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".
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.