Hanf Numbers and Presentation Theorems in AECs

TL;DR

This paper establishes that strongly compact cardinals set upper bounds for Hanf numbers related to amalgamation in Abstract Elementary Classes (AECs), introducing a functorial presentation theorem to facilitate transfer of amalgamation properties.

Contribution

It introduces a new functorial relational presentation theorem for AECs and demonstrates its use in bounding Hanf numbers via strongly compact cardinals.

Findings

Strongly compact cardinals bound Hanf numbers for amalgamation in AECs

A new functorial presentation theorem aids in transferring amalgamation properties

Semantic and syntactic methods are used to establish these bounds

Abstract

We prove that a strongly compact cardinal is an upper bound for a Hanf number for amalgamation, etc. in AECs using both semantic and syntactic methods. To syntactically prove non-disjoint amalgamation, a different presentation theorem than Shelah's is needed. This relational presentation theorem has the added advantage of being {\it functorial}, which allows the transfer of amalgamation.