On forcing without the continuum hypothesis

Abstract

ℵ2. One of the first examples of the forcing method is cardinal collapsing (A. Levy, see [5]), for example, collapsing ℵ2 to ℵ1: the poset P is the collection of all countable functions from a countable ordinal into ℵ2. As is well known, in VP, , and because P is closed under union of countable chains, remains a cardinal and, in fact, no new countable sets are added. But to prove that remains a cardinal we need to conclude ∣P∣ ≤ ℵ2 and hence ℵ3 is not collapsed. If it has been observed that is actually collapsed in VP. Hence the following theorem, which makes no assumptions on the continuum, is relevant.1. Theorem. There is a poset R such that in VR ℵ2 becomes of cardinality ℵ1, but ℵ1 and the cardinals above ℵ2 are not collapsed.