Editing Gödel's incompleteness theorems (section)

=== The Hilbert&ndash;Bernays conditions ===

The standard proof of the second incompleteness theorem assumes that the provability predicate {{math|''Prov''<sub>A</sub>(''P'')}} satisfies the [[Hilbert–Bernays provability conditions]]. Letting {{math|#(''P'')}} represent the Gödel number of a formula {{mvar|P}}, the provability conditions say:

# If {{mvar|F}} proves {{mvar|P}}, then {{mvar|F}} proves {{math|''Prov''<sub>A</sub>(#(''P''))}}.
# {{mvar|F}} proves 1.; that is, {{mvar|F}} proves {{math|''Prov''<sub>A</sub>(#(''P'')) → ''Prov''<sub>A</sub>(#(''Prov''<sub>A</sub>(#(''P''))))}}.
# {{mvar|F}} proves {{math|''Prov''<sub>A</sub>(#(''P'' → ''Q'')) ∧ ''Prov''<sub>A</sub>(#(''P'')) → ''Prov''<sub>A</sub>(#(''Q''))}} &nbsp; (analogue of [[modus ponens]]).

There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert&ndash;Bernays conditions. Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than Peano arithmetic.