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Consistency of strictly impredicative NF and a little more …

  • Sergei Tupailo (a1)

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).

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References

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[1]Boffa, M., ZFJ and the consistency problem for NF, Jahrbuch der Kurt Gödel Gesellschaft, vol. 1 (1988), pp. 102106.
[2]Crabbé, M., On the consistency of an impredicative subsystem of Quine's NF, this Journal, vol. 47 (1982), pp. 131136.
[3]Forster, T. E., Set theory with a universal set, second ed., The Clarendon Press, Oxford, 1995.
[4]Hailperin, T., A set of axioms for logic, this Journal, vol. 9 (1944), pp. 119.
[5]Holmes, M. R., Subsystems of Quine's “New Foundations” with predicativity restrictions, Notre Dame Journal of Formal Logic, vol. 40 (1999), no. 2, pp. 183196.
[6]Jech, T., Set theory, the Third Millennium ed., Springer-Verlag, 2002.
[7]Jensen, R. B., On the consistency of a slight(?) modification of Quine's NF, Synthese, vol. 19 (1969), pp. 250263.
[8]Kunen, K., Set theory. An introduction to independence proofs, Elsevier, 1980.
[9]Quine, W. V., New Foundations for Mathematical Logic, The American Mathematical Monthly, vol. 44 (1937), pp. 7080.
[10]Specker, E. P., Typical ambiguity, Logic, Methodology and Philosophy of Science (Nagel, E., editor), Stanford University Press, Stanford, California, 1962, pp. 116124.
[11]Tupailo, S., Monotone inductive definitions and consistency of New Foundations, Proceedings of logic Colloquium 2005 (Dimitracopoulos, C., Newelski, L., Normann, D., and Steel, J. R., editors), Lecture Notes in Logic, vol. 28, Cambridge University Press, 2008, pp. 255272.
[12]Tupailo, S., NF and indiscernables in ZF, Proceedings of the 70th anniversary NF meeting in Cambridge (Crabbé, M. and Forster, T., editors), Cahiers du Centre de logique, vol. 16, Academia-Bruylant, Louvain-la-Neuve (Belgique), 2009, pp. 99107.

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Consistency of strictly impredicative NF and a little more …

  • Sergei Tupailo (a1)

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