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A reduction of the NF consistency problem. (English) Zbl 1116.03046
Summary: 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 \(n\geq 1\) 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.

03E70 Nonclassical and second-order set theories
03E35 Consistency and independence results
Full Text: DOI
[1] Proceedings of the International Congress, Stanford, 1960 pp 116–124– (1962)
[2] Set theory with a universal set 20 (1992)
[3] Soviet Mathematics Doklady 10 pp 1387–1390– (1969)
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