A reduction of the NF consistency problem

Abstract

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.