The nonaxiomatizability of \(L(Q^ 2_{\aleph_ 1})\) by finitely many schemata (Q918971)

The logic \(L(Q^ n_{\aleph_ 1})\) is obtained from first order logic by adding the n-ary quantifier \(Q^ n_{\aleph_ 1}x_ 1,...,x_ n\mu\) saying that there exists a set of cardinality \(\aleph_ 1\) homogeneous for \(\mu\). Under suitable set theoretic hypotheses, Magidor and Malitz established a completeness theorem for \(L(Q^ n_{\aleph_ 1})\). In the present paper, without additional set theoretic hypotheses, it is shown that \(L(Q^ 2_{\aleph_ 1})\) cannot be axiomatized by any collection of axiom schemata of bounded quantifier depth.