[FOM] From theorems of infinity to axioms of infinity
Monroe Eskew
meskew at math.uci.edu
Fri Mar 15 16:23:33 EDT 2013
On Mar 15, 2013, at 10:57 AM, Nik Weaver <nweaver at math.wustl.edu> wrote:
> If
> the standard of judgement is based on formalizing normal mathematics,
> then we are better served by the more elementary systems which are
> fully adequate to this task, involve coding machinery which is no
> more complicated than what we encounter in ZFC, and avoids the
> metaphysical extravagence of ZFC which has no clear philosophical
> justification and merely makes room for pathological structures we
> don't care about.
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.
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