In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays, are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224). These conditions are used in many proofs of Kurt Gödel's second incompleteness theorem. They are also closely related to axioms of provability logic. The conditions Let be a formal theory of arithmetic with a formalized provability predicate , which is expressed as a formula of with one free number variable. For each formula in the theory, let be the Gödel number of . The Hilbert–Bernays provability conditions are: If proves a sentence then proves . For every sentence , proves proves that and imply Note that is predicate of numbers, and it is a provability predicate in the sense that the intended interpretation of is that there exists a number that codes for a proof of . Formally what is required of is the above three conditions. In the more concise notation of provability logic, letting denote " proves " and denote : Use in proving Gödel's incompleteness theorems The Hilbert–Bernays provability conditions, combined with the diagonal lemma, allow proving both of Gödel's incompleteness theorems shortly. Indeed the main effort of Godel's proofs lied in showing that these conditions (or equivalent ones) and the diagonal lemma hold for Peano arithmetics; once these are established the proof can be easily formalized. Using the diagonal lemma, there is a formula such that . Proving Godel's first incompleteness theorem For the first theorem only the first and third conditions are needed. The condition that is ω-consistent is generalized by the condition that if for every formula , if proves , then proves . Note that this indeed holds for an -consistent because means that there is a number coding for the proof of , and if is -consistent then going through all natural numbers one can actually find such a particular number , and then one can use to construct an actual proof of in . Suppose T could have proven . We then would have the following theorems in : (by construction of and theorem 1) (by condition no. 1 and theorem 1) Thus proves both and . But if is consistent, this is impossible, and we are forced to conclude that does not prove . Now let us suppose could have proven . We then would have the following theorems in