Stratified Formulas Are Not Context-Free

Stratified formulas were introduced by Quine as an alternative way to attack Russell’s paradox. Instead of limiting comprehension by size (as in ZF set theory, using its axiom scheme of separation), unlimited comprehension is given to formulas that are in some sense descended from formulas of typed set theory. By keeping variables in a stratified structure, the most common candidates for inconsistency such as { x|x ∉ x } are eliminated. Under the usual syntax of set theory, the set of stratified formulas form a formal language. We show that, unlike the full class of well-formed formulas of set theory, this language is not context-free, and extend the result to its complement. Therefore, much like the axioms of PA and ZF (under their usual axiomatizations), the theory NF (New Foundations) as a formal language is not context-free. We then introduce a nonstandard syntax of set theory and show that with this syntax there is a restricted class of formulas, the exo-stratified formulas, that is context-free and full (up to relabeling of variables).