Authors: Wes Hansen
In an earlier paper, “The Q-Naturals: A Recursive Arithmetic Which Extends the ‘Standard’ Model,” we developed a set of non-standard naturals called q-naturals and demonstrated, by construction, the existence of a recursive arithmetical structure which extends the “standard” model. In this paper we extend the q-naturals out to algebraic closure and explore the properties of the various sub-structures along the way. In the process of this development, we realize that the “standard” model of arithmetic and the Q-Natural model are simply the zeroth-order and first-order recursive arithmetics, respectively, in a countable subsumption hierarchy of recursive arithmetics; there exist countably many recursive arithmetical structures.
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