Re:Set Theory and Foundations of Mathematics
by Shaniqua on Friday July 31, 2015 @11:52AM
(#9982465)
It's no big deal if you don't admit universal set or anything other than ZFC.
Re:Set Theory and Foundations of Mathematics
by johntb86 on Friday July 31, 2015 @03:05PM
(#9983663)
It took 16 years (1901-1917) to get from russell's paradox to a set theory which didn't allow it, but which was able to create a lot of interesting and useful sets (ZF). So it seems like a big deal. And we still can't talk about "the set/collection/whatever of all sets" in the language of ZFC, so we're still missing out.
If V = {x | x not in x}, then if V contains itself, V is not in V (and vice versa) is an obvious contradiction. Your new set S doesn't help in the slightest.
Fixing it is emphatically "not easy" and mathematicians generally rely on the ZFC axiomation (although several other possibilities were proposed).