Vaughan writes: An awful lot of mathematicians are under the impression that their definitions are grounded in set theory. Note, please, that I was writing about _core_ mathematics. There's an aperture problem here. What do we take as the universe of "mathematicians"? If we stick to those parts of mathematics as described by the US National Science Foundation as "core mathematics" then I will stand by my statement. A semantics is unneeded. I must agree, of course, that there have been proofs in core mathematics that have used structures for which the semantics became questionable. Because of their technical nature (that is, because they were needed in proofs, not theorems) consistency arguments -- in lieu of clear semantics -- were acceptable. There's a reasonably clear historical argument in favor of ZF for purposes of such arguments. (One example: In what John Thompson described as his best work he proved that certain finite groups -- such as the Monster -- appear as the Galois groups of number fields; in his proof he used the existence of infinitely many automorphisms of the complex numbers. Whether or not your semantics allows more than two automorphisms of the complex numbers, there's no question that the consistency of ZFC implies that the groups in question do appear as the Galois groups of number fields.)