[FOM] Deflationism and the Godel phenomena

[Grant Olney Passmore]

> I'm afraid that this operation of extending the induction schema to
> cover all formulae of the extended language is a manner of doing second
> order logic under first order notation. Jeffrey Ketland says that set
> theory is used, and set theory can't be reduced to first order logic
> (predicate calculus).

I don't think that anyone would argue against ZF and ZFC being first-order 
theories.  Perhaps the "second order logic under first order notation" 
you're referring to is the fact that ZF is not finitely axiomatized, and 
one might confuse the axiom scheme of comprehension with a second order 
axiom quantifying over predicates.  Just represent weak second-order 
models (with two universes: one for classes, one for sets) a la the 
first-order clauses of Von Neumann-Bernays-G\"odel Set Theory, and you 
most certainly have a first-order finite axiomatization of set theory with 
the wonderful property that a set-restricted model of vNGB is a model for 
ZF (and the converse as well).

(Alternatively, if the many-sorted nature of the universe is problematic, 
just add a predicate IsASet(x) to the language, augment the vNBG axioms 
accordingly, and you may stay in traditional/single-sorted logic).

Best,
Grant