[FOM] From theorems of infinity to axioms of infinity

Nik Weaver nweaver at math.wustl.edu
Sat Mar 23 20:16:54 EDT 2013

Tim Chow wrote, quoting me:

> >So if you're wedded to the language of set theory I don't object.
> >Personally I feel that if one is working at this level of specificity
> >then third order arithmetic is a little nicer.
>  The way you phrased this makes it sound like sets can be dispensed with 
> in favor of pure arithmetic.

Come again?  How do you get that??

I explained that ordinary mathematics can be developed in a predicativly
valid way, either using the language of pure set theory (with variables
for sets), or using the language of third order arithmetic (with variables
for numbers, sets of numbers, and classes of sets of numbers).  How does
that "sound like" I am saying sets can be dispensed with in favor of
pure arithmetic?

> Your arguments would make more sense if you were to direct your 
> criticism against ZFC in particular.  As it stands, you are making 
> blanket statements against "set theory" in general that amount to 
> shooting yourself in the foot.

I'm not sure if you've been following the previous discussion, but
I am arguing that the power set axiom is philosophically dubious and
not needed for ordinary mathematics.  I don't have any idea what you're
arguing about.


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