Foundations of Mathematics - Partial Resolution of The Crisis

Partial Resolution of The Crisis

Starting in 1935, the 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 among working mathematicians, due to difficulties of constructive mathematics.

We may consider that Hilbert's program has been partially completed, so that the crisis is essentially resolved, satisfying ourselves with lower requirements than Hibert's original ambitions. His ambitions were expressed in a time when nothing was clear: we did not know if mathematics could have a rigorous foundation at all. Now we can say that mathematics has a clear and satisfying foundation made of set theory and model theory. Set theory and model theory are clearly defined and the right foundation for each other.

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 don't have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF. But it is still possible to justifiy the consistency of ZF in informal ways.

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.

Toward the middle of the 20th century it turned out that set theory (ZFC or otherwise) was inadequate as a foundation for some of the emerging new fields, such as homological algebra, and category theory was proposed as an alternative foundation by Samuel Eilenberg and others.