Completeness of infinitary heterogeneous logic

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 alternate sequences of quantifiers) within the language $\mathcal{L}_{κ^+, κ}$, where there are conjunctions and disjunctions of at most $κ$ any formulas and quantification (including the heterogeneous one) is applied to less than $κ$ many variables. This type of quantification is interpreted in $\mathcal{S}et$ 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. Within this axiom system we prove the soundness and completeness theorem with respect to a class of set-valued structures that we call well-determined. Although this class is more restrictive than the class of determined structures in Takeuti's determinate logic, the completeness theorem works in our case for a wider variety of formulas of $\mathcal{L}_{κ^+, κ}$, and the category of well-determined models of heterogeneous theories is accessible. Our system is formulated within the sequent style of categorical logic and we do not need to impose any specific requirements on the proof trees, disregarding thus the eigenvariable conditions needed in Takeuti's system. We also investigate intuitionistic systems with heterogeneous quantifiers and prove analogously a completeness theorem with respect to well-determined structures in categories in general, in $κ$-Grothendieck toposes 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.