| | The empty set is the set with no members. Because sets are uniquely defined by membership, the empty set is unique. The empty set is usually denoted by {} or ∅. | | The empty set is the set with no members. Because sets are uniquely defined by membership, the empty set is unique. The empty set is usually denoted by {} or ∅. |
| − | In the axiomatic set theory, '''any''' other set is build over the empty set. An example of a set with one element is the set { ∅ } - the set whose single element is the empty set. Von Neumann called the empty set ''0'', this set { ∅ } = ''1'', and proceeded in defining each ''natural number'' ''n'' as the set with the ''n'' elements from ''0'' to ''n-1''. In this way, it's possible to define the (infinite) set of natural numbers. Ordered pairs, functions, relations, etc, are all defined as sets with some properties.<ref>[https://www.britannica.com/science/set-theory/Schema-for-transfinite-induction-and-ordinal-arithmetic ''Axiomatic set theory''], retrieved from [[Enciclopedia Britannica]] on Apr 15, 2022.</ref><ref name=stanford /> | + | In the axiomatic set theory, '''any''' other set is build over the empty set. An example of a set with one element is the set { ∅ } - the set whose single element is the empty set. Von Neumann called the empty set ''0'', this set { ∅ } = ''1'', and proceeded in defining each ''natural number'' ''n'' as the set with the ''n'' elements from ''0'' to ''n-1''. In this way, it's possible to define the (infinite) set of natural numbers. Ordered pairs, functions, relations, etc, are all defined as sets with some properties.<ref>[https://www.britannica.com/science/set-theory/Schema-for-transfinite-induction-and-ordinal-arithmetic ''Axiomatic set theory''], retrieved from [[Encyclopedia Britannica]] on Apr 15, 2022.</ref><ref name=stanford /> |