In Set Theory
In New Foundations (NF) and related set theories, a formula in the language of first-order logic with equality and membership is said to be stratified if and only if there is a function which sends each variable appearing in (considered as an item of syntax) to a natural number (this works equally well if all integers are used) in such a way that any atomic formula appearing in satisfies and any atomic formula appearing in satisfies .
It turns out that it is sufficient to require that these conditions be satisfied only when both variables in an atomic formula are bound in the set abstract under consideration. A set abstract satisfying this weaker condition is said to be weakly stratified.
The stratification of New Foundations generalizes readily to languages with more predicates and with term constructions. Each primitive predicate needs to have specified required displacements between values of at its (bound) arguments in a (weakly) stratified formula. In a language with term constructions, terms themselves need to be assigned values under, with fixed displacements from the values of each of their (bound) arguments in a (weakly) stratified formula. Defined term constructions are neatly handled by (possibly merely implicitly) using the theory of descriptions: a term (the x such that ) must be assigned the same value under as the variable x.
A formula is stratified if and only if it is possible to assign types to all variables appearing in the formula in such a way that it will make sense in a version TST of the theory of types described in the New Foundations article, and this is probably the best way to understand the stratification of New Foundations in practice.
The notion of stratification can be extended to the lambda calculus; this is found in papers of Randall Holmes.
Read more about this topic: Stratification (mathematics)
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