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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?

Floridus Floridi
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  • That wasn't consistent. Thanks for this observation. – Floridus Floridi Feb 02 '21 at 23:00
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    This is the Frege approach; no, you can't really define an "equivalence relation" on the collection of all sets: equivalence relations are defined to be certain subsets of cartesian squares, and you don't have a cartesian square of the collection of all sets. Moreover, even the collection of all singletons is not a set, so the "equivalence classes" you would get are not actually sets and thus not subject to your set theory. – Arturo Magidin Feb 02 '21 at 23:01
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    You can think of the equipotent/equipollent relation as a two-variable property/formula: $\Phi(X,Y)$ is true if $X$ is equipotent to $Y$, and false if not. It's not really a "relation" or "equivalence relation" in the sense of set theory, even though we recognize it as one in the meta-theory. That's why you prove things like "If $A\sim B$ and $B\sim C$, then $A\sim C$" instead of saying "The relation 'equipotent' is transitive." You could formalize it as an honest-to-goodness equivalence relation among small sets (sets that are in a particular universe), but then you need universes. – Arturo Magidin Feb 02 '21 at 23:03
  • @ArturoMagidin.- Thanks! – Floridus Floridi Feb 02 '21 at 23:05

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