Wide Gaps with Short Extenders

The paper is a continuation of [Gi]. Extending the methods of [Gi] we show the following: Let κ = ⋃ n<ω κn, κ0 < κ1 < · · · < κn < · · · (1) if each κn carries an extender of the length of the first Mahlo above κn, then for every λ > κ there is a generic extension satisfying 2 ≥ λ; (2) if each κn carries an extender of the length of the first fixed point of the א-function above κn of order n, then for every λ, κ < λ < least inaccessible above κ there is a generic extension satisfying 2 ≥ λ.