User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-30279966-20200118174447/@comment-35470197-20200118233942

The terminology of "computable number" in googology is different from that in mathematics. The latter one is a property of a real number formalised in mathematics, while the former one is not.

I think that when we refer to the computability of a natural number, then we are talking about the defining formula rather than the resulting number itself. Roughly speaking, I think that we are considering the following ideas:
 * 1) Every defining formula of a natural number x given by x=n, where n is an explicitly written decimal expression is a "valid" defining formula of a comutable number.
 * 2) Every defining formula of a natural number x given by x=f(n) equipped with "valid" defining formulae of f and n are "valid" formulae of a computable number. ("Equipped with" means "a tuple of".)
 * 3) Every definig formula of a computable function f given by the code equipped with a "valid" defining formula of the code is a "valid" formula of a computable function.

In that case, there can be infinitely many "valid" defining formulae, while there are finitely many "valid" definig formula which have been essentially written down in the world.