The non-unique Universe

TL;DR

This paper explores the logical and mathematical conditions under which a multiverse is necessary in physics, discusses the implications of Godel's theorem for a final theory, and classifies types of multiverses based on model theory.

Contribution

It provides a logical framework for understanding multiverse existence, linking mathematical logic with physical theories and classifying multiverses using model theory.

Findings

The universe's structure might be an undecidable theory, limiting epistemology but not ontologically.

Two types of multiverses are identified: elementarily equivalent and inequivalent classes.

A unique final theory would require a finite model, negating multiverse possibilities.

Abstract

The purpose of this paper is to elucidate, by means of concepts and theorems drawn from mathematical logic, the conditions under which the existence of a multiverse is a logical necessity in mathematical physics, and the implications of Godel's incompleteness theorem for theories of everything. Three conclusions are obtained in the final section: (i) the theory of the structure of our universe might be an undecidable theory, and this constitutes a potential epistemological limit for mathematical physics, but because such a theory must be complete, there is no ontological barrier to the existence of a final theory of everything; (ii) in terms of mathematical logic, there are two different types of multiverse: classes of non-isomorphic but elementarily equivalent models, and classes of model which are both non-isomorphic and elementarily inequivalent; (iii) for a hypothetical theory of everything to have only one possible model, and to thereby negate the possible existence of a multiverse, that theory must be such that it admits only a finite model.