Dear Cristi,
I also read your entry with interest and pleasure and good luck to you too.
I need to clarify on one misunderstanding about my approach. You say: "If the Universe can be described by a mathematical structure (which you are trying to axiomatize)". This is not what I am trying to do. This is Tegmark's approach and I do not think it has a good change of success for several reasons. The first reason is Goedel's theorem. Tegmark's response to this challenge is to consider computability as a requirement of nature, but I have serious reservations with computability as a core feature of nature. My second reason to doubt it is that nature seems to be infinite and mathematicians (who are part of nature) are discovering new axioms all the time. This is at odds with Chaitin's results which limit the amount of complexity one can derive from a given set of axioms. If Nature is one huge mathematical structure then we should see some sort of maximum information which can be extracted from it, and this is at odds with what we observe, particularly regarding the orthogonal groups describing space-time as it was shown by Jochen Rau.
So if this in not my approach, then what is exactly my approach? Axiomatization is possible, but not as a massive single mathematical structure. Realty is made of all mathematical structures, but only some of them play distinguished roles. How can we identify them? Why is space-time 4 dimensional? Why are complex numbers appearing in QM, etc, etc? Mathematical structures are relational and abstract, devoid of ontological meaning. But physics can also be put in relational format. A trivial example is speed. More realistic example include Rovelli's relational interpretation of QM, and LQG's background independent approach. If there are no supernatural explanations of reality (like transcendental or God arguments), all that remains are mathematical structures as the building blocks to construct ontology. Therefore by looking at the differenced between math and reality one should extract precisely the core features of reality which in turn would select all the universal distinguished mathematical structures we see in nature: Minkowski space, quantum mechanics, etc.
The 3 physics principles: universal truth, composability, deformability (or events, interaction, infinite complexity) acts as filters of selecting the distinguished mathematical structure physicists use and experimentally confirm every day. I am not deriving them from the 3 principles, I am only eliminating all the other mathematical structures and therefore the axiomatization of physics does not result in a mathematical closed form "super structure" from which all reality is derived. This is a double negation trick which eliminates the closed form of the answer and avoids the Goedel and Chaitin roadblocks.
Regarding the global hyperbolicity, this a long discussion, but I can give you the gist. The critical observation is that there is a link between incompleteness theorem and the Minkowski space. From this one can translate the Goedel's proof almost one to one into the proof of global hyperbolicity on arbitrary event manifolds. But event manifolds are decidable and at the end of the proof one is force to conclude that one has inconsistency for event manifolds for which Godel's theorem translation holds (precisely the ones where global hyperbolicity is violated). This is different from Goedel because he derives incompleteness instead of inconsistency, but because of decidability, incompleteness is ruled out. The sensitive part of the proof is to prove the godelization and this works only in classical mechanics and therefore the proof is valid only in the region of the validity of the correspondence principle, and not at Plank scale. Global hyperbolicity and time are ill defined in that range anyway.
PS: I had completed the basic hyperbolicity proof right after the first FQXi essay contest and I would have loved to enter the nature of time contest and explain all this in detail, but that is life.
