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operators and alternating sentences in arithmetic

  • Larry Manevitz (a1) and Jonathan Stavi (a1)

Extract

Determining the truth value of self-referential sentences is an interesting and often tricky problem. The Gödel sentence, asserting its own unprovability in P (Peano arithmetic), is clearly true in N(the standard model of P), and Löb showed that a sentence asserting its own provability in P is also true in N (see Smorynski [Sm, 4.1.1]). The problem is more difficult, and still unsolved, for sentences of the kind constructed by Kreisel [K1], which assert their own falsity in some model N* of P whose complete diagram is arithmetically defined. Such a sentence χ has the property that N ⊨ iff N* ⊭ χ (note that ¬χ has the same property).

We show in §1 that the truth value in N of such a sentence χ, after a certain normalization that breaks the symmetry between it and its negation, is determined by the parity of a natural number, called the rank of N, for the particular construction of N* used. The rank is the number of times the construction can be iterated starting from N and is finite for all the usual constructions. We also show that modifications of, e.g., Henkin's construction (in his completeness proof of predicate calculus) allow arbitrary finite values for the rank of N. Thus, on the one hand the truth value of χ in N, for a given “nice” construction of N*, is independent of the particular (normalized) choice of χ, and we shall see that χ is unique up to (provable) equivalence in P. On the other hand, the truth value in question is sensitive to minor changes in the definition of N* and its determination seems to be largely a combinatorial problem.

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[EK]Ehrenfeucht, A. and Kreisel, G., Strong models of arithmetic, Bulletin de l'Académie Polonaise des Sciences, vol. 14 (1966), pp. 107110.
[F]Friedman, H., Uniformly defined descending sequences of degrees, this Journal, vol. 41 (1976), pp. 363367.
[K1]Kreisel, G., Note on arithmetic models for consistent formulae of the predicate calculus, Fundamenta Mathematicae, vol. 37 (1950), pp. 265285.
[K2]Kreisel, G., Survey of proof theory, this Journal, vol. 33 (1968), pp. 321388.
[K.T]Kreisel, G. and Takeuti, G., Formally self-referential propositions in cut-free classical analysis and related systems, Dissertationes Mathematicae, no. 118, pp. 150.
[Sc]Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Dekker, J.C.E., Editoi), Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, R.I., 1962.
[Sm]Smorynski, C., The incompleteness theorems, Handbook of mathematical logic (Barwise, K.J., Editor), North-Holland, Amsterdam, 1977, pp. 821866.

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