[FOM] Re: AXIOM SCHEMATA
Dmytro Taranovsky
dmytro at mit.edu
Wed Jul 14 17:06:24 EDT 2004
Matthew Frank wrote:
> Given that we generally prefer finitely axiomatized theories to infinitely
> axiomatized theories, why do we tend to use ZF instead of NBG? --Matt
Unlike NBG, ZFC is reflexive over predicate calculus, that is for every formula
phi with one free variable, ZFC proves "for all finite ordinals n, if phi(n) is
provable, then it is true". ZFC also proves a much stronger reflection
principle: For every formula phi with one free variable, ZFC proves that for a
proper class of ordinals kappa, for all x in V(kappa), phi(x) is true in
V(kappa) iff it is true in V.
A reflexive (over predicate calculus) theory is not finitely axiomatizable,
so one needs an infinite schema of axioms to fully exploit the language of the
theory.
Also, most mathematicians regard some statements in the language of NBG as
meaningless because proper classes do not actually exist; they are used as a
figure of speech.
Dmytro Taranovsky
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