extensionality

Extensionality: Classes having the same members are the same class.

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

Extensionality axiom: Two sets are identical if they have the same members.

This set is unique by the axiom of extensionality.

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).

(A1) (extensionality) - Classes that have the same elements are the same.

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

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

Extensionality for a general overview.

However additional problems raise from propositional extensionality and proof irrelevance [1].

The extensionality of causation and causal-explanatory contexts.

Extensionality.

Identical to Extensionality above.

In logic, extensionality, or extensional equality refers to principles that judge objects to be equal if they have the same external properties.

The representativeness heuristic violates one of the fundamental properties of probability: extensionality.

Boolos's name for Zermelo set theory minus extensionality was Z-.

The axiom of extensionality: .

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

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

In lambda calculus, extensionality is expressed by the eta-conversion rule, which allows conversion between any two expressions that denote the same function.

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.

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

Mereological systems in which Extensionality holds are termed extensional, a fact denoted by including the letter E in their symbolic names.