Adding a lot of Cohen reals by adding a few

1995/07/05 by Moti Gitik, Gitik, Moti · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.math/9507209

openalex publication_date 1995/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of the paper is to produce models V₁ ⊂ V₂ such that adding kappa-many Cohen reals to V₂ adds lambda Cohen reals to V₁. Some of the results: 1. Suppose that V satisfies GCH, kappa = ∪ kappa_n= ∪ o(kappa_n). Then there is a cardinal preserving generic extension V₁ of V satisfying GCH and having the same reals as V does , so that adding kappa many Cohen reals over V₁ produces kappa⁺ Cohen reals over V. 2. Suppose that V is a model of GCH. Then there is a cofinality preserving extension V₁ satisfying GCH so that adding a Cohen real to V₁ produces aleph₁ Cohen reals over V. 3. There is a pair (W,W₁) of generic cofinality preserving etensions of L such that W is contained in W₁ and W₁ contains a perfect set of W-reals which is not in W. The last statement is a slight improvement of a result of B.Velickovic and H.Woodin on the Prikry problem.

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