Maximal models up to the first measurable in ZFC

TL;DR

The paper constructs a complete sentence in infinitary logic with maximal models existing in all cardinals below a first measurable but none at or above it, revealing complex model-theoretic behavior around measurable cardinals.

Contribution

It introduces a specific complete sentence in $L_{_1,}$ demonstrating the precise boundary of maximal models at the first measurable cardinal.

Findings

Maximal models exist in a cofinal set of cardinals below the first measurable.

No maximal models are present at or above the first measurable.

The result clarifies the structure of models in relation to large cardinal thresholds.

Abstract

Theorem: There is a {\em complete sentence} $\phi$ of $L_{\omega_1,\omega}$ such that $\phi$ has maximal models in a set of cardinals $\lambda$ that is cofinal in the first measurable $\mu$ while $\phi$ has no maximal models in any $\chi \geq \mu$.