# RE: Cardinality of the MW

From: Higgo James <james.higgo.domain.name.hidden>
Date: Wed, 14 Jul 1999 08:58:58 +0100

But even such an 'identical' human has different spatial co-ordinates and is
therefore different, no?

> -----Original Message-----
> From: Christopher Maloney [SMTP:dude.domain.name.hidden]
> Sent: Wednesday, July 14, 1999 6:25 AM
> To: everything-list
> Subject: Re: Cardinality of the MW
>
> Let me add my information to this confusing brew:
>
> hal.domain.name.hidden wrote:
> >
> > Russell Standish, <R.Standish.domain.name.hidden>, writes:
> > > numbers. However, it seems to me that c \leq \aleph_1. \aleph_1 is the
> > > cardinality of the set of all sets of cardinalilty \leq\aleph_0. Since
> c
> > > is the cardinality of the set of all subsets of N, which is a subset
> > > of the set of all sets of cardinality \leq\aleph_0.
>
> This is wrong, from what I know. I agree with Hal below that aleph-1
> is defined to be the "next" cardinal after aleph-0. That is, by
> definition, there is no transfinite number with cardinality > aleph-0
> and < aleph-1.
>
> > >
> > > What has never been proven is that c=\aleph_1, although it is widely
> > > suspected.
>
> In fact it has been shown that c == aleph-1 is not provable by the
> axioms of Zermelo-Fraenkel set theory. This is known as the
> continuum hypothesis (CH). CH is also not disprovable,
> which means that it is independent of those axioms. Thus it is
> possible to construct set theories which assume that ~CH, and these
> are known as non-Cantorian set theories.
>
> On the other hand, in standard set theory, assuming the CH does no
> harm.
>
> [More below]
>
> >
> > This is pretty much over my head. As I understand it aleph 1 is
> > defined to be the next cardinal number after aleph 0, and can be
> > shown to be the cardinality of the set of all countable ordinals
> > (1...w...w+1...2w...w^2...). Since the elements of this set are all
> > ordered sets (i.e. ordinals), while the subsets of N don't have an
> > ordering requirement, this gives more flexibility to c and so you can't
> > compare them as simply as you have shown here.
> >
> > See http://www.ii.com/math/ch/ for a detailed discussion of these
> > matters.
> >
> > Actually on further thought I think I was wrong to suggest that the
> number
> > of TM programs is c, since that would allow for infinite length
> programs,
> > which is perhaps outside the spirit of a TM. If we require only finite
> > length programs then the number of TMs is aleph 0 since we can enumerate
> > all the programs, and that would be the number of universes as well.
> > Not so many after all.
> >
> > Hal
>
> I think you are right that the cardinality of the set of all programs
> is aleph-0.
>
> But neither of you (nor anyone else) has addressed my reason for
> conjecturing that the set of branches in our structure must be
> aleph-(aleph-0), which is based on the SSA.
>
> To tell you the truth, I'm certainly not convinced of it, but I
> think it's worth considering. To discard the conclusion, I would
> think that you'd have to assume "the identity of indiscernables".
> My reasoning is illustrated if you only assume, for the moment,
> that some observable (say, x) can take a continuum of possible
> values when measured. Forget about the Plank length, for now.
> That would mean that the set of all possible humans would have
> cardinality c (at least). Thus it would be impossible to map
> that set onto the set of all programs.
>
> But if you believe in the computationalist hypothesis, then you'd
> have to assume that at some point, a simulation of a human becomes
> "close enough" to be identical. That is (to oversimplify) when
> each particle is simulated to within a Plank length, then the
> simulation becomes indiscernable from the original, and thus
> identical. If this is true, then I would no longer expect the
> physical laws to give rise to ever-increasing cardinality of
> universes, since that could never increase the cardinality of
> the set of humans past aleph-0, anyway.
>
>
>
> --
> Chris Maloney
> http://www.chrismaloney.com
>
> "Knowledge is good"
> -- Emil Faber
Received on Wed Jul 14 1999 - 01:04:44 PDT

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