Complete Infinitary Type Logics

Abstract

For each regular cardinal κ, we set up three systems of infinitary type logic, in which the length of the types and the length of the typed syntactical constructs are < κ. For a fixed κ, these three versions are, in the order of increasing strength: the local system Σ_(κ), the global system gΣ_(κ) (the difference concerns the conditions on eigenvariables) and the τ-system τΣ_(κ) (which has anti-selection terms or Hilbertian τ-terms, and no conditions on eigenvariables). A full cut elimination theorem is proved for the local systems, and about the τ-systems we prove that they admit cut-free proofs for sequents in the τ-free language common to the local and global systems. These two results follow from semantic completeness proofs. Thus every sequent provable in a global system has a cut-free proof in the corresponding τ-systems. It is, however, an open question whether the global systems in themselves admit cut elimination.