TY - JOUR
T1 - INFINITARY LOGICS and ABSTRACT ELEMENTARY CLASSES
AU - Shelah, Saharon
AU - Villaveces, Andres
N1 - Publisher Copyright:
2021 American Mathematical Society
PY - 2022
Y1 - 2022
N2 - We prove that every abstract elementary class (a.e.c.) with Löwenheim–Skolem–Tarski (LST) number κ and vocabulary τ of cardinality ≤ κ can be axiomatized in the logic L2(κ)+++,κ+(τ). An a.e.c. K in vocabulary τ is therefore an EC class in this logic, 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 canonical tree S = SK of an a.e.c. 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.
AB - We prove that every abstract elementary class (a.e.c.) with Löwenheim–Skolem–Tarski (LST) number κ and vocabulary τ of cardinality ≤ κ can be axiomatized in the logic L2(κ)+++,κ+(τ). An a.e.c. K in vocabulary τ is therefore an EC class in this logic, 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 canonical tree S = SK of an a.e.c. 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.
UR - http://www.scopus.com/inward/record.url?scp=85121979630&partnerID=8YFLogxK
U2 - 10.1090/proc/15688
DO - 10.1090/proc/15688
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AN - SCOPUS:85121979630
SN - 0002-9939
VL - 150
SP - 371
EP - 380
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 1
ER -