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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.

##### MSC:
 3e+70 Nonclassical and second-order set theories 3e+35 Consistency and independence results
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##### References:
 [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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