The number of normal measures, revisited

TL;DR

This paper introduces a new, simplified forcing technique to control the number of normal measures on a measurable cardinal in extensions, extending the Friedman-Magidor theorem without relying on inner model assumptions.

Contribution

It provides a novel, non-structural forcing method to precisely manipulate normal measures and extends the theorem to extenders, broadening its applicability.

Findings

For every measurable cardinal and specified number, a forcing extension can produce exactly that many lifts of a normal measure.

The method does not use inner model or fine-structural assumptions, enabling application beyond current inner model limits.

The technique generalizes to extenders and employs simple nonstationary support product forcing.

Abstract

We present a new version of the Friedman-Magidor theorem: for every measurable cardinal $\kappa$ and $\tau\leq\kappa^{++}$, 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 $\tau\leq \kappa^+$, 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.