Truth definitions without exponentiation and the Σ1 collection scheme
| Autor: | Leszek Aleksander Kołodziejczyk, Jeff B. Paris, Zofia Adamowicz |
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| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | The Journal of Symbolic Logic. 77:649-655 |
| ISSN: | 1943-5886 0022-4812 |
| DOI: | 10.2178/jsl/1333566643 |
| Popis: | 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translation of Σ1 sentences into boolean combinations of sentences.We also present an old but previously unpublished proof of the consistency of IΔ0 + ¬exp + ¬BΣ1 under the assumption that the size parameter in Lessan's Δ0 universal formula is optimal.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. |
| Databáze: | OpenAIRE |
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