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Re: A natural theory proving Con(ZFC)
Posted:
Feb 9, 2013 1:49 AM


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 nonmathematicians 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



