Infinitary Logics and Abstract Elementary Classes

TL;DR

This paper demonstrates that abstract elementary classes can be fully characterized within a specific infinitary logic, improving the definability scope beyond previous results and introducing a canonical tree structure for analysis.

Contribution

It shows that any a.e.c. with certain size constraints can be axiomatized in a powerful infinitary logic, advancing the understanding of their definability and introducing the canonical tree as a new combinatorial tool.

Findings

A.e.c. can be axiomatized in ${ m L}_{eth_2( abla)^{+++}, abla^+}( au)$.

The canonical tree provides new combinatorial insights into a.e.c.

A connection is established between a.e.c. definitions and the logic $L^1_ abla$.

Abstract

We prove that every abstract elementary class (a.e.c.) with LST number $\kappa$ and vocabulary $\tau$ of cardinality $\leq \kappa$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(\kappa)^{+++},\kappa^+}(\tau)$. In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \emph{canonical tree} $\mathcal S={\mathcal S}_{\mathcal K}$ of an a.e.c. $\mathcal K$. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_\lambda$.}