On 9/22/2011 1:05 AM, Russell Standish wrote:
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.
[SPK]
Thank you for this explanation. There is just something about this
that is still unsettling to me. I will ponder it further.
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.
[SPK]
How do we extend this to a countable number of seperate observers
and their interactions and communications with each other. It seems to
me, and I may be wrong, that this generates a diagonalization. The
bisimulation algebra that was developed is not closed in non-symmetric
cases.
**
Summary of basic properties:
A = A ~ A real identity bisimulation rule
B ~ C not= C ~ B non-commutativity rule; conjugate of
bisimulation not equal to itself
A ~ A = A ~ B ~ A law of real identity bisimulation (when
conjugate equal to itself)
Corollary:
A ~ A not= A ~ B ~ C ~ A by law of real identity bisimulation
A ~ A = A ~ B ~ C ~ B ~ A retractable path independence; by law of
real identity bisimulation
A ~ C not= A ~ B ~ C non-closure
**
Am I missing something?
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.
[SPK]
e and pi are Computable by an endless computation... Here I have a
problem, the computation involves the use of an indefinite quantity of
resources... Not a very 'physical' situation.
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).
[SPK]
I hope that he has a comment on this point.
Onward!
Stephen
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