infinity

Infinity vs unboundedness

You can have unboundedness in the sense of a computation that doesn’t stop yielding results. But you cannot take the last result of such a computation, you cannot have a computation that relies on knowing the last digit of pi before it goes to the next step → You cannot have infinity in that sense, where inifnities are about the conclusions of functions. Unboundedness means that you always get something new, unexpected, that you couldn’t have predicted from before (relation to open-endedness & computational irreducibility?).

If there was any way to get the result of an infinite computation, it could not be expressed in any mathematical language that doesn’t have contradictions (incompleteness theorems).
→ We can only build languages in which we have to assume that infinities cannot be built. Infinity is meaningless because we cannot make it.

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There are no infinities in the known universe.
Unbounded / unlimited on the other hand simply means that there is always more.
→ There can be no universal computer, because they need unlimited–potentially infinite–storage.

The only thing that is possible are finite state automata.