Axiom of regularity
Axiom ensuring no set contains itself and no infinite descending chains.
Wikipedia / Wikimedia Commons
The axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory. It states that every non-empty set A contains an element that is disjoint from A. The axiom was originally formulated by von Neumann and adopted in a formulation closer to contemporary textbooks by Zermelo.
- field
- Mathematics (set theory)
- known_for
- Axiom of regularity (foundation), preventing sets from being elements of themselves, and eliminating infinite descending membership chains
- formulated_by
- von Neumann
- adopted_by
- Zermelo
Lore & Background
The axiom of regularity, together with the axiom of pairing, implies that no set is an element of itself and that there is no infinite sequence where each term is an element of the previous. With the axiom of dependent choice, this result can be reversed: if no such infinite sequences exist, then the axiom of regularity is true. In this context, the axiom is equivalent to the sentence that there are no downward infinite membership chains.
Reader's Guide
The axiom of regularity is a foundational principle in Zermelo–Fraenkel set theory. It was originally formulated by von Neumann and later adopted in a form closer to contemporary textbooks by Zermelo. Virtually all results in branches of mathematics based on set theory hold even in the absence of regularity. However, regularity makes some properties of ordinals easier to prove and allows induction on well-founded relational structures such as certain lexicographical orderings. Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of regularity in intuitionistic theories, where the two axioms are not equivalent. Non-standard set theories have postulated the existence of sets that are elements of themselves, omitting the axiom of regularity.
Did You Know?
- The axiom of regularity together with the axiom of pairing implies that no set is an element of itself.
- With the axiom of dependent choice, the axiom of regularity is equivalent to the statement that there are no infinite descending membership chains.
- The axiom was originally formulated by von Neumann and adopted in a formulation closer to contemporary textbooks by Zermelo.
- Non-standard set theories have postulated the existence of sets that are elements of themselves, omitting the axiom of regularity.
Frequently Asked Questions
Who is Axiom of regularity?
Axiom of regularity, also called the axiom of foundation, is a structural rule in Zermelo–Fraenkel set theory requiring that every non-empty set possess at least one element disjoint from the set itself. It was originally formulated by von Neumann and later taken up by Zermelo in a form closer to what modern textbooks present.
What are Axiom of regularity's powers/role?
Its job is to block two kinds of pathology: a set being a member of itself and the existence of infinite descending membership chains. By enforcing a well-founded membership relation, it lets the universe of sets be organized into a clean hierarchy of ranks.
How does Axiom of regularity's story end?
In the standard ZF axiom list it appears near the end, serving as the final structural constraint after the other Zermelo–Fraenkel axioms have been stated. It is independent of the remaining axioms, so one can consistently work in set theories that omit it, yet most working mathematicians include it as a matter of convention.
Why is Axiom of regularity important?
Without it, self-membership and infinite loops in the ∈ relation become possible, which undermines the intuitive picture of sets being built up from simpler sets and complicates inductive arguments. It is what justifies rank-based induction and keeps the foundational landscape free of circular reference.
Who created Axiom of regularity?
John von Neumann is credited with the original formulation of the axiom in the early twentieth century. Paul Zermelo subsequently adopted and refined the statement into the version that appears in most contemporary set-theory textbooks.
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