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Empty set - Axiomatic set theory | | Empty set - Axiomatic set theory: Encyclopedia II - Empty set - Axiomatic set theory | | | In the axiomatization of set theory known as Zermelo-Fraenkel set theory, the existence of the empty set is assured by the axiom of empty set. The uniqueness of the empty set follows from the axiom of extensionality.
Any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation. For example, if A is a set then the axiom schema of separation allows the construction of the set B = {x in A | x ≠ See also: Empty set, Empty set - Notation, Empty set - Properties, Empty set - Common problems, Empty set - Axiomatic set theory, Empty set - Does it exist or is it necessary?, Empty set - Operations on the empty set, Empty set - Bounds, Empty set - The empty set and zero, Empty set - Category theory | | | Empty set, Empty set - Axiomatic set theory, Empty set - Bounds, Empty set - Category theory, Empty set - Common problems, Empty set - Does it exist or is it necessary?, Empty set - Notation, Empty set - Operations on the empty set, Empty set - Properties, Empty set - The empty set and zero | | |
| | | Empty set: Encyclopedia II - Empty set - Axiomatic set theory
Empty set - Axiomatic set theory
In the axiomatization of set theory known as Zermelo-Fraenkel set theory, the existence of the empty set is assured by the axiom of empty set. The uniqueness of the empty set follows from the axiom of extensionality.
Any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation. For example, if A is a set then the axiom schema of separation allows the construction of the set B = {x in A | x ≠ x}, which can be defined to be the empty set.
Other related archivesAndré Weil, Bourbaki group, Cartesian product, For any, George Boolos, Greek letter, Jonathan Lowe, O, Set theory, Zermelo-Fraenkel set theory, axiom of empty set, axiom of extensionality, axiom schema of separation, axiomatic set theory, axiomatization of set theory, boundary points, cardinality, category, circle, closed, closure, compact set, continuous, empty function, empty product, extended reals, finite, finite set, function, identity element, infimum, initial object, interior points, intersection, mathematical symbols, mathematics, measure theory, null set, nullary, one, open, ordered set, product, quantifying plurally, real number line, reifying, set, set theory, set-theoretic definition of natural numbers, subset, sum, supremum, topological space, trivially, union, unions, vacuous truth, zero, Øø, Φ
 Adapted from the Wikipedia article "Axiomatic set theory", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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