On 06 Oct 2010, at 17:43, Brian Tenneson wrote:
I figure this is especially of interest because of the references to
Tegmark's works.
From a logician's standpoint, it may be of interest that I show that
there is a structure U such that all structures, regardless of
symbol set, can be elementarily embedded within it.
From a physicist's point of view, at least one who might subscribe
to Tegmark's 4-level hierarchy of parallel universes, a structure
with this property might be of interest under the hypothesis that
reality is a mathematical structure. If we suppose that reality is
something which is all encompassing, then the structure with the
aforementioned property could be said to be all encompassing.
Now that I have this structure in hand, I can try to go further by
looking at the structure from a model-theoretic point of view. This
task to further the investigation will be undertaken soon.
Here is a link
http://www.alphaomegadimension.info/media/A_Mathematical_Structure_Isomorphic_to_Reality_ver_5-12_anon.pdf
Any feedback is encouraged, critical or otherwise.
Let us call universe, the ultimate reality.
Then I agree with this: if the universe is a mathematical object, then
NF is the best tool to attempt a description of that universal object.
The universe, when being a mathematical object, has to belong to
itself, so we need a theory à-la Quine, instead of the usual Zermelo-
Franekek or Von Neuman Bernays Gödel. In that sense it improves the
raw description Tegmark makes of level 4.
I got a shock at first reading your definition of 'mathematical
structure" page 2, as formal system, but you make it clear later that
you meant the models, or semantics, of those system (yet interpreted
themselves in NFU).
My reading of Ebbinghaus Flum Thomas did make me feel less guilty to
listen more to lobian machine talking only first order language, but
not up to the point of believing that the universal mathematical
universe could be a first order structure, even in theory like NF, at
least not in in the genuine sense needed to address the mind body
problem, or the consciousness/reality problem.
A theory of everything should explain both the physical and the
mental, conceptually.
If you assume that the brain works like a universal machine (universal
with respect to computability notion, and thus with Church thesis),
you get a theory of mind, as what universal machine can prove and
infer about themselves and their possible consistent extensions.
But then, in term of Tegmark levels, The relation between mind and
reality becomes a partially mathematical phenomenon which blurs the
level 3 and the level 4.
Such universal machine cannot know in which computational history she
would belong, still less in which mathematical structure she belongs,
but below its level of substitution, she belongs to an infinity of
universal history (number relations, combinators relation, Horn clause
relations) 'competing' in term of a measure of credibility.
So with mechanism the physical is not something mathematical among the
mathematical, it is a very special structure which sums on all
mathematical structures is a way specified by computer science and the
logic of self-references. It is based on distinction of different
internal sel-referential views.
Also, I am not convinced by your argument that from the premise "there
exists a reality completely independent of us human" it follows that
reality is a mathematical structure". You beg the question by
identifying a baggage free description with a mathematical structure.
A physicalist argues in general that baggage-free description is what
him provides: particles, waves, fields, and that mathematics is an
approximate language conveying human ideas on those things. Your proof
seems to me just a platonist act of faith.
You miss the importance of the consciousness problem, concerning
physics.
Perhaps you follow Tegmark. Quoting him from "The Multiverse Hierarchy":
"Indeed, the standard mental picture of what the physical world
is corresponds to a third intermediate viewpoint that could be
termed the consensus view. From your subjectively perceived
frog perspective, the world turns upside down when you stand
on your head and disappears when you close your eyes, yet you
subconsciously interpret your sensory inputs as though there is
an external reality that is independent of your orientation, your
location and your state of mind. It is striking that although
this third view involves both censorship (like rejecting dreams),
interpolation (as between eye-blinks) and extrapolation (say at-
tributing existence to unseen cities) of your inside view, inde-
pendent observers nonetheless appear to share this consensus
view. Although the inside view looks black-and-white to a cat,
iridescent to a bird seeing four primary colors, and still more dif-
ferent to bee a seeing polarized ligh
