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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:
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*> > > My memory is fading somewhat about transfinite cardinal
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*> > > numbers. However, it seems to me that c \leq \aleph_1. \aleph_1 is the
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*> > > cardinality of the set of all sets of cardinalilty \leq\aleph_0. Since
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*> c
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*> > > is the cardinality of the set of all subsets of N, which is a subset
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*> > > 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
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*> definition, there is no transfinite number with cardinality > aleph-0
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*> and < aleph-1.
*

*>
*

*> > >
*

*> > > What has never been proven is that c=\aleph_1, although it is widely
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*> > > 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,
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*> which means that it is independent of those axioms. Thus it is
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*> possible to construct set theories which assume that ~CH, and these
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*> are known as non-Cantorian set theories.
*

*>
*

*> On the other hand, in standard set theory, assuming the CH does no
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*> harm.
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*>
*

*> [More below]
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*>
*

*> >
*

*> > This is pretty much over my head. As I understand it aleph 1 is
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*> > defined to be the next cardinal number after aleph 0, and can be
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*> > shown to be the cardinality of the set of all countable ordinals
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*> > (1...w...w+1...2w...w^2...). Since the elements of this set are all
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*> > ordered sets (i.e. ordinals), while the subsets of N don't have an
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*> > ordering requirement, this gives more flexibility to c and so you can't
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*> > compare them as simply as you have shown here.
*

*> >
*

*> > See http://www.ii.com/math/ch/ for a detailed discussion of these
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*> > matters.
*

*> >
*

*> > Actually on further thought I think I was wrong to suggest that the
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*> number
*

*> > of TM programs is c, since that would allow for infinite length
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*> programs,
*

*> > which is perhaps outside the spirit of a TM. If we require only finite
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*> > length programs then the number of TMs is aleph 0 since we can enumerate
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*> > 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
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*> is aleph-0.
*

*>
*

*> But neither of you (nor anyone else) has addressed my reason for
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*> conjecturing that the set of branches in our structure must be
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*> aleph-(aleph-0), which is based on the SSA.
*

*>
*

*> To tell you the truth, I'm certainly not convinced of it, but I
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*> think it's worth considering. To discard the conclusion, I would
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*> think that you'd have to assume "the identity of indiscernables".
*

*> My reasoning is illustrated if you only assume, for the moment,
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*> that some observable (say, x) can take a continuum of possible
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*> values when measured. Forget about the Plank length, for now.
*

*> That would mean that the set of all possible humans would have
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*> cardinality c (at least). Thus it would be impossible to map
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*> that set onto the set of all programs.
*

*>
*

*> But if you believe in the computationalist hypothesis, then you'd
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*> have to assume that at some point, a simulation of a human becomes
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*> "close enough" to be identical. That is (to oversimplify) when
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*> each particle is simulated to within a Plank length, then the
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*> simulation becomes indiscernable from the original, and thus
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*> identical. If this is true, then I would no longer expect the
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*> physical laws to give rise to ever-increasing cardinality of
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*> universes, since that could never increase the cardinality of
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*> 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

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?

Received on Wed Jul 14 1999 - 01:04:44 PDT

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