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 a

predicate must have a superset. this was Zermelo's brainchild to

remove the biconditional from naive set theory.

http://en.wikipedia.org/wiki/Axiom_schema_of_specification

ALL(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 transitive

closure has no utility.and ALL(p) is missing.

I would write it the other way around, ALL(S) under contention must

abide with the axiom (of having some EXISTING(SS))

Herc

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