Complete $\mathcal{L}_{\omega_1,\omega}$-Sentences with Maximal Models in Multiple Cardinalities

TL;DR

This paper constructs complete $L_{_1,}$-sentences that have maximal models across multiple and even uncountably many cardinalities, advancing understanding of model sizes in logic.

Contribution

It provides the first examples of complete $L_{_1,}$-sentences with maximal models in multiple cardinalities, including uncountably many.

Findings

Constructed complete sentences with maximal models in various cardinalities.

Demonstrated the existence of sentences with maximal models in uncountably many cardinalities.

Extended previous work on incomplete sentences to complete ones.

Abstract

In [BKS15] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper we give examples of complete $L_{\omega_1,\omega}$-sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of $\kappa$ we construct sentences with maximal models in $\kappa$ and in one of $\kappa^+, \kappa^\omega, 2^\kappa$ and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.