Remarks On The Axioms
- Although different kinds of variables are used for classes and sets, the language is not many-sorted; sets are identified with classes having the same extension. Small case variables are used as mere abbreviations for various contexts; e.g.,
- Since the quantification in A2 ranges over classes, i.e., is not set-bound, A2 is the comprehension scheme of Morse–Kelley set theory, not that of Von Neumann–Bernays–Gödel set theory. This extra strength of A2 is employed in the definition of the ordinals (not presented here).
- Since there is no axiom of pairing, it must be proved that for any two sets x and y, the Kuratowski pair {{x},{x,y}} exists and is a set. Hence proving that there exists a one-to-one correspondence between two classes does not prove that they are equinumerous.
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