From: Perry E. Metzger (perry@piermont.com)
Date: Sun Apr 27 2003 - 09:51:45 MDT
Simon Gordon <sim_dizzy@yahoo.com> writes:
> --- "Perry E. Metzger" <perry@piermont.com> wrote:
> >
> > the number of Level IV universes is
> > countable.
>
> You try counting all the level IV universes, lol.
>
> Seriously if you can provide me with a link to a proof
> that the set of all formal systems is countable i'd be
> more than happy
All formal systems are expressable as finite strings of symbols (by
definition), and thus trivially mappable to integers. QED.
Do you still need a link? If you don't understand why that's trivially
so, you might want to re-read about formal systems and about Cantorian
set theory.
> but in any case i think the level IV
> is much broader than the ensemble of all formal
> systems. Having not read the Tegmark level
> classification paper yet i cannot yet be sure.
It is literally the set of all formal systems, because it is the set
of all consistent mathematical systems.
> Either way i see no problem with equating sets to
> universes though.
Well, Tegmark makes the point that if you don't have a consistent
mathematical system underlying you don't have enough structure to get
self aware substructures and thus only they can be physically realized
given his operational definition of "has physical reality". Of course,
you might yourself have differing operational definitions of physical
reality (in Tegmark's definition, a detailed simulation has physical
reality) but that's another story -- if we're using Tegmark's
terminology then we should be consistent.
Of course, if you're going to bring things down to the level of "set"
you start getting into a much deeper problem. "Sets" don't
interact. They don't form a structure capable of self aware
substructures at all. Our universe, for example, is pretty trivially
observable to not be a set -- it might contain sets, but it is a lot
richer than that.
And if you were willing to accept that we aren't in a universe and
move on to "sets" as your fundamental universes, what sort of set
theory are you going to adopt as "fundamental"? After all, in some set
theories, you get the axiom of choice, in some you don't. In some, the
continuum hypothesis is an axiom, in some it isn't. The list goes on
and on.
Sets, you see, aren't as primitive as formal systems at all, even
though they are much less powerful, just as integers aren't as
primitive as formal systems although they are much less powerful.
(You can, of course, express formal systems as sets just as you can
express them as integers (and just as you can express integers as sets
for that matter) but without the external interpretation of your
representational system there isn't much "there" there. In the end,
formal systems are the most primitive object...)
-- Perry E. Metzger perry@piermont.com
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