Within certain parameters, that's right.

And that probably makes as much sense (if not more) as the alternative for all I know. There is a way to think of mathematics as corresponding to computation in a way where anything that cannot be computed cannot be proven. If you cannot "construct" or "compute" infinity, then you cannot say that it exists.

I think what I am describing corresponds to the "constructivist" view of mathematics, but tbh I don't know enough about this to say whether the framework allows for infinities.