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Virgil
Posts:
8,833
Registered:
1/6/11


Re: Antifoundation axiom
Posted:
Mar 9, 2013 7:58 PM


In article <60eb0d021d73401199cf4b30f376a718@9g2000yqy.googlegroups.com>, CharlieBoo <shymathguy@gmail.com> wrote:
> On Mar 6, 7:59 am, Zuhair <zaljo...@gmail.com> 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) > > + > > AntiFoundation 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 nonEuclidean, for example. 



