On Sun, 11 Feb 2001, John Duskin wrote:
Bill Lawvere, in contrast, noticed that when the working axioms of set-theory were rephrased in purely "category-theoretic" terms, that they, amazingly, all became "first order" statements , thereby raising the question of an entirely new way to look at foundational questions in which the pesky membership paradoxes could not arise nor even be formally expressible. He, in contrast to Grothendieck, "pushed" the much more radical move of, effectively, "banning all use of Hom-sets" and thereby made the divide crystal clear.
Can someone give a reference to an elaboration of this point of view? Since I am not a real category theorist and have at best a weak understanding of foundations, something written for the mainstream mathematician would be even better. Thanks in advance. Jim Borger