Are Sets All There is?
Our next goal is to determine whether the basic results of naive set theory in Chapter 2 can be proved on the basis of the axioms of Zermelo. Right at the start we hit a snag: to define the crucial notion of equinumerosity we need functions; to define countable sets we need the specific set N of natural numbers; the fundamental theorem 2.21 of Cantor is about the set 𝓡 of real numbers, etc. Put another way, the results of Chapter 2 are not only about sets, but about points, numbers, functions, Cartesian products and many other mathematical objects which are plainly not sets. Where will we find these objects in the axioms of Zermelo which speak only about sets?
KeywordsEquivalence Relation Binary Relation Pairwise Disjoint Mathematical Object Definite Condition
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