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

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Frederick Williams

Posts: 2,166
Registered: 10/4/10
Re: Anti-foundation axiom
Posted: Mar 13, 2013 1:14 PM
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K_h wrote:
>
> "Charlie-Boo" wrote in message
> news:60eb0d02-1d73-4011-99cf-4b30f376a718@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.


In ZF, "everything is a set" is usually true in as much as there is just
one type (or is the word "sort"?) of variable and they range over sets
and nothing else. I write "usually" because Suppes' set theory is ZF,
but he has "things" (those are scare quotes, I'm not quoting Suppes)
only some of which are sets. I don't know if Suppes calls his set
theory ZF because I don't have the book to hand at the moment.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting



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