New Foundations

In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled ""New Foundations for Mathematical Logic""; hence the name. Much of this entry discusses NFU, an important variant of NF due to Jensen (1969) and exposited in Holmes (1998). In 1940 and 1951 Quine introduced an extension of NF sometimes called ""Mathematical Logic"" or ""ML"", that included classes as well as sets. New Foundations has a universal set, so it is a non well founded set theory. That is to say, it is a logical theory that allows infinite descending chains of membership such as…xn ∈ xn-1 ∈ …x3 ∈ x2 ∈ x1. It avoids Russell's paradox by only allowing stratifiable formulae in the axiom of comprehension. For instance x ∈ y is a stratifiable formula, but x ∈ x is not (for details of how this works see below).