Today there is no doubt that the empty set for the whole of mathematics is as reasonable and useful as zero for arithmetic. This however was not always the case, and surprisingly even Zermelo, who based his infinite set on the empty set, was unsure. The following quotes shed some light on this mostly forgotten development. My question is, when had these doubts become completely forgotten? I would also be interested in further quotes doubting the "justifyability" of the empty set.

Bernard Bolzano, the inventor of the notion set (Menge) in mathematics would not have named a nothing an empty set. In German the word set has the meaning of many or great quantity. Often we find in German texts the expression große (great or large) Menge, rarely the expression kleine (small) Menge. Therefore Bolzano apologizes for using this word in case of sets having only two elements: "Allow me to call also a collection containing only two parts a set." [J. Berg (ed.): B. Bolzano, Einleitung zur Grössenlehre, Friedrich Frommann Verlag, Stuttgart (1975) p. 152]

Also Richard Dedekind discarded the empty set. But he accepted the singleton, i.e., the non-empty set of less than two elements: "For the uniformity of the wording it is useful to permit also the special case that a system S consists of a single (of one and only one) element a, i.e., that the thing a is element of S but every thing different from a is not an element of S. The empty system, however, which does not contain any element, shall be excluded completely for certain reasons, although it might be convenient for other investigations to fabricate such." [R. Dedekind: "Was sind und was sollen die Zahlen?" Vieweg, Braunschweig (1887), 2nd ed. (1893) p. 2]

Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member." [Bertrand Russell: "On some difficulties in the theory of transfinite numbers and order types", Proc. London Math. Soc. (2) 4 (1906) p. 47]

Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O. P = O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such." [Cantor, p. 146]

And even Ernst Zermelo who made the "Axiom II There is an (improper) set, the 'null-set' 0 which does not contain any element" [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908) p. 263], this same Zermelo himself said in private correspondence: "It is not a genuine set and was introduced by me only for formal reasons." [E. Zermelo, letter to A. Fraenkel (1 Mar 1921)] "I increasingly doubt the justifiability of the 'null set'. Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification." [E. Zermelo, letter to A. Fraenkel (9 May 1921)]


The answer and comments have shown me that it is hard if not impossible to answer this question. Zermelo for instance expressed his doubts only in private correspondence. What about McAtee or Peano or Young mentioned by Gerad Edgar? Therefore I will take the alternative: Who was the last recognized mathematician to utter doubts with respect to the existence of the empty set? Kanamori, mentioned by Dave L Renfro gives a hint to 1943.

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    $\begingroup$ Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member." What you stated about Bertrand Russell, regardless of its truth, does not follow from the quoted statement. $(P \implies Q$ does not imply $Q \implies P.)$ Regarding your question, maybe The empty set, the singleton, and the ordered pair by Akihiro Kanamori (2003) will help. $\endgroup$ Apr 29, 2018 at 9:54
  • $\begingroup$ Thank you for the hint to Kanamori. Concerning your criticism, I would formalize Russell's statement: Def: Existing class <==> class with 1 or more members. $\endgroup$
    – Franz Kurz
    Apr 29, 2018 at 10:03
  • $\begingroup$ The question is oddly phrased, all the quoted authors viewed empty set as useful verbiage, just as number zero, they could not care less about its "existence". Just as nobody does today. They only doubted whether the verbiage is useful enough to keep. $\endgroup$
    – Conifold
    Apr 29, 2018 at 21:05

1 Answer 1


In MathWords we find

EMPTY SET. A JSTOR search found the term used--without explanation--in J. E. McAtee "Modular Invariants of a Quadratic Form for a Prime Power Modulus," American Journal of Mathematics, 41, (1919), p. 237. The term began to become common in the 1930s, although in these early days it was nothing like as common as null set.


NULL CLASS is found in Bertrand Russell’s Principles of Mathematics (1903). G. Peano’s "Dizionario di Matematica," Revue de mathématiques, 7, (1900-1), 160-172 has an entry under Nulla: "La classe nulle, classe non contenente individui ..."


NULL SET. Null-set appears in 1906 in Theory of Sets and Points by W. H. and G. C. Young [OED].

  • $\begingroup$ Some early systems of set theory used $V$ for the "universal class" and $\Lambda$ for the "empty class" ... but I did not find where that started. $\endgroup$ Apr 29, 2018 at 11:51
  • $\begingroup$ Thank you for this answer +1, but what about doubts? I have recognized that it is impossible to answer my original question. Therefore I have taken the alternative side, see the edit. $\endgroup$
    – Franz Kurz
    Apr 29, 2018 at 18:04
  • $\begingroup$ Ref to Russell (1903), page 22: "A propositional function is said to be null when it is false for all values of $x$; and the class of $x$’s satisfying the function is called the null-class, being in fact a class of no terms. Either the function or the class, following Peano, I shall denote by $\Lambda$." $\endgroup$ May 1, 2018 at 18:01

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