There is a lot of interesting debate over whether a "theory of everything" (ToE) is allowed to exist in the mathematical sense, see Does Gödel preclude a workable ToE?, Final Theory in Physics: a mathematical existence of proof? and Arguments Against a Theory of Everything. Since as far as I can tell this is still an open question, let's assume for now that formulating a ToE is possible. I'm curious about what can be said about the uniqueness of such a theory.
Now the hard part, where I'm pretty sure I'm about to back myself into a logical corner. To clarify what I mean by unique: physical theories are formulated mathematically. It is possible to test whether two mathematical formalisms are equivalent (right? see bullet point 3 below). If so, it is possible to test the mathematical equivalence of various theories. From the Bayesian point of view, any theory which predicts a set of observables is equally valid, but the degree of belief in a given theory is modulated by observations of the observables and their associated errors. So now consider the set of all possible formulations of theories predicting the set of all observables - within this set live subsets of mathematically equivalent formulations. The number of these subsets is the number of unique ToEs. Now the question becomes how many subsets are there?
- It can be proven that if a ToE exists, it is necessarily unique ($1$ subset).
- It can be proven that if a ToE exists, it is necessarily not unique ($>1$ subset).
- It can be proven that it is impossible to say anything about the uniqueness of a ToE, should it exist (it is impossible to test mathematical equivalence of theories).
- We don't know if we can say anything about the uniqueness of a ToE, should it exist.
So this is really asking about the ensemble of closed mathematical systems (physical theories) of an arbitrary (infinite?) number of variables (observables). This is honestly a pure math question, but here strongly physically motivated.
I suspect the answer is probably the fourth bullet, but surely there has been some research on the topic? Hopefully someone familiar with the ToE literature can shed some light on the question.