Did Zermelo in 1914 miss the fact, known in 1905 already, that there are only countably many finite strings, for instance polynomials with defined parameters? He wrote: Every real or complex number $\alpha$ satisfies at least one algebraic equation of the form
$g(x, \eta) = g(x, \eta_1, \eta_2, \eta_3, ..., \eta_t) = 0$
where $g$ is a polynomial with integer coefficients, and $\eta_1, \eta_2, \eta_3, ..., \eta_t$ are basis numbers (a Hamel basis), and the largest power of $x$ is at least 1.
[E. Zermelo: "Über ganze transzendente Zahlen", Math. Annalen 75 (1914) pp. 434-442]
In 1920 Skolem introduced his "normal form ", showed that every satisfiable well formed formula of first order predicate calculus has a satisfiable Skolem normal form (and vice versa), and improved and generalized the proof of Löwenheim's theorem: Every proposition in normal form either is a contradiction or is already satisfiable in a finite or denumerably infinite domain.
"Zermelo regarded Skolem's position as a real danger for mathematics and, therefore, saw 'a particular duty' to fight against it. [...] His remedy consisted of infinitary languages. [...] Skolem had considered such a possibility, too, but had discarded it because of a vicious circle." [Heinz-Dieter Ebbinghaus: "Ernst Zermelo, an approach to his life and work", Springer (2007) p. 200ff]
Additional question relevant for the main question: What is the first paper where Zermelo explicitly considers uncountable alphabets?