In the area of mathematics called set theory, a specific construction due to John von Neumann^{[39]}^{[40]} defines the natural numbers as follows:

- Set 0 = { }, the empty set,
- Define
*S*(*a*) =*a*∪ {*a*} for every set*a*.*S*(*a*) is the successor of*a*, and*S*is called the successor function. - By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be
*inductive*. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms. - It follows that each natural number is equal to the set of all natural numbers less than it:

- 0 = { },
- 1 = 0 ∪ {0} = {0} = {{ }},
- 2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},
- 3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},
*n*=*n*−1 ∪ {*n*−1} = {0, 1, ...,*n*−1} = {{ }, {{ }}, ..., {{ }, {{ }}, ...}}, etc.

With this definition, a natural number *n* is a p

In the area of mathematics called set theory, a specific construction due to John von Neumann^{[39]}^{[40]} defines the nat

In the area of mathematics called set theory, a specific construction due to John von Neumann^{[39]}^{[40]} defines the natural numbers as follows:

- Set 0 = { }, the empty set,
- Define
*S*(*a*) =*a*∪ {*a*} for every set*a*.*S*(*a*) is the successor of*n*is a particular set with*n*elements, and*n*≤*m*if and only if*n*is a subset of*m*. The standard definition, now called definition of**von Neumann ordinals**, is: "each ordinal is the well-ordered set of all smaller ordinals."Also, with this definition, different possible interpretations of notations like ℝ

^{n}(*n*-tuples versus mappings of*n*into ℝ) coincide.Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

#### Zermelo ordinals

Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo's construction goes as follows:

^{[40]}- Set 0 = { }
- Define
*S*(*a*) = {*a*}, - It then follows that

- 0 = { },
- 1 = {0} = {{ }},
- 2 = {1} = {{{ }}},
*n*= {*n<*Also, with this definition, different possible interpretations of notations like ℝ

^{n}(*n*-tuples versus mappings of*n*into ℝ) coincide.Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo's construction goes as follows:

^{[40]}- Set 0 = { }
- Define
*S*(*a*) = {*a*}, - It then follows that

- 0 = { },
- 1 = {0} = {{ }},
- 2 = {1} = {{{ }}}