This is an imaginary conversation between me and a Bayesian. His answers are in parenthesis. Do you find this line of argument convincing?
Consider all possible worlds consistent with your memories and current experiences. In other words, all possible worlds that contain at least one observer with memories and current experiences exactly identical to yours. Are there more than one such world?
Is every one of these worlds isomorphic to some mathematical structure?
(How do you define "mathematical structure"?)
A set class.
Why not a category? It can be bigger. Why not the category CAT of all categories? It is much bigger. You will meet here the problem of defining mathematically the class of all mathematical structures. A very old insoluble problem ... It is one of the major problem in Tegmark approach.
The other problem which I see in your argument, and which is common in both Tegmark and Schmidhuber (and not mentionned in Hal Finney recent answer to your post) is that you are implicitly associating mind and structure/universe using some form of psycho-parallelism. Such association are incompatible with just quantum mechanics without collapse (I think Zeh has seen this point). Actually such association is completely forbidden with only the comp hyp as I have argued at length before. Even the ontologically large modal realism of David Lewis makes such association at least not-obvious. Keeping just the class of all sets is also ambiguous by itself: which theory of sets will you choose? If you take a theory with an extensionality axiom (where sets are defined completely by their elements)? In that case I don't know any "physical" object which could be seen as a set. Do you accept the axiom of choice for non countable sets? Are you allowing higher infinities? Which one? More generally, how will you relate the worlds and the sets? Is a chair described by its wave function or by more palatable observer memories entangled with that wave function? How do you intend to relate first and third person point of view?
If you go back and look at how those principles of reasoning were derived or justified, it was on the basis of simplicity and avoiding absurd actions ("absurd" being defined by intuition or common sense). The assumption that the actual world is the class of all sets is equally justified on the basis of avoiding absurd actions and is simpler than having a prior over possible worlds, so why not?
I have no clues what you mean by "absurd" here.