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Models with second order properties in successors of singulars

Published online by Cambridge University Press:  12 March 2014

Rami Grossberg
Rami Grossberg*
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
*
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
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Abstract

Let L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)).

Theorem 1. Let λ be a singular cardinal; assume □λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits p( = p has no support, to be defined in §1), then T has a model of cardinality λ+ in the λ+ interpretation which omits p.

Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ+ such that a formula φ(x) is realized by λ+ elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming λ = λ<λ. It is new when λ is singular or when ∣T∣ = λ is regular.

Theorem 3. Let λ be singular. If Con(ZFC + GCH + ∃κ) [κ is a strongly compact cardinal]), then the following is consistent: ZFC + GCH + the conclusions of all above theorems are false.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 122 - 137
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[BD]Ben-David, S., On Shelah's compactness of cardinals, Israel Journal of Mathematics, vol. 31 (1976), pp. 34–56, 394. MR 80f: 03060.CrossRefGoogle Scholar
[BDG]Ben-David, S. and Grossberg, R., On the strength of gap one principle for successors of singulars, Abstracts of Papers Presented to the American Mathematical Society, vol. 5 (1984), p. 229 (abstract 84T-03-272).Google Scholar
[BDM]Ben-David, S. and Magidor, M., The weak □* is really weaker than the full □, this Journal, vol. 51 (1986), pp. 1029–1033.Google Scholar
[C]Chang, C. C., A note on the two cardinal problem, Proceedings of the American Mathematical Society, vol. 16(1965), pp. 1148–1155. MR 33 # 1238(G. Fodor).CrossRefGoogle Scholar
[CK]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973. MR 53 #12927 (B. Weglorz).Google Scholar
[De]Devlin, K. J., Aspects of constructibility, Lecture Notes in Mathematics, vol. 354, Springer-Verlag, Berlin, 1973. MR 51 # 12527 (F. R. Drake).CrossRefGoogle Scholar
[D]Dodd, A. J., The core model, London Mathematical Society Lecture Note Series, vol. 61, Cambridge University Press, Cambridge, 1982. MR 84a: 03062 (T. J. Jech).CrossRefGoogle Scholar
[G]Gregory, J., Higher Souslin trees and the generalized continuum hypothesis, this Journal, vol. 41 (1976), pp. 663–671. MR 58 # 5208.Google Scholar
[Gr1]Grossberg, R., Omitting types theorem for L(Q), Abstracts of Papers Presented to the American Mathematical Society, vol. 3 (1982), pp. 592–593 (abstract 82T-03-568).Google Scholar
[Gr2]Grossberg, R., Chang Shelah two cardinal theorem for successors of singulars, Abstracts of Papers Presented to the American Mathematical Society, vol. 5 (1984), p. 199 (abstract 84T-03-167).Google Scholar
[Ke]Keisler, H.J., Logic with the quantifier “there exist uncountably many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 1–93. MR 41 #8217 (K. J. Barwise).CrossRefGoogle Scholar
[L]Litman, A., Ph.D. thesis, The Hebrew University, Jerusalem, 1982. (Hebrew; English abstract)Google Scholar
[Sh0]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978. MR 81a: 03030 (D. Lascar).Google Scholar
[Sh1]Shelah, S., Models with second order properties. I, Annals of Mathematical Logic, vol. 14 (1978), pp. 57–72. MR 80b: 03047a (M. Weese).CrossRefGoogle Scholar
[Sh2]Shelah, S., Models with second order properties. II, Annals of Mathematical Logic, vol. 14 (1978), pp. 73–87. MR 80b: 03047b (M. Weese).CrossRefGoogle Scholar
[Sh3]Shelah, S., Models with second order properties. III, Archiv für Mathematische Logik und Grundlagenforschung, vol. 21 (1980), pp. 1–11. MR 83a: 03031 (M. Weese).CrossRefGoogle Scholar
[Sh4]Shelah, S., Models with second order properties. IV, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 183–212. MR 85j: 03056 (M. Weese).CrossRefGoogle Scholar
[Sh5]Shelah, S.et al., Models with second order properties. V, Annals of Pure and Applied Logic (to appear).Google Scholar
[Sh6]Shelah, S., Gap one two-cardinal principle and the omitting type theorem for L(Q), Israel Journal of Mathematics (to appear).Google Scholar
[Va]Vaught, R., A Löwenheim-Skolem theorem for cardinals far apart, Theory of models (Addison, J. W.et al., editors), North-Holland, Amsterdam, 1965, pp. 390–401. MR 35 # 4060 (M. Morley).Google Scholar