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Topic: A natural theory proving Con(ZFC)
Replies: 1   Last Post: Feb 9, 2013 1:49 AM

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Graham Cooper

Posts: 4,280
Registered: 5/20/10
Re: A natural theory proving Con(ZFC)
Posted: Feb 9, 2013 1:49 AM
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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
--

www.BLoCKPROLOG.com



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