Definition:Natural Numbers/Axiomatization

Definition

The natural numbers $\N$ can be axiomatised in the following ways:

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 $P$

Naturally Ordered Semigroup

The concept of a naturally ordered semigroup is intended to capture the behaviour of the natural numbers $\N$, addition $+$ and the ordering $\le$ as they pertain to $\N$.

Naturally Ordered Semigroup Axioms

A naturally ordered semigroup is a (totally) ordered commutative semigroup $\struct {S, \circ, \preceq}$ satisfying:

\((\text {NO} 1)\)	$:$		$S$ is well-ordered by $\preceq$	\(\ds \forall T \subseteq S:\)	\(\ds T = \O \lor \exists m \in T: \forall n \in T: m \preceq n \)
\((\text {NO} 2)\)	$:$		$\circ$ is cancellable in $S$	\(\ds \forall m, n, p \in S:\)	\(\ds m \circ p = n \circ p \implies m = n \)
					\(\ds p \circ m = p \circ n \implies m = n \)
\((\text {NO} 3)\)	$:$		Existence of product	\(\ds \forall m, n \in S:\)	\(\ds m \preceq n \implies \exists p \in S: m \circ p = n \)
\((\text {NO} 4)\)	$:$		$S$ has at least two distinct elements	\(\ds \exists m, n \in S:\)	\(\ds m \ne n \)

1-Based Natural Numbers

The following axioms capture the behaviour of $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations $+$ and $\times$ as they pertain to $\N_{>0}$:

\((\text A)\)	$:$			\(\ds \exists_1 1 \in \N_{> 0}:\)	\(\ds a \times 1 = a = 1 \times a \)
\((\text B)\)	$:$			\(\ds \forall a, b \in \N_{> 0}:\)	\(\ds a \times \paren {b + 1} = \paren {a \times b} + a \)
\((\text C)\)	$:$			\(\ds \forall a, b \in \N_{> 0}:\)	\(\ds a + \paren {b + 1} = \paren {a + b} + 1 \)
\((\text D)\)	$:$			\(\ds \forall a \in \N_{> 0}, a \ne 1:\)	\(\ds \exists_1 b \in \N_{> 0}: a = b + 1 \)
\((\text E)\)	$:$			\(\ds \forall a, b \in \N_{> 0}:\)	\(\ds \)Exactly one of these three holds:\( \)
					\(\ds a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \)
\((\text F)\)	$:$			\(\ds \forall A \subseteq \N_{> 0}:\)	\(\ds \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \)