[FOM] Potential and Actual Infinity

[katzmik at macs.biu.ac.il]

John,

Thanks for this interesting contribution.  Perhaps you could mention that the
arrows are inclusion maps rather than identity maps.  If "inclusion" does not
make sense in this topos, still a different term seems appropriate for the
arrows. MK

On Thu, March 12, 2015 01:31, John Bell wrote:
> A nice way of representing potential infinity is to allow sets to undergo
> explicit variation over time, as in the topos E of of sets varying over the
> natural numbers. The objects of this topos are all sequences of maps between
> sets
>
> A0 –-> A1 –>  A2 –> .....An –> ....
>
> Such an object may be thought of as a set A “varying over discrete time: An is
> the “state” of A at time n.
> Now consider the temporally varying set , call it K,
>
>
> {0} –-> {0, 1} –> {0,1,2}  -->  ...  --> {0,1,2, ..., n}  --> ...
>
> in which all the arrows are identity maps. In E, K “grows” indefinitely and
> hence potentially infinite. On the other hand at each specific time K’s s
> state  is finite and so K is not actually infinite. In short, in E, K is
> potentially, but not actually infinite..
>
> In the internal logic of E, K fails be finite in that it  is not equipollent
> with any natural number. On the other hand  K is not transfinite in that the
> set of natural numbers cannot be injected into it.
>
> -- John Bell
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