Axiom:Axiomatization of 1-Based Natural Numbers

Axioms
The following can be used as an axiomatization of the (non-zero) natural numbers as an algebraic structure $\left({\N_{> 0}, +, \times}\right)$:

Note
The above axiom schema specifies the old-fashioned definition of the natural numbers as:
 * $\text{The set of natural numbers} = \left\{{1, 2, 3, \ldots}\right\}$

as opposed to the more modern approach which defines them as:
 * $\text{The set of natural numbers} = \left\{{0, 1, 2, 3, \ldots}\right\}$

In order to eliminate confusion, on the set $\left\{{1, 2, 3, \ldots}\right\}$ will be denoted as $\N_{> 0}$ or $\N_{\ne 0}$.

When $\N$ is used, $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ is to be understood.