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Topic: A natural theory proving Con(ZFC)
Replies: 3   Last Post: Feb 11, 2013 12:25 PM

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: A natural theory proving Con(ZFC)
Posted: Feb 8, 2013 8:27 AM
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Zuhair wrote:
> I see the following theory a natural one that proves the consistency
> of ZFC.
> Language: FOL(=,in)

How do you express Con(ZFC) in that language? I know one can encode it
using names of sets rather as one can encode Con(PA) using numerals, but
isn't it rather hard work and is your claim justified without at least
an outline?

> Define: set(x) iff Exist y. x in y
> Axioms: Identity axioms +
> (1) Extensionality: (for all z. z in x iff z in y) -> x=y
> (2) Comprehension: if phi is a formula in which x is not free then
> (Exist x. for all y. y in x iff set(y) & phi) is an axiom.
> (3) Pairing: (for all y. y in x -> y=a or y=b) -> set(x)
> (4) Hereditary size limitation: set(x) <-> Exist y. set(y) & for all m
> in x (m << y)
> (5) Simple size limitation: set(x) & y < x -> set(y)
> where relations <, << are defined as:
> x < y iff Exist z. z suclass_of y & Exist f. f:z --> x & f is a
> surjection.
> where z subclass_of y iff for all m. m in z -> m in y.
> x << y iff x < y & for all z in TC(x). z < y
> TC(x) is defined as:
> TC(x)=y iff [for all z. z in y iff (for all s. x subclass_of s & s is
> transitive -> z in s)]
> where transitive is defined as:
> x is transitive iff (for all y,z. z in y & y in x -> z in x)
> /
> In nutshell there are mainly two scenarios here essential to prove
> ZFC, that of Unique Construction of classes, and Size criteria.
> Zuhair

When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

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