Category:Axioms/Peano's Axioms

This category contains axioms related to Peano's Axioms.

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 $P$

Pages in category "Axioms/Peano's Axioms"

The following 5 pages are in this category, out of 5 total.

\((\text P 1)\)	$:$				\(\ds 0 \in P \)			$0$ 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 $P$

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