I can illustrate the question better in specific context:

Axiom for the existence of successor operation that is capable of "counting" the set of natural numbers. Is there some phenomenon that corresponds to counting as such. (A measure for other processes of engineering significance.)

In the Turing machine, the computation can evolve at each step by subsuming further finite amounts from inexhaustable pool of memory. How should I understand the "memory" metaphor here? Is this something technological, natural, or conceptual? For example, the domain of our algorithmic ideas (e.g. addition of positional numerals can be applied to arbitrarily large numbers in principle).

In denotational semantics of recursive programs, a certain rule (operator) extends each operation from a finite domain to a slightly larger finite domain. The transition into infinity is straightforward mentally, as generalization of the finite cases, but what are the technical provisions? I mean, computers are finite-state automaton, and the construction of such operator is only conceptual there.

In summary. What is the contemporary point of view on the subject?

- That our computations might actually scale indefinitely, since countable infinity is conceivable in practice

- That our computational methods and algorithms provide us only with convenient approximations of solutions to problems that may arise, albeit knowing that nothing in our social, mental, or physical reality is countably infinite

(I am a programmer.)

Thanks for your attention