[FOM] Potential infinity
Daniel Leivant
leivant at indiana.edu
Mon Mar 9 20:04:59 EDT 2015
A formalism related to potential infinity is described in
http://www.sciencedirect.com/science/article/pii/S0890540184710388
One starts there with a comprehension principle well below PRA.
Daniel
On 03/09/2015 05:40 AM, Arnon Avron wrote:
> 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.
>
> Arnon
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