The number of normal measures, revisited

Abstract

We present a new version of the Friedman-Magidor theorem: for every measurable cardinal $\kappa$ and $\tau \le {\kappa }^{++}$, there exists a forcing extension $V\subseteq V[G]$ such that any normal measure $U\in V$ on $\kappa$ has exactly $\tau$ distinct lifts in $V[G]$, and every normal measure on $\kappa$ in $V[G]$ arises as such a lift. This version differs from the original Friedman-Magidor theorem in several notable ways. First, the new technique does not involve forcing over canonical inner models or rely on any fine-structural tools or assumptions, allowing it to be applied in the realm of large cardinals beyond the current reach of the inner model program. Second, in the case where $\tau \le {\kappa }^{+}$, all lifts of a normal measure $U\in V$ on $\kappa$ to $V[G]$ have the same ultrapower. Finally, the technique generalizes to a version of the Friedman-Magidor theorem for extenders. An additional advantage is that the forcing used is notably simple, relying only on nonstationary support product forcing.

arXiv commentA new version with relatively minor corrections and simplifications