[FOM] From theorems of infinity to axioms of infinity

[Nik Weaver]

Mart Dowd wrote:

> It is necessary to have the set of all functions from the reals to
> the reals, even for calculus; then subsets such as the continuous
> functions,and differentiable functions can be defined.

Mart, I don't agree that you need all functions, including crazy
non-measurable functions, to do calculus.  More to the point, you
don't need "all functions from R to R" *to be a set*.

> This illustrates that the power set of a set is a self-evident and
> useful concept;

I believe that people think the power set axiom is self-evident
because they lack a sharp conceptual distinction between sets and
proper classes.  Once one understands that the defining feature of
sets is surveyability, it is not at all self-evident that power
"sets" preserve this property.  I spelled out this objection in
my last message to you, and to be fair, I don't think you've really
addressed it.

Nik