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A natural theory proving Con(ZFC)
Posted:
Feb 8, 2013 7:27 AM


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
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



