Deduction Systems for Coalgebras Over Measurable Spaces

A theory of infinitary deduction systems is developed for the modal logic of coalgebras for measurable polynomial functors on the category of measurable spaces. These functors have been shown by Moss and Viglizzo to have final coalgebras that represent certain universal type spaces in game-theoretic economics. A notable feature of the deductive machinery is an infinitary Countable Additivity Rule. A deductive construction of canonical spaces and coalgebras leads to completeness results. These give a proof-theoretic characterization of the semantic consequence relation for the logic of any measurable polynomial functor as the least deduction system satisfying Lindenbaums Lemma. It is also the only Lindenbaum system that is sound. The theory is additionally worked out for Kripke polynomial functors, on the category of sets, that have infinite constant sets in their formation.