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.
Let T be a formal theory of arithmetic with a formalized provability predicate Prov(n), which is expressed as a formula of T with one free number variable.
The Hilbert–Bernays provability conditions are: Note that Prov is predicate of numbers, and it is a provability predicate in the sense that the intended interpretation of Prov(#(φ)) is that there exists a number that codes for a proof of φ.
Formally what is required of Prov is the above three conditions.
In the more concise notation of provability logic, letting
: 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.
Note that this indeed holds for an ω-consistent T because Prov(#(φ)) means that there is a number coding for the proof of φ, and if T is ω-consistent then going through all natural numbers one can actually find such a particular number a, and then one can use a to construct an actual proof of φ in T. Suppose T could have proven
But if T is consistent, this is impossible, and we are forced to conclude that T does not prove
But if T is consistent, this is impossible, and we are forced to conclude that T does not prove
Using Rosser's trick, one needs not assume that T is ω-consistent.
This follows from the fact that for every formula φ, Prov(#(φ)) holds if and only if ProvR holds.
An additional condition used is that T proves that ProvR(#(φ)) implies ¬ProvR(#(¬φ)).
This condition holds for every T that includes logic and very basic arithmetics (as elaborated in Rosser's trick#The Rosser sentence).
Using Rosser's trick, ρ is defined using Rosser's provability predicate, instead of the naive provability predicate.
The first part of the proof remains unchanged, except that the provability predicate is replaced with Rosser's provability predicate there, too.
The second part of the proof no longer uses ω-consistency, and is changed to the following: Suppose T could have proven
But if T is consistent, this is impossible, and we are forced to conclude that T does not prove
We assume that T proves its own consistency, i.e. that: For every formula φ: It is possible to show by using condition no.