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

Author:
Robert A. Di Paola

Journal:
Proc. Amer. Math. Soc. **91** (1984), 291-297

MSC:
Primary 03G25; Secondary 03F30

MathSciNet review:
740189

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

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0740189-9

Article copyright:
© Copyright 1984
American Mathematical Society