Element

Set

Axiom

Theorem

Two sets are equal if and only if they have the same elements.

The empty set is a set.

The union of two sets is a set.

The intersection of two sets is a set.

Every non-empty set contains a unique element that is not a member of itself.

There exists a set that contains all sets.

The union of two sets is a set.

The intersection of two sets is a set.

The cardinality of the set of real numbers is (\aleph_0).

The cardinality of the set of real numbers is (\aleph_1).

The cardinality of the set of real numbers is (\aleph_2).

The cardinality of the set of real numbers is (\aleph_3).

Every first-order theory with a countable model has a model of every cardinality.

Every first-order theory with a finite model has a model of every cardinality.

Every first-order theory with an uncountable model has a model of every cardinality.

Every first-order theory with a model of cardinality (\aleph_0) has a model of every cardinality.

An axiomatic set theory developed by Kurt Gödel and Paul Bernays.

An axiomatic set theory developed by John von Neumann.

An axiomatic set theory developed by Zermelo and Fraenkel.

An axiomatic set theory developed by Ernst Zermelo.

An axiomatic set theory developed by Zermelo and Fraenkel.

An axiomatic set theory developed by John von Neumann.

An axiomatic set theory developed by Kurt Gödel and Paul Bernays.

An axiomatic set theory developed by Ernst Zermelo.

An axiomatic set theory developed by John von Neumann, Paul Bernays, and Kurt Gödel.

An axiomatic set theory developed by Zermelo and Fraenkel.

An axiomatic set theory developed by Kurt Gödel and Paul Bernays.

An axiomatic set theory developed by Ernst Zermelo.

An axiomatic set theory developed by Willard Van Orman Quine.

An axiomatic set theory developed by John von Neumann.

An axiomatic set theory developed by Kurt Gödel and Paul Bernays.

An axiomatic set theory developed by Ernst Zermelo.

Every set can be well-ordered.

Every non-empty set contains a unique element that is not a member of itself.

The union of two sets is a set.

The intersection of two sets is a set.

A paradox in set theory that arises from the definition of the set of all sets that do not contain themselves.

A paradox in set theory that arises from the definition of the set of all sets.

A paradox in set theory that arises from the definition of the set of all sets that contain themselves.

A paradox in set theory that arises from the definition of the set of all sets that are not members of themselves.

A paradox in set theory that arises from the definition of the set of all sets.

A paradox in set theory that arises from the definition of the set of all sets that contain themselves.

A paradox in set theory that arises from the definition of the set of all sets that are not members of themselves.

A paradox in set theory that arises from the definition of the set of all sets that do not contain themselves.

A paradox in set theory that arises from the definition of the set of all sets that contain themselves.

A paradox in set theory that arises from the definition of the set of all sets.

A paradox in set theory that arises from the definition of the set of all sets that are not members of themselves.

A paradox in set theory that arises from the definition of the set of all sets that do not contain themselves.

An axiomatic set theory developed by Zermelo, Fraenkel, and Skolem.

An axiomatic set theory developed by John von Neumann.

An axiomatic set theory developed by Kurt Gödel and Paul Bernays.

An axiomatic set theory developed by Ernst Zermelo.

An axiomatic set theory developed by Zermelo, Fraenkel, and Gödel.

An axiomatic set theory developed by John von Neumann.

An axiomatic set theory developed by Kurt Gödel and Paul Bernays.

An axiomatic set theory developed by Ernst Zermelo.