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# Download PDF A cardinal preserving extension making the set of points of countable V cofinality nonstationary

Magidor M. Changing cofinality of cardinals. Mitchell W. Applications of the covering lemma for sequences of measures.

## Prof. Itay Neeman

Mitchell, W. In: Foreman, M. Handbook of set theory to appear Google Scholar. The core model for sequences of measures. Shelah S. Proper and improper forcing Perspectives in mathematical logic. Springer, Berlin Google Scholar. We are now ready to finish the proof of our Theorem. This ends the proof of our Theorem. We mention some related previous results.

Also it is implicit in B. Velickovic and H. The following question is still open. The authors are indebted to Hiroshi Sakai for valuable comments and suggestions. National Center for Biotechnology Information , U. Sponsored Document from. Topol Appl. Author information Article notes Copyright and License information Disclaimer.

Keywords: Strong Chang's Conjecture, Tree property. Fact 1. Lemma 2. Definition 2. Main Theorem In this section we prove our main result.

Theorem 3. Definition 3. Proposition 3. Lemma 3.

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We need the following Claim: Claim 3. Claim 3. Some final remarks We mention some related previous results.

Corollary 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. An argument similar to the one in Magidor  establishes that the combined poset P does not add reals and does not collapse cardinals.

### 1. Introduction

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. 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.

## Ramsey cardinal - Cantor's Attic

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. This contradiction completes the proof. Gitik, M.

Symbolic Logic 64 1 , 1—12 3. Jech, T. Springer Monographs in Mathematics.