Natural Number

Definition

$\mathbb{N}$

Definition by Peano Axioms

Peano Axioms

Definition

Axioms for the Natural Number

Axioms

$\exists 1\in \mathbb{N}$

1 as a natural number

$\forall n\in \mathbb{N},\exists n^{\prime }\in \mathbb{N}$

For every natural number $n$, the successor $n^{\prime }$ is a natural number

$\nexists n\in \mathbb{N},n^{\prime }=1$

For every natural number $n$, $n^{\prime }=1$ is false. There is no natural number whose successor is 1

$\forall m,n\in \mathbb{N},m^{\prime }=n^{\prime }\Rightarrow m=n$

For all natural numbers $m,n$, if $m^{\prime }=n^{\prime }$, then $m=n$

$\forall S\subset \mathbb{N},(1\in S)\wedge (\forall n,(n\in S)\Rightarrow (n^{\prime }\in S))\Rightarrow S=\mathbb{N}$

If $S$ is a set such that: $0\in S$ and for every natural number $n$, $n$ being in $S$ implies that $n^{\prime }$ is in $S$, then $S$ contains every natural number.

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Definition by Set Theory

$1:=\varnothing \in \mathbb{N}$

Let an empty set be $1$

$n^{\prime }=a\cup \{a\}\in \mathbb{N}$

Define the successor of any set $a$

$\forall m,n\in \mathbb{N}$

• $n⊊m\Rightarrow n<m$
• $(n\subset m)\wedge (n\supset m)\Rightarrow n=m$

$\forall m,n\in \mathbb{N}$

• $n+1=n^{\prime }$
• $n+m^{\prime }=(n+m{)}^{\prime }$
• $(n+m)-n=m$
• $n\times 1=n$
• $n\times m^{\prime }=n\times m+n$

Properties

Order

$\mathbb{N}$ set is Well-Ordered Set Every non-empty subset of $\mathbb{N}$ always has minimum element

Facts

$\mathbb{N}$ is not bounded above