In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty...

replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed...

The axiom of foundation (or regularity) demands that every set be well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems...

then S + Regularity is consistent. S + Regularity implies the axiom of limitation of size. Since this is the only axiom of his 1925 axiom system that...

Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity...

without the axiom of regularity) that well-foundedness implies regularity. In variants of ZFC without the axiom of regularity, the possibility of non-well-founded...

The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory...

comprehension, or the axiom of regularity and axiom of pairing. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any...

axiom is identical to the axiom of regularity in ZF. This axiom is conservative in the sense that without it, we can simply use comprehension (axiom schema...

of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of...

x } {\displaystyle x=\{x\}} from the Axiom of regularity. The axiom of pairing also allows for the definition of ordered pairs. For any objects a {\displaystyle...

polyhedron Axiom of Regularity, also called the Axiom of Foundation, an axiom of set theory asserting the non-existence of certain infinite chains of sets Partition...

of the cumulative hierarchy. The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for...

0} . Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a...

the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty...

contradicts the axiom of regularity, as {a, c} has no minimal element under the relation "element of." If {a, b} = {c, d}, then a is an element of a, from a...

The axiom of induction in the context of KP is stronger than the usual axiom of regularity, which amounts to applying induction to the complement of a set...

relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all...

by axiom of infinity, and is now included as part of it. Zermelo set theory does not include the axioms of replacement and regularity. The axiom of replacement...

systems of set theory that include the axiom of regularity, but they can exist in non-well-founded set theory. ZF set theory with the axiom of regularity removed...

principle is an axiom schema, granting an axiom for any predicate (i.e. class). In contrast, the axiom of regularity is a single axiom, formulated with...

space, but these examples provide more insight on the T0 axiom than on regularity. An example of a regular space that is not completely regular is the Tychonoff...

of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at...

V} . L {\displaystyle L} is a model of ZFC, which means that it satisfies the following axioms: Axiom of regularity: Every non-empty set x {\displaystyle...

The axiom of regularity is of a restrictive nature as well. Therefore, one is led to the formulation of other axioms to guarantee the existence of enough...

Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the axiom of regularity is true in the model. There exists...

well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other...

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962...

set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any...

foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational...