Adding a lot of Cohen reals by adding a few

Abstract

The purpose of the paper is to produce models V_1 \subset V_2 such that adding kappa-many Cohen reals to V_2 adds lambda Cohen reals to V_1. Some of the results: 1. Suppose that V satisfies GCH, kappa = \cup kappa_n= \cup o(kappa_n). Then there is a cardinal preserving generic extension V_1 of V satisfying GCH and having the same reals as V does , so that adding kappa many Cohen reals over V_1 produces kappa^+ Cohen reals over V. 2. Suppose that V is a model of GCH. Then there is a cofinality preserving extension V_1 satisfying GCH so that adding a Cohen real to V_1 produces aleph_1 Cohen reals over V. 3. There is a pair (W,W_1) of generic cofinality preserving etensions of L such that W is contained in W_1 and W_1 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.