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Topic: Anti-foundation axiom
Replies: 11   Last Post: Mar 15, 2013 6:00 AM

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Posts: 419
Registered: 4/12/07
Re: Anti-foundation axiom
Posted: Mar 12, 2013 5:28 PM
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"Charlie-Boo" wrote in message

> 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 free-for-all
> 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 non-constructible 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 "free-for-all" in what set
theory is.


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