Journal of Symbolic Logic 85 (3):1224-1253 (2020)

We investigate relationships between versions of derivability conditions for provability predicates. We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by derivability conditions. First, we classify already known versions of the second incompleteness theorem, and exhibit some new sets of conditions which are sufficient for unprovability of Hilbert–Bernays’ consistency statement. Secondly, we improve Buchholz’s schematic proof of provable $\Sigma_1$ -completeness. Then among other things, we show that Hilbert–Bernays’ conditions and Löb’s conditions are mutually incomparable. We also show that neither Hilbert–Bernays’ conditions nor Löb’s conditions accomplish Gödel’s original statement of the second incompleteness theorem.
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DOI 10.1017/jsl.2020.33
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References found in this work BETA

Derivability Conditions on Rosser's Provability Predicates.Toshiyasu Arai - 1990 - Notre Dame Journal of Formal Logic 31 (4):487-497.
Transductions in Arithmetic.Albert Visser - 2016 - Annals of Pure and Applied Logic 167 (3):211-234.
Self-Reference and Modal Logic.[author unknown] - 1987 - Studia Logica 46 (4):395-398.

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On the Depth of Gödel’s Incompleteness Theorems.Yong Cheng - forthcoming - Philosophia Mathematica.

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Derivability Conditions on Rosser's Provability Predicates.Toshiyasu Arai - 1990 - Notre Dame Journal of Formal Logic 31 (4):487-497.
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