[FOM] From theorems of infinity to axioms of infinity

[Nik Weaver]

Monroe Eskew wrote:

> Set theory is the only branch of mathematics currently capable of 
> addressing classical questions which turned out to be independent such 
> as:
> 1) Is the continuum hypothesis true?
> 2) Is the axiom of choice needed to build a nonmeasurable set of reals?
> 3) Can there be a probability measure on R which measures all subsets?
>
> These are things we do care about (if "we" means the general 
> mathematical community when these questions were asked classically). 
> This should serve as justification for using set theory and all its 
> metaphysical extravagance.

Monroe,

You seem to be confusing working within ZFC and reasoning about ZFC.
You cite relative consistency results *which are theorems of Peano
arithmetic* as evidence that *we should use set theory*.  For
instance, question number (2) certainly did not "turn out to be
independent", it is a theorem of PA that if inaccessible cardinals
are consistent then the axiom of choice is needed to build nonmeasurable
sets.

But how on earth can the fact that various questions *cannot* be
answered within standard set theory, be a reason for using set theory?

This phenomenon, where seemingly fundamental questions like the
continuum hypothesis are independent of ZFC, should rather raise
suspicions that something is wrong with set theory as a foundational
system.  All the more so when these seemingly fundamental questions
turn out to have little or no relevance to mainstream mathematical
concerns.

Nik