> All this ado cannot veil the fact that there is no actual infinity
Then all WM's set theories must have some maximum finite set size and thus lack anything like a set of all natural numbers or an inductive set of any sort.
> Every mentioning of anything, including real numbers, belongs to a countable > set (given "countable" was a sensible notion).
While there may be only contably many mentions, there is no limit on the number of objects one can refer to in a single mentioning.
The sets mentioned in the finite expession of the Peano postulates, for example, is a 'single mentoin" which "mentions" infinitely many objects.
> If mathematics is defined, as > many assert, to be what can appear in mathematical discourse, dialogue, or > monologue, then undefinable objects do not belong to mathematics.
One can easily finitely define sets, like |N, or |Z or |N, or |R, or 2^|N, and so on, each of which is actually infinite and some of which are uncountalby infinite, as long as one is free from the corruptions of WM's wild weird world of WMytheology.
> They belong to matheology.
Then that is merely WM's habitual misspelling of "mathematics" --