[FOM] Potential and Actual Infinity

[John Bell]

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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