Date: Sun Apr 27 2003 - 08:29:54 MDT
"Perry E. Metzger" <firstname.lastname@example.org> wrote:
> At Level II, I must confess I don't understand the whole "chaotic
> perpetual inflation" thing well enough to grok but I'd guess
Tegmark mentions in his paper that the number of Level II universes is
countable if inflation is not past-eternal, and uncountable if inflation
is past-eternal. I think the former is considered more likely.
However, as Eliezer points out, even if there are only countably many
universes in our Level IV universe, there is probably at least one Level
IV universe with uncountably many universes in it (for example, one
described by past-eternal inflation).
Eliezer also mentions a diversity of aleph_2, but this would depend on
whether the continuum hypothesis ("the cardinality of the continuum
(reals) is aleph_1") is true. This made me think about a confusing issue.
If every Level IV universe is a formal system, and the continuum
hypothesis is independent of the other axioms of set theory, then it
might be that our universe is described by set theory with CH, or by set
theory without CH. If we could somehow measure experimentally (using
esoteric math-tech) the cardinality of the continuum in our world, then
if the CH was among the axioms, we'd find aleph_1. If one of the axioms
was "c == aleph_5", we'd find aleph_5. But the possibility of doing such
an experiment should not depend on whether there is such an axiom -- so
what happens if our universe is described by set theory with no axioms
that determine the cardinality of the continuum?
I'm sure I'm overlooking many subtle interpretational issues here -- when
discussing this sort of thing I always feel as if carefully avoiding a
giant murky swamp of decision theory and probability theory and anthropic
reasoning and reference classes and SSAs and SIAs and mathematical logic
and set-theoretical paradoxes.
Keep it complex,
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