On the Consistency of a Slight (?) Modification of Quine’s New Foundations

Quine’s system of set theory, New Foundations (NF), can be conveniently formalized as a first-order theory containing two predicates ≡ (identity) and e (set membership). One of the most attractive features of NF is its simplicity. Apart from the rules and axioms of first-order identity logic, we need only two specifically set theoretical axiom schemes: the extensionality axiom: (Ext) \(\wedge z(z \in x \leftrightarrow z \in y) \to x \equiv y,\) and the set abstraction schema: (Abst) \(\vee y\, \wedge x(x \in y \leftrightarrow \mathfrak{A}),\) where y is not free in \(\mathfrak{A}\) and \(\mathfrak{A}\) is stratified (as usual, we call a formula stratified if indices can be assigned to its variables in such a manner that it becomes a formula of simple type theory).