# Definition:Non-Successor Element

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## 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 Peano's Axiom $\text P 4$: $0 \notin \Img s$, 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**.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 4$: The natural numbers