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 pre-measurement uncertainty 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 a probability measure. A universe that is a set of universes that are macroscopically isomorphic and span all possible configurations of local beäbles fulfills that role. If objective probability is a property of branches, then a successful Deutsch–Wallace decision-theoretic argument would justify the Principal Principle and be part of probability theory rather than specific to many-worlds theory. 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 the many-interacting-worlds theory, but the observer inhabits the set of worlds, not an individual world. An observer has many elemental bodies.

Personal Identity and Uncertainty in the Everett Interpretation of Quantum Mechanics

“…Moreover, Papineau [17] and Tappenden [18] reject Ignorance as the foundation for understanding probability in EQM. (Instead, in a recent publication, Tappenden [19] embraces pre-measurement uncertainty. However, it bears more similarity to Tappenden's previous approach that rejects Ignorance, and remains to be justified whether it truly qualifies as a "pre-measurement uncertainty" approach.…”

“…To assert that there are two persons present before the branching simply means that there are two different ways to combine this particular three-dimensional person-stage with other person-stages to constitute a four-dimensional person. (Tappenden [19] misconstrues Saunders and Wallace's approach as it "reject(s) the concept of splitting, which is arguably Everett's key idea". Everett is not concerned about personhood or personal identity.…”

The Many-Worlds Interpretation of Quantum Mechanics: Current Status and Relation to Other Interpretations

“…Tappenden [23] instead suggests ... interpreting the universal wavefunction as representing a set of interacting deterministic universes which contain microscopic local beables. Objects in our environment become sets of objects which are macroscopically isomorphic but differ in their microscopic configurations.…”