[FOM] Finite axiomatizations extending ZFC

Ali Enayat ali.enayat at gmail.com
Fri May 9 13:28:59 EDT 2008

This is a reply to the following query of J.Pax:

"Can someone give an example of a *finite* set of sentences T (in the
language { \epsilon })
with consequences (necessarily strictly) containing all instances of axioms
of ZFC ?
No schemas are allowed, and nothing like 0=1, of course.
The lesser the total number of symbols in T is, the better."

ANSWER 1: There is no such theory T, unless ZFC is inconsistent.

This was first proved by Montague in the late 1950's, as a corollary of his
Reflection Theorem, and Godel's Second Incompleteness Theorem.

[The Reflection Theorem shows that ZF, as well as any extension of ZF by
finitely many axioms, proves the formal consistency of each of its own
finitely axiomatized subtheories].

ANSWER 2: There are finitely axiomatizable theories T in the usual language
of set theory {epsilon) that *interpret* ZFC.

For example, one can choose T to be GBC + Con(GBC).

Here GBC is Godel-Bernays theory of classes with the class form of the axiom
of choice, and Con(GBC) is the arithmetical statement asserting the formal
consistency of GBC.

Note that GBC can be formulated in the usual language of set theory
{epsilon} [some expositions of GBC opt for a richer language in which the
class/set distinction is built-in].

It is well-known that (1) GBC is finitely axiomatizable, and (2) GBC can
(faithfully) interpret ZFC.

Best regards,

Ali  Enayat
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