From: Perry E. Metzger (email@example.com)
Date: Sun Apr 27 2003 - 10:04:00 MDT
"Eliezer S. Yudkowsky" <firstname.lastname@example.org> writes:
> Perry E. Metzger wrote:
> > In the Level I universe (using the terminology we've been
> > discussing),
> > there are countably many universes for obvious reasons. In the
> > Level III universe, there would also (I think) be countably many
> > universes, because there are only finitely many quantum states any
> > given volume of space could assume. At Level II, I must confess I
> > don't understand the whole "chaotic perpetual inflation" thing well
> > enough to grok but I'd guess "countable". At Level IV, if
> > metamathematics follows the rules we've established for it in the last
> > 120 years or so (i.e. we are talking about formal systems), by
> > definition the set of all possible formal systems is countable.
> But a *single* Level IV Process could conceivably contain uncountably
> many universes - aleph-one, aleph-two, or beyond - while still being
> just one formal system according to your enumeration.
Possibly, just as there are a countably infinite number of formal
systems capable of expressing real numbers, which are themselves not
countable. I suspect this is not the case (see below) but it may be --
I don't have anything approaching a proof.
Tegmark makes the interesting point, you might note, that the vast
complexity of our universe might evolve from essentially no
information at all. Any given one of our universes has vast
Chaitin-Kolmogrov information even though the ensemble has nearly
none, just as a given 20,000,000 digit prime has large
Chaitin-Kolmogrov information compared to the entire set of integers!
We should not be surprised that a given formal system is capable of
containing things "bigger than itself". :) Formal systems are somewhat
like TARDISes in this regard. :)
> There may be uncountably many universes contained in other Level IV
> Processes - for example, a Process might contain universes that run
> on reals or fields of reals.
Well, merely because the universes have as part of their mathematical
structure real numbers doesn't mean that you end up with a transfinite
number of universes within the Level IV container.
Consider that you need to express initial conditions within your
formal system itself. That means that it appears that you would need
to use only constructible entities -- Pi is fine, but arbitrary reals
are not, because no formal system can capture them specifically within
> Actually, Bayesia itself appears to be made up of fields of real and
> complex numbers, but in a way that leads to only countably many
That may very well be a consequence of the above -- that you can make
the reals part of the operation of the formal system but not part of
its initial axioms because (of course) axioms have to be finite.
> Thus, Bayesia consists of datums of possible diversity aleph-two (the
> quantum fields),
Wait, why are you assuming the quantum fields have a diversity of
aleph two? It isn't even clear to me that they have a diversity of
aleph one -- they might very well be countable. I followed what you
were saying fine up until there, but it suddenly skipped.
-- Perry E. Metzger email@example.com
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