Notre Dame Journal of Formal Logic

The Arithmetics of a Theory

Albert Visser

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Abstract

In this paper we study the interpretations of a weak arithmetic, like Buss’s theory S21, in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of the framework of the arithmetics of U. Finally, we provide a deeper study of the arithmetics of a finitely axiomatized sequential theory.

Article information

Source
Notre Dame J. Formal Logic Volume 56, Number 1 (2015), 81-119.

Dates
First available in Project Euclid: 24 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1427202975

Digital Object Identifier
doi:10.1215/00294527-2835029

Mathematical Reviews number (MathSciNet)
MR3326590

Zentralblatt MATH identifier
1350.03045

Subjects
Primary: 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25]
Secondary: 03F30: First-order arithmetic and fragments 03F40: Gödel numberings and issues of incompleteness 03F25: Relative consistency and interpretations

Keywords
interpretation weak arithmetic provability logic $\Sigma_{1}$-sentence

Citation

Visser, Albert. The Arithmetics of a Theory. Notre Dame J. Formal Logic 56 (2015), no. 1, 81--119. doi:10.1215/00294527-2835029. https://projecteuclid.org/euclid.ndjfl/1427202975


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