The Axiom of Replacement finds its most important applications in von Neumann’s beautiful theory of Ordinal Numbers, and in the construction of the Cumulative Hierarchy of pure, grounded sets. One can live without knowing the ordinals, to be sure, but not as well: they bring many gifts, among them true cardinal numbers which give substance to the “virtual” theory of equinumerosities with which we have been making do. The Cumulative Hierarchy extends the iteration of the power operation we have used to construct HF “as far as it will go” and presents the pure, grounded sets as the most compelling intuitive understanding of what sets really are. It is not so clear one can live without knowing that, not among set theorists, at any rate.
KeywordsOrdinal Number Definite Operation Cardinal Number Intuitive Conception Ordinal Property
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