Infinitary Logics and Abstract Elementary Classes
Saharon Shelah, Andr\'es Villaveces

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 and vocabulary of cardinality can be axiomatized in the logic . 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} of an a.e.c. . 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 .}
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Taxonomy
TopicsAdvanced Algebra and Logic · Logic, Reasoning, and Knowledge · Advanced Topology and Set Theory
