Editing Axiom of extensionality (section)

== In NF set theory ==
[[Willard Van Orman Quine|Quine]]'s [[New Foundations]] (NF) set theory, in Quine's original presentations of it, treats the symbol <math>=</math> 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). That is, <math>x=y</math> is treated as shorthand either for <math>\forall z \, \left.(x \in z\right. \rightarrow \left. y\in z\right.)</math>, as in the original 1937 paper, or for <math>\forall z \, \left.(z \in x\right. \leftrightarrow \left. z\in y\right.)</math>, as in Quine's ''Mathematical Logic'' (1951). The second version of the definition is exactly equivalent to the [[Antecedent (logic)|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 <math>=</math> 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, <math>( ( x \subset y ) \supset ( ( y \subset x ) \supset (x = y) ) )</math>,<ref name=":2">{{Cite journal |last=Quine |first=W. V. |date=1937 |title=New Foundations for Mathematical Logic |url=https://www.jstor.org/stable/2300564 |journal=The American Mathematical Monthly |volume=44 |issue=2 |pages=74, 77 |doi=10.2307/2300564 |jstor=2300564 |issn=0002-9890|url-access=subscription }}</ref> which, for readability, may be restated as <math> x \subset y \rightarrow ( y \subset x \rightarrow x = y )</math>. The definition D8, which defines the symbol <math>=</math> for identity or equality, defines <math>(\alpha = \beta)</math> as shorthand for <math>(\gamma) \, (\left.(\alpha \in \gamma\right.) \supset (\left. \beta\in \gamma\right.))</math>.<ref name=":2" /> In his ''Mathematical Logic'' (1951), having already developed [[quasi-quotation]], Quine defines <math>\ulcorner \zeta=\eta \urcorner</math> as shorthand for <math>\ulcorner (\alpha) \, (\left.\alpha \in \zeta\right. \; . \equiv  \, .  \, \left. \alpha \in \eta\right.) \urcorner</math> (definition D10), and does not define an axiom or principle "of extensionality" at all.<ref>{{Cite journal |last=Quine |first=W. V. |date=1951-12-31 |title=Mathematical Logic |url=http://dx.doi.org/10.4159/9780674042469 |journal=DeGruyter |pages=134–136 |doi=10.4159/9780674042469|isbn=978-0-674-04246-9 |url-access=subscription }}</ref>

[[Thomas Forster (mathematician)|Thomas Forster]], however, ignores these fine distinctions, and considers NF to accept the axiom of extensionality in its ZF form.<ref>{{Citation |last=Forster |first=Thomas |title=Quine's New Foundations |date=2019 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/quine-nf/ |access-date=2024-11-24 |edition=Summer 2019 |publisher=Metaphysics Research Lab, Stanford University}}</ref>