Consistency of strictly impredicative NF and a little more

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. ∗Acknowledgements: (1) Research partly supported by the Estonian Science Foundation grant ETF7134 and the Institute of Cybernetics at TUT, Tallinn, Estonia; (2) The main body of this work was done at Stanford University, U.S.A., during 2007-08 academic year. The visit was made possible by the Fulbright grant # 07-31122. The result was reported to a conference ”NF in the Bay Area”, Stanford University, June 25–27, 2008, http://www.cs.ioc.ee/∼sergei/nfbayarea08.html; (3) Research partly supported by the U.K. Royal Society International Joint Projects award 2007/R1; (4) Anonymous referee report and suggestions are gratefully acknowledged.