Source : Moses Richardson, Fundamentals Of Modern Mathematics. ( Available at Archive.org)
- This is a basic question regarding the definition of cardinal numbers using the equipotent relation or equinumericity .
(1) From what I have understood so far, the strategy is to define the cardinal number of a set as its equivalence class under the relation " being equipotent" .
(2) It is also possible to use some ( arbitrarily chosen) representant in such an equivalence class, say the set $\{$ Peter, Mary, John$\}$ , and to define the number $3$ as " the equivalence class of the set $\{$ Peter, Mary, John$\}$".
Note : points (1) and (2) are taken from Richardon; what follows is personal reflexion and doubts
(3) There is however a point I'd need to be clarified : can one say that the equipotent relation induces a partition of the set of sets? it would be convenient to say that each cardinal number is a cell in this partition of such a " set of sets" , but is it correct?
(4) The reason that causes trouble for me is that the " set of sets " is a problematic object.
(5) If the equipotent relation does not " act" on the " set of sets ", what does it act on?