# Quine's New Foundations (NF)

[https://www.wikiwand.com/en/New\_Foundations](https://www.wikiwand.com/en/New_Foundations) [https://plato.stanford.edu/entries/quine-nf/](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).
