The nonaxiomatizability of $L(Q^2_{\aleph_1})$ by finitely many schemata.
Saharon Shelah, Charles I. Steinhorn · Notre Dame Journal of Formal Logic · 1989
Under set-theoretic hypotheses, it is proved by Magidor and Malitz that logic with the Magidor-Malitz quantifier in the Ki -interpretation is recursively axiomatizable.It is shown here, under no additional settheoretic hypotheses, that this logic cannot be axiomatized by finitely many schemata.Magidor and Malitz [2] introduced the ^-variable-binding quantifiers Q n .The language L(Q") is formed by adding Q n to first-order predicate logic.For an infinite cardinal K, Q n X\X 2 .. .xn φ may be assigned the so-called /c-interpretation in a structure ΐPίί, wherein Q n X\.. .xn φ is satisfied if there exists anAc ΐPίί of power K that is homogeneous for φ, i.e., for any a u ... ,a n E A, φ(ctι,... 9 a n ) holds in 9K.Among many other results Magidor and Malitz establish, under the set-theoretic axiom 0 Kl , a completeness theorem for L(Q") in the K t -interpretation (hereafter LίQ^)).Unfortunately, the complete axiom system for L(Qκj) exhibited in [2] lacks the simplicity of, e.g., Keisler's set of axioms for L(Q^) (cf.[1]).This paper, a sequel to [3], demonstrates that this failure of simplicity is not without reason.It will be shown here, without additional set-theoretic hypotheses, that LίQ^) cannot be axiomatized by finitely many schemata.Even more strongly, we prove: Theorem 1No collection of axiom schemata of bounded quantifier depth suffices to axiomatize L(Q^).