As long as we agree on what direction gravity has inside matter, which should be to some sort of a center, it seems most probable that a black hole won't split into two. To get it to split you need to introduce some 'force', able to split it, as for example tidal forces, overwhelming the gravitational 'infinite force' inside that event horizon, splitting it. That should then be a analogue to the idea of greater and smaller infinities, acting upon each other. So you need to make the angular momentum of a black hole infinite, relative the infinity of directional gravity described inside the event horizon. And if you really want to insist on it, making the angular momentum of a greater infinity

to get to one infinity ripping another infinity apart

I know there is a theorem of lesser and greater infinity's, and that it seems logically sound, or mathematically as it is. But in a one to one correspondence, lifting out the 'bits', it then must end in a lesser infinity disappearing before a greater. Which to my eyes make for a definition in where a infinity has an end, either making a one to one correspondence meaningless as a foundation for describing a infinity, or making the ideas of infinity meaningless, as we always can introduce a greater infinity, then using a one to one correspondence to kill of the lesser, lifting 'bits' out.

Either way, using a one to one correspondence my way, it should never end, no matter what size of infinity you define. If you want there to exist infinity's? Defining it that way there is no way to define a magnitude of 'force' to a infinity, because if there was, there would be no infinity's.