Complete Infinitary Type Logics

For each regular cardinal , 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 < . For a fixed , these three versions are, in the order of increasing strength: the local system ₍₎, the global system g₍₎ (the difference concerns the conditions on eigenvariables) and the -system ₍₎ (which has anti-selection terms or Hilbertian -terms, and no conditions on eigenvariables). A full cut elimination theorem is proved for the local systems, and about the -systems we prove that they admit cut-free proofs for sequents in the -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 -systems. It is, however, an open question whether the global systems in themselves admit cut elimination.