From: Perry E. Metzger (firstname.lastname@example.org)
Date: Sun Apr 27 2003 - 10:10:37 MDT
> "Perry E. Metzger" <email@example.com> 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
> > "countable".
> 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.
I missed that -- could you give me a quick quote from it so I can do a
search through the paper for the section?
> 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?
Eeek! A fascinating new question! Indeed, it opens up an entire raft
of new questions.
> 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.
Mmmm. Well, if the universe was too simple, what would we do for fun
in coming millennia? :)
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