Notre Dame Journal of Formal Logic

Internal Categoricity in Arithmetic and Set Theory

Jouko Väänänen and Tong Wang

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We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of models for these systems. In both cases we give a consistency proof, but naturally we have to assume more than the mere comprehension axioms.

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Notre Dame J. Formal Logic, Volume 56, Number 1 (2015), 121-134.

First available in Project Euclid: 24 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C85: Second- and higher-order model theory
Secondary: 03B15: Higher-order logic and type theory 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35]

second-order logic arithmetic categoricity set theory


Väänänen, Jouko; Wang, Tong. Internal Categoricity in Arithmetic and Set Theory. Notre Dame J. Formal Logic 56 (2015), no. 1, 121--134. doi:10.1215/00294527-2835038.

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