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

Published online by Cambridge University Press:  12 March 2014

Chaz Schlindwein
Chaz Schlindwein*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 18602,, E-mail:[email protected]
*
Dept. of Mathematical Sciences, University of Nevadaat Las Vegas Las Vegas, Nevada 89154
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Extract

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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 1 - 29
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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