Editing Naive set theory (section)

== Sets, membership and equality ==
In naive set theory, a '''set''' is described as a well-defined collection of objects. These objects are called the '''elements''' or '''members''' of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even [[integer]]s. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.

[[File:Passage with the set definition of Georg Cantor.png|thumb|upright=1.15|Passage with the original set definition of Georg Cantor]]
The definition of sets goes back to [[Georg Cantor]]. He wrote in his 1915 article ''[https://web.archive.org/web/20141020034245/http://gdz.sub.uni-goettingen.de/index.php?id=pdf&no_cache=1&IDDOC=36218 Beiträge zur Begründung der transfiniten Mengenlehre]'':

{{quote|Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.|Georg Cantor}}

{{quote|A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.|Georg Cantor}}

[[File:First usage of the symbol ∈.png|thumb|upright=1.15|First usage of the symbol ϵ in the work ''[https://archive.org/details/arithmeticespri00peangoog Arithmetices principia nova methodo exposita]'' by [[Giuseppe Peano]]]]

=== Note on consistency ===
It does ''not'' follow from this definition ''how'' sets can be formed, and what operations on sets again will produce a set. The term "well-defined" in "well-defined collection of objects" cannot, by itself, guarantee the consistency and unambiguity of what exactly constitutes and what does not constitute a set. Attempting to achieve this would be the realm of axiomatic set theory or of axiomatic '''class theory'''.

The problem, in this context, with informally formulated set theories, not derived from (and implying) any particular axiomatic theory, is that there may be several widely differing formalized versions, that have both different sets and different rules for how new sets may be formed, that all conform to the original informal definition. For example, Cantor's verbatim definition allows for considerable freedom in what constitutes a set. On the other hand, it is unlikely that Cantor was particularly interested in sets containing cats and dogs, but rather only in sets containing purely mathematical objects. An example of such a class of sets could be the [[von Neumann universe]]. But even when fixing the class of sets under consideration, it is not always clear which rules for set formation are allowed without introducing paradoxes.

For the purpose of fixing the discussion below, the term "well-defined" should instead be interpreted as an ''intention'', with either implicit or explicit rules (axioms or definitions), to rule out inconsistencies. The purpose is to keep the often deep and difficult issues of consistency away from the, usually simpler, context at hand. An explicit ruling out of ''all'' conceivable inconsistencies (paradoxes) cannot be achieved for an axiomatic set theory anyway, due to Gödel's second incompleteness theorem, so this does not at all hamper the utility of naive set theory as compared to axiomatic set theory in the simple contexts considered below. It merely simplifies the discussion. Consistency is henceforth taken for granted unless explicitly mentioned.

=== Membership ===
If ''x'' is a member of a set ''A'', then it is also said that ''x'' '''belongs to''' ''A'', or that ''x'' is in ''A''. This is denoted by ''x''&nbsp;∈&nbsp;''A''. The symbol ∈ is a derivation from the lowercase Greek letter [[epsilon]], "ε", introduced by [[Giuseppe Peano]] in 1889 and is the first letter of the word [[:wikt:ἐστί|ἐστί]] (means "is"). The symbol ∉ is often used to write ''x''&nbsp;∉&nbsp;''A'', meaning "x is not in A".

=== Equality ===
Two sets ''A'' and ''B'' are defined to be '''[[Equality (mathematics)|equal]]''' when they have precisely the same elements, that is, if every element of ''A'' is an element of ''B'' and every element of ''B'' is an element of ''A''. (See [[axiom of extensionality]].) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all [[prime number]]s less than 6.
If the sets ''A'' and ''B'' are equal, this is denoted symbolically as ''A''&nbsp;=&nbsp;''B'' (as usual).

=== Empty set ===
The [[empty set]], denoted as <math>\varnothing</math> and sometimes <math>\{\}</math>, is a set with no members at all. Because a set is determined completely by its elements, there can be only one empty set. (See [[axiom of empty set]].){{sfn|Halmos|1974|p=9}} Although the empty set has no members, it can be a member of other sets. Thus <math>\varnothing\neq\{\varnothing\}</math>, because the former has no members and the latter has one member.{{sfn|Halmos|1974|p=10}}