Journal of Symbolic Logic

Consistency of strictly impredicative NF and a little more …

Sergei Tupailo

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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).

Article information

J. Symbolic Logic, Volume 75, Issue 4 (2010), 1326-1338.

First available in Project Euclid: 4 October 2010

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Tupailo, Sergei. Consistency of strictly impredicative NF and a little more …. J. Symbolic Logic 75 (2010), no. 4, 1326--1338. doi:10.2178/jsl/1286198149.

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