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Truth definitions without exponentiation and the Σ1 collection scheme

Published online by Cambridge University Press:  12 March 2014

Zofia Adamowicz
Affiliation:
Mathematical Institute of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland, E-mail: zosiaa@impan.pl
Leszek Aleksander Kołodziejczyk
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, E-mail: lak@mimuw.edu.pl
J. Paris
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK, E-mail: jeff.paris@manchester.ac.uk

Abstract

We prove that:

  1. • 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,

  2. • 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 Σ n truth definition for Σ n sentences.

The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ1 sentences into boolean combinations of 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We also present an old but previously unpublished proof of the consistency of IΔ0 + ¬exp + ¬BΣ1 under the assumption that the size parameter in Lessan's Δ0 universal formula is optimal.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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2012

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References

[AKZ03] Adamowicz, Z., Kołodziejczyk, L. A., and Zbierski, P., An application of a reflection principle, Fundamenta Mathematicae, vol. 180 (2003), pp. 139159.CrossRefGoogle Scholar
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