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It is proved that there is a single unary formula F of Peano arithmetic PA and a fixed infinite set S of fixed points 0 of F in PA with the following property. Let T be any recursively enumerable, lo-sound extension of PA. Then (i) almost all 4 in S are undecidable in T, and (ii) for all such 4 and all equivalence relations E satisfying reasonable conditions and refining provable equivalence in T (but not depending on 4 or T) there is a sentence 4 equivalent to 4 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.

A uniformly, extremely nonextensional formula of arithmetic with many undecidable fixed points in many theories