Why is the class of all sets denoted $V$?

Question

In NBG set theory, why is the class of all sets denoted $V$? $S$ seems to me to be the natural designation.

Answer 1

The most likely source is Giuseppe Peano, Arithmetices Principia Novo Methodo Exposita (1889), page viii :

Signum $\text V$ significat verum [...] Signum $\Lambda$ significat falsum, sive absurdum.

But the same symbols are used by Peano in the "calculus of classes" (page xi) :

Signum $\Lambda$ indicat classem quae nullum continet individuum [i.e. the empty set].

Signo $\text V$, quod classem ex omnibus individuis constituam, [i.e. the universal class].

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From Peano, the symbol arrived to Alfred North Whitehead & Bertrand Russell, Principia Mathematica, I (2nd ed 1927), page 216 :

The universal class, denoted by $\text V$, is the class of all objects [...] Its definition is as follows :

*24.01. $\text V = \hat x(x = x)$ Df [in modern symbols : $\text V = \{ x \mid (x = x) \}$].

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Finally, the current usage is due to Kurt Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis With the Axioms of Set Theory (1940), page 8 :

By means of the axioms on intersection and complement, it is possible to prove the existence of a universal class $\text V$ and a null class $O$.

Answer 2

What my professor told me is that V stands for Volk, the German word for people/masses. Since both Zermelo and Fraenkel were German I do not doubt that the "V" stands for the initial letter of a German word, but I cannot provide a solid reference for this version.

Answer 3

I'd argue that it's because of the Von Neumann hierarchy, which tells us that the class of all sets consists of "levels" which are increasing in size, starting with the empty set, so V represents this hierarchy in a way https://en.m.wikipedia.org/wiki/Von_Neumann_universe