ZFC

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

AllAfrica News: Latest 2009

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).

Skolem's Paradox Bays, Timothy 2009

Shouldn't the fact that M satisfies ZFC ensure that

Skolem's Paradox Bays, Timothy 2009

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

Skolem's Paradox Bays, Timothy 2009

From a proof-theoretic 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).

Skolem's Paradox Bays, Timothy 2009

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

Skolem's Paradox Bays, Timothy 2009

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

Set Theory: Constructive and Intuitionistic ZF Crosilla, Laura 2009

In particular, then, if M is a model for second-order ZFC and if mˆ

Skolem's Paradox Bays, Timothy 2009

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

Skolem's Paradox Bays, Timothy 2009

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

Skolem's Paradox Bays, Timothy 2009