关于可导性条件的注记
A note on derivability conditions
摘要：
我们研究可证明性谓词的可导性条件的版本之间的关系。我们给出了条件之间的几种蕴涵和非蕴涵关系，并讨论了由可导性条件引起的相容性陈述的不可证明性。首先，我们对第二不完全性定理的已知版本进行分类，并展示了一些新的条件集，这些条件足以证明希尔伯特-伯奈斯一致性陈述的不可证明性。其次，我们改进了Buchholz关于可证$Σ_1$ -完备性的图解证明。然后，除其他事项外，我们表明，希尔伯特-伯奈斯的条件和勒布的条件是相互不可比较的。我们还表明，无论是希尔伯特-伯奈斯的条件，也没有Löb的条件完成哥德尔的第二不完全性定理的原始声明。
英文摘要：
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 $Σ_1$-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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