```Date: Feb 9, 2013 1:49 AM
Author: Graham Cooper
Subject: Re: A natural theory proving Con(ZFC)

On Feb 9, 2:35 pm, George Greene <gree...@email.unc.edu> wrote:> On Feb 8, 7:27 am, Zuhair <zaljo...@gmail.com> wrote:>> > I see the following theory a natural one that proves the consistency> > of ZFC.>> > Language: FOL(=,in)>> > Define: set(x) iff  Exist y. x in y>> This is ALREADY NOT "natural".> In the first place, = is ELIMINABLE in extensional set theory.> That (a language with =)  is NOT the appropriate language if it is> going to be "natural".> x=y is just an abbreviation for Az[zex<->zey], or equivalently, x is a> subset of y and y is a subset of x.>> Less trivially, the limitation of size principle here IS NOT natural.> That is TOTALLY counter-> intuitive.  The definition of set you are giving here  IS THE OPPOSITE> of the NATURAL one.> If you ask anyone whose mind has not been corrupted by studying> mathematical philosophy> what A SET is, they will tell you that it is A COLLECTION.  With the> possible exception of the> empty set, sets are sets by virtue of CONTAINING things, NOT by virtue> OF BEING contained> in things!  Indeed, there are ALL KINDS of "natural" sets -- the set> of children in a family,> the set of planets, a set of plates or silverware, a set of matching> cards, AD NAUSEAM,> where the members of these sets ARE NOT sets and are therefore COUNTER-> examples> to your definition!  You will plead that these are concrete and not> mathematical objects,> but in NATURAL treatments, the abstract mathematical objects behave> ANALOGOUSLY TO> concrete mathematical objects with which non-mathematicians are> familiar.then maybe ZFC isn't your thing because all sets defined by apredicate must have a superset.   this was Zermelo's brainchild toremove the biconditional from naive set theory.http://en.wikipedia.org/wiki/Axiom_schema_of_specificationALL(SS)  EXIST(S)ALL(x)  x e S   <->  ( x e SS  &  p(x,SS, a,b,c..))although I don't like this Axiom FORMAT as it's exhaustive transitiveclosure has no utility.and ALL(p) is missing.I would write it the other way around, ALL(S) under contention mustabide with the axiom (of having some EXISTING(SS))Herc--www.BLoCKPROLOG.com
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