## Journal of Symbolic Logic

### On the Expressibility Hierarchy of Magidor-Malitz Quantifiers

#### Abstract

We prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for $PC$ classes. Let $M \models Q^nx_1 \cdots x_n \varphi(x_1 \cdots x_n)$ mean that there is an uncountable subset $A$ of $|M|$ such that for every $a_1, \ldots, a_n \in A, M \models \varphi\lbrack a_1, \ldots, a_n\rbrack$. Theorem 1.1 (Shelah) $(\diamond_{\aleph_1})$. For every $n \in \omega$ the class $K_{n + 1} = \{\langle A, R\rangle \mid \langle A, R\rangle \models \neg Q^{n + 1} x_1 \cdots x_{n + 1} R(x_1, \ldots, x_{n + 1})\}$ is not an $\aleph_0$-PC-class in the logic $\mathscr{L}^n$, obtained by closing first order logic under $Q^1, \ldots, Q^n$. I.e. for no countable $\mathscr{L}^n$-theory $T$, is $K_{n + 1}$ the class of reducts of the models of $T$. Theorem 1.2 (Rubin) $(\diamond_{\aleph_1}).^3$. Let $M \models Q^E x y\varphi(x, y)$ mean that there is $A \subseteq |M|$ such that $E_{A, \varphi} = \{\langle a, b \rangle \mid a, b \in A$ and $M \models \varphi\lbrack a, b\rbrack\}$ is an equivalence relation on $A$ with uncountably many equivalence classes, and such that each equivalence class is uncountable. Let $K^E = \{\langle A, R\rangle\mid \langle A, R\rangle\models \neg Q^Exy R(x, y)\}$. Then $K^E$ is not an $\aleph_0$-PC-class in the logic gotten by closing first order logic under the set of quantifiers $\{Q^n \mid n \in \omega\}$ which were defined in Theorem 1.1.

#### Article information

Source
J. Symbolic Logic, Volume 48, Issue 3 (1983), 542-557.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183741312

Mathematical Reviews number (MathSciNet)
MR716614

Zentralblatt MATH identifier
0537.03026

JSTOR