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

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Posts: 8,833
Registered: 1/6/11
Re: Anti-foundation axiom
Posted: Mar 9, 2013 7:58 PM
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In article
Charlie-Boo <> wrote:

> On Mar 6, 7:59 am, Zuhair <> wrote:
> > The following theory Violates foundation in preference to somewhat
> > plausible
> > axioms.
> >
> > ZF - foundation
> > +
> > For all x. Exist H(x)
> > +
> > for all x. x subnumerous to H(x)
> > +
> > Anti-Foundation Axiom: An infinite Dedekindian finite set exists.
> >
> > Where H(x) is the set of all sets hereditarily subnumerous to x.
> >
> > Zuhair

> You can't really have axioms for set theory. Axioms are a way to
> formalize our understanding of a subject e.g. number theory or
> geometry. They are so well developed and understood that we try to
> codify our great understanding with a formal system.
> But how well do we understand sets? There are a dozen versions of set
> theory. You end up with conflicting theorems from one system to
> another!! Can you imagine 2 different versions of Geometry

I can imagine at least 2 versions of geometry, Euclidean and at least
two kinds of non-Euclidean, for example.

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