Notre Dame Journal of Formal Logic

Interpolation and the Interpretability Logic of PA

Evan Goris

Abstract

In this paper we will be concerned with the interpretability logic of PA and in particular with the fact that this logic, which is denoted by ILM, does not have the interpolation property. An example for this fact seems to emerge from the fact that ILM cannot express Σ₁-ness. This suggests a way to extend the expressive power of interpretability logic, namely, by an additional operator for Σ₁-ness, which might give us a logic with the interpolation property. We will formulate this extension, give an axiomatization which is modally complete and arithmetically complete (although for proofs of these theorems we refer to an earlier paper), and investigate interpolation. We show that this logic still does not have the interpolation property.

Article information

Source
Notre Dame J. Formal Logic, Volume 47, Number 2 (2006), 179-195.

Dates
First available in Project Euclid: 25 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1153858645

Digital Object Identifier
doi:10.1305/ndjfl/1153858645

Mathematical Reviews number (MathSciNet)
MR2240618

Zentralblatt MATH identifier
1114.03048

Subjects
Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}
Secondary: 03F30: First-order arithmetic and fragments

Keywords
provability logic interpretability logic interpolation

Citation

Goris, Evan. Interpolation and the Interpretability Logic of PA. Notre Dame J. Formal Logic 47 (2006), no. 2, 179--195. doi:10.1305/ndjfl/1153858645. https://projecteuclid.org/euclid.ndjfl/1153858645


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References

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