Devin Harris wrote:
> Again, the question of how infinite is the Universe that
> contains MWs. Or said otherwise, how vast or how ruled is
> the possible world, assuming all possibilities exist?
I posted about this before, in
http://www.escribe.com/science/theory/index.html?mID=674.
I think that what Devin describes in his last post is
correct: that the cardinality of the structure in which
we find ourselves must be, in some sense, be infinitely
infinite. It must be Aleph-infinity, if this makes any
sense. A trivial application of the SSA indicates this.
I've thought about this some, but I am not a mathemetician,
so I'd appreciate any feedback.
For any arbitrary (finite or infinite) set S, the Power
Set of S is denoted *S, and is the set of all subsets of
S. For example, if S contains {1, 2, 3}, then *S would
be { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }.
*S is called the "Power Set" because, for finite sets:
|*S| = 2 ^ |S|,
where |S| is the cardinality of S.
For infinite sets, I believe that this power equation is
defined in terms of |*S|.
Cantor's theorem states that, for any set (finite or
infinite), |*S| > |S|. This provides a mechanism for
generating infinite sets with ever increasing cardinality.
defined in terms of |*S|. Now, the cardinality of the
power set of integers is c, the cardinality of the real
numbers. I think of this intuitively by imagining that
every real number can be expressed as an infinite number
of binary digits, for example,
...1101001000101110110101.11101010001010101001...
Each digit can have one of two values, and there are
|Z| digits, so c = 2^|Z|.
Does anyone know what the cardinality of the branches in
the traditional MWI is? By traditional, I mean that from
the Schrodinger's Equation. It seems plausible to me
that it's 2^c, by an intuitive reasoning similar to the
above. I think it depends on whether the universe is
finite or infinite in the number of particles it holds.
Now, if the original line of reasoning is correct, then
I predict that further breakthroughs in physics will
exhibit, by some mechanism (this is obviously highly
speculative) that the cardinality of the "branches" is
of a new order. By the SSA, we must find ourselves in
universes with physical laws that admit the greatest
number of SASs. Perhaps this feature has already been
exhibited by some of the more advanced quantum theories
out there - I admit ignorance of QFT in general.
The only mechanism that I can imagine is some sort of
infinite regress with regards to the structure of the
universe. That is, I would guess that no matter how
far we dig into the underlying structure of matter, we
will always uncover deeper layers.
--
Chris Maloney
http://www.chrismaloney.com
"Knowledge is good"
-- Emil Faber
Received on Sun Jul 11 1999 - 20:17:55 PDT