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Anyone wishing to found arithmetic on the theory of the natural numbers as the finite cardinal numbers faces the task, first and foremost, of defining finite sets; for cardinal number is by nature a property of sets, and every statement about finite cardinal numbers can always be expressed as one about finite sets. In what follows we shall try to derive the most important property of the natural numbers, namely the principle of mathematical induction, from as simple a definition as possible of finite sets, and to show, at the same time, that the various other existing definitions are equivalent to the one assumed here. For these developments we shall have to rely on the basic concepts and methods of general set theory created by G. Cantor and R. Dedekind. We shall, however, not use the assumption that there exists an "infinite set", i.e., one which is equivalent to one of its parts as Dedekind still does in his fundamental essay: "Was sind und was sollen die Zahlen? ".

DOI: 10.1007/978-3-540-79384-7_8

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