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Truth definitions without exponentiation and the Σ1 collection scheme

  • Zofia Adamowicz (a1), Leszek Aleksander Kołodziejczyk (a2) and J. Paris (a3)


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We also present an old but previously unpublished proof of the consistency of IΔ0 + ¬exp + ¬BΣ1 under the assumption that the size parameter in Lessan's Δ0 universal formula is optimal.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.



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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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