Full Idea
A problem with type theory is that there are only finitely many individuals, and finitely many sets of individuals, and so on. The hierarchy may be infinite, but each level is finite. Mathematics required an axiom asserting infinitely many individuals.
Gist of Idea
Type theory has only finitely many items at each level, which is a problem for mathematics
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
Book Reference
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.47
A Reaction
Most accounts of mathematics founder when it comes to infinities. Perhaps we should just reject them?