδ,η , p ˜ ξδ,η are incompatible”.) Next wechoose a tree of conditions ¯ p δ ∗ = h p δ ∗ ,t : t ∈ T δ i such that for each t ∈ T δ : • dom( p δ ∗ ,t ) = (cid:0) dom( p ) ∪ S α<δ dom( r α ) ∪ w δ (cid:1) ∩ rk δ ( t ) and • for ξ ∈ dom( p δ ∗ ,t ) \ w δ , p δ ∗ ,t ( ξ ) is the < ∗ χ –first P ξ –name for a condition in Q ˜ ξ such that p δ ∗ ,t ↾ ξ (cid:13) P ξ “ if the set { r α ( ξ ) : α < δ } ∪ { p ( ξ ) } has an upper bound in Q ˜ ξ , then p δ ∗ ,t ( ξ ) is such an upper bound ”, • p δ ∗ ,t ( ξ ) = ¯ p ˜ ξδ, ( t ) ξ for ξ ∈ dom( p δ ∗ ,t ) ∩ w δ .Because of ( ∗ ) we may use Lemma 2.6 to pick a tree of conditions ¯ q δ ∗ = h q δ ∗ ,t : t ∈ T δ i such that • ¯ p δ ∗ ≤ ¯ q δ ∗ , • if t ∈ T δ , ξ ∈ w δ ∩ rk δ ( t ), then q δ ∗ ,t ↾ ξ (cid:13) P ξ p δ ∗ ,t ( ξ ) ≤ δ pr q δ ∗ ,t ( ξ ), • if t ∈ T δ , rk δ ( t ) = γ and α < δ and there is a condition q ∈ P γ such that(a) q δ ∗ ,t ≤ q , and(b) q ↾ ξ (cid:13) P ξ q δ ∗ ,t ( ξ ) ≤ δ pr q ( ξ ) for all ξ ∈ w δ and(c) q forces a value to τ ˜ α ,then q δ ∗ ,t forces a value to τ ˜ α .Note that if ξ ∈ w δ , t ∈ T δ , rk δ ( t ) = ξ and η , η ∈ Y ξδ are distinct, then q δ ∗ ,t (cid:13) P ξ “ the conditions q δ ∗ ,t ∪{h ξ,η i} ( ξ ) , q δ ∗ ,t ∪{h ξ,η i} ( ξ ) are incompatible ”. Therefore we may choose P ξ –names q ˜ ξδ,η (for ξ ∈ w δ ) such that • (cid:13) P ξ “¯ q ˜ δ,ξ = h q ˜ ξδ,η : η ∈ Y ξδ i is a system of conditions in Q ˜ ξ indexed by Y ξδ ”, • (cid:13) P ξ “ ( ∀ η ∈ Y ξδ )( p ˜ ξδ,η ≤ δ pr q ˜ ξδ,η ) ”, • if t ∈ T δ , rk δ ( t ) > ξ , then q δ ∗ ,t ↾ ξ (cid:13) P ξ q δ ∗ ,t ( ξ ) = q ˜ ξδ, ( t ) ξ .Finally, we define r − δ , r δ ∈ P γ so thatdom( r − δ ) = dom( r δ ) = [ t ∈ T δ dom( q δ ∗ ,t )and • r − (0) = r (0) = p (0), • if ξ ∈ w α +1 , α < δ , then r − δ ( ξ ) = r δ ( ξ ) = r α ( ξ ), • if ξ ∈ dom( r − δ ) \ w δ , then r − δ ( ξ ) is the < ∗ χ –first P ξ –name for an element of Q ˜ ξ such that r − δ ↾ ξ (cid:13) P ξ “ r − δ ( ξ ) is an upper bound of { r α ( ξ ) : α < δ } ∪ { p ( ξ ) } andif t ∈ T δ , rk δ ( t ) > ξ, and q δ ∗ ,t ↾ ξ ∈ G ˜ P ξ and the set { r α ( ξ ) : α < δ } ∪ { q δ ∗ ,t ( ξ ) , p ( ξ ) } has an upper bound in Q ˜ ξ , then r − δ ( ξ ) is such an upper bound ”,and r δ ( ξ ) is the < ∗ χ –first P ξ –name for an element of Q ˜ ξ such that r δ ↾ ξ (cid:13) P ξ “ r δ ( ξ ) is given to Complete by st ˜ ξ as the answer to h r − α ( ξ ) , r α ( ξ ) : α < δ i ⌢ h r − δ ( ξ ) i in the game a λ ( Q ˜ ξ , ∅ ˜ Q ˜ ξ ) ”.It follows from ( ∗ ) + ( ∗ ) from the previous stages that r − δ , r δ are well defined and p, r α ≤ r − δ ≤ r δ for α < δ (using induction on ξ ∈ dom( r δ )).This completes the description of the inductive definition of the objects listed in( ⊗ ) δ ; it should be clear from the construction that demands ( ∗ ) –( ∗ ) are satisfied.For each ξ ∈ w β +1 \ w β , β < λ , look at the sequence h ¯ p ˜ δ,ξ , ¯ q ˜ δ,ξ : δ < λ i and use ( ∗ ) to choose a P ξ –name q ( ξ ) for a condition in Q ˜ ξ such that (cid:13) P ξ “ q ( ξ ) ≥ r β ( ξ ) is aux-generic over h q ˜ ξδ,η : δ < λ & η ∈ Y ξδ i and D V P ξ ”(if ξ = 0 then q (0) ≥ r (0) is aux-generic over h q δ,η : δ < λ & η ∈ Y δ i , D ). Thisdetermines a condition q ∈ P γ with dom( q ) = N ∩ γ . It follows from ( ∗ ) that p ≤ r β ≤ q for all β < λ .Let us argue that q is ( N, P γ )–generic. Let τ ˜ ∈ N be a P γ –name for an ordinal,say τ ˜ = τ ˜ α ∗ , α ∗ < λ , and let us show that q (cid:13) τ ˜ ∈ N . So suppose towardscontradiction that q ′ ≥ q , q ′ (cid:13) τ ˜ = ζ , ζ / ∈ N . For each ξ ∈ N ∩ γ fix a P ξ –name st ˜ + ξ such that (cid:13) P ξ “ st ˜ + ξ is a winning strategy of COM in a aux¯ Y ξ (cid:0) q ( ξ ) , h q ˜ ξδ,η : δ < λ & η ∈ Y ξδ i , Q ˜ ξ , ≤ , ¯ ≤ pr , D V P ξ (cid:1) ”.Construct inductively a sequence h r + α , r ′ α , η α ( ξ ) , η ˜ α ( ξ ) , h A iα ( ξ ) , A ˜ iα ( ξ ) : i < λ i , A ˜ α ( ξ ) : α < λ & ξ ∈ N ∩ γ i such that the following demands ( ∗ ) –( ∗ ) are satisfied.( ∗ ) r + α , r ′ α ∈ P γ , r +0 = q , r ′ ≥ q ′ and r + β ≤ r ′ β ≤ r + α for β < α < λ . ORE ABOUT λ –SUPPORT ITERATIONS 11 ( ∗ ) For each ξ ∈ N ∩ γ and α < λ we have that η α ( ξ ) ∈ α λ , A iα ( ξ ) ∈ D , η ˜ α ( ξ )is a P ξ –name for a member of α λ , A ˜ iα ( ξ ) is a P ξ –name for a member of D and A ˜ α ( ξ ) is a P ξ –name for a member of D V P ξ , and (cid:13) P ξ “ h ( r + α ( ξ ) , A ˜ α ( ξ ) , η ˜ α ( ξ ) , r ′ α ( ξ )) : α < λ i is a result of a play of a aux¯ Y ξ (cid:0) q ( ξ ) , h q ˜ ξδ,η : δ < λ & η ∈ Y ξδ i , Q ˜ ξ , ≤ , ¯ ≤ pr , D V P ξ (cid:1) in which COM follows the strategy st ˜ + ξ ”.( ∗ ) For j, β ≤ α < λ and ξ ∈ w α we have r ′ α ↾ ξ (cid:13) “ η ˜ α ( ξ ) = η α ( ξ ) & △ i<λ A ˜ iα ( ξ ) ⊆ A ˜ α ( ξ ) & A ˜ jβ ( ξ ) = A jβ ( ξ ) ”.(It should be clear how to carry out the construction; remember P γ is ( <λ )–strategically complete, so in particular it does not add new members of α λ for α < λ .) Take a limit ordinal ε > α ∗ such that ε ∈ T ξ ∈ w ε T i,j<ε A ij ( ξ ). Then, by( ∗ ) –( ∗ ) , for each ξ ∈ w ε we have r + ε ↾ ξ (cid:13) P ξ “ ε ∈ △ α<λ A ˜ α ( ξ ) and η ˜ ε ( ξ ) = [ α<ε η α ( ξ ) = η ε ( ξ ) ∈ Y ξε ”and consequently, by ( ∗ ) ,( ∗ ) r + ε ↾ ξ (cid:13) P ξ “ q ˜ ξε,η ε ( ξ ) ≤ ε pr r + ε ( ξ ) ” for each ξ ∈ w ε .Also note that( ∗ ) p ≤ r δ ≤ q ≤ r + ε for all δ < λ .Let t ∈ T ε be such that rk ε ( t ) = γ and ( t ) ξ = η ε ( ξ ) for ξ ∈ w ε . By inductionon ξ ≤ γ , ξ ∈ N , we show that q ε ∗ ,t ↾ ξ ≤ P ξ r + ε ↾ ξ . So let us assume that ξ < γ and we have shown that q ε ∗ ,t ↾ ξ ≤ P ξ r + ε ↾ ξ . If ξ ∈ w ε then by ( ∗ ) + ( ∗ ) we have q ε ∗ ,t ↾ ( ξ + 1) ≤ P ξ +1 r + ε ↾ ( ξ + 1). So assume ξ / ∈ w ε . Now, by ( ∗ ) , p ε ∗ ,t ↾ ξ ≤ r + ε ↾ ξ , so r + ε ↾ ξ (cid:13) P ξ “ r α ( ξ ) ≤ p ε ∗ ,t ( ξ ) for all α < ε ”(remember ( ∗ ) + ( ∗ ) ), and hence r + ε ↾ ξ (cid:13) P ξ “ r α ( ξ ) ≤ q ε ∗ ,t ( ξ ) for all α < ε ”.Consequently, it follows from ( ∗ ) that r + ε ↾ ξ (cid:13) P ξ “ q ε ∗ ,t ( ξ ) ≤ r − ε ( ξ ) ≤ r ε ( ξ ) ≤ r + ε ( ξ ) ”and thus q ε ∗ ,t ↾ ( ξ + 1) ≤ P ξ +1 r + ε ↾ ( ξ + 1).Now, since q ε ∗ ,t ≤ r + ε and ( ∗ ) holds, we may use the condition ( ∗ ) to concludethat q ε ∗ ,t (cid:13) P γ τ ˜ = ζ (remember q ′ ≤ r + ε , α ∗ < ε ) and consequently ζ ∈ N , acontradiction. (cid:3) Remark . Semi–pure properness is very similar to being reasonably merry of[11, Section 6]. Despite of some differences in the parameters involved, one maysuspect that the games are essentially the same if ≤ δ pr = ≤ . This would suggest thatsemi–pure properness is a weaker condition than being reasonably merry. However,the index sets Y δ here are known before the master game starts, while in [11] theindex sets I δ are decided at the stage δ of the game. This makes our present notionsomewhat stronger. Note that in our proof of the Iteration Theorem 2.7 we reallyhave to know Y δ ’s in advance – we cannot decide names for them and take care of ( ∗ ) + ( ∗ ) at the same time. (This obstacle was not present in the proof of [11,Theorem 6.4] as there we did not deal with the auxilary relations ≤ δ pr .)It should be noted that some of the λ –semi-purely proper forcing notions dis-cussed in the next section (see Proposition 3.6) are not reasonably merry as theydo not have the bounding property of [11, Theorem 6.4(b)]. Problem 2.9.
Are there any relationships between semi–pure properness and theproperties introduced in [10, Definition A.3.6], [11, Defnitions 2.2, 6.3] ?3.
The Forcings
In this section we will show that our “last forcing standing” Q λ and some of itsrelatives fit the framework of semi–pure properness (so their λ –support iterationspreserve λ + ). A slight modification of Q λ was used in iterations in Friedman andZdomskyy [4] and Friedman, Honzik and Zdomskyy [3]. It was called Miller( λ )there and the main difference between the two forcings is in condition [4, Definition2.1(vi)].The filter D from the previous section will be the club filter, so it is not men-tioned; also until Proposition 3.9 the auxiliary relations ≤ α pr do not depend on α ,so instead of ¯ ≤ pr we have just ≤ pr and f ( α ) = λ so we write λ instead of f (seeDefinition 2.3(6)).For our results we have to assume that λ is strongly inaccessible; the case ofsuccessor λ remains untreated here (we will deal with it in a subsequent paper). Definition 3.1. (1) Let T club be the family of all complete λ –trees T ⊆ <λ λ such that • if t ∈ T , then | succ T ( t ) | = 1 or succ T ( t ) is a club of λ , and • ( ∀ t ∈ T )( ∃ s ∈ T )( t ⊳ s & | succ T ( s ) | > Q λ as follows: a condition in Q λ is a tree T ∈ T club such that • if h t i : i < j i ⊆ T is ⊳ –increasing, | succ T ( t i ) | > i < j and t = S i
1, and • if t ∈ T and lh( t ) / ∈ C , then | succ T ( t ) | = 1. Observation 3.2.
Let T ∈ T club . Then T ∈ Q , ∗ λ if and only if there exists asequence h F α : α < λ i of fronts of T such that • if α < β < λ , t ∈ F β , then there is s ∈ F α such that s ⊳ t , • if δ < λ is limit, t α ∈ F α (for α < δ ) are such that t α ⊳ t β whenever α < β < δ , then S α<δ t α ∈ F δ , • for each t ∈ T , | succ T ( t ) | > if and only if t ∈ S α<λ F α . Observation 3.3. Q λ ⊆ Q , ∗ λ ⊆ Q λ = Q , ∗ λ ⊆ Q λ and Q , ∗ λ ⊆ Q λ ⊆ Q λ , and Q , ∗ λ is a dense subforcing of Q λ . Observation 3.4.
Let ℓ ∈ { , , , } . (1) ( Q ℓλ , ≤ , ≤ pr ) is a forcing notion with λ –complete semi-purity. (2) Moreover, the relations ( Q ℓλ , ≤ ) and ( Q ℓλ , ≤ pr ) are ( < λ ) –complete. Lemma 3.5.
Let < ℓ ≤ . Assume that T δ ∈ Q ℓλ and F δ ⊆ T δ (for δ < λ ) aresuch that (i) F δ is a front of T δ , T δ +1 ⊆ T δ , and F δ ⊆ T δ +1 , (ii) if δ is limit, then T δ = T i<δ T i and F δ = (cid:8) t ∈ T δ : ( ∀ ξ < δ )( ∃ i < lh( t ))( t ↾ i ∈ F ξ ) and ( ∀ i < lh( t ))( ∃ ξ < δ )( ∃ j < lh( ν ))( i < j & ν ↾ j ∈ F ξ ) (cid:9) , (iii) ( ∀ t ∈ F δ +1 )( ∃ s ∈ F δ )( s ⊳ t ) , (iv) if t ∈ F δ and | succ T δ ( t ) | > , then | succ T δ +1 ( t ) | > .Then S def = T δ<λ T δ ∈ Q ℓλ .Proof. Plainly, S is a tree closed under unions of ⊳ –chains shorter than λ , and by(i)–(iii) we see that for each t ∈ S there is s ∈ S such that t ⊳ s . Hence S is acomplete λ –tree.Also, for each α < λ we have(v) F α is a front of S and for all β ≥ α { t ∈ S : ( ∃ s ∈ F α )( t E s ) } = { t ∈ T β : ( ∃ s ∈ F α )( t E s ) } . Hence every splitting node in S splits into a club. Suppose now that s ∈ S andlet η ∈ lim λ ( S ) be such that s ⊳ η . Since T i ∈ Q λ (remember 3.3), the set { α < λ : | succ T i ( η ↾ α ) | > } contains a club (for each i < λ ). Also the set { α < λ : η ↾ α ∈ F α } is a club (remember (iii)+(ii)). So we may pick a limitordinal δ < λ such that lh( s ) < δ , η ↾ δ ∈ F δ and | succ T i ( η ↾ δ ) | > i < δ .Then (by (ii)) also | succ T δ ( η ↾ δ ) | > | succ S ( η ↾ δ ) | > s ⊳ η ↾ δ ). So we may conclude that S ∈ T club . We will argue that S ∈ Q ℓλ considering the four possible values of ℓ separately. Case ℓ = 1Suppose η ∈ lim λ ( S ). Then for each δ < λ the set { α < λ : | succ T δ ( η ↾ α ) | > } contains a club and thus the set A def = (cid:8) α < λ : α is limit and ( ∀ δ < α )( | succ T δ ( η ↾ α ) | >
1) and η ↾ α ∈ F α (cid:9) contains a club. But if α ∈ A , then also | succ T α ( η ↾ α ) | > | succ S ( η ↾ α ) | > Case ℓ = 2Suppose that a sequence h s i : i < j i ⊆ S is ⊳ –increasing and | succ S ( s i ) | > i < j . Let s = S i
1. By (v)+(iv)+(iii)+(i) we easily conclude | succ S ( s ) | > s E t for some t ∈ F δ ). Case ℓ = 3Let C δ ⊆ λ be a club such that α ∈ C δ & t ∈ T δ ∩ α λ ⇒ | succ T δ ( t ) | > . Set C = △ δ<λ C δ . Then for each limit α ∈ C and t ∈ S ∩ α λ we have that | succ T δ ( t ) | > δ < α , and hence also | succ T α ( t ) | > | succ S ( t ) | > t ∈ S , lh( t ) ∈ C is limit. Case ℓ = 4If root( S ) ⊳ s ∈ S , then | succ T δ ( s ) | > δ < λ and hence | succ S ( s ) | > (cid:3) Proposition 3.6.
Let λ be a strongly inaccessible cardinal, Y δ = δ δ for δ < λ and ¯ Y = h Y δ : δ < λ i . Then the forcing notions ( Q ℓλ , ≤ , ≤ pr ) for ℓ ∈ { , , } are λ -semi–purely proper over ¯ Y .Proof. Let 1 < ℓ ≤ T ∈ Q ℓλ . Consider the following strategy st of Generic in thegame a main¯ Y ( T, Q ℓλ , ≤ , ≤ pr ).In the course of the play, in addition to her innings h T δ,η : η ∈ Y δ i , Genericchooses also sets A δ ⊆ Y δ and conditions T δ ∈ Q ℓλ so that T δ is decided before thestage δ of the game. Suppose that the two players arrived to a stage δ < λ . If δ = 0 then Generic lets T = T and if δ is limit, then she puts T δ = T i<δ T i (in bothcases T δ ∈ Q ℓλ ). Now Generic determines A δ and h T δ,η : η ∈ Y δ i as follows. Shesets A δ = T δ ∩ Y δ and then she lets h T δ,η : η ∈ Y δ i ⊆ Q ℓλ be a system of pairwiseincompatible conditions chosen so that • if η ∈ A δ then T δ,η = ( T δ ) η .Generic’s inning at this stage is h T δ,η : η ∈ Y δ i . After this Antigeneric answers witha system h S δ,η : η ∈ Y δ i ⊆ Q ℓλ such that T δ,η ≤ pr S δ,η , and then Generic writesaside T δ +1 def = (cid:8) t ∈ T δ : (cid:0) ∃ η ∈ A δ (cid:1)(cid:0) η E t & t ∈ S δ,η (cid:1) or (cid:0) ∀ α ≤ lh( t ) (cid:1)(cid:0) t ↾ α / ∈ A δ (cid:1)(cid:9) . It should be clear that T δ +1 is a condition in Q ℓλ .After the play is finished and sequences h T δ,η , S δ,η : δ < λ & η ∈ Y δ i and h A δ , T δ : δ < λ i have been constructed, Generic lets S = \ δ<λ T δ ⊆ T. Claim 3.6.1. S ∈ Q ℓλ is aux-generic over ¯ S = h S δ,η : δ < λ & η ∈ Y δ i .Proof of the Claim. First note that the sequence h T δ , F δ = T δ ∩ δ λ : δ < λ i satisfiesthe assumptions of Lemma 3.5 and hence S ∈ Q ℓλ .Now we consider the three possible cases separately. ORE ABOUT λ –SUPPORT ITERATIONS 15 Case ℓ = 2.Let us describe a strategy st ∗ of COM in the game a aux¯ Y ( S, ¯ S, Q λ , ≤ , ≤ pr ). It in-structs COM to play as follows. Aside, COM picks also ordinals ξ δ < λ so thatafter arriving at a stage δ < λ , when a sequence h ( S α , A α , η α , S ′ α ) , ξ α : α < δ i hasbeen already constructed, she answers with S δ , A δ , η δ (and ξ δ ) chosen so that thefollowing demands are satisfied.(A) S = S , ξ = lh(root( S )) + 942, A = [ ξ , λ ) and η = hi .(B) If δ is a successor ordinal, say δ = α + 1, then η α ⊳ η δ ∈ S ′ α ∩ δ λ, ξ δ = ξ α + sup( η δ ( i ) : i < δ ) + lh (cid:0) root (cid:0) ( S ′ α ) η δ (cid:1)(cid:1) + 942 ,A δ = [ ξ δ , λ ) and S δ = ( S ′ α ) η δ . (Note: then we will also have η δ ∈ S ′ δ .)(C) If δ is a limit ordinal, then η δ = S α<δ η α , ξ δ = sup( ξ α : α < δ ) + 942, A δ = [ ξ δ , λ ) and S δ = T α<δ S α = T α<δ S ′ α = T α<δ (cid:0) S ′ α (cid:1) η δ . (Note: then we willalso have η δ ∈ S ′ δ .)Note that if h ( S α , A α , η α , S ′ α ) : α < λ i is a play in which COM follows st ∗ and δ ∈ △ α<λ A α is a limit ordinal, then η δ ∈ Y δ ∩ S and it is a limit of splitting points in S ′ α , so also | succ S ′ α ( η δ ) | > α < δ . Therefore, by (C), η δ is a splitting nodein S δ (and in S as well). It follows from the description of the δ th move of Genericin a main¯ Y ( T, Q λ , ≤ , ≤ pr ), that( T δ ) η δ ≤ pr S δ,η δ = ( T δ +1 ) η δ ≤ pr ( S ) η δ ≤ pr S δ . Consequently, st ∗ is a winning strategy for COM. Case ℓ = 3.The winning strategy st ∗ of COM in the game a aux¯ Y ( S, ¯ S, Q λ , ≤ , ≤ pr ) is almostexactly the same as in the previous case. The only difference is that now COMshrinks the answers S ′ α of INC to members of Q , ∗ λ pretending they were playedin the game. The argument that this is a winning strategy is exactly the same asbefore (as Q , ∗ λ ⊆ Q λ ). Case ℓ = 4. Similar. (cid:3)(cid:3) The forcing notions considered above can be slightly generalized by allowing theuse of filters other than the club filter on λ . The forcing notions Q ¯ EE of [11, Definition1.11] and P ¯ EE of [11, Definition 4.2] follow this pattern. However, to apply theiteration theorems of [11] we need to assume that the filter E controlling splittingsalong branches is concentrated on a stationary co-stationary set. Therefore thecase of E being the club filter seems to be of a different character. Putting generalfilters on the splitting nodes only and controlling the splitting levels by the clubfilter leads to Definition 3.7.The forcing notion Q , ¯ E was studied by Brown and Groszek [1] who describedwhen this forcing adds a generic of minimal degree. Definition 3.7.
Suppose that ¯ E = h E t : t ∈ <λ λ i is a system of ( <λ )–completefilters on λ . (These could be principal filters.) We define forcing notions Q ℓ, ¯ E for ℓ = 1 , , , A condition in Q , ¯ E is a complete λ –tree T ⊆ <λ λ such that(a) if t ∈ T , then | succ T ( t ) | = 1 or succ T ( t ) ∈ E t , and (b) ( ∀ t ∈ T )( ∃ s ∈ T )( t ⊳ s & | succ T ( s ) | > if h t i : i < j i ⊆ T is ⊳ –increasing, | succ T ( t i ) | > i < j and t = S i
Let ¯ E = h E t : t ∈ <λ λ i be a system of ( <λ ) –complete filters on λ and ℓ ∈ { , , , } . (1) ( Q ℓ, ¯ E , ≤ , ≤ pr ) is a forcing notion with λ –complete semi-purity. Moreover,the relations ( Q ℓ, ¯ E , ≤ ) and ( Q ℓ, ¯ E , ≤ pr ) are ( < λ ) –complete. (2) If λ is strongly inaccessible, Y δ = δ δ for δ < λ and ¯ Y = h Y δ : δ < λ i , thenthe forcing notions ( Q ℓ, ¯ E , ≤ , ≤ pr ) for ℓ ∈ { , , } are λ -semi–purely properover ¯ Y .Proof. Same as 3.4, 3.6. (cid:3)
Close relatives of the forcing notions Q ℓ, ¯ E were considered in [10, Section B.8]and [11, Definition 4.6]. The modification now is that we consider trees branchinginto less than λ successor nodes (but there are many successors from the point ofview of suitably complete filters). Definition 3.10.
Assume that • λ is strongly inaccessible, f : λ −→ λ is a increasing function such that each f ( α ) is a regular uncountable cardinal and Q ξ<α f ( ξ ) | α | < f ( α ) (for α < λ ), • ¯ F = h F t : t ∈ S α<λ Q ξ<α f ( ξ ) i where F t is a
Assume λ, f, ¯ F are as in 3.10. (1) ( Q ℓf, ¯ F , ≤ , ¯ ≤ pr ) is a forcing notion with f –complete semi-purity. (2) If Y δ = Q ξ<δ f ( ξ ) for δ < λ and ¯ Y = h Y δ : δ < λ i , then ¯ Y is an indexingsequence and the forcing notions ( Q ℓf, ¯ F , ≤ , ¯ ≤ pr ) for ℓ ∈ { , , } are f -semi–purely proper over ¯ Y .Proof. Similar to 3.4, 3.6. (cid:3)
Observation 3.12.
Let η ∈ λ λ and Y α = { η ↾ α } for α < λ . Suppose that ( Q , ≤ ) is a strategically ( ≤ λ ) –complete forcing notion and let ≤ α pr be ≤ (for α < λ ). Then Q is λ –semi-purely proper over h Y ξ : ξ < λ i and the club filter with h≤ α pr : α < λ i witnessing this. Corollary 3.13.
Let λ be a strongly inaccessible cardinal. Suppose that ¯ E is asin 3.7 and f, ¯ F are as in 3.10. Let ¯ Q = h P ξ , Q ˜ ξ : ξ < γ i be a λ –support iterationsuch that for every ξ < γ the iterand Q ˜ ξ is either strategically ( ≤ λ ) –complete, orit is one of Q λ , Q λ , Q λ , Q , ¯ E , Q , ¯ E , Q , ¯ E , Q f, ¯ F , Q f, ¯ F , Q f, ¯ F . Then P γ = lim( ¯ Q ) is λ –proper in the standard sense. Are Q λ , Q λ very different? The forcing notions Q λ and Q λ appear to be very close. In this section we willshow that, consistently, they are equivalent, but also consistently, they may bedifferent. Lemma 4.1.
Assume T ∈ T club and consider ( T, ⊳ ) as a forcing notion. Let η ˜ bea T –name for the generic λ –branch added by T . Suppose that (cid:13) T “ the set { α < λ : | succ T ( η ˜ ↾ α ) | > } contains a club ”.Then there is T ∗ ⊆ T such that T ∗ ∈ Q λ .Proof. Let C ˜ be a T –name for a club of λ such that (cid:13) T “ (cid:0) ∀ α ∈ C ˜ (cid:1)(cid:0) | succ T ( η ˜ ↾ α ) | > (cid:1) ”, and put S = (cid:8) t ∈ T : lh( t ) is a limit ordinal and t (cid:13) T “ (cid:0) ∀ α < lh( t ) (cid:1)(cid:0) C ˜ ∩ ( α, lh( t )) = ∅ (cid:1) } ”.One easily verifies that(i) if t ∈ S , then t (cid:13) T “ lh( t ) ∈ C ˜ ” and hence | succ T ( t ) | > h t α : α < α ∗ i ⊆ S is ⊳ –increasing, α ∗ < λ , then S α<α ∗ t α ∈ S ,(iii) ( ∀ t ∈ T )( ∃ s ∈ S )( t ⊳ s ).Consequently we may choose T ∗ ⊆ T so that T ∗ ∈ T club and for some fronts F α of T ∗ (for α < λ ) we have • F α ⊆ S , and if α < β < λ , t ∈ F β , then for some s ∈ F α we have s ⊳ t , • if δ < λ is limit and t α ∈ F α for α < δ are such that α < β < δ ⇒ t α ⊳ t β ,then S α<δ t α ∈ F δ , • | succ T ∗ ( t ) | > t ∈ S α<λ F α .Then also T ∗ ∈ Q , ∗ λ = Q λ (remember Observations 3.2, 3.3). (cid:3) Definition 4.2. (1) The λ –Cohen forcing notion C λ is defined as follows: a condition in C λ is a sequence ν ∈ <λ λ , the order ≤ of C λ is the extension of sequences (i.e., ν ≤ ν if and onlyif ν E ν ).(2) The axiom Ax + C λ is the following statement:if S ˜ is a C λ –name and (cid:13) C λ “ S ˜ is a stationary subset of λ ”, and O α ⊆ C λ are open dense sets (for α < λ ) then there is a E –directed E –downwardclosed set H ⊆ C λ such that • H ∩ O α = ∅ for all α < λ , and • the interpretation S ˜ [ H ] of the name S ˜ is a stationary subset of λ . Lemma 4.3.
Let T ∈ T club . Then the following conditions are equivalent: (a) there is T ∗ ⊆ T such that T ∗ ∈ Q λ , (b) there is T ∗ ⊆ T such that T ∗ ∈ T club and (cid:13) C λ ( ∀ η ∈ lim λ ( T ∗ ))( the set { δ<λ : | succ T ∗ ( η ↾ δ ) | > } contains a club of λ ) . Proof.
Assume (a). By the ( <λ )–completeness of C λ we see that (cid:13) C λ T ∗ ∈ Q λ ,and hence (cid:13) C λ T ∗ ∈ Q λ (remember Observation 3.3). Consequently (b) follows.Now assume (b). Since ( T ∗ , E ) (as a forcing notion) is isomorphic with C λ wehave (cid:13) T ∗ ( ∀ η ∈ lim λ ( T ∗ ))(the set { δ<λ : | succ T ( η ↾ δ ) | > } contains a club of λ ) , so in particular (cid:13) T ∗ “ the set { δ < λ : | succ T ( η ˜ ↾ δ ) | > } contains a club of λ ”,where η ˜ is a T ∗ –name for the generic λ –branch. It follows now from Lemma 4.1that (a) holds. (cid:3) Proposition 4.4.
Assume Ax + C λ . Then Q λ is a dense subset of Q λ (so the forcingnotions Q λ , Q λ are equivalent). ORE ABOUT λ –SUPPORT ITERATIONS 19 Proof.
Let T ∈ Q λ and let us consider ( T, E ) as a forcing notion. Let S ˜ be a T –name given by (cid:13) T S ˜ = { δ < λ : | succ T ( η ˜ ↾ δ ) | > } where η ˜ is a T –name for the generic λ –branch. Ask the following question • Does (cid:13) T “ S ˜ contains a club of λ ” ?If the answer is “yes”, then by Lemma 4.1 there is T ∗ ⊆ T such that T ∗ ∈ Q λ .So assume that the answer to our question is “not”. Then there is t ∈ T suchthat t (cid:13) T “ λ \ S ˜ is stationary ”.Let S ˜ ′ = { (ˇ α, s ) : s ∈ T and α = lh( s ) and | succ T ( s ) | = 1 } . Then S ˜ ′ is a T –namefor a subset of λ and (cid:13) T S ˜ ′ = λ \ S ˜ . Therefore, t (cid:13) T “ S ˜ ′ is stationary ” and sincethe forcing notion T above t is isomorphic with C λ , we may use the assumption ofAx + C λ to pick a E –directed E –downward closed set H ⊆ T such that t ∈ H and • H ∩ { s ∈ T : lh( s ) > α } 6 = ∅ for all α < λ , and • S ˜ ′ [ H ] is stationary in λ .Then for each α < λ the intersection H ∩ α λ is a singleton, say H ∩ α λ = { η α } , and • if α < β then η α ⊳ η β , and • α ∈ S ˜ ′ [ H ] if and only if | succ T ( η α ) | = 1.Let η = S α<λ η α . Then η ∈ lim λ ( T ) and the set { α < λ : | succ T ( η ↾ α ) | = 1 } isstationary, contradicting T ∈ Q λ . (cid:3) Proposition 4.5.
It is consistent that Q λ is not dense in Q λ .Proof. We will build a ( <λ )–strategically complete λ + –cc forcing notion forcingthat Q λ is not dense in Q λ . It will be obtained by means of a ( <λ )–supportiteration of length 2 λ . First, we define a forcing notion Q : a condition in Q is a tree q ⊆ <λ λ such that | q | < λ ; the order ≤ = ≤ Q of Q is defined by q ≤ q ′ if and only if q ⊆ q ′ and ( ∀ η ∈ q )( | succ q ( η ) | = 1 ⇒ | succ q ′ ( η ) | = 1).Plainly, Q is a ( <λ )–complete forcing notion of size λ . Let T ˜ be a Q –namesuch that (cid:13) Q “ T ˜ = S G ˜ Q ”. Then (cid:13) Q “ T ˜ ∈ T club and (cid:0) ∀ η ∈ T ˜ (cid:1)(cid:0) | succ T ˜ ( η ) | > ⇒ succ T ˜ ( η ) = λ (cid:1) ”.For a set A ⊆ λ let Q A be the forcing notion shooting a club through A . Thus a condition in Q A is a closed bounded set c included in A , the order ≤ = ≤ Q A of Q A is defined by c ≤ c ′ if and only if c = c ′ ∩ (cid:0) max( c ) + 1 (cid:1) .Now we inductively define a ( <λ )–support iteration ¯ Q = h P ξ , Q ˜ ξ : ξ < λ i anda sequence h A ˜ ξ , η ˜ ξ : ξ < λ i so that the following demands are satisfied.(i) Q is the forcing notion defined above, T ˜ is the Q –name for the generictree added by Q .(ii) P ξ is strategically ( <λ )–complete, satisfies λ + –cc and has a dense subset ofsize 2 λ (for each ξ ≤ λ ).(iii) η ˜ ξ , A ˜ ξ are P ξ –names such that (cid:13) P ξ “ η ˜ ξ ∈ lim λ ( T ˜ ) and A ˜ ξ = { α < λ : | succ T ˜ ( η ˜ ξ ↾ α ) | > } ”.(iv) (cid:13) P ξ “ Q ˜ ξ = Q A ˜ ξ ”. (v) If η ˜ is a P λ –name for a member of lim λ ( T ˜ ), then for some ξ < λ we have (cid:13) P ξ “ η ˜ = η ˜ ξ ”.Clause (ii) will be shown soon, but with it in hand using a bookkeeping devicewe can take care of clause (v). Then the iteration ¯ Q will be fully determined. So letus argue for clause (ii) (assuming that the iteration is constructed so that clauses(i), (iii) and (iv) are satisfied).For 0 < ξ ≤ λ we let P ∗ ξ consist of all conditions p ∈ P ξ such that 0 ∈ dom( p )and for some limit ordinal δ p < λ , for each i ∈ dom( p ) \ { } we have:(a) p (0) ⊆ ≤ δ p +1 λ and for some η p,i ∈ δ p λ ∩ p (0), p ↾ i (cid:13) P i “ η ˜ i ↾ δ p = η p,i ”,(b) p ( i ) is a closed subset of δ p + 1 (not just a P i –name) and δ p ∈ p ( i ),(c) if β ∈ p ( i ), then | succ p (0) ( η p,i ↾ β ) | > Claim 4.5.1. (1) If p, p ′ ∈ P ∗ ξ , then p ≤ P ξ p ′ if and only if dom( p ) ⊆ dom( p ′ ) , p (0) ≤ Q p ′ (0) and ( ∀ i ∈ dom( p ) \ { } )( p ( i ) = p ′ ( i ) ∩ ( δ p + 1)) . (2) | P ∗ ξ | = λ · | ξ | <λ . (3) P ∗ ξ is a ( <λ ) –complete λ + –cc subforcing of P ξ . Moreover, if h p α : α < γ i is a ≤ P ξ –increasing sequence of members of P ∗ ξ , γ < λ , then there is p ∈ P ∗ ξ such that p α ≤ P ξ p for all α < λ and δ p = sup( δ p α : α < γ ) . (4) P ∗ ξ is dense in P ξ . Moreover, for every p ∈ P ξ and α < λ there is q ∈ P ∗ ξ such that p ≤ P ξ q and δ q > α .Proof of the Claim. (1), (2), (3) Straightforward.(4) Induction on ξ ∈ (0 , λ ]. Case ξ = ξ + 1Let p ∈ P ξ . Construct inductively a sequence h p n : n < ω i ⊆ P ∗ ξ such that for each n < ω we have • p ↾ ξ ≤ P ξ p n ≤ P ξ p n +1 and α < δ p n < δ p n +1 , • for some closed set c ⊆ δ p we have p (cid:13) P ξ “ p ( ξ ) = c ”, • for some sequence ν n ∈ δ pn λ ∩ p n +1 (0) we have p n +1 (cid:13) P ξ “ η ˜ ξ ↾ δ p n = ν n ”.(The construction is clearly possible by our inductive hypothesis.) Now we definea condition q ∈ P ∗ ξ . We declare that dom( q ) = S n<ω dom( p n ) ∪ { ξ } and for i ∈ dom( q ) \ { , ξ } we set η q,i = S { η p n ,i : i ∈ dom( p n ) , n ∈ ω } and we also put η q,ξ = S n<ω ν n . We define • δ q = sup n<ω δ p n , q (0) = S n<ω p n (0) ∪ { η q,i , η q,i⌢ h i , η q,i⌢ h i : i ∈ dom( q ) \ { }} , • q ( i ) = S n<ω p n ( i ) ∪ { δ q } for i ∈ dom( q ) \ { , ξ } and • q ( ξ ) = c ∪ { δ q } .One easily verifies that q ∈ P ∗ ξ and it is stronger than p . Case ξ is limit and cf( ξ ) < λ Let p ∈ P ξ . Fix an increasing sequence h ξ ε : ε < cf( ξ ) i ⊆ ξ cofinal in ξ and thenuse the inductive assumption (and properties of an iteration) to choose inductivelya sequence h p ε : ε < cf( ξ ) i such that for each ε < ε ′ < cf( ξ ) we have p ε ∈ P ∗ ξ ε , α < δ p ε < δ p ε ′ and p ↾ ξ ε ≤ P ξε p ε ≤ P ξε p ε ′ ↾ ξ ε . Then define a condition q ∈ P ∗ ξ as follows. Declare that dom( q ) = S ε< cf( ξ ) dom( p ε )and for i ∈ dom( q ) \ { } set η q,i = S { η p ε ,i : i ∈ dom( p ε ) , ε < cf( ξ ) } . Put ORE ABOUT λ –SUPPORT ITERATIONS 21 • δ q = sup ε< cf( ξ ) δ p ε , • q (0) = S ε< cf( ξ ) p ε (0) ∪ { η q,i , η q,i⌢ h i , η q,i⌢ h i : i ∈ dom( q ) \ { }} , • q ( i ) = S ε< cf( ξ ) p ε ( i ) ∪ { δ q } for i ∈ dom( q ) \ { } . Case ξ is limit and cf( ξ ) ≥ λ Immediate as then P ξ = S ζ<ξ P ζ . (cid:3) It follows from 4.5.1 that the clause (ii) of the construction of the iteration issatisfied. In particular, the limit P λ is strategically ( <λ )–complete λ + –cc and hasa dense subset of size 2 λ . It should be also clear that (cid:13) P λ “ T ˜ ∈ Q λ ” (remember(iii)–(v)). Claim 4.5.2. (cid:13) P λ “ T ˜ contains no tree from Q λ ”.Proof of the Claim. Suppose towards contradiction that T ˜ is a P λ –name such that p (cid:13) P λ “ T ˜ ∈ Q λ and T ˜ ⊆ T ˜ ” for some p ∈ P λ . Note that, by 4.5.1(3,4),( ∗ ) if p ≤ P λ q , ν ∈ <λ λ , q (cid:13) P λ ν ∈ T ˜ , and κ < λ and ρ ˜ i are P λ –names formembers of λ λ (for i < κ ),then there are q ∗ ∈ P ∗ λ and ν ∗ ∈ q ∗ (0) such that q ≤ P λ q ∗ , ν ⊳ ν ∗ and q ∗ (cid:13) P λ “ ν ∗ ∈ T ˜ & | succ T ˜ ( ν ∗ ) | > ∀ i < κ )( ρ ˜ i ↾ lh( ν ∗ ) = ν ∗ ) ”.Using ( ∗ ) repeatedly ω times we may construct a sequence h p n , ν ∗ n : n < ω i suchthat • p n ∈ P ∗ λ , p ≤ P λ p n ≤ P λ p n +1 , δ p n < δ p n +1 , and • ν ∗ n ∈ <λ λ , ν ∗ n ∈ p n +1 (0), ν ∗ n ⊳ ν ∗ n +1 , and p n +1 (cid:13) P λ “ ν ∗ n ∈ T ˜ & | succ T ˜ ( ν ∗ n ) | > ∀ i ∈ dom( p n ) \ { } )( η ˜ i ↾ lh( ν ∗ n ) = ν ∗ n ) ”.Then we define a condition q ∈ P ∗ ξ : we declare that dom( q ) = S n<ω dom( p n ) andfor i ∈ dom( q ) \ { } we set η q,i = S { η p n ,i : i ∈ dom( p n ) , n ∈ ω } . We also put ν ∗ = S n<ω ν ∗ n , δ q = sup n<ω δ p n , and then: • q (0) = S n<ω p n (0) ∪{ η q,i , η q,i⌢ h i , η q,i⌢ h i : i ∈ dom( q ) \{ }}∪{ ν ∗ , ν ∗ ⌢ h i} , • q ( i ) = S n<ω p n ( i ) ∪ { δ q } for i ∈ dom( q ) \ { } .Note that ν ∗ / ∈ { η q,i ↾ lh( ν ∗ ) : i ∈ dom( q ) \ { }} and lh( ν ∗ ) ≤ δ q = lh( η q,i ) for i ∈ dom( q ) \ { } . Now we easily check that q ∈ P ∗ λ is stronger than p and it forcesthat ν ∗ ∈ T ˜ is a limit of splitting points of T ˜ , but itself it is not a splitting point(even in T ˜ ). A contradiction with p (cid:13) T ˜ ∈ Q λ . (cid:3)(cid:3) Proposition 4.6.
Assume that the complete Boolean algebras
RO( Q λ ) and RO( Q λ ) are isomorphic. Then Q λ is a dense subset of Q λ .Proof. Since RO( Q λ ) and RO( Q λ ) are isomorphic, we may find Q − ℓλ –names H ˜ ℓ , η ˜ ℓ (for ℓ = 1 ,
2) such that ( ⊡ ) (cid:13) Q ℓλ “ H ˜ − ℓ ⊆ Q − ℓλ is generic over V and η ˜ − ℓ ∈ λ λ is the correspondinggeneric branch ”,( ⊡ ) if G ℓ ⊆ Q ℓλ is generic over V and G − ℓ = H ˜ − ℓ [ G ℓ ], then G ℓ = H ˜ ℓ [ G − ℓ ].Consider Q λ × Q λ with the product order and for ℓ = 1 , R ℓ = { ( T , T ) ∈ Q λ × Q λ : T ℓ (cid:13) Q ℓλ T − ℓ ∈ H ˜ − ℓ } and R = R ∩ R . Claim 4.6.1. R is a dense subset of both R and R .Proof of the Claim. First note that( ⊡ ) if ( T , T ) ∈ Q λ × Q λ and T Q λ “ T / ∈ H ˜ ”, then there is T ∗ ≥ T such that ( T ∗ , T ) ∈ R (and symmetrically when the roles of 1 and 2 areinterchanged).Also,( ⊡ ) if ( T , T ) ∈ R ℓ , ℓ = 1 ,
2, then T − ℓ Q − ℓλ T ℓ / ∈ H ˜ ℓ .[Why? Assume towards contradiction that T − ℓ (cid:13) Q − ℓλ T ℓ / ∈ H ˜ ℓ . Let G ℓ ⊆ Q ℓλ be a generic over V such that T ℓ ∈ G ℓ . Put G − ℓ = H ˜ − ℓ [ G ℓ ]. Then G − ℓ ⊆ Q − ℓλ is generic over V and H ˜ ℓ [ G − ℓ ] = G ℓ . Since ( T , T ) ∈ R ℓ we know T − ℓ ∈ H ˜ − ℓ [ G ℓ ] and hence (by our assumption towards contradiction) T ℓ / ∈ H ˜ ℓ [ G − ℓ ] = G ℓ , contradicting the choice of G ℓ .]Now suppose ( T , T ) ∈ R ℓ , ℓ ∈ { , } . Choose inductively a sequence h ( T n , T n ) : n < ω i such that ( T , T ) = ( T , T ) for all n < ω : • if n is even, then ( T n , T n ) ∈ R ℓ , • if n is odd, then ( T n , T n ) ∈ R − ℓ , • ( T n , T n ) ≤ ( T n +11 , T n +12 ).By ( ⊡ ) + ( ⊡ ) there are no problems with carrying out the inductive process. Put T ω = \ n<ω T n and T ω = \ n<ω T n . Then T ωℓ is the least upper bound of h T nℓ : n < ω i and hence easily ( T ω , T ω ) ∈ R ,( T , T ) ≤ ( T ω , T ω ). (cid:3) Claim 4.6.2.
Let ℓ ∈ { , } , T ∈ Q ℓλ . Then there is T ∗ ≥ T such that for some ν ∈ <λ λ we have lh( ν ) = lh(root( T ∗ )) and T ∗ (cid:13) Q ℓλ “ ν ⊳ η ˜ − ℓ ”.Proof of the Claim. By induction on α < λ choose a sequence h T α : α < λ i so thatfor all α < β < λ we have( ⊡ ) T α ≤ T β , root( T α ) ⊳ root( T β ) and( ⊡ ) T α +1 forces a value to η ˜ − ℓ ↾ lh(root( T α )), and( ⊡ ) if α is limit then T α = T ξ<α T ξ .Let η = S α<λ root( T α ) ∈ λ λ . Then η ∈ lim λ ( T α ) for each α < λ so the sets { δ < λ : | succ T α ( η ↾ δ ) | > } contain clubs (for each α < λ ). Consequently we maypick limit δ < λ such that | succ T α ( η ↾ δ ) | > α < δ . Then also, by ( ⊡ ) , η ↾ δ = root( T δ ) and clearly (by ( ⊡ ) ) T δ forces a value to η ˜ − ℓ ↾ δ . (cid:3) ORE ABOUT λ –SUPPORT ITERATIONS 23 Claim 4.6.3.
Let ℓ ∈ { , } , T ∈ Q ℓλ . Then there is T ∗ ≥ T such that for every t ∈ T ∗ , for some ν ∈ <λ λ we have lh( ν ) = lh( t ) and ( T ∗ ) t (cid:13) Q ℓλ “ ν ⊳ η ˜ − ℓ ”.Proof of the Claim. We choose inductively conditions T α ∈ Q ℓλ and fronts F α of T α so that for all α < β < λ :( ⊡ ) T = T , F = {hi} ,( ⊡ ) T α ≤ T β , F α ⊆ T β and ( ∀ t ∈ F β )( ∃ i < lh( t ))( t ↾ i ∈ F α ),( ⊡ ) if α is limit, then T α = T ξ<α T ξ and F α = (cid:8) t ∈ T α : ( ∀ ξ < α )( ∃ i < lh( t ))( t ↾ i ∈ F ξ ) and ( ∀ i < lh( t ))( ∃ ξ < α )( ∃ j < lh( t ))( i < j & t ↾ j ∈ F ξ ) (cid:9) ,( ⊡ ) if t ∈ F α , ζ ∈ succ T α ( t ) then t ⌢ h ζ i ∈ T α +1 and for some s = s t,ζ ∈ F α +1 we haveroot (cid:0) ( T α +1 ) t ⌢ h ζ i (cid:1) = s and ( T α +1 ) s forces a value to η ˜ − ℓ ↾ lh( s ) , ( ⊡ ) F α +1 = { s t,ζ : t ∈ F α & ζ ∈ succ T α ( t ) } (where s t,ζ are determined by( ⊡ ) ).It should be clear that the construction is possible (at successor stages use 4.6.2).Set T ∗ = T α<λ T α . By Lemma 3.5 we know that T ∗ ∈ Q ℓλ and F α ⊆ T ∗ are fronts of T ∗ (for α < λ ). Moreover,( ⊡ ) if s ∈ T ∗ and | succ T ∗ ( s ) | >
1, then s ∈ S α<λ F α .It follows from ( ⊡ ) + ( ⊡ ) + ( ⊡ ) that for every t ∈ F α the condition ( T ∗ ) t forces a value to η ˜ − ℓ ↾ lh( t ). If t ∈ T ∗ \ S α<λ F α , then choose the shortest s ∈ S α<λ F α such that t ⊳ s . Then ( T ∗ ) t = ( T ∗ ) s (remember ( ⊡ ) ) and hence in particular thecondition ( T ∗ ) t forces a value to η ˜ − ℓ ↾ lh( t ). (cid:3) Now suppose that T ∈ Q λ . Use Claim 4.6.3 to choose a condition T ∗ ∈ Q λ suchthat T ≤ T ∗ and( ⊡ ) T ∗ , for every t ∈ T ∗ the condition ( T ∗ ) t forces a value to η ˜ ↾ lh( t ).Then use Claim 4.6.1 to pick ( T ′ , T ′ ) ∈ R so that T ∗ ≤ T ′ . Note that then also( ⊡ ) T ′ , holds. Apply Claim 4.6.3 to T ′ and ℓ = 2 to find a condition T ′′ ≥ T ′ suchthat the suitable demand ( ⊡ ) T ′′ , holds, and then use Claim 4.6.1 again to choose( T +1 , T +2 ) ∈ R such that T +1 ≥ T ′ and T +2 ≥ T ′′ . Note that then( ⊡ ) T + ℓ , − ℓ for every t ∈ T + ℓ the condition ( T + ℓ ) t forces a value to η ˜ − ℓ ↾ lh( t ).For ℓ = 1 , t ∈ T + ℓ let ϕ ℓ ( t ) ∈ <λ λ be such that lh( ϕ ℓ ( t )) = lh( t ) and( T + ℓ ) t (cid:13) “ ϕ ℓ ( t ) ⊳ η ˜ − ℓ ”. Since ( T +1 , T +2 ) ∈ R we know that( ⊡ ) ℓ ϕ ℓ ( t ) ∈ T +3 − ℓ for each t ∈ T + ℓ and by ( ⊡ ) we also have( ⊡ ) ϕ ◦ ϕ is the identity on T +2 and ϕ ◦ ϕ is the identity on T +1 . Moreover,if t ∈ T +1 , then (( T +1 ) t , ( T +2 ) ϕ ( t ) ) ∈ R andif s ∈ T +2 , then (( T +1 ) ϕ ( s ) , ( T +2 ) s ) ∈ R .Thus ϕ ℓ : T + ℓ −→ T +3 − ℓ is a bijection preserving levels and the extension relation ⊳ ,and ϕ is the inverse of ϕ . Consequently, t ∈ T + ℓ is a splitting of T + ℓ if and onlyif ϕ ℓ ( t ) is a splitting in T +3 − ℓ . Therefore we may conclude that T +1 ∈ Q λ . (cid:3) Conclusion . It is consistent that the forcing notions Q λ , Q λ are not equivalent. Proof.
By Propositions 4.5 and 4.6. (cid:3)
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Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-0243, USA
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
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