This paper introduces Hilbert systems for λ-calculus, called sequent combinators, addressing many of the problems of Hilbert systems that have led to the more widespread adoption of natural deduction systems in computer science. This suggests that Hilbert systems, with their uniform approach to meta-variables and substitution, may be a more suitable framework than λ-calculus for type theories and programming languages. Two calculi are introduced here. The calculus SKIn captures λ-calculus reduction faithfully, is confluent even in the presence of meta-variables, is normalizing but not strongly normalizing in the typed case, and standardizes. The sub-calculus SKInT captures λ-reduction in slightly less obvious ways, and is a language of proof-terms not directly for intuitionistic logic, but for a fragment of S4 that we name near-intuitionistic logic. To our knowledge, SKInT is the first confluent, first-order calculus to capture λ-calculus reduction fully and faithfully and be strongly normalizing in the typed case. In particular, no calculus of explicit substitutions has yet achieved this goal.

The λ-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of λ-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is λ-definable. These results subsume earlier ones using cartesian closed categories, as well as those employing so-called Henkin and Kripke λ-models.