Dave Raub asks: > For those of you who are familiar with Max Tegmark's TOE, could someone tell > me whether Georg Cantor's " Absolute Infinity, Absolute Maximum or Absolute > Infinite Collections" represent "mathematical structures" and, therefore have > "physical existence".

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I don't know the answer to this, but let me try to answer an easier question which might shed some light. That question is, "is a Tegmarkian mathematical structure *defined* by an axiomatic formal system?" I got the ideas for this explanation from a recent discussion with Wei Dai. Russell Standish on this list has said that he does interpret Tegmark in this way. A mathematical structure has an associated axiomatic system which essentially defines it. For example, the Euclidean plane is defined by Euclid's axioms. The integers are defined by the Peano axioms, and so on. If we use this interpretation, that suggests that the Tegmark TOE is about the same as that of Schmidhuber, who uses an ensemble of all possible computer programs. For each Tegmark mathematical structure there is an axiom system, and for each axiom system there is a computer program which finds its theorems. And there is a similar mapping in the opposite direction, from Schmidhuber to Tegmark. So this perspective gives us a unification of these two models. However we know that, by Godel's theorem, any axiomatization of a mathematical structure of at least moderate complexity is in some sense incomplete. There are true theorems of that mathematical structure which cannot be proven by those axioms. This is true of the integers, although not of plane geometry as that is too simple. This suggests that the axiom system is not a true definition of the mathematical structure. There is more to the mathematical object than is captured by the axiom system. So if we stick to an interpretation of Tegmark's TOE as being based on mathematical objects, we have to say that formal axiom systems are not the same. Mathematical objects are more than their axioms. That doesn't mean that mathematical structures don't exist; axioms are just a tool to try to explore (part of) the mathematical object. The objects exist in their full complexity even though any given axiom system is incomplete. So I disagree with Russell on this point; I'd say that Tegmark's mathematical structures are more than axiom systems and therefore Tegmark's TOE is different from Schmidhuber's. I also think that this discussion suggests that the infinite sets and classes you are talking about do deserve to be considered mathematical structures in the Tegmark TOE. But I don't know whether he would agree. Hal Finney