I don't see a distinction myself between unbounded and infinite, but the Typing system might be unbounded but it's being run on a computer with bounded memory so it becomes bounded. But that isn't a property of the Typing system, it's a property of the environment it's being executed in.
Unless you allow unbounded input there is technically a limit to the amount of memory an N-state Turing machine could possibly allocate (and still halt).
Although we don't know what that limit is, and I believe it's uncomputable.
yes, but for any given real machine with a memory limit, i can come up with a turing machine it can't simulate
When you look at the space complexity of an algorithm, it's never ‘infinity’.
Nonsense, here's an implementation (in shell, so you can easily paste and run it in your terminal) of such an algorithm:
:(){ :|:& };:
It's a ... uh ... a recursive parallel processing algorithm. Yeah.