Can we use a Turing machine with infinite tape as a basis to prove anything disregarding the fact that such a thing can never exist? Do we have the right to regard a machine (a construct) in the same way we regard a number or a set (an abstract)?
If yes could I similarly use a Turing angel with infinite scroll instead, would that be ok?
EDIT: since I cannot comment.
This question is somewhat different from infinity in mathematics because a set even infinite can have a finite representation. On the other hand there is no finite representation for an infinite tape That is the algorithm that generates the output.
Computation requires resources, that means that it takes place in a time-space, not in an abstract timeless dimension. While it could be ok to draw on demand more time for the computation pushing on to infinity, space has a cap on the amount of information it can hold, and before reaching that point the tape would revert to write only, that is implode into a black hole. So even if an algorithm is "finite" its output could be big enough to become "unreadable". For example lets say we have a machine that counts all numbers at some Number H when trying to add more information on the tape the tape colapses forms a black hole and number H+1 becomes unreadable. Counting up to H+2 is a finite process but cannot be completed. One could argue that space and time restrictions should not concern us but I m not convinced that is true when dealing with objects embedded in spacetime
I could accept a definition of a machine with sufficiently large tape. But I think that the size does matter. Can for example 2 Turing machines with different tape-size considered equivalent?