A Uniformly, Extremely Nonextensional Formula of Arithmetic with Many Undecidable Fixed Points in Many Theories

It is proved that there is a single unary formula F of Peano arithmetic PA and a fixed infinite set S of fixed points ^ of F in PA with the following property. Let T be any recursively enumerable, 2"-sound extension of PA. Then (i) almost all $ in & are undecidable in T, and (ii) for all such ¡j> and all equivalence relations E satisfying reasonable conditions and refining provable equivalence in T (but not depending on or T) there is a sentence \¡/ equivalent to via E which is not a fixed point of F in T. The theorem furnishes an extreme instance of the difficulties encountered in trying to introduce quantification theory into the diagonalizable algebras of Magari, and yet preserve a central theorem about these structures, the De Jongh-Sambin fixed point theorem. The construction is designed for further applications.