Notre Dame Journal of Formal Logic

Nonstandard Functional Interpretations and Categorical Models

Amar Hadzihasanovic and Benno van den Berg

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Abstract

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models.

Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 3 (2017), 343-380.

Dates
Received: 5 February 2014
Accepted: 28 October 2014
First available in Project Euclid: 19 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1492567509

Digital Object Identifier
doi:10.1215/00294527-3870348

Mathematical Reviews number (MathSciNet)
MR3681099

Zentralblatt MATH identifier
06761613

Subjects
Primary: 03H15: Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]
Secondary: 03F25: Relative consistency and interpretations 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10]

Keywords
nonstandard arithmetic functional interpretations categorical logic Dialectica interpretation proof mining constructive mathematics

Citation

Hadzihasanovic, Amar; van den Berg, Benno. Nonstandard Functional Interpretations and Categorical Models. Notre Dame J. Formal Logic 58 (2017), no. 3, 343--380. doi:10.1215/00294527-3870348. https://projecteuclid.org/euclid.ndjfl/1492567509


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