New Sigma^1_3 facts

Abstract

We use ``iterated square sequences&#39;&#39; to show: There is an L-definable partition n: L-singulars --> omega such that if M is an inner model without 0#: (a) For some n, M satisfies that {alpha | n(alpha)=n} is stationary. (b) For each n there is a generic extension of M in which 0# does not exist and {alpha | n(alpha)<n} is non-stationary. The above result is then applied to show that if M is an inner model without 0# then some Sigma^1_3 sentence not true in M can be forced over M.