Abstract

Given a regular cardinal κ such that κ<κ=κ (e.g., if the generalized continuum hypothesis holds), we develop a proof system for classical infinitary logic that includes heterogeneous quantification (i.e., infinite alternating sequences of quantifiers) within the language Lκ+,κ, where there are conjunctions and disjunctions of at most κ many formulas and quantification (including the heterogeneous one) is applied to less than κ many variables. This type of quantification is interpreted in Set using the usual second-order formulation in terms of strategies for games, and the axioms are based on a stronger variant of the axiom of determinacy for game semantics. With respect to this axiom system we prove the soundness and completeness theorem with respect to a class of set-valued structures that we call well determined. We also investigate intuitionistic systems with heterogeneous quantifiers for Lκ+,κ,κ (when only conjunctions of less than κ many formulas are allowed), and prove analogously a completeness theorem with respect to well-determined structures in categories in general, in κ-Grothendieck topoi in particular, and, when κ<κ=κ, also in Kripke models. Finally, we consider an extension of our system in which heterogeneous quantification with bounded quantifiers is expressible, and extend our completeness results to that case.