# Quine's New Foundations (NF)

<https://www.wikiwand.com/en/New_Foundations> <https://plato.stanford.edu/entries/quine-nf/>

Quine's system of axiomatic set theory, NF, takes its name from the title "New Foundations for Mathematical Logic", which was a 1937 article which introduced it. The axioms of NF are extensionality together with stratified comprehension.

**New Foundations** (NF) is an axiomatic set theory, proposed in a 1937's article, "*New Foundations for Mathematical Logic*" by Willard Van Orman Quine, as a simplification of the **theory of types** of *Principia Mathematica*.

In 1940 and in a revision of 1951 Quine introduced an extension of NF sometimes called "*Mathematical Logic*" or "**ML**", that included proper classes as well as sets.

NF has a universal set, so it's a non-well founded set theory i.e. it's an axiomatic set theory that allows infinite descending chains of membership such as $$\dots x\_n \in x\_{n-1} \in \dots x\_3\in x\_2 \in x\_1$$.

It avoids Russell's paradox by permitting only stratifiable formulae to be defined using the axiom (schema) of comprehension. For instance $$x \in y$$ is a stratifiable formula, but $$x \in x$$ is not.

Much of this entry discusses NFU, an important variant of NF due to Jensen (1969) and exposited in Holmes (1998).
