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Ordering Relation Intersection Set Zermelo Fraenkel Axioms Number Theory Set Theory Empty Set Zero Elements Monoidal Category Cardinal Arithmetic Inductive Set

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Natural Number

Given the Zermelo-Fraenkel axioms of set theory, one can prove that there exists an inductive set such that $\emptyset \in X$ . The natural numbers $\mathbb{N}$ are then defined to be the intersection of all subsets of which are inductive sets and contain the empty set as an element.

The first few natural numbers are:

$0 := \emptyset$
$1 := 0' = \{0\} = \{ \emptyset \}$
$2 := 1' = \{0,1\} = \{\emptyset, \{ \emptyset \} \}$
$3 := 2' = \{0,1,2\} = \{\emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \}$

Note that the set 0 has zero elements, the set has one element, the set has two elements, etc. Informally, the set is the set consisting of the elements $0, 1, \dots, n-1$ , and is both a subset of $\mathbb{N}$ and an element of $\mathbb{N}$ .

In some contexts (most notably, in number theory), it is more convenient to exclude 0 from the set of natural numbers, so that $\mathbb{N} = \{1,2,3,\dots\}$ . When it is not explicitly specified, one must determine from context whether 0 is being considered a natural number or not.

Addition of natural numbers is defined inductively as follows:

for all $a \in \mathbb{N}$
for all $a,b \in \mathbb{N}$

Multiplication of natural numbers is defined inductively as follows:

$a \cdot 0 := 0$ for all $a \in \mathbb{N}$
$a \cdot b' := (a\cdot b) + a$ for all $a,b \in \mathbb{N}$

The natural numbers form a monoid under either addition or multiplication. There is an ordering relation on the natural numbers, defined by: $a \leq b$ if $a \subseteq b$ .

1	Empty Set
1	Set Theory
1	Set
1	Ordering Relation
1	Zermelo Fraenkel Axioms
1	Intersection
1	Number Theory
1	Zero Elements
2	Monoidal Category
2	Inductive Set
2	Cardinal Arithmetic