The Non-unique Universe

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 Gödel’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.