Kevin Karn wrote: ... > (In terms of the theory of computation, I would state the position this > way: There are no infinite tapes. Therefore, it is sufficient to study > finite automata. This doesn't mean we have to establish a certain m, > which is the maximum size of any possible FA. It just means we don't > concern ourselves with impossible things, like infinite tapes. You can > reject infinity without specifying m. You just ignore them both. What > this looks like to the casual observer is a scientist simply studying > real machines, and avoiding metaphysical tangents.)
The problem with limiting ourselves to finite automata is that it forces us to deal with running out of memory when we just want to look at time complexity without considering space complexity.
I think theory of computation should provide easy-to-analyze abstractions, so that we don't have to deal with all the complexities of real machines at the same time.
Are you assuming there is an unspecified bound on the natural numbers, or that the natural numbers are unbounded but we are not allowed to mention or think about the set of all natural numbers?