Set Theory and Many Worlds

Abstract

The 2022 Tel Aviv conference on the Many Worlds interpretation of quantum mechanics highlighted many differences between theorists. A very significant dichotomy is between Everettian fission (splitting) and Saunders-Wallace-Wilson divergence. For fission, an observer may have multiple futures, whereas for divergence they always have a single future. Divergence was explicitly introduced to resolve the problem of pre-measurement uncertainty for Everettian theory, which is universally believed to be absent for fission. Here, I maintain that there is indeed uncertainty about future observations prior to fission, so long as objective probability is a property of Everettian branches. This is made possible if the universe is a set and branches are subsets with probability measure. A universe which is a set of universes which are macroscopically isomorphic and span all possible configurations of microscopic local be\"ables fulfils that role. If objective probability is a property of branches, a successful Deutsch-Wallace decision-theoretic argument would justify the Principal Principle and be part of probability theory rather than being specific to Many Worlds. Any macroscopic object in our environment becomes a set of isomorphs with different microscopic configurations, each in an elemental universe (elemental in the set-theoretic sense). This is similar to Many Interacting Worlds theory but the observer inhabits the set of worlds, not an individual world. An observer has many elemental bodies.