How Menge became Set?

(intended as a comment, but too big)

In set theory one now speaks of both sets and classes which have different meanings.

The early development of set theory, from Cantor through to Von Neumann seems to be dominated by German publications, in which the term "mengenlehre" is used.
English translations of these works use the term "Set Theory" for mengenlehre.

The distinction between set and class originates with a distinction made by Cantor between consistent and inconsistent multiplicities, but does not seem to appear in axiomatic set theory until Von Neuman in 1925 published an axiomatic set theory in which a similar distinction appears.

Ontologically this is a strange set theory, since the entities involved are functions rather than sets. Nevertheless the paper, in german, again uses the term mengenlehre in the title.
It may however be the source of the migration of mathematical logicians from talk of classes (which is the term used by Russell) to sets (with classes as collections which do not qualify as sets).

Von Neumann's introduction into axiomatic set theory of Cantor's distinction between consistent and inconsistent multiplicities leads eventually to the NBG (Neumann, Bernays, Goedel) set theory in which the ontology involves two kinds of multiplicity called classes and sets.  The sets are the ones which correspond to the multiplicities in theories which do not make the distinction (they lack the distinction because their ontologies lack the entities which are known in NBG as classes).

It is possibly because of the need to make this distinction that we have two terms for menge, and the (possibly arbitrary) choice to use "class" for the "inconsistent" multiplicities (which are rendered consistent in the formal theories in which both kinds occur), seems to have forced a migration of terminology in the English speaking world from talk of classes to talk of sets (bearing in mind that the set theories in which the distinction appears, and in which there are classes in this new sense, have largely disappeared from active use, so far as I am aware).  The term class, once use as menge, was appropriate for use specifically for a kind of collection which we don't really need, and mengenlehre became set theory.

How this played out in German I have no idea.
Presumably German set theorists do now have to make the same distinction, at least when they speak of systems like NBG, but how they do it I have no idea.
Possibly by talking of sets and classes!

This seems to me a case where prior usage is of marginal influence. If one had considered prior usage of the words "set" and "class" one might come up with grounds for chosing which to use for (let us say) consistent or inconsistent multiplicities (or grounds perhaps connected with the fact that classes are larger than sets) but it is doubtful that any such exercise was ever undertaken or that mathematicians ever go into such matters when chosing their terms.
(another example one might speculate about is the choice of the term "category theory", which actually seems to me quite appropriate).






Apparently the word set used as a noun for collection dates back to 15C, long before the mathematicians got in on the act, and possibly originates in the old French "sette".

Roger Jones