Notre Dame Journal of Formal Logic

Internal Categoricity in Arithmetic and Set Theory

Jouko Väänänen and Tong Wang

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 24 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1427202976

Digital Object Identifier
doi:10.1215/00294527-2835038

Mathematical Reviews number (MathSciNet)
MR3326591

Zentralblatt MATH identifier
1372.03088

Subjects
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]

Keywords
second-order logic arithmetic categoricity set theory

Citation

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. https://projecteuclid.org/euclid.ndjfl/1427202976


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References

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Duke University Press

Notre Dame Journal of Formal Logic

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