On Hanf numbers of the infinitary order property

TL;DR

This paper investigates cardinal and ordinal functions related to Hanf numbers in model theory, focusing on the infinitary order property and defining new functions to analyze their behavior across theories and formulas.

Contribution

It introduces and studies new cardinal and ordinal-valued functions related to Hanf numbers and the infinitary order property, extending understanding of their behavior in model theory.

Findings

Defined functions mu_T^*(gamma, kappa) and mu^*(gamma, kappa) to analyze order properties.

Established bounds and properties of these functions in relation to theories and formulas.

Provided insights into the structure of Hanf numbers in infinitary logic.

Abstract

We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{kappa^+,omega}}.