User:Dfeuer/Ordering on Natural Numbers/Peano

Definition

Let $\left({\N, 0, s}\right)$ be a Peano structure, where $\N$ is a set.

This article, or a section of it, needs explaining. In particular: At this stage, has $\N$ been shown to be the natural numbers? If so, declare it here - if not, use a different symbol so as to remove that unspoken assumption in the reader's mind. $\omega$ is a suggestion but has similar connotations. $P$ is another which has been used in this context. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Let $<$ be the transitive closure of $s$.

Let $\le$ be the reflexive closure of $<$.

Then:

$<$ is the usual strict ordering of the natural numbers.

$\le$ is the usual ordering of the natural numbers.

This article, or a section of it, needs explaining. In particular: There appears to be a tacit assumption that $\N$ is the set of natural numbers. Clarification is needed as to where in the chain of thought between first positing a Peano structure and subsequently declaring that the natural numbers do indeed meet the criteria to be such a structure that the assumption needs to be made. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.