I don't think you are taking seriously what I mean by "sensory data for which the simplest possible explanation is X".
The simplest possible explanation cannot be something self-contradictory; a self-contradictory statement is not an explanation, any more than a random series of grunts is an explanation.
You cannot imagine sensory data in which Haltcast solves the halting problem, for two reasons: 1) you cannot imagine running Haltcast over "all" programs, because that would take an infinite amount of time; you can only running Haltcast over "a whole bunch" of programs, which would not prove anything; and 2) if you imagine running Haltcast over itself and it returning "yes, I would halt", then the simplest explanation is not that you have solved the halting problem, but that Haltcast has a bug.
You cannot actually imagine, in detail, working through a known mathematical proof and coming to a different conclusion without making a mistake. You can imagine imagining it, in a fuzzy lyrical way, maybe, but that's not what I mean by "sensory data for which the simplest possible explanation is X".
Similarly, you cannot imagine computing "kilroy was here" in pi any more than you can imagine writing down "1 + 1 =" and then writing down "3" and being right. That is, of course it's easy to imagine if you simply redefine 3 (or make up an arbitrary encoding system for which the digits of pi do encode "kilroy was here"); but that is just imagining yourself cheating.
This is because math is a game. Imagining the same game, played the same way, yielding different results, is not imagining something empirically wrong; it is imagining nonsense.
Math does not say anything about the physical universe unless it does, if you see what I mean. You cannot look at math and say "ah, now I know that such-and-such an event in the physical universe is impossible." You can only look at math and say "ah, now I realize that if such-and-such an event in the physical universe occurs, this would be the wrong math to describe it."
If you can imagine *making an observation*, then science must be broad enough to encompass the possibility that you *will in fact make that observation*. You cannot imagine making an observation that 1=2, or of solving the Halting Problem, because these are not *observations* -- they are incorrect (because self-contradictory) *conclusions* about observations. Other than being complex, imagining a computer program seeming to solve the Halting Problem leading you to conclude that Turing machines can solve the Halting Problem (and therefore that all Cretans really are liars and that the set of all sets that do not contain themselves contains itself, since these problems are all isomorphic), is not any different than seeing conjoined twins and concluding that 1=2.
Observations do not prove anything about math, other than, perhaps, that you are using the wrong math. When we observe cosmic microwave background radiation that suggests that the universe is flat, when we thought it was round, we do not say "oh, the math must be wrong, flat must really mean round." No, the math is fine; we were using the wrong math to describe physical reality. Roundness has not changed any; it just turns out we are living in flatness.
If one believes otherwise -- if one thinks that just by looking at the math, without the previous assumption that this particular math happens to fit this universe, one can prove that certain facts of physical reality cannot be true (because, presumably, it has been revealed to one which math is the special privileged math) then one is living on the island of Hayy Ibn Yaqzan.
Now, it may be that I am misreading Tegmark in a sense, and that he really means some subset of *all possible* mathematical models, so that it is not merely things which are self-contradictory which are excluded, but some other set of things, so that if I posit a particular exception such as "on Tuesdays in Idaho", etc., that may turn out to be excluded by the constraints on his definition of "mathematical model". I doubt it, but it doesn't matter.
Here's why. Consider this equation:
F = MA
I assume you accept that this is a formalizable equation. :-) But note that it has some "externals", if you will. Take "mass". Mass is not derived from other things in Newtonian mechanics. It is simply a quality that things have. You measure it, and plug it into the equations. It is, if you will, arbitrary.
Now consider this:
F = MA + K(d^(1/2))
Where K is an arbitrary constant and d is the distance of the center of mass from a "special point" at the center of the universe (so that the force on an object is equal to mass times acceleration plus a constant times the square root of the distance from the center of the universe).
The "special point",like mass, is an external -- it is axiomatic. It does not arise from the theory any more than mass does. The second equation happens to be wrong. But it seems absurd to me to say that it is "not physics". There is nothing self-contradictory about it. A contemporary of Newton could have proposed this equation, and it would have been a coherent, falsifiable, wrong hypothesis.
Note that if K is very, very small, Newton's contemporaries would probably not have been able to make the determination for lack of appropriate data (especially if they were far from the center of the universe).
Now consider this equation:
F = MA + K(d^(1/2)) + K2(d2^(1/3))
Here the above equation is adjusted by K2 (a constant) times the cube root of d2, which is the distance to another special point, this one hovering somewhere near Jupiter.
This equation, surely, is equally admissable as a mathematical model.
Now consider adding a fourth root term, a fifth root term, and so on, so that the equation becomes an arbitrarily long polynomial relating an arbitrary number of special points in space to the fundamental laws of nature.
We could be living in such a universe, especially if the "extra term" effects are very small.
Occam's razor certainly suggests that we are not. But there is no Occam's razor, at least on a local-universe level, in Tegmark's model. Tegmark's multiverse, by including all non-self-contradictory, admissible mathematical models, contains an infinite number of universes in which each coherent simple equation of the form F = MA is adjusted by an arbitrary number of wacky outlier terms such as the above.
(If for some reason you think adding such terms to F=MA would break the relationship with the other laws, rendering the whole self-contradictory -- though I can't see why, since the other laws would be derived from this first law anyway -- you can imagine that instead of that, the gravitational constant G is a function of the distance to an arbitrary number of special points. Slightly weaker, but makes the point just as well).
Occam's razor is the foundation of science. You can use Occam's razor as an Aristotelian pragmatist ("since we are never going to know the true laws, we might as well choose the simplest ones that work") or as a religious Platonist ("I feel deeply that the universe must be elegant").
But you can't claim that Occam's razor is true and that you can prove it, or its likelihood, and that thus we are definitely living in a Platonic universe. This is, one way or another, just an updated version of medieval proofs of the existance of God, resting on the same "it must be so because I cannot bear to imagine otherwise".