[ExI] [MATH] Infinite Mappings

[Lee Corbin]

Samantha writes

> No it doesn't have to be the case if the universe is infinite.  If the 
> universe is countably infinite then there is no way to map on to that 
> infinite variations of everything within that infinity.

You *could* have what is known as a dense set of infinite variations,
but probably you mean the uncountable set of possible variations.
More specifically, "infinite variations of everything" would include all
binary choices, e.g., assuming the number of spaces countable,
then 1 for the first space, 0 for the second, 0 for the third, 1 for
the fourth, and so on, in all possible arrangements, which *does* 
provide uncountably many possibilities, as you say.

On the other hand (going back to a dense set), if our universe
were to consist of volumes containing a certain string, 1, 0, 1, 1, 
..., then there is a countable set of universes that come 
arbitrarily close to it.  (For example, all of those which are
identical to ours out to however many places you'd like to
go, but from then on end in a string of 1's, or a string of 0's.)

Therefore, for all practical purposes   :-)    if there are countably
many parallel universes, then they might still be arranged to 
resemble ours however closely as is desired.

For what it is worth, however, David Deutsch has opined that
the cardinality of the Everett multiverse is the cardinality of
the continuum, which is, of course, uncountable.

Lee