New $\Sigma^1_3$ facts

We use ``iterated square sequences'' to show that there is an $L$-definable partition $n:L{\text-}Singulars \to \omega$ such that if $M$ is an inner model not containing $0^\#$:

(a)

For some $k, M \models \{\alpha|n(\alpha)\leq k\}$ is stationary.

(b)

For each $k$ there is a generic extension of $M$ in which $0^\#$ does not exist and $\{\alpha|n(\alpha)\leq k\}$ is non-stationary.

This 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 $M$.