Russell's definition of finite cardinals

Question

whether the thought had been previously adumbrated, perhaps confusedly, i know not, but the name of Bertrand Russell has become associated with the assertion that:

the number $2$ is the set of all sets which contain exactly two elements.

i think there is even today some adherence to this rather neat definition.

but my elementary knowledge of these matters suffices to give rise to two questions:

a) should not Russell have said: the proper class of sets having exactly two elements?

b) does this imply that to understand the meaning of the symbol $2$ it is necessary to have a model of the entire class of transfinite cardinals, and to accept something like the Axiom of Choice? and is the theory of these infinities in fact research into the Kolmogorov Complexity of the concept of a finite cardinal?

Answer 1

Let me answer your first question, since I'm not quite sure I understand the second one at its current form.

Russell's definition was given much before von Neumann introduced the term "proper class".

In addition to that, von Neumann suggested we choose a representative from each equivalence class, and using the axiom of choice and the definition of ordinals he suggested, we have the modern definition of cardinals instead.

One of the first serious mathematical definitions of cardinal was the one devised by Gottlob Frege and Bertrand Russell, who defined a cardinal number |A| as the set of all sets equipollent to A. (Moore 1982, p. 153; Suppes 1972, p. 109). Unfortunately, the objects produced by this definition are not sets in the sense of Zermelo-Fraenkel set theory, but rather "proper classes" in the terminology of von Neumann.

(Weisstein, Eric W. "Cardinal Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CardinalNumber.html)