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...

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...

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...

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...

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...

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...

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...

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...