[FOM] Potential infinity

Arnon Avron aa at tau.ac.il
Mon Mar 9 05:40:41 EDT 2015

On Fri, Mar 06, 2015 at 06:11:45PM -0500, Timothy Y. Chow wrote:
> There is of course a long tradition in philosophy of distinguishing
> between "potential infinity" and "actual infinity."  In modern
> mathematics, this distinction doesn't seem to exist.  

I disagree.

The terminology used by many might indeed have changed, but not the need to
distinguish between the two types of infinity. Thus in set theory we
do distinguish between infinite collections like N, which are objects
themselves, and so are taken as "complete", and proper classes like the
universe of ZF (whatever this means) which is not a "complete"
object and is "absolutely infinite" (in Cantor's terminology). So instead
of talking about potential versus actual infinity, platonists speak now
about non-absolute versus absolute infinity, and push the limit
strongly higher. Still, I believe that many (perhaps most) mathematicians
view V not as a well-determined object, but as a potential collection
of actual objects (exactly like the way most predicativists see N).

I take this opportunity to note about the false claim that quantifying 
over N in general, and PA in particular, are justified only if one
accepts N as a complete object which is actually infinite.
According to this "logic", the quantification
that is done in ZF over the universe of sets  is
justified  only if one accepts V as a complete object which is absolutely infinite.

As I have hinted in my previous posting, for seeing  the validity of the axioms of PA 
a potential understanding of the collection of the natural numbers suffices.


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