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A Reduction of the NF consistency Problem

  • Athanassios Tzouvaras (a1)


We give a necessary and sufficient condition in order that a type-shifting automorphism be constructed on a model of the Theory of Simple Types (TST) by forcing. Namely it is proved that, if for every n ≥ 1 there is a model of TST in the ground model M of ZFC that contains an n-extendible coherent pair, then there is a generic extension M[G] of M that contains a model of TST with a type-shifting automorphism, and hence M[G] contains a model of NF. The converse holds trivially. It is also proved that there exist models of TST containing 1-extendible coherent pairs.



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[1]Forster, T. E., Set theory with a universal set, Oxford Logic Guides, vol. 20, Oxford University Press, 1992.
[2]Grishin, V. N., Consistency of a fragment of Quine's NF system, Soviet Mathematics Doklady, vol. 10 (1969), no. 6, pp. 1387–1390.
[3]Grishin, V. N., The method of stratification in set theory, Ph.D. thesis, Moscow University, 1972, in Russian.
[4]Specker, E., Typical ambiguity, Proceedings of the International Congress, Stanford, 1960, Logic Methodology and Philosophy of Science, Stanford University Press, 1962, pp. 116–124.



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