> No. Zermelo's AC requires that one name can be written > with certainty.
This statement is not Zermelo's original statement. It can be proven to hold, iff it was possible to choose, in practice, one element from every subset of T. If this was do-able.
There have been many mathematicians criticizing Zermelo's axiom (Borel, Peano, Poincaré and others). Zermelo discusses a lot of objections in another 1908 paper. And the most amazing fact is, that at that time none of the arguments aims at the fact, that there are only countably many choices possible by theoretical reasons.
Zermelo agrees that the AC is not provable. He did not know, at that time, that it is disprovable by theoretical mathematics.