Axiom:Peano's Axioms

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Peano's Axioms are a set of properties which can be used to serve as a basis for logical deduction of the properties of the natural numbers.

Peano's Axioms are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: s \left({n}\right) = n + 1$ and $0$ as an element of $\N$.

Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$.

Historically, the existence of $s$ and the existence of $0$ were considered the first two of Peano's Axioms. The other three are as follows:

Formulation 1

\((P3)\)   $:$     \(\displaystyle \forall m, n \in P:\) \(\displaystyle s \left({m}\right) = s \left({n}\right) \implies m = n \)    $s$ is injective             
\((P4)\)   $:$     \(\displaystyle \forall n \in P:\) \(\displaystyle s \left({n}\right) \ne 0 \)    $0$ is not in the image of $s$             
\((P5)\)   $:$     \(\displaystyle \forall A \subseteq P:\) \(\displaystyle \left({0 \in A \land \left({\forall z \in A: s \left({z}\right) \in A}\right)}\right) \implies A = P \)    Principle of Mathematical Induction:             
Any subset $A$ of $P$, containing $0$ and             
closed under $s$, is equal to $P$             

Formulation 2

\((P3)\)   $:$     \(\displaystyle \forall m, n \in P:\) \(\displaystyle s \left({m}\right) = s \left({n}\right) \implies m = n \)    $s$ is injective             
\((P4)\)   $:$   \(\displaystyle \operatorname{Im} \left({s}\right) \ne P \)    $s$ is not surjective             
\((P5)\)   $:$     \(\displaystyle \forall A \subseteq P:\) \(\displaystyle \left({\left({\exists x \in A: \neg \exists y \in P: x = s \left({y}\right)}\right) \land \left({\forall z \in A: s \left({z}\right) \in A}\right)}\right) \)    Principle of Mathematical Induction:             
\(\displaystyle \implies A = P \)    Any subset $A$ of $P$, containing an element not             
in the image of $s$ and closed under $s$,
is equal to $P$   


Successor Mapping

Let $\left({P, s, 0}\right)$ be a Peano structure.

Then the mapping $s: P \to P$ is called the successor mapping on $P$.

Non-Successor Element

Let $\left({P, s, 0}\right)$ be a Peano structure.

Then the element $0 \in P$ is called the non-successor element.

This is justified by Axiom $(P4)$, which stipulates that $0$ is not in the image of the successor mapping $s$.

Peano Structure

Such a set $P$, together with the successor mapping $s$ and non-successor element $0$ as defined above, is known as a Peano structure.

Also defined as

Some treatments of this topic 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

Peano's Axioms are also known as the Peano Axioms, the Dedekind-Peano Axioms and the Peano Postulates.

Also see

  • Results about Peano's axioms can be found here.

Source of Name

This entry was named for Giuseppe Peano.

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

Historical Note

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 minimal infinite successor 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.