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UBC Theses and Dissertations

On some definitions of a finite set Gutteridge, Lance


Six definitions of a finite set are studied; and each implication between the definitions is shown to be either derivable from the Zermelo-Fraenkel axioms or independent of them. The method of Boolean-valued models, as studied by D. Scott, is used to show that it is consistent with the Zermelo-Fraenkel axioms to deny some of these implications by constructing appropriate models. The six definitions are shown to satisfy a list of properties which a definition of finite should reasonably possess. It is also shown that any definition which satisfies these properties must encompass a larger class of sets than the first definition and a smaller class than the sixth. Three more definitions, which do not possess these properties, are mentioned, and the differences between these definitions and the first six are discussed.

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