# [FOM] The semantics of set theory (Set Theology)

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Wed Oct 2 21:38:42 EDT 2002



>From: JoeShipman at aol.com
>Thu Oct  3 02:54:19 2002

>Let's expand the language of set theory to include a constant
>/kappa, and the axiom scheme
>phi iff V_/kappa satisfies phi
>, for ALL sentences phi in the original language
>(not including the symbol /kappa)..

Nelson 1977 refers to this method as well known and typical,
I wonder does anybody know an original reference?

Anyway, let us call the theory ZFCVk, it contains all ZFC
axioms (which may or may not contain \kappa, does not matter)
together with the equivalence scheme as described above.
This is easily a conservative extension of ZFC: anything
\kappa-free provable in ZFCVk is provable in ZFC alone.

>/kappa shall henceforth be referred to as an "infallible cardinal"

>If an infallible cardinal exists

This is meaningless. In ZFCVk it "exists", of course,
in ZFC such a concept just canot be defined.

Generally, there is no way to define ZFC-truth other than to
extend the language of ZFC.
Three typical methods are known.
First, as ZFCVk.
Second, add the truth predicate (another atomic unary predicate)
with appropriate supporting axioms.
Third, consider a second-order impredicative theory of classes.
Of those, the first method is of somewhat different nature than
the other two.