Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun set theory
Initialism of ZermeloFraenkel set theory with Choice, the standardaxiomatization ofset theory , including theaxiom of choice .
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Examples

"Banks, CBZ in particular, and fertilizer companies such as ZFC and Windmill should be complimented for their efforts in assisting wheat growers.

Here, the fact that this model satisfies ZFC is supposed to ensure that it satisfies all of the theoretical constraints which come from set theory itself, while the richness of ZFC ensures that the model also has the resources to code up our best scientific theories (and thereby to satisfy all of the theoretical constraints which come from natural science).

Shouldn't the fact that M satisfies ZFC ensure that

Then, as noted in section the Transitive Submodel Theorem says that if we start with any transitive model of ZFC, then we can find a transitive model whose domain is countable

From a prooftheoretic standpoint, for example, there is a difference between unrelativized quantification and quantification which has been explicitly relativized to some formula in our language (where this formula is one that, from an intuitive perspective, serves to “pick out” the domain of countable model of ZFC).

(For convenience, this entry will focus on the case where T is ZFC, but any standard axiomatization of set theory would do equally well.)

One usually considers ZFC (the axiom system ZF plus the axiom of choice) as a foundation for mathematics (see the entry on set theory).

In particular, then, if M is a model for secondorder ZFC and if mË

Suppose, then, that M is a countable transitive model of ZFC.

And, as we noted in section 2, secondorder versions of ZFC do not give rise to Skolem's Paradox.
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