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Some Results in the Extension with a Coherent Suslin Tree Some results in the extension with a coherent Suslin tree Teruyuki Yorioka (Shizuoka University) Winter School in Abstract Analysis section Set Theory & Topology 28th January { 4th February, 2012 Motivation Theorem (Kunen, Rowbottom, Solovay, etc). implies K : Every ccc forcing MA@1 2 has property K. Question (Todorˇcevi´c). Does K imply ? 2 MA@1 Theorem (Todorˇcevi´c). PID +p > @1 implies no S-spaces. Question (Todorˇcevi´c). Under PID, does no S-spaces imply p > @1? Definition (Todorˇcevi´c). PFA(S) is an axiom that there exists a coherent Suslin tree S such that the forcing axiom holds for every proper forcing which preserves S to be Suslin. Theorem (Farah). t = @1 holds in the extension with a Suslin tree. Proof. Suppose that T is a Suslin tree, and take π : T ! [!]@0 such that ∗ s ≤T t ! π(s) ⊇ π(t) and s ?T t ! π(s) \ π(t) finite: Then for a generic branch G through T , the set fπ(s): s 2 Gg is a ⊆∗-decreasing sequence which doesn't have its lower bound in [!]@0. Motivation Theorem (Kunen, Rowbottom, Solovay, etc). implies K : Every ccc forcing MA@1 2 has property K. Question (Todorˇcevi´c). Does K imply ? 2 MA@1 Theorem (Todorˇcevi´c). PID +p > @1 implies no S-spaces. Question (Todorˇcevi´c). Under PID, does no S-spaces imply p > @1? Definition (Todorˇcevi´c). PFA(S) is an axiom that there exists a coherent Suslin tree S such that the forcing axiom holds for every proper forcing which preserves S to be Suslin. Question (Todorˇcevi´c). Under PFA(S), does S force K2? Question (Todorˇcevi´c). Under PFA(S), does S force no S-spaces? Question. What statements can be forced by S under PFA(S)? Motivation Theorem (Kunen, Rowbottom, Solovay, etc). implies K : Every ccc forcing MA@1 2 has property K. Question (Todorˇcevi´c). Does K imply ? 2 MA@1 Theorem (Todorˇcevi´c). PID +p > @1 implies no S-spaces. Question (Todorˇcevi´c). Under PID, does no S-spaces imply p > @1? Definition (Todorˇcevi´c). PFA(S) is an axiom that there exists a coherent Suslin tree S such that the forcing axiom holds for every proper forcing which preserves S to be Suslin. Question (Todorˇcevi´c). Under PFA(S), does S force K2? Question (Todorˇcevi´c). Under PFA(S), does S force no S-spaces? Question. What statements can be forced by S under PFA(S)? PFA(S) was introduced to combines many of the consequences of the two contra- dictory set theoretic axioms, the weak diamond principle, and PFA. Theorem (Consequences from the weak }). A Suslin tree forces the following. (Farah) t = @1. (Farah) It doesn't hold that all @1-dense subsets of the reals are isomorphic. (Larson{Todorˇcevi´c) Every ladder system has an ununiformized coloring. (Larson{Todorˇcevi´c) There are no Q-sets. (Moore{Hruˇs´ak{Dˇzamonja) }(R; R; =)6 holds. Theorem (Consequences from PFA). Under PFA(S), S forces the following. @ (Todorˇcevi´c) 2 0 = @2 = h = add(N ). (Farah) The open graph dichotomy. (Todorˇcevi´c) The P -ideal dichotomy. (Todorˇcevi´c) There are no compact S-spaces. Today, we see the following. Theorem. Under PFA(S), S forces the following. x1. Every forcing with rectangle refining property has precaliber @1. x2. There are no !2-Aronszajn trees. x3. All Aronszajn trees are club-isomorphic. x4. The weak club guessing and f fail. x1. Every forcing with rec. ref. has precaliber @1 in the ext. with S under PFA(S). Definition. FSCO0 is the collection of forcing notions P such that • conditions of P are finite sets of countable ordinals, • P is uncountable, and • ≤P is equal to the superset relation ⊇, that is, for any σ and τ in P, σ ≤P τ iff σ ⊇ τ. E.g., a specialization of an Aronszajn tree, freezing an (!1;!1)-gap, adding an uncountable homogeneous set of a partition. Definition (Y.). A forcing notion P in FSCO0 has the rectangle refining property (REC) if P is uncountable and for any I and J 2 [P]@1, if I [ J forms a ∆-system, then 0 @ 0 @ 0 0 there are I 2 [I] 1 and J 2 [J] 1 s.t. for every p 2 I and q 2 J , p 6?P q. Note that REC implies CCC. FSCO2 ⊆ FSCO0 is defined (omitted here). Lemma. For any ladder system and its colorig, there is a forcing with REC in FSCO2 which adds a function uniformizing the coloring. Theorem (Larson{Todorˇcevi´c). In the extension with a coherent Suslin tree, every ladder system has a coloring which cannot be uniformized. So, forces that ( in ) fails. S MA@1 REC FSCO2 Lemma. Under ( ), forces that every forcing with in has pre- MA@1 S S REC FSCO2 caliber @1. Therefore, Theorem. Under ( ), forces that every forcing with in has MA@1 S S REC FSCO2 precaliber @ and ( in ) fails. 1 MA@1 REC FSCO2 Compare with the following. Theorem (Todorˇcevi´c{Veliˇckovi´c). Every ccc forcing has precaliber @ iff 1 MA@1 holds. x2. There are no !2-Aronszajn trees in the extension with S under PFA(S). This proof is quite standard. Claim. For a σ-closed forcing P and an S-name T_ for an !2-tree, P adds no new S-names for cofinal chains through T_ whenever c > @1 holds. Claim. For an S-name T_ for a tree of size @1 and of height !1 which doesn't have uncountable (i.e. cofinal) chains through T_, there exists a ccc forcing notion which preserves S to be Suslin and forces T_ to be special (i.e. to be a union of countably many antichains through T_). The following is the motivation of this work. Theorem (Todorˇcvi´c). PFA implies the failure of κ,ω1 for any unctbl κ. Theorem (Magidor). It is consistent that PFA and κ,ω2 hold for any unctbl κ. Theorem (Magidor). It is consistent that PDFA and κ,ω1 hold for any unctbl κ. Theorem (Raghavan). PID implies the failure of κ,ω, for any unctbl κ, and @ PID +b > 1 implies the failure of κ,ω1 for any κ with cf(κ) > !1. @ Question. Does PID +p > 1 imply the failure of !1;!1? We note that !1;!1 holds iff there exists a special !2-Aronszajn tree. Claim. For an S-name T_ for a tree of size @1 and of height !1 which doesn't have uncountable (i.e. cofinal) chains through T_, there exists a ccc forcing notion which preserves S to be Suslin and forces T_ to be special (i.e. to be a union of countably many antichains through T_). Sketch. Assume that <_ T is an S-name such that S \ T_ = h!1; <_ T i " and for any 2 6? _ s S and α, β in !1, if s S \ α T_ β " and α < β, then s S \ α <T_ β ". Take a club C on !1 s.t. for every δ 2 C, every node of Sδ decides <_ T \ (δ × δ). [ ( ) <@ σ P consists of finite partial functions p : S ! [!1] 0 such that σ2[!]<@0 • for every s 2 dom(p) and n 2 dom(p(s)), p(s)(n) ⊆ sup(C \ lv(s)) and s S \ p(s)(n) is an antichain in T_ "; • for every s and t in dom(p), if s <S t, then for every n 2 dom(p(s))\dom(p(t)), t S \ p(s)(n) [ p(t)(n) is an antichain in T_ "; p ≤P q : () p ⊇ q: Note that P adds an S-name which witnesses that T_ to be special in the extension with S. It is proved that if P×S has an uncountable antichain, then some node of S forces that T_ has an uncountable chain. x3. All Aronszajn trees are club-isomorphic in the extension with S under PFA(S). _ _ _ _ ⊆ <!1 ⊆ Let T and US-names for Aronszajn trees s.t. S \ T; U ! & <T_ =<U_ = ": P consists of the functions p such that • dom(p) is a finite 2-chain of countable elementary submodels of H(@2) with S, T_ and U_ , D E • 2 p p 2 for each M dom(p), p(M) = tM ; fM , where tM S and p p p p p αM ! αM fM : ! ! ; non-empty finite partial injection for some αM < ht(tM ), • for each M; M0 2 dom(p) with M0 2 M, p 62 , p 2 , p 62 and p 2 tM M tM0 M αM M αM0 M; • for each M 2 dom(p), p ≤ p ≤ p _ \ αM _ \ αM { tM decides the S-names T ! and U ! , p p ⊆ _ p ⊆ _ { tM S \ dom(fM ) T & ran(fM ) U "; and [ { p \ p [ p is an order-preserving map whose domain is tM S fM0 fM M02dom(p)\M with tp < tp M0 S M a subtree of T_ in which every maximal n chain is of height o 0 2 dom( ) \ ; p p + 1 ", M p M tM0 <S tM DD E E DD E E p p 2 q q 2 P and for each p = tM ; fM ; M dom(p) and q = tM ; fM ; M dom(q) in , ( ) ≤ () ⊇ 8 2 p q p ⊇ q p P q : dom(p) dom(q)& M dom(q) tM = tM & fM fM : P _ _ _ For a -generic GP, define S-names IGP and fGP such that, letting GS be a canonical S-generic name over the extension by GP, n o _ p 2 2 p 2 _ S \ IGP := αM ; p GP & M dom(p)& tM GS " and [ _ p S \ fGP := fM ": p2GP M2dom(p) p 2 _ with tM GS Note that _ is an -name for an uncountable subset of and IGP S n o !1n o _ _ 2 _ 2 _ ! 2 _ 2 _ fGP is an S-name for an isomorphism x T ; ht(x) IGP y U; ht(y) IGP .
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