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?


2 Answers 2


The assumption on which this question is based is incorrect, so, no, Zermelo did not miss this. The set of basis numbers, as defined by Zermelo, is itself uncountable, and the set of finite strings over an uncountable alphabet is of course uncountable as well.

  • $\begingroup$ Please do not post comments as answers. $\endgroup$
    – Conifold
    Commented Apr 18, 2018 at 1:54
  • 3
    $\begingroup$ @Conifold If you consider this as a comment, rather than an answer, then there is no answer, so the question should be deleted. $\endgroup$
    – Uwe
    Commented Apr 18, 2018 at 5:21
  • $\begingroup$ @Uwe: If the set of basis numbers is uncountable, then it is impossible to choose most of them. Hamel in 1905 did not yet know that, but in 1914 it should have become general knowledge. Re Uncountable alphabets: By what should the letters of uncountable alphabets be defined? Certainly not by finite strings over countable alphabets! But if uncountable alphabets are accepted by what "reasons" ever, then finite strings are not required at all. Simply choose a "letter" for everything that you want to express. (Of course it would be easier to delete my question than to answer it.) $\endgroup$
    – Franz Kurz
    Commented Apr 18, 2018 at 7:55
  • $\begingroup$ That the question should be closed is also a comment and could have been added to the "answer" remark. $\endgroup$
    – Conifold
    Commented Apr 18, 2018 at 23:27

The above equation has as parameters the so-called "basis numbers" $\eta$ belonging to $\Omega$, where $\Omega$ is an arbitrary well-ordering of [the continuum] $\mathfrak c$.

Regarding Zermelo's "infinitary" logic, see :

Proceeding from the assumption that it should be possible to represent all mathematical concepts and theorems by means of a fixed finite system of signs, we inevitably run into the well-known “Richard antinomy” already in the case of the arithmetical continuum. [...] a healthy “metamathematics”, a true “logic of the infinite”, will only become possible once we have definitively renounced the assumption characterized above, which I would like to call the “finitistic prejudice”. Mathematics, generally speaking, is not really concerned with “combinations of signs”, as some assume, but with conceptually ideal relations among the elements of a conceptually posited infinite manifold.

See also :

  • $\begingroup$ Thank you. But can you also determine the step when Zermelo first talked about uncountable alphabets? I am not sure whether he did at all. His Göttingen colleague (via Hilbert) Wilhelm F. Ackermann wrote as late as 1950, three yers before Zermelo's death: "The reviewer however cannot follow the author when he speaks of the possibility of a more than countable set of primitive symbols since such a system of names cannot exist." [W. Ackermann: "Review of Leon Henkin: 'The completeness of the first-order functional calculus'", J. Symbolic Logic 15,1 (1950) p. 68] $\endgroup$
    – Franz Kurz
    Commented Apr 18, 2018 at 11:29
  • $\begingroup$ @Wilhelm - as said, Zermelo rejects the "finitistic" symbolic approach. Of course, is "infinitary" logic is flawed (according to modern standards) but we may imagine to use very real as a "symbol" for itself, like in some approach to model theory. $\endgroup$ Commented Apr 18, 2018 at 15:10
  • $\begingroup$ I know. You can read this in hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 87. But "infinitary language" does not mean that he committed the fallacy to accept uncountable alphabets. If Zermelo consciously used every real as its own symbol, where did he explicitly mention this for the first time? I have never heard that he did, but Uwe's answer has aroused my curiosity. $\endgroup$
    – Franz Kurz
    Commented Apr 18, 2018 at 17:45

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