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Extender based forcings, fresh sets and Aronszajn trees (2011)
| Citations: | 1 - 0 self |
BibTeX
@MISC{Gitik11extenderbased,
author = {Moti Gitik},
title = {Extender based forcings, fresh sets and Aronszajn trees},
year = {2011}
}
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Abstract
Extender based forcings are studied with respect of adding branches to Aronszajn trees. We construct a model with no Aronszajn tree over ℵω+2 from the optimal assumptions. This answers a question of Friedman and Halilovic ́ [1]. The reader interested only in Friedman and Halilovic ́ question may skip the first section and go directly to the second. 1 No branches to κ+- Aronszajn trees. We deal here with Extender Based Prikry forcing, Long and short extenders Prikry forcing. Let us refer to [2] for definitions. Theorem 1.1 Extender based Prikry forcing over κ cannot add a cofinal branch to a κ+-Aronszajn tree. Proof. Let 〈T,≤T 〉 be a κ+-Aronszajn tree. Denote by P the extender based Prikry forcing over κ. Suppose that P adds a cofinal branch through T. Let b ∼ be a name of such branch and 0P b ∼ is a κ+-branch through T. Let p, q ∈ P and n < ω. We say that qis an n-extension of p iff q ≥ p and q is obtained
