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 $s: \N \to \N: \map s n = n + 1$ 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:

\((\text P 1)\)   $:$   \(\ds 0 \in P \)    $0$ is an element of $P$      
\((\text P 2)\)   $:$     \(\ds \forall n \in P:\) \(\ds \map s n \in P \)    For all $n \in P$, its successor $\map s n$ is also in $P$      

The other three are as follows:

Formulation 1

\((\text P 3)\)   $:$     \(\ds \forall m, n \in P:\) \(\ds \map s m = \map s n \implies m = n \)    $s$ is injective      
\((\text P 4)\)   $:$     \(\ds \forall n \in P:\) \(\ds \map s n \ne 0 \)    $0$ is not in the image of $s$      
\((\text P 5)\)   $:$     \(\ds \forall A \subseteq P:\) \(\ds \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P \)    Principle of Mathematical Induction:      
Any subset $A$ of $P$, containing $0$ and      
closed under $s$, is equal to $P$      

Formulation 2

\((\text P 3)\)   $:$     \(\ds \forall m, n \in P:\) \(\ds \map s m = \map s n \implies m = n \)    $s$ is injective      
\((\text P 4)\)   $:$   \(\ds \Img s \ne P \)    $s$ is not surjective      
\((\text P 5)\)   $:$     \(\ds \forall A \subseteq P:\) \(\ds \paren {\paren {\exists x \in A: \neg \exists y \in P: x = \map s y} \land \paren {\forall z \in A: \map s z \in A} } \)    Principle of Mathematical Induction:      
\(\ds \implies A = P \)    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 $V$ be a basic universe.

The successor mapping $s$ is the mapping on $V$ defined and denoted:

$\forall x \in V: \map s x := x \cup \set x$

where $x$ is a set in $V$.

Peano Structure

Let $\struct {P, s, 0}$ be a Peano structure.

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

Successor Mapping on Natural Numbers

Let $\N$ be the set of natural numbers.

Let $s: \N \to \N$ be the mapping defined as:

$s = \set {\tuple {x, y}: x \in \N, y = x + 1}$

Considering $\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping.

Non-Successor Element

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 $s$.

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 presented as

Some sources present the axioms in a different order.

For example, 1964: J. Hunter: Number Theory places the induction axiom $(\text P 5)$ as axiom $(3)$ and moves $(\text P 3)$ and $(\text P 4)$ down to be $(4)$ and $(5)$ respectively.

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

Peano's axioms are also known as:

the Peano axioms
the Dedekind-Peano axioms (for Richard Dedekind)
the Peano postulates
Peano's postulates

and so on.

Also see

  • Results about Peano's axioms 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.