The consistency of some 4-stratified subsystem of NF including NF3

As is well known, NF is a first-order theory whose language coincides with that of ZF. The nonlogical axioms of the theory are: Extensionality. (x)(y)[(z)(z ∈ x ↔ z ∈ y) → x = y].Comprehension. (Ex)(y)(y ∈ x ↔ ψ) for every stratified ψ in which x does not occur free (a formula of NF is said to be stratified if it can be turned into a formula of the simple theory of types by adding type indices (natural numbers ≥ 0) to its variables).Before stating our result, a few preliminaries are in order. Let T be the simple theory of types. If ψ is a formula of T, we denote by ψ+ the formula obtained from ψ by raising all type indices by 1. T* is the result of adding to T every axiom of the form ψ ↔ ψ+. A formula of T is n-stratified (n > 0) if it does not contain any type index ≥ n. A formula of NF is n-stratified if it can be turned into an n-stratified formula of T by adding type indices to its variables. (In practice, we shall allow ourselves to confuse an n-stratified formula of T with the corresponding n-stratified formula of NF). For n > 0, Tn (resp. ) is the subtheory of T (resp. T*) containing only n-stratified formulae. For n > 0, NFn is the subtheory of NF generated by those axioms of NF which are n-stratified. Let = 〈M0, M1,…,=, ∈〉 be a model of T.