The Axiom of Foundation
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The Axiom of Foundation book
The notion of a well-ordering is one of the key concepts of set theory. It emerged in Chapter 3, when we introduced natural numbers; the fact that the natural numbers are well-ordered by size is essentially equivalent to the Induction Principle. We studied well-orderings in full generality in Chapter 6, and we saw many applications of them in subsequent chapters. It is thus of great interest to consider whether the concept allows further useful generalizations. It turns out that, for many purposes, the “ordering” stipulation is unimportant; it is. the “well-” part, i.e., the requirement that every nonempty subset has a minimal element, that is crucial. This leads to the basic definition of this section.