axiom of extensionality

We can use the axiom of extensionality to show that there is only one empty set.

By the axiom of extensionality this set is unique.

According to the axiom of extensionality, the identity of a set is determined by its elements.

The axiom of extensionality implies the empty set is unique (does not depend on w).

In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the empty set.

Axiom of extensionality: Two sets are the same if and only if they have the same elements.

An axiom of extensionality for this simulated set theory follows from E's extensionality.

Note that in this case, the axiom of extensionality must be formulated to apply only to objects that are not urelements.

The axiom of extensionality: .

Bisimilar sets are considered indistinguishable and thus equal, which leads to a strengthening of the axiom of extensionality.

To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.

That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory.

Includes no axiom of extensionality because the usual extensionality principle follows from the definition of collection and an easy lemma.

Over the course of his career, Quine proposed three variants of axiomatic set theory, each including the axiom of extensionality:

The usual axiom of extensionality for sets, as well as one for functions, and the usual axiom of union.

Naive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent due to Russell's paradox.

I would be identical to the axiom of extensionality in ZFC, except that the scope of I includes proper classes as well as sets.

The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

In mathematical logic, equality is formalized with axioms (e.g. the first few Peano axioms, or the axiom of extensionality in ZF set theory).

Ur-elements can be treated as a different logical type from sets; in this case, makes no sense if is an ur-element, so the axiom of extensionality simply applies only to sets.

In this case, one should replace the usual axiom of extensionality, , by , i.e. if x and y have the same elements, then they belong to the same sets.

Abian and LaMacchia (1978) studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice.

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality.

Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo-Fraenkel set theory (ZF).