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On the expressibility hierarchy of Magidor-Malitz quantifiers

Published online by Cambridge University Press:  12 March 2014

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


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

Let MQnx1xnφ(x1xn) mean that there is an uncountable subset A of ∣M∣ such that for every a1 …, anA, Mφ[a1, …, an].

Theorem 1.1 (Shelah) (♢ℵ1). For every nωthe classKn+1 = {‹A, R› ∣ ‹A, R› ⊨ ¬ Qn+1x1xn+1R(x1, …, xn+1)} is not an0-PC-class in the logicn, obtained by closing first order logic underQ1, …, Qn. I.e. for no countablen-theory T, isKn+1the class of reducts of the models of T.

Theorem 1.2 (Rubin) (♢ℵ1). Let MQE x yφ(x, y) mean that there is A ⊆ ∣Msuch thatEA, φ = {‹a, b› ∣ a, bA and Mφ[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.

Research Article
Copyright © Association for Symbolic Logic 1983

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