On Wed, Sep 21, 2011 at 10:08:55AM -0400, Stephen P. King wrote:
> On 9/21/2011 6:41 AM, Russell Standish wrote:
> >On Mon, Sep 19, 2011 at 01:14:04PM -0400, Stephen P. King wrote:
> >>     Exactly why are there not a continuum of OMs? It seems to me if
> >>we parametrize the cardinality of distinct OMs to *all possible*
> >>partitionings of the tangent spaces of physical systems (spaces
> >>wherein the Lagrangians and Hamiltonians exist) then we obtain at
> >>least the cardinality of the continuum. It is only if we assume some
> >>arbitrary coarse graining that we have a countable set of OMs.
> >I do not assume an arbitrary coarse graining, but do think that each OM
> >must contain a finite amount of information. This implies the set of
> >OMs is countable.
> [SPK]
>     Umm, how does the finiteness of the elements of a set X  induce
> finiteness of X? I may have missed this in my studies of set theory.

That is not what I said. Firstly, I said the set of OMs are countable, which
includes the lowest transfinite cardinal aleph_0. Also, there is more
to it. Perhaps I wasn't explicit about the fact that I consider two
OMs with the same information content to be identical. Ie, the
contained information uniquely identifies the OM.

In that case, the set of all OM can be mapped 1-1 to the set of finite
binary strings [0,1]* (I think that's how it is written). That set is
countable, so the set of all OMs must be too.

> >>     I must dispute this claim because that reasoning in terms of
> >>'two integer' encoding of rationals ignores the vast and even
> >>infinite apparatus required to decode the value of an arbitrary pair
> >>of 'specified by two integers' values.
> >Both the human brain, and computers are capable of handling rational
> >numbers exactly. Neither of these are infinite apparatuses. If you're
> >using an arbitrary precision integer representation (eg the software
> >GMP), the only limitation to storing the rational number (or decoding
> >it, as you put it) is the amount of memory available on the computer.
> [SPK]
>     True, but that misses my point. Brains and Computers are not
> entities existing in an otherwise empty universe; we have to
> consider a multiplicity of mutually observing and measuring entities
> and the internal interpretational and representational structures
> thereof.  Consider a simple digital camera. The images that the
> camera can capture are limited by the pixel resolution of the
> camera, this is a constraint induced by the physical design of the
> camera. The camera itself, as a physical object, is not limited in
> the detail of its properties by those intrinsic constraints. We must
> take care to not assume that the limits of the observational or
> measurement process is not assumed to be that of the system that is
> making the observations/measurement.
> 

Since the observable world is defined by the observer, one can't
really not take the observer into account. One can perhaps get higher order
cardinalities by looking at the boundary of that which is common to
all observers. For concreteness, consider the UD trace UD* in Bruno's
work. UD* is isomorphic to the reals - you would have to define
something like that to be your world to get uncountable things.

> 
> >The amount of information needed to represent any rational number is
> >finite (although may be arbitarily large, as is the case for any
> >integer). Only real numbers, in general, require infinite
> >information. Such numbers are known as uncomputable numbers.
> [SPK]
>     Surely Reality is not limited to the rationals! Are we to be
> crypto-Pythagoreans, claiming to believe that only the rationals
> exist, yet still using pi, e and other irrationals without
> question??? If Nature is computational, does it not make sense that
> its computations /information accessing and processing might not be
> limited to the rationals?
> 

No, again, I didn't say that. I think of reality as being the set of
knowable things, which is necessarily countable. Various computable
numbers such as e, pi etc are definitely knowable.

John Eastmond was the one to bring up the rationals by means of a
bijection from a set of OMs. I was pointing to flaws in his use of
rational numbers (they're still countable, for instance).


-- 

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Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au
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