Axiom:Peano's Axioms/Formulation 1

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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:

\((\text P 3)\)   $:$     \(\ds \forall m, n \in P:\) \(\ds \map s m = \map s n \implies m = n \)    $s$ 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 $P$      

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.