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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关于派生条件的注释

我们研究可证明谓词的可推导条件版本之间的关系。我们展示了条件之间的几种含义和非含义，并且我们讨论了由可推导条件引起的一致性陈述的不可证明性。首先，我们对第二不完备定理的已知版本进行分类，并展示一些足以证明 Hilbert-Bernays 一致性陈述的新条件。其次，我们改进了 Buchholz 的可证明的示意图$\Sigma_1$-完整性。然后除其他外，我们表明希尔伯特-伯奈斯条件和 Löb 条件是相互无法比较的。我们还表明，希尔伯特-伯奈斯条件和 Löb 条件都不能完成哥德尔关于第二不完备性定理的原始陈述。