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 NF with urelements (NFU), an important variant of NF due to Jensen and clarified by Holmes. 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. New Foundations is closely related to Russellian unramified typed set theory (TST), a streamlined version of the theory of types of Principia Mathematica with a linear hierarchy of types. The primitive predicates of TST 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.