From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Sat Apr 26 2003 - 19:13:47 MDT
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. 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.
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 universes.
Furthermore, the measure of these countably many universes appears to be
divided up among a finite number of distinguishable computational
processes at any finite time. After any finite amount of time within any
finite volume of space, there are a finite number of different possible
material configurations, a finite causal history (past light cone) for any
spacetime event, a finite cognitive embodiment for any distinguishable
thinking entity, and a finite number of finite subjective experiences that
are possible given a finite thinking time. Even if time grows without
upper bound, the t coordinate always has a finite value at any given
point; it never assumes a specific number called "infinity". Though
perhaps chaotic perpetual inflation means that any universe has an
infinite past history from "before" its Big Bang... still, it doesn't look
like this materially alters the point.
Thus, Bayesia consists of datums of possible diversity aleph-two (the
quantum fields), which interact in such way as to create a countable
number of universes, with their measure divided up among an enumerable
(but perhaps unbounded) number of distinguishable computational processes.
Perhaps much the same relation would likely hold of any Level IV Process
capable of containing conscious observers, regardless of how high an order
of infinity the number of universes was. And if so, perhaps we should
just expect to find ourselves in one of the simpler or most commonly
occurring Processes that gives rise to conscious observers, without having
an anthropic prejudice in favor of explanations that predict "finitely" or
"countably" or "uncountably" many observers.
-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence
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