A complete axiomatization of infinitary first-order intuitionistic logic over Lκ+,κ

Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa }$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel in [8] for intuitionistic logic over $L_{{\omega }_1,\omega }$. When κ is a regular cardinal such that ${\kappa }^{<\kappa }=\kappa$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa ,\kappa }$, the language with disjunctions of at most κ formulas, conjunctions of less than κ formulas and quantification on less than κ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.