Editing Foundations of mathematics (section)

== Toward resolution of the crisis ==
Starting in 1935, the [[Nicolas Bourbaki|Bourbaki]] group of French mathematicians started publishing a series of books to formalize many areas of mathematics on the new foundation of set theory.

The intuitionistic school did not attract many adherents, and it was not until [[Errett Bishop|Bishop]]'s work in 1967 that [[constructivism (mathematics)|constructive mathematics]] was placed on a sounder footing.<ref>{{citation | title = Five stages of accepting constructive mathematics | author = Andrej Bauer | journal = Bull. Amer. Math. Soc. | volume = 54 | issue = 3 | year = 2017 | doi = 10.1090/bull/1556 | page = 485 | doi-access = free }}</ref>

One may consider that [[Hilbert's program#Hilbert's program after Gödel|Hilbert's program has been partially completed]], so that the crisis is essentially resolved, satisfying ourselves with lower requirements than Hilbert's original ambitions. His ambitions were expressed in a time when nothing was clear: it was not clear whether mathematics could have a rigorous foundation at all.

There are many possible variants of set theory, which differ in consistency strength, where stronger versions (postulating higher types of infinities) contain formal proofs of the consistency of weaker versions, but none contains a formal proof of its own consistency. Thus the only thing we do not have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF.

In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of [[ZFC]], generally their preferred axiomatic system. In most of mathematics as it is practiced, the incompleteness and paradoxes of the underlying formal theories never played a role anyway, and in those branches in which they do or whose formalization attempts would run the risk of forming inconsistent theories (such as logic and category theory), they may be treated carefully.

The development of [[category theory]] in the middle of the 20th century showed the usefulness of set theories guaranteeing the existence of larger classes than does ZFC, such as [[Von Neumann–Bernays–Gödel set theory]] or [[Tarski–Grothendieck set theory]], albeit that in very many cases the use of large cardinal axioms or Grothendieck universes is formally eliminable.

One goal of the [[Reverse Mathematics|reverse mathematics]] program is to identify whether there are areas of "core mathematics" in which foundational issues may again provoke a crisis.