Re: Question on the power set axiom



> > does a proper class of ordered
> > pairs really "count as" a function, in NBG?

Barb Knox wrote:
> Yes.

How do you know?
Seriously, it matters what you can vs. can't say
INside NBG vs. in a meta-language.
You very notoriously CAN say that something is a set
(which you can't in ZFC); there is a class of all sets.
If what you say here is true, though, then there is NOT
a class of all functions (since "functions" that are proper
classes cannot be members of that class, since they,
as proper classes, cannot be members of ANY class).
You CAN define a class (call it Func) that is the class
of all SETS of ordered pairs. But it seems to me that
members of this class (which are clearly functions) and
Those Other Bigger Things are different in some important
way, and that you don't get to just call them all by the same
name, even if they ARE all functions, from the viewpoint of
the meta-language.

There is a unary predicate-schema-over-classes
phi(c)<=df=> Ax[xec-> x is an ordered pair], with the property
that some c's satisfy phi and some don't, but translating this
as "c is a function" really seems to me to be against the
spirit of NBG, where you are SUPPOSED to be equating
classes with predicates over SETS.

.

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