Introduction. In [Sh, Chapter IX], Shelah constructs a model of set theory in which Suslin's hypothesis is true, yet there is an Aronszajn tree which is not special. In his model, we have . He asks whether the same result could be obtained consistently with CH. In this paper, we answer his question in the affirmative.Let us say that a tree T is S-st-special iff there is a function ƒ with dom(f) = {t ∈ T: rank(t) ∈ S} and for every t1 < t2 both in dom(t) we have f(t2) ≠ f(t1) < rank(t1). In Shelah's model, every tree is S-st-special for some fixed stationary costationary set S. Also, there is some tree T such that T is not S′-st-special whenever S′ – S is stationary. These properties, which are sufficient to ensure that Suslin's hypothesis holds and that T is not special, also hold in the model constructed in this paper. These properties also ensure that every Aronszajn tree has a stationary antichain (i.e., an antichain A such that {rank(t): t ∈ A} is stationary). Hence, it is natural to ask whether there is a model of Suslin's hypothesis in which some Aronszajn tree has no stationary antichain. We answer in the affirmative in [S].The construction we use owes much to Shelah's approach to the theorem, due to Jensen (see [DJ]), that CH is consistent with Suslin's hypothesis. This is given in [Sh, Chapter V].
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic.?1. Introduction. One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing co, which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semiproper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[ W] (the forcing for collapsing a stationary co-stationary subset of {fa
There are two versions of the Proper Iteration Lemma. The stronger (but less well-known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ω ω -bounding property.In this paper we give an account of the following theorem of Shelah:Theorem A Suppose P η : η ≤ κ is a countable support forcing iteration of ω ω -bounding proper forcing notions. Then P κ is ω ω -bounding.The proof we give is a simplified version of [9, Conclusion VI.2.8D] building on the main idea of [9, Theorem VI.1.12]. The proof of Theorem A given in [9] is also given in [1]. We also note [3], which gives a slightly weaker result. The argument of [5] is not correct ([6]).There is an interesting reason why we are able to give a simpler argument. The reason is that there are two versions of the Proper Iteration Lemma in the literature. The version which is better known, which we may call the "Weak Proper Iteration Lemma", is given in [1, Lemma 2.6], [2, Lemma 6.1.3], [3, Lemma 3.18], [8, 9, Theorem III.3.2], and [9, Lemma III.3.3H]. The stronger version of the Proper Iteration Lemma uses a weaker hypothesis. A proof of this Lemma is given in [7, Lemma 25]. Here we merely give the statement of the Lemma and refer the reader to [7] for a proof.Proper Iteration Lemma Suppose P η : η ≤ κ is a countable support forcing iteration based on Q η : η < κ and for every η < κ we have that 1 Pη "Q η is proper". Suppose also that α < κ, λ is a sufficiently large regular cardinal, N is a countable elementary submodel of H λ , {P κ , α} ∈ N , p ∈ P α is N -generic, and p "q ∈Ṗ α,κ ∩ N [G Pα ]". Then there is r ∈ P κ such that r is N -generic, r α = p, p "r [α, κ) ≤q", and supp(r) ⊆ α ∪ N .We now turn to the ω ω -bounding property. Definition Suppose f and g are elements of ω ω.P r o o f o f T h e o r e m A . We show that P κ is ω ω -bounding by induction on κ. We may assume that κ is a limit ordinal. Suppose p ∈ P κ and p "f ∈ ω ω". We show that there is an h ∈ ω ω and r ≤ p such that r "f ≤ * ȟ ".Let λ be a sufficiently large regular cardinal (in particular, the set H(λ) contains the power set of P κ , the power set of ω 1 , and the name f ) and let N be a countable elementary substructure of H λ containing {P
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.