# A note on derivability conditions

## TL;DR

This paper analyzes various derivability conditions for provability predicates, clarifies their logical relationships, and explores their implications for the second incompleteness theorem and related consistency statements.

## Contribution

It classifies known and new derivability conditions, clarifies their implications, and improves proof techniques for provable $oldsymbol{ extSigma_1}$-completeness.

## Key findings

- Hilbert--Bernays' and L"ob's conditions are mutually incomparable.
- Neither Hilbert--Bernays' nor L"ob's conditions suffice for G"odel's original second incompleteness theorem.
- New sets of conditions are identified that ensure unprovability of Hilbert--Bernays' consistency statement.

## Abstract

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 L\"ob's conditions are mutually incomparable. We also show that neither Hilbert--Bernays' conditions nor L\"ob's conditions accomplish G\"odel's original statement of the second incompleteness theorem.