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I have two questions concerning the history of set theory, both related to the distinction between the notion of a set and the notion of a class:

  1. Who was the first mathematician to make this distinction explicit?

  2. Did the terminology for this distinction immediately settle to the current terms "set" and "class", or was there some variance in the terminology used to make this distinction in the beginning?

Question 2 can also be extended to other languages, e.g. whether the terminology immediately settled to "Menge" and "Klasse" in German.

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2 Answers 2

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As far as I understand it (although I am no historian) Cantor himself came to make an explicit distinction between "Absolute Infinities" or "Inconsistent Multiplicites" on the one hand and the things that could form a 'completed' whole, that is sets. Of the two former terms he seemed to change which term he used depending on who he was talking to. He also had the unfortunate habit on occasion of using the word "Menge" when he really meant one of these proper class forms. Nevertheless he was quite aware of the difference. (See his letter to Dedekind reprinted in the van Heijenoort "From Frege to Goedel" volume.)

von Neumann made the distinction completely explicit: he gave a functional axiomatisation of what would become the "N" part in "NBG". This was in two papers in the late 20's. His was a thorough-going mathematisation of the concepts and had three types of function with arguments depending on whether they were 'sets' ,'classes' (or either).

But earlier than this Zermelo's axiomatisation was about which classes could be regarded as sets.

(The words 'set' and 'class' in English were used early on by Russell - pre-Principles of Mathematics. Cantor said that he like the idea of "ensemble" (French) rather than the German Menge - presumably because it carried overtones of being gathered together into a completed whole).

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A detailed discussion of the early history of set theory is in :

The starting point are Dedekind and Cantor.

Dedekind, in his exposition of the theory of fields (1871) writes [§159] :

By a field [Körper] we mean an infinite system [System] of real or complex numbers [...].

In a draft for Zahlen, written between 1872 and 1878, he introduces the concepts of system and of mapping ["Die Begriffe des Systems, der Abbildung"].

In Was sind und was sollen die Zahlen? (1888) we find also : "zusammengesetze System" and "Gemeinheit der Systeme" ["compounded system" (union - Def.8), and "community of systems" (intersection - Def.17)].


Cantor in the years 1863-64 introduced his theory od point-sets, and in 1872 speaks of set of points [Punktmenge].


Up to now, we have "sets of" something : numbers or points.

The "pure" notion of set has been introduced by Cantor in 1879 :

I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined ["Eine Mannichfaltigkeit (ein Inbegriff, eine Menge) von Elementen, die irgend welcher Begriffsphare angehoren"]

followed in 1883 by the "mature" concept :

By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic εἶδος or ἰδέα. ["Unter einer Mannichfaltigkeit oder Menge ...].


We can find class into the claculus of class of The Algebra of Logic Tradition : from Boole to Schröder.

Peano in Arithmetices principia (1889, page x) uses classis, followed by Russell (Principles, 1903). Of course, also Frege did [Appendix to vol.2 of Grundgesetze, discussing Russell's paradox, also if class is a "derived" notion for Frege, "concept" [Begriffe] being the primitive one] :

I cannot see how arithmetic could be given a scientific foundation, how numbers could be conceived as logical objects and introduced, if it is not allowed - at least conditionally - to go from a concept over to its extension. Can I always speak of the extension of a concept, of a class?


Zermelo (1908) starts from Cantor's definition of set (1895) as "a collection, gathered into a whole, of certain well-distinguished objects of our perception or our thought" [Unter einer 'Menge' verstehen wir jede Zusammenfassung ...]

He introduce also the notion of "Klassenaussage" ('class-statement') meaning a logical expression with one variable, what Russell called a propositional function.

Finally, von Neumann (1925) speaks of : set [Menge] and class [Bereich] :

"sets" are the sets that (in the earlier terminology) are "not too big", and "classes" are all totalities [Gesamtheiten] irrespective of their "size".



See also The Early Development of Set Theory.

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