13:42 A modality is a Monad on Propositions (and the theory of a monad is decidable! See Oracle modalities by Andrew W Swan and Andrej Bauer and Ronald Brown On Monads and Groupoids and Alexander Grothendieck on his idea for a Science Fiction Novel on Motives ). To understand this idea you need to be able to understand Kleene's notion of Realizability and Hyland's Effective Topos which depends heavily upon Tripos Theory , a tripos here is "a topos-representing, indexed, pre-ordered set". Kleene's attempts to interpret intuitionistic logic were all more than a little weird, I suspect he had dubious motives. See Markov's principle and Realizability . Intuitionistic logic is in a sense strictly more expressive than classical logic. See Emily Riehl on Univalent Foundations . Subscribe to Types and Topology Workshop . Dr. Martin Escardo on Searchable sets and ordinals in system T given at a six month 2012 programme called...