Natural Number Arithmetic in the Theory of Finite Sets

Details Natural Number Arithmetic in the Theory of Finite Sets Natural Number Arithmetic in the Theory of Finite Sets

2007-11-19

arxiv.org/abs/0711.2922v2

Mathematics

Logic

53 pages; second version; section 6 added; section 12 revised; material added on connection with bounded arithmetic

Scientific paper

We describe a theory of finite sets, and investigate the analogue of Dedekind's theory of natural number systems (simply infinite systems) in this theory. Unlike the infinitary case, in our theory, natural number systems come in differing lengths and with different closure properties. We give examples of natural number systems incomparable in length; we define hierarchies of natural number systems closed under increasingly powerful functions; and we describe a method by which to construct natural number systems with given closure properties. These natural number systems form natural models for various systems of weak arithmetic.

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