TL;DR
This paper clarifies the distinctions between proof by induction and recursion, explores the construction of natural numbers as ordinals, and generalizes to algebraic signatures, culminating in a class ON_S with free algebra properties.
Contribution
It introduces a generalized construction of ordinal classes for any algebraic signature, extending the understanding of natural numbers and ordinals.
Findings
Natural numbers as a special kind of ordinal
Construction of ON_S class for algebraic signatures
Existence of a free algebra subclass within ON_S
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 ON_S that is to the class of ordinals as S is to {0,s}. In particular, ON_S has a subclass that is a free algebra in the signature S.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Logic, programming, and type systems · Advanced Algebra and Logic