# Definition:Non-Successor Element

## Definition

Let $\struct {P, s, 0}$ be a Peano structure.

Then the element $0 \in P$ is called the non-successor element.

This is justified by Axiom $(\text P 4)$, which stipulates that $0$ is not in the image of the successor mapping $s$.

## Also defined as

Some treatments of Peano's axioms define the non-successor element (or primal element) to be $1$ and not $0$.

The treatments are similar, but the $1$-based system results in an algebraic structure which has no identity element for addition, and so no zero for multiplication.

## Also known as

It would be nice if there were a name for this element more terse than non-successor element and more general than zero.

A suggestion coined at $\mathsf{Pr} \infty \mathsf{fWiki}$ is primal element.