On the Expressibility Hierarchy of Magidor-Malitz Quantifiers
Rubin, Matatyahu ; Shelah, Saharon
J. Symbolic Logic, Tome 48 (1983) no. 1, p. 542-557 / Harvested from Project Euclid
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.
Publié le : 1983-09-14
Classification: 
@article{1183741312,
     author = {Rubin, Matatyahu and Shelah, Saharon},
     title = {On the Expressibility Hierarchy of Magidor-Malitz Quantifiers},
     journal = {J. Symbolic Logic},
     volume = {48},
     number = {1},
     year = {1983},
     pages = { 542-557},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1183741312}
}
Rubin, Matatyahu; Shelah, Saharon. On the Expressibility Hierarchy of Magidor-Malitz Quantifiers. J. Symbolic Logic, Tome 48 (1983) no. 1, pp.  542-557. http://gdmltest.u-ga.fr/item/1183741312/