Hanf numbers for Extendibility and related phenomena

TL;DR

This paper proves that in certain extensions of ZFC, there exists a complete sentence in $L_{1,}$ with maximal models cofinal in the first measurable cardinal, highlighting phenomena related to Hanf numbers.

Contribution

It provides a detailed proof of the existence of a complete sentence with maximal models cofinal in the first measurable cardinal within suitable ZFC extensions.

Findings

Existence of a complete $L_{1,}$ sentence with maximal models cofinal in the first measurable cardinal.

Maximal models occur only in cardinals cofinal in the first measurable cardinal.

The result connects Hanf numbers with extendibility phenomena in set theory.

Abstract

This paper contains portions of Baldwin's talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of $L_{\omega_1,\omega}$ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.