On Sat, Sep 21, 2002 at 09:20:26PM -0400, [EMAIL PROTECTED] wrote:
> 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'm not sure what "Absolute Maximum" and "Absolute Infinite Collections"
refer to (a search for "Absolute Infinite Collections" on Google gave no
hits), but I'll take the question to mean whether proper classes (i.e.
collections that are bigger than any set, for example the class
of all sets) have "physical existence".
I think the answer is yes, or at least I don't see a reason to rule it
out. To make the statement meaningful, we need (at least) two things, (1)
a way to assign probabilities to proper classes so you can say "I am
observer-moment X with probability p" where X is a proper class, and (2)
a theory of consciousness of proper classes, so you can know what it feels
like to be X when X is a proper class.
(2) seems pretty hopeless right now. We don't even have a good theory of
consciousness for finite structures yet. Once we have that, we would still
have to go on to a theory of consciousness for countably infinite sets,
and then to uncountable sets, before we could think about what it feels
like to be proper classes. But still, it may not be impossible to work it
As to (1), Tegmark doesn't tell us how to assign probabilities to observer
moments. (He says to use a uniform distribution, but gives no proposal for
how to define one over all mathematical structures.) However, it does not
seem difficult to come up with a reasonable one that applies to proper
classes as well as sets.
Here's my proposal. Consider a sentence in set theory that has one unbound
variable. This sentence defines a class, namely the class of sets that
make the sentence true when substituted for the unbound variable. It may
be a proper class, or just a set. Call the classes that can be defined by
finite sentences of set theory describable classes. Any probability
distribution P over the sentences of set theory, translates to a
probability distribution Q over describable classes as follows:
Q(X) = Sum of P(s), over all s that define X
Take P to be the universal a priori probability distribution (see Li
and Vitanyi's book) over the sentences of set theory, and use the
resulting Q as the distribution over observer moments.
Of course this distribution is highly uncomputable, so in
practice one would have to use computable approximations to it. However,
computability is relative to one's resources. We have access to certain
computational resources now, but in the future we may have more. We may
even discover laws of physics that allow us to compute some non-recursive
functions, which in turn would allow us to better approximate this Q. The
point is that by using Q, instead of a more computable but less dominant
distribution (such as ones suggested by Schmidhuber), in our theory of
everything, we would not have to revise the theory, but only our
approximations, if we discover more computational resources.