[FOM] From theorems of infinity to axioms of infinity
paul at mtnmath.com
Mon Mar 18 12:31:33 EDT 2013
On 03/15/2013 01:23 PM, 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.
Set theory as an interesting formalism can be separated from its
"metaphysical extravagance" at least in the mind of Paul J. Cohen who
derived part of these independence results.
"Does set theory, once we get beyond the integers, refer to an existing
reality, or must it be regarded, as formalists would regard it, as an
interesting formal game? ... Through the years I have sided more firmly
with the formalist position. This view is tempered with a sense of
reverence for all mathematics which has used set theory as a basis, and
in no way do I attack the work which has been done in set theory." From
Paul J. Cohen, "Skolem and Pessimism about Proof in Mathematics"
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