Definition:Peano Structure

Definition

A Peano structure $\mathcal P = \left({P, 0, s}\right)$ (also known as a Dedekind-Peano structure) is a set $P$ together with:

• A mapping $\exists s: P \to P$ which is:
  • injective
  • specifically not surjective.

• An element (usually denoted $0$ or a variant) such that $0 \in P \setminus s \left({P}\right)$, where:
  • $\setminus$ denotes set difference;
  • $s \left({P}\right)$ denotes the image of the mapping $s$.

Such a structure fulfils the Peano axioms.

In Non-Successor Element of Peano Axiom Schema is Unique, we see that any two elements in $P \setminus s \left({P}\right)$ are the same element.

Thus we are justified in singling out $0$ as a specifically distinguished element of $P$.

Source of Name

This entry was named for Giuseppe Peano and Richard Dedekind.

They were formulated by Peano, and were later refined by Dedekind.