[ExI] [MATH] Infinite Mappings

[Lee Corbin]

Eliezer writes

> Lee Corbin wrote:
>> 
>> 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.
> 
> How the devil did Deutsch arrive at *that* idea?

I don't know, but quantum fields can take on continuously
many values at a point, and, furthermore, the time at which
a certain quantum event (e.g. a disintegration) may occur
at any of continuously many times between, say  t0 and t1.
(Deutsch did once mention in a post on Fabric of Reality
that he does not know (or it is not known) whether the angle
at which a photon comes away from an emission is discrete
or continuous, though I may be misremebering that and it
could have been "countably infinite" for one of those two
---but I doubt it.  Merely aleph-zero of anything doesn't
seem to accord with QFT.)

> That sounds really really bizarre.  Are you sure this is Deutsch?

Yes. I've quoted page 211 many times, but maybe not here.
On p. 211 of "The Fabric of Reality" Deutsch writes

"Let us start by imagining some parallel universes stacked like a 
pack of cards...(which greatly understates the complexity of
the multiverse)...  Now let us alter the model to take account
of the fact that the multiverse is not a discrete set of
universes but a continuum, and that not all the universes are 
different.  In fact, for each universe that is present there is
also a continuum of identical universes present, comprising
a certain tiny but non-zero proportion of the multiverse. In our model, this
proportion may be represented by the thickness of a card, where
each card now represents all the universes of a gven type. However, 
unlike the thickness of a card, the proportion of each type of universe
changes with time under quantum-mechanical laws of motion.  
Consequently, the proportion of universes having a given
property also changes, and it changes continuously. In the
case of a discrete variable changing from 0 to 1, suppose that
the variable has the value 0 in all universes before the change
begins, and that after the change, it has the value 1 in all universes.
During the change, the proportion of universes in which the value
is 0 falls smoothly from 100 percent to zero."

Lee