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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 14, 2013 12:27 AM
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"Shmuel (Seymour J.)Metz" wrote in message

In <>, on 03/12/2013
at 02:28 PM, "K_h" <> said:

> >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.

> There is no "everything" in modern set theory. What you have described
> applies to GBN but not to ZFC, where the universe of discourse
> includes only sets.

Contrary to what you are assuming, my remarks were not limited to ZFC. My
remarks were about mathematics generally. In the generally accepted
approach, the class of ordinals is a proper class (not a set). All of the
ordinals exist in the cumulative hierarchy, just like all sets exist in the
CH, and the cumulative hierarchy is itself a proper class. You are correct
that the discourse of ZFC does not include proper classes.

> >Controversy exists over whether or not there exists
> >non-constructible sets,

> Controversy exists over whether to allow axiom systems that permit
> non-constructible sets, but there is no controversy over whether there
> are non-constructible sets in, e.g., ZFC.

ZFC is agnostic on the issue of the existence of non-constructible sets. In
a theory like ZFC+(V=L), the theory says that only constructible sets exist.
ZFC+(V=/=L) says non-constructible sets exist. For mathematical Platonists,
there is disagreement over whether or not (V=L) is true. It is not the case
that ZFC-->(V=/=L).


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