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K_h
Posts:
403
Registered:
4/12/07


Re: Antifoundation axiom
Posted:
Mar 12, 2013 5:28 PM


"CharlieBoo" wrote in message news:60eb0d021d73401199cf4b30f376a718@9g2000yqy.googlegroups.com...
> > Do you consider relationships besides sets (relations)? Right now we > have: > > A. Everything is a set. > B. x ~e x is not a set.
This is not a correct characterization of set theory. In the generally accepted approach, not everything is a set and no collection can be a member of itself. Collections are bifurcated into two types and they are sets and proper classes. So "Everything is a set" just isn't part of modern theory.
> Mathematics is supposed to be about the truth. With this freeforall > at declaring what set theory is  rather than looking at the whole
Historically, there has been controversy over what a set is. Even today the text books acknowledge that it's not an easy task to say what a set is. In the generally accepted approach, there are a list of axioms and the axioms themselves provide the best current answer as to what sets are. Look at the ZFC axioms and you will see that each one is helping to answer the question as to what a set is. (examples, axiom of pairs says that for any two sets there is a set whose members are precisely those sets) Controversy exists over whether or not there exists nonconstructible sets, whether or not there are sets of Mahlo cardinality, and so forth. Because those things are controversial, there are no axioms for them in the generally accepted theory of sets (i.e. ZFC). Therefore, there is no "freeforall" in what set theory is.
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