Axiom:Peano's Axioms/Formulation 2

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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.

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

Historically, these two presuppositions were considered the first two of Peano's Axioms. The other three can be formulated as follows:

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

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 see

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

Source of Name

This entry was named for Giuseppe Peano and Richard Dedekind.

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.

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.