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Consistency of strictly impredicative NF and a little more …

Published online by Cambridge University Press:  12 March 2014

Sergei Tupailo
Affiliation:
Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Avenida Professor Gama Pinto 2, 1649-003 Lisboa, Portugal. E-mail: sergei@cs.ioc.ee
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Abstract

An instance of Stratified Comprehension

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).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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