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On the expressibility hierarchy of Magidor-Malitz quantifiers

Published online by Cambridge University Press:  12 March 2014

Matatyahu Rubin  and
Saharon Shelah
Matatyahu Rubin
Affiliation:
Ben Gurion University of the Negev, Beer Sheva, Israel
Saharon Shelah
Affiliation:
The Hebrew University of Jerusalem, Jerusalem, Israel
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Abstract

We prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for PC classes.

Let M ⊨ Qnx1 … xnφ(x1 … xn) mean that there is an uncountable subset A of ∣M∣ such that for every a1 …, an ∈ A, M ⊨ φ[a1, …, an].

Theorem 1.1 (Shelah) (♢ℵ1). For every n ∈ ωthe classKn+1 = {‹A, R› ∣ ‹A, R› ⊨ ¬ Qn+1x1 … xn+1R(x1, …, xn+1)} is not an ℵ0-PC-class in the logic ℒn, obtained by closing first order logic underQ1, …, Qn. I.e. for no countable ℒn-theory T, isKn+1the class of reducts of the models of T.

Theorem 1.2 (Rubin) (♢ℵ1). Let M ⊨ QE x yφ(x, y) mean that there is A ⊆ ∣M∣ such thatEA, φ = {‹a, b› ∣ a, b ∈ A and M ⊨ φ[a, b]) is an equivalence relation on A with uncountably many equivalence classes, and such that each equivalence class is uncountable. Let KE = {‹A, R› ∣ ‹A, R› ⊨ ¬ QExyR(x, y)}. Then KE is not an ℵ0-PC-class in the logic gotten by closing first order logic under the set of quantifiers {Qn ∣ n ∈ ω) which were defined in Theorem 1.1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 542 - 557
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[F]Fodor, G., Proof of a conjecture of P. Erdos, Acta Universitatis Szegediensis, Acta Scientiarum Mathematicarum, vol. 14 (1952), pp. 219–227.Google Scholar
[G]Garavaglia, S., Relative strength of Malitz quantifiers, Notre Dame Journal of Formal Logic, vol. 19 (1978), pp. 495–503.CrossRefGoogle Scholar
[H]Hajnal, A., Proof of a conjecture of S. Ruziewicz, Fundamenta Mathematicae, vol. 50 (1961), pp. 123–128.CrossRefGoogle Scholar
[Kr]Keisler, H. J., Logic with the quantifier “there exist uncountably many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 1–93.CrossRefGoogle Scholar
[KT]Kunen, K. and Tall, F.Google Scholar
[Ma, Re]Malitz, J. and Reinhardt, W., Maximal models in the language with quantifier “there exist uncountably many”, Pacific Journal of Mathematics, vol. 40 (1972), pp. 139–155.CrossRefGoogle Scholar
[MM]Magidor, M. and Malitz, J., Compact extensions of L(Q)(part 1a), Annals of Mathematical Logic, vol. 2 (1977), pp. 217–261.CrossRefGoogle Scholar
[MR]Malitz, J. and Rubin, M., Fragments of higher order logic, Mathematical Logic in Latin America, North-Holland Studies in Logic, vol. 99, North-Holland, Amsterdam, 1980, pp. 219–238.Google Scholar
[Ra]Rabin, M.O., Arithmetical extensions with prescribed cardinality, Koninklijke Nederlandse Akademie von Wetenschappen, Proceedings, Series A, Mathematical Sciences, vol. 21 (1959), pp. 439–446.Google Scholar
[RS]Rubin, M. and Shelah, S., Combinatorial problems on trees. Partitions, ⊿-systems and large free subsets, Annals of Mathematical Logic (to appear).Google Scholar
[Sl]Shelah, S., Models with second order properties. III, Omitting types in λ+ for L(Q), Proceedings of the “Berlin Workshop in Logic July 1977”, Archiv fur Mathematische Logik und Grundlagenforschung, vol. 21 (1980), pp. 1–11.Google Scholar