An Infinitude of Finitudes
نویسنده
چکیده
Definitions of finiteness are set-theoretical properties P such that, if the axiom of choice (AC) is assumed, a set satisfies P if and only if it is finite. It is well known that the equivalence of many definitions of finiteness (to one another, and to “ordinary” finiteness, defined below) requires some (perhaps weaker) form of AC. Thus, it is of interest to study how various definitions of finiteness relate to one another in theories without AC (see [1] for a comprehensive overview of previous work done in this area). For this and other reasons, a great many—but nevertheless, finitely many— definitions of finiteness have been invented and investigated over the years, beginning with the work of Tarski [5] in 1924. In this note, I show how a natural generalization of a particularly well-studied notion of finiteness yields infinitely many new, non-equivalent notions.
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