Gödel showed the connection between the Lewis Modal System S4 and Intuitionistic Logic. Many have written about an algebraic formulation using as models topological spaces with the interior operator as necessitation, and the open subsets as modeling intuitionistic propositional logic. Topological models were also used to model many first-order logics. After Cohen's proofs were recast using Boolean-valued models, topological models for modal higher-order logic have been studied. For Boolean-valued logic, the complete Boolean algebra M = Meas([0,1])/Null of measurable subsets of [0, 1] modulo sets of measure zero gives every proposition a probability. Note that the measure algebra also carries a nontrivial S4 modality defined via the sublattice Open([0,1])/Null of open sets modulo null sets. Working by analogy to the Boolean-valued models for ZF, we construct over M a model for a modal ZF (MZF) where membership and equality predicates have interesting modal properties, and where real numbers correspond to random variables. Some theorems of Ergodic Theory then become principles about the proposed MZF.

Bio: Dana Stewart Scott is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. His research career has spanned computer science, mathematics, and philosophy, and has been characterized by a marriage of a concern for elucidating fundamental concepts in the manner of informal rigor, with a cultivation of mathematically hard problems that bear on these concepts. His work on automata theory earned him the ACM Turing Award in 1976, while his collaborative work with Christopher Strachey in the 1970s laid the foundations of modern approaches to the semantics of programming languages. He has worked also on modal logic, topology, and category theory. He is the editor-in-chief of the new journal Logical Methods in Computer Science.