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A NOTE ON DERIVABILITY CONDITIONS
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 3 / September 2020
- Published online by Cambridge University Press:
- 07 September 2020, pp. 1224-1253
- Print publication:
- September 2020
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- Article
- Export citation
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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.
