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  text: '[summary: \nA provability predicate of a theory $T$ is a formula $P(x)$ with one free variable $x$ such that:\n\n1. If $T\\vdash S$, then $T\\vdash P(\\ulcorner S \\urcorner)$\n2. $T\\vdash P(\\ulcorner A\\rightarrow B \\urcorner)\\rightarrow (P(\\ulcorner A \\urcorner)\\rightarrow P(\\ulcorner B \\urcorner))$\n3. $T\\vdash P(\\ulcorner S \\urcorner)\\rightarrow P(\\ulcorner P(\\ulcorner S \\urcorner) \\urcorner)$\n]\n\nA provability predicate is a formula $P$ of a theory satisfying the Hilbert-Bernais derivability conditions. If the diagonal theorem is applicable in the theory as well, then [55w] and [ Gödel's second incompleteness theorem] are provable for $P$.\n\nThe Hilbert-Bernais derivability conditions are as follows:\n\n1. (**Necessitation**) If $T\\vdash S$, then $T\\vdash P(\\ulcorner S \\urcorner)$\n2. (**Provability of modus ponens / distributive axioms**) $T\\vdash P(\\ulcorner A\\rightarrow B \\urcorner)\\rightarrow (P(\\ulcorner A \\urcorner)\\rightarrow P(\\ulcorner B \\urcorner))$\n3. (**Provability of renecessitation**) $T\\vdash P(\\ulcorner S \\urcorner)\\rightarrow P(\\ulcorner P(\\ulcorner S \\urcorner) \\urcorner)$\n\nThe derivability conditions are tighlty related to the axioms and rules of inference of [-534]. In fact, the normal systems of provability are defined as those that have necessitation as a rule and the distributive axioms. %%note:They also have to be closed under substitution%% On the other hand, D3 is the axiom that defines the system [ K4], and it is also satisfied by [GL].\n\n##Examples\n\nThe **verum** formula $x=x$ trivially satisfies the derivability conditions.\n\nThe [-5gt] of arithmetic is a provability predicate.',
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