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