Numbers
Abstract
This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a construction of natural numbers as a special kind of ordinal. In any case, the natural numbers can be understood as composing a free algebra in a certain signature, 0,s. The paper here culminates in a construction of, for each algebraic signature S, a class ONS that is to the class of ordinals as S is to 0,s. In particular, ONS has a subclass that is a free algebra in the signature S.
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