By using our site, you agree to our collection of information through the use of cookies. To learn more, view our Privacy Policy. Log In Sign Up. A cardinal preserving extension making the set of points of countable V cofinality nonstationary Archive for Mathematical Logic, Moti Gitik.

Dima Sinapova. Itay Neeman. A cardinal preserving extension making the set of points of countable V cofinality nonstationary. Logic — DOI Finally we show that our large cardinal assumption is optimal. Sinapova e-mail: sinapova math.

### 1. Introduction

Gitik et al. We prove that this assumption is optimal.

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But preservation of cardinals was not an issue in that context, and the extensions involved did not in fact preserve cardinals. There has also been work on forcing to add clubs consisting of regulars in V, see Gitik [2]. Proof First we construct a model M by forcing over V with a restricted product of the posets defined in Magidor [4]. Then we construct W by forcing over M to shoot a club C through the complement of A. Let M be obtained by forcing with P over V. This fact will be used in the proof of Claim 3 below.

## Prof. Itay Neeman

An argument similar to the one in Magidor [4] establishes that the combined poset P does not add reals and does not collapse cardinals. It remains to show that the forcing preserves cardinals and reals. H is therefore an elementary sub- structure of V M satisfying conditions 1 and 2 of the claim. Equality 1 follows from the fact that R is normal and belongs to V, and equality 2 follows since the points in ran g are inaccessible and in fact measurable cardinals of V.

Let G be Q-generic over M.

V and M have the same cardinals and the same reals. M and W therefore have the same cardinals and the same reals.

W has a club disjoint from A. This completes the proof of Theorem 1. Proof Let W be the model given by Theorem 1.

## Glossary of set theory

It follows that the iteration preserves reals and preserves cardinals. The following theo- rem shows that this consequence already requires precisely the large cardinal assumed in Theorem 5. Otherwise stop the construction.

This assumption always holds in our case, as K and W have the same cardinals. Assume for contradiction that the construction stops at a finite stage. Syntax Advanced Search. About us. Editorial team. Raffaella Cutolo. Mathematical Logic Quarterly 65 1 Edit this record. Mark as duplicate. Find it on Scholar.

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