Authors: Wes Hansen
In what follows we develop foundations for a set of non-standard natural numbers we call q-naturals, where q stands for quanta, by the recursive generation of reflexive sets. From the practical perspective, these q-naturals correspond to ordered pairs of natural numbers with the lexicographic ordering, hence, they are isomorphic to ω^2. In addition, we demonstrate a novel definition of the arithmetical operation, multiplication, which turns out to be recursive. This operation, together with lexicographic order and coordinate-wise addition, defines an arithmetical structure which extends the “standard” model but yet has a recursive order relation and recursive arithmetical operations defined on the entire domain.
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