Editing Gödel's incompleteness theorems (section)

=== Consequences for logicism and Hilbert's second problem ===

The incompleteness theorem is sometimes thought to have severe consequences for the program of [[logicism]] proposed by [[Gottlob Frege]] and [[Bertrand Russell]], which aimed to define the natural numbers in terms of logic.{{sfn|Hellman|1981|pp=451–468}} [[Bob Hale (philosopher)|Bob Hale]] and [[Crispin Wright]] argue that it is not a problem for logicism because the incompleteness theorems apply equally to first-order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.

Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to [[David Hilbert]]'s [[Hilbert's second problem|second problem]], which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see "[[Hilbert's second problem#Modern viewpoints on the status of the problem|Modern viewpoints on the status of the problem]]").