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Categoricity-like Properties in the First Order Realm
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By classical results of Dedekind and Zermelo, second order logic imposes categoricity features on Peano Arithmetic and Zermelo-Fraenkel set theory. However, we have known since Skolem’s anti-categoricity theorems that the first order formulations of Peano Arithmetic and Zermelo- Fraenkel set theory (i.e., PA and ZF) are not categorical. Here we investigate various categoricity-like properties (including tightness, solidity, and internal categoricity) that are exhibited by a distinguished class of first order theories that include PA and ZF, with the aim of understanding what is special about canonical foundational first order theories.
Title: Categoricity-like Properties in the First Order Realm
Description:
By classical results of Dedekind and Zermelo, second order logic imposes categoricity features on Peano Arithmetic and Zermelo-Fraenkel set theory.
However, we have known since Skolem’s anti-categoricity theorems that the first order formulations of Peano Arithmetic and Zermelo- Fraenkel set theory (i.
e.
, PA and ZF) are not categorical.
Here we investigate various categoricity-like properties (including tightness, solidity, and internal categoricity) that are exhibited by a distinguished class of first order theories that include PA and ZF, with the aim of understanding what is special about canonical foundational first order theories.
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