Geodesic
Structural logic and abstract elementary classes with intersections
Will Boney1 ; Sebastien Vasey2
1 Department of Mathematics Harvard University Cambridge, MA 02138, U.S.A. <a href="http://math.harvard.edu/~wboney/">http://math.harvard.edu/~wboney/</a>
2 Department of Mathematics Harvard University Cambridge, MA 02138, U.S.A. <a href="http://math.harvard.edu/~sebv/">http://math.harvard.edu/~sebv/</a>
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 1-17

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We give a syntactic characterization of abstract elementary classes (AECs) closed under intersections using a new logic with a quantifier for isomorphism types that we call structural logic: we prove that AECs with intersections correspond to classes of models of a universal theory in structural logic. This generalizes Tarski’s syntactic characterization of universal classes. As a corollary, we prove that any AEC closed under intersections with countable Löwenheim–Skolem–Tarski number is axiomatizable in $\mathbb {L}_{\infty , \omega } (Q)$, where $Q$ is the quantifier “there exist uncountably many”.
DOI : 10.4064/ba8178-12-2018
Keywords: syntactic characterization abstract elementary classes aecs closed under intersections using logic quantifier isomorphism types call structural logic prove aecs intersections correspond classes models universal theory structural logic generalizes tarski syntactic characterization universal classes corollary prove aec closed under intersections countable wenheim skolem tarski number axiomatizable mathbb infty omega where quantifier there exist uncountably many

Will Boney 1 ; Sebastien Vasey 2

1 Department of Mathematics Harvard University Cambridge, MA 02138, U.S.A. <a href="http://math.harvard.edu/~wboney/">http://math.harvard.edu/~wboney/</a>
2 Department of Mathematics Harvard University Cambridge, MA 02138, U.S.A. <a href="http://math.harvard.edu/~sebv/">http://math.harvard.edu/~sebv/</a> 
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Will Boney; Sebastien Vasey. Structural logic and abstract elementary classes with intersections. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 1-17. doi: 10.4064/ba8178-12-2018

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