An instance of Stratified Comprehension
∀ x1…∀ xn∃ y∀ x (x∈ y ↔ φ(x,x1,…,xn))
is called strictly impredicative iff, under minimal stratification, the type of x is 0. Using the technology of forcing, we prove that the fragment of NF based on strictly impredicative Stratified Comprehension is consistent. A crucial part in this proof, namely showing genericity of a certain symmetric filter, is due to Robert Solovay.
As a bonus, our interpretation also satisfies some instances of Stratified Comprehension which are not strictly impredicative. For example, it verifies existence of Frege natural numbers.
Apparently, this is a new subsystem of NF shown to be consistent. The consistency question for the whole theory NF remains open (since 1937).
"Consistency of strictly impredicative NF and a little more …." J. Symbolic Logic 75 (4) 1326 - 1338, December 2010. https://doi.org/10.2178/jsl/1286198149