[FOM] Potential and Actual Infinity
katzmik at macs.biu.ac.il
katzmik at macs.biu.ac.il
Thu Mar 12 04:55:21 EDT 2015
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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