Large continuum, oracles

Abstract

Our main theorem is about iterated forcing for making the continuum larger than ℵ₂. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ₁,ℵ₂ by λ, λ ⁺ (starting with λ = λ ^<λ > ℵ₁). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ ⁺ but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.