The Absorption Law, or How to Kreisel a Hilbert-Bernays-L\"ob

In this paper, we show how to construct for a given consistent theory $U$ a ${\mathrm{\Sigma }}_1^0$-predicate that both satisfies the L"ob Conditions and the Kreisel Condition —even if $U$ is unsound. We do this in such a way that $U$ itself can verify satisfaction of an internal version of the Kreisel Condition.