User blog comment:Ubersketch/The Ultrafinitists are Right!!! Mohamed Ababou/@comment-35470197-20181221025646

The notion of a number in mathematics is related to the notions of a "number" in natural language, daily counting by human being, meta-theory, and ultrafinitism, but is not actually the same one. It is just an axiomatic formalisation using strings called formulae of meta-theoretically finite length.

Since the majourity of googology is concerned to numbers in mathematics, it is irrelevant to the "logic" which you refer to.