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 $$P$$.

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