New Sigma(1)(3) facts

We use "iterated square sequences" to show that there is an L-definable par tition n : L-Singulars --> omega such that if M is an inner model not conta ining 0(#): (a) For some k, M satisfies {alpha\n(alpha) less than or equal to k} is sta tionary. (b) For each k there is a generic extension of M in which 0(#) does not exist and {alpha\n(alpha) less than or equal to k} is non-sta tionary. This result is then applied to show that if M is an inner model without 0(# ), then some Sigma(3)(1) sentence not true in M can be forced over M.