2025/07/28 by Eyal Kaplan, Kaplan, Eyal · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2507.20466
openalex publication_date 2025/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new version of the Friedman-Magidor theorem: for every measurable cardinal κ and τ≤κ++, there exists a forcing extension V⊆ V[G] such that any normal measure U∈ V on κ has exactly τ distinct lifts in V[G], and every normal measure on κ 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 τ≤ κ+, all lifts of a normal measure U∈ V on κ 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.