I think "recursion" simpy said, allows for algorithms which end as much as "not ending algorithms".

but in practice, an algorithm has gotta end, otherwise it's not very useful. I think some academics would go as far as insisting a function or process which never ends is not even a proper "algorithm"; but I digress.

recursion does allow us to "reach the infinite"; however, philosophically, we can only ever grasp the finite.