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Some highly saturated models of Peano arithmetic

  • James H. Schmerl (a1)

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Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.

Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal and is a λ-saturated model of PA such that ∣M∣ > λ, then has an elementary extension of the same cardinality which is also λ-saturated and which, in addition, is rather classless. The construction in [10] produced a model for which cf() = λ+. We asked in Question 5.1 of [10] what other cofinalities could such a model have. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].

Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.

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[1]Chang, C. C. and Keisler, H. J., Model theory, 3rd ed., North-Holland, Amsterdam, 1990.
[2]Gregory, J., Higher Souslin trees and the generalized continuum hypothesis, this Journal, vol. 41 (1976), pp. 663671.
[3]Jin, R., Cuts in the hyperfinite time lines, this Journal, vol. 57 (1992), pp. 522527.
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[6]Kaye, R., Models of Peano arithmetic, Oxford University Press, Oxford, 1991.
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[9]Schmerl, J. H., Recursively saturated, rather classless models of Peano arithmetic, Logic year 1979–80 (Lerman, M.et al., editor), Lecture Notes in Mathematics, no. 859, Springer-Verlag, Berlin, 1981, pp. 268282.
[10]Schmerl, J. H., Rather classless, highly saturated models of Peano arithmetic, Logic colloquium '96 (Larrazabal, J. M.et al., editor), Lecture Notes in Logic, no. 12, Springer-Verlag, Berlin, 1998, pp. 237246.
[11]Shelah, S., Vive la Difference II – the Ax-Kochen isomorphism theorem, Israel Journal of Mathematics, vol. 85 (1994), pp. 351390.

Some highly saturated models of Peano arithmetic

  • James H. Schmerl (a1)

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