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[[John von Neumann]], considered by many to have been the most brilliant man of the 20th century (and a convert to [[Roman Catholic Church|Catholicism]], did his best work in set theory.  He developed a definition of an ordinal number as the set of all smaller ordinal numbers, the definition that is commonly accepted today.<ref>https://www.britannica.com/biography/John-von-Neumann</ref>  That approach helps circumvent issues caused by the transfinite numbers of Georg Cantor.

[[John von Neumann]], considered by many to have been the most brilliant man of the 20th century (and a convert to [[Roman Catholic Church|Catholicism]], did his best work in set theory.  He developed a definition of an ordinal number as the set of all smaller ordinal numbers, the definition that is commonly accepted today.<ref>https://www.britannica.com/biography/John-von-Neumann</ref>  That approach helps circumvent issues caused by the transfinite numbers of Georg Cantor.

The language of set theory is based on a single fundamental relation, called membership. We say that <math>x</math> is a member of <math>A</math> (in symbols <math>x \in A</math>), or that the set <math>A</math> contains <math>x</math> as an element.<ref>The mathematician Giuseppe Peano created the symbol ∈ in 1889 to mean "is an element of," from the Greek epsilon that is the first letter of εἰμί, which means "I am."</ref> The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements,<ref>http://plato.stanford.edu/entries/set-theory/</ref> or, equivalently, if each is a [[subset]] of the other.

The language of set theory is based on a single fundamental relation, called membership. We say that <math>x</math> is a member of <math>A</math> (in symbols <math>x \in A</math>), or that the set <math>A</math> contains <math>x</math> as an element.<ref>The mathematician Giuseppe Peano created the symbol ∈ in 1889 to mean "is an element of," from the Greek epsilon that is the first letter of εἰμί, which means "I am."</ref> The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements,<ref name=stanford>[https://plato.stanford.edu/entries/set-theory/ ''Set Theory''], retrieved from [[Stanford Encyclopedia of Philosophy]] on Apr 15, 2022.</ref> or, equivalently, if each is a [[subset]] of the other.

==History of set theory==

==History of set theory==

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. For example, 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'', and 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>[https://plato.stanford.edu/entries/set-theory/ ''Set Theory''], retrieved from [[Stanford Encyclopedia of Philosophy]] on Apr 15, 2022.</ref>

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 />

== See also ==

== See also ==