Consistency of Suslin's hypothesis, a nonspecial Aronszajn tree, and GCH

Abstract

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].

A short proof of the preservation of the ωω‐bounding property

“…The good news is that this Lemma is indeed true provided that we are very careful in our conventions regarding names for names. In particular, the Lemma (and hence the proof of the Proper Iteration Lemma) can be repaired by replacing the last sentence of [7, Definition 70] with the following definition: With this technical change, the proof of the Proper Iteration Lemma given in [7] is correct.…”

“…

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: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].

…”

“…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.…”

“…The version which is better known, which we may call the "Weak Proper Iteration Lemma", is given in [ 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.…”

“…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.…”