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 NF with urelements (NFU), an important variant of NF due to Jensen (1969) and clarified by Holmes (1998). In 1940 and in a revision in 1951, Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included proper 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 an axiomatic set theory that allows infinite descending chains of membership, such as … xn ∈ xn-1 ∈ … ∈ x2 ∈ x1. It avoids Russell's paradox by permitting only stratifiable formulas to be defined using the axiom schema of comprehension. For instance, x ∈ y is a stratifiable formula, but x ∈ x is not. The Type Theory TST The primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality () and membership (). TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-) natural number n, type n+1 objects are sets of type n objects; sets of type n have members of type n-1. Objects connected by identity must have the same type. When writing formulas in a many-sorted theory such as TST, some annotations are usually added to variables to denote their types. In TST it is customary to write the type indices as superscripts: denotes a variable of type n. Thus the following two atomic formulas succinctly describe the typing rules: and . (Quinean set theory seeks to eliminate the need to write out these type indices explicitly.) The axioms of TST are: Extensionality: sets of the same (positive) type with the same members are equal; An axiom schema of comprehension, namely: If is a formula, then the set exists. In other words, given any formula , the formula is an axiom where represents the set and is not free in . This type theory is much less complicated than the one first set out in the Principia Mathematica, which included types for relations whose arguments were not necessarily all of the same types. In