Axiom of extensionality

The axiom of extensionality,[1][2] also called the axiom of extent,[3][4] is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory.

[5][6] The axiom defines what a set is.

[1] Informally, the axiom means that the two sets A and B are equal if and only if A and B have the same members.

The axiom's usefulness can be seen from the fact that, if one accepts that

doesn't, then the axiom assures that there is a unique set

whose members are precisely whatever objects (urelements or sets, as the case may be) satisfy the formula

, follows from the substitution property of equality.

Despite this, the axiom is sometimes given directly as a biconditional, i.e., as

[1] Quine's New Foundations (NF) set theory, in Quine's original presentations of it, treats the symbol

for equality or identity as shorthand either for "if a set contains the left side of the equals sign as a member, then it also contains the right side of the equals sign as a member" (as defined in 1937), or for "an object is an element of the set on the left side of the equals sign if, and only if, it is also an element of the set on the right side of the equals sign" (as defined in 1951).

, as in Quine's Mathematical Logic (1951).

The second version of the definition is exactly equivalent to the antecedent of the ZF axiom of extensionality, and the first version of the definition is still very similar to it.

By contrast, however, the ZF set theory takes the symbol

for identity or equality as a primitive symbol of the formal language, and defines the axiom of extensionality in terms of it.

(In this paragraph, the statements of both versions of the definition were paraphrases, and quotation marks were only used to set the statements apart.)

In Quine's New Foundations for Mathematical Logic (1937), the original paper of NF, the name "principle of extensionality" is given to the postulate P1,

The definition D8, which defines the symbol

for identity or equality, defines

[10] In his Mathematical Logic (1951), having already developed quasi-quotation, Quine defines

(definition D10), and does not define an axiom or principle "of extensionality" at all.

[11] Thomas Forster, however, ignores these fine distinctions, and considers NF to accept the axiom of extensionality in its ZF form.

[12] In the Scott–Potter (ZU) set theory, the "extensionality principle"

is given as a theorem rather than an axiom, which is proved from the definition of a "collection".

In the Zermelo–Fraenkel axioms, there are no ur-elements, but they are included in some alternative axiomatisations of set theory.

Ur-elements can be treated as a different logical type from sets; in this case,

is an ur-element, so the axiom of extensionality simply applies only to sets.

Alternatively, in untyped logic, we can require

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

To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads: That is: Yet another alternative in untyped logic is to define

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