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June 2012 Truth definitions without exponentiation and the Σ₁ collection scheme
Zofia Adamowicz, Leszek Aleksander Kołodziejczyk, Jeff Paris
J. Symbolic Logic 77(2): 649-655 (June 2012). DOI: 10.2178/jsl/1333566643

Abstract

We prove that:

  • if there is a model of IΔ₀ + ¬exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then IΔ₀ + ¬exp + ¬ BΣ₁ is consistent,

  • there is a model of IΔ₀ + Ω₁ + ¬exp with cofinal Σ₁-definable elements, both a Σ₂ and a Π₂ truth definition for Σ₁ sentences, and for each n ≥ 2, a Σn truth definition for Σn sentences.

The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations of ∃Σb₀ sentences.

We also present an old but previously unpublished proof of the consistency of IΔ₀ + ¬exp + ¬ BΣ₁ under the assumption that the size parameter in Lessan's Δ₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of IΔ₀ + ¬exp + ¬ BΣ₁ unconditionally has turned out to be so difficult.

Citation

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Zofia Adamowicz. Leszek Aleksander Kołodziejczyk. Jeff Paris. "Truth definitions without exponentiation and the Σ₁ collection scheme." J. Symbolic Logic 77 (2) 649 - 655, June 2012. https://doi.org/10.2178/jsl/1333566643

Information

Published: June 2012
First available in Project Euclid: 4 April 2012

zbMATH: 1245.03058
MathSciNet: MR2963027
Digital Object Identifier: 10.2178/jsl/1333566643

Rights: Copyright © 2012 Association for Symbolic Logic

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Vol.77 • No. 2 • June 2012
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