Definition:Peano Structure
Definition
A Peano structure $\struct {P, 0, s}$ comprises a set $P$ with a successor mapping $s: P \to P$ and a non-successor element $0$.
These three together are required to satisfy Peano's axioms.
Successor Mapping
Let $\struct {P, s, 0}$ be a Peano structure.
Then the mapping $s: P \to P$ is called the successor mapping on $P$.
Also known as
A Peano structure is also known as a Dedekind-Peano structure, for Richard Dedekind.
Also see
- Results about Peano structures can be found here.
Source of Name
This entry was named for Giuseppe Peano.
Historical Note
A set of axioms on the same topic as Peano's axioms was initially formulated by Richard Dedekind in $1888$.
Giuseppe Peano published them in $1889$ according to his own formulation, in a more precisely stated form than Dedekind's.
Bertrand Russell pointed out that while Peano's axioms give the key properties of the natural numbers, they do not actually define what the natural numbers actually are.
According to 1960: Paul R. Halmos: Naive Set Theory:
- [These] assertions ... are known as the Peano axioms; they used to be considered as the fountainhead of all mathematical knowledge.
It is worth pointing out that the Peano axioms can be deduced to hold for the minimally inductive set as defined by the Axiom of Infinity from the Zermelo-Fraenkel axioms.
Thus they are now rarely considered as axiomatic as such.
However, in their time they were groundbreaking.