On 12/28/2010 1:13 PM, Bruno Marchal wrote:
On 28 Dec 2010, at 19:46, Brian Tenneson wrote:
Thank you, happy new year to you, too!
Thanks.
On Dec 27, 8:36 am, Bruno Marchal <[email protected]> wrote:
On 26 Dec 2010, at 22:51, Brian Tenneson wrote:
"Limits To Science: God, Godel, Gravity"
http://www.science20.com/hammock_physicist/limits_science_god_godel_g...
Here is my comment:
An important question is whether or not a TOE will be finite in
length.
Of course, this is a matter of definition.
Indeed.
I am taking 'TOE' to be, as a working definition, a complete
description of reality or a complete description of everything that
exists.
That does not exist. Arithmetical truth is already not recursively
enumerable.
If a complete description of arithmetical truth is not possible, what
exactly are we talking about?
We talk about the set of all true arithmetical statements; written in
the (first order logic) language of arithmetic.
That set contains the verifiable atomic formula like the usual taught
in school: 0 + 1 = 1, 45 + 9 = 54, 3*7 + 3 = 24, etc. Where 3
abbreviates s(s(s(0))), etc.
Then the sigma_1 true sentence like Ex( x = s(0)), or ExEy(x = 2y).
The existence of a computational state relatively to a universal
number is sigma_1. Then the true pi_1 sentences like Ax(x = 0 V not(x
= 0)).
Here we can already show that for any (proving/believing) machine
there is an arithmetical pi_1 proposition which is true but not
provable by that machine. Note that the Riemann hypothesis can be
shown to be equivalent with an arithmetical pi_1 sentence, and is an
example of a pi_1 sentence that we suspect to be true but that we can
still not prove.
Then the sigma_2 sentences, equivalent to a proposition with the shape
ExAyP(x, y, a, b, ...), then the pi_2 (AxEyP(x, y, a, b, ...) with P a
decidable (sigma_0) propositions, etc. Most of them are beyond the
human mind.
Arithmetical truth is the set of all true propositions in arithmetic.
That set is not recursively enumerable (cannot be defined by a sigma_1
proposition, and actually is not arithmetical (cannot be defined by
*any* arithmetical proposition!). That follows from Gödel and Tarski.
We, humans, have a rather good intuition of what is a true
arithmetical sentence, independently of the fact that we have to
recognize that it can be quite tricky to decide if this or that
arithmetical proposition is true or not.
Then one *might* consider this program which generates a TOE to
arbitrary precision to be "the" TOE, a compression of an infinitely
long document into a finitely long document, thus showing that reality
at its core does not possess the trait of Kolmogorov randomness. Being
that reality contains the uncomputable, it *seems* unlikely that
everything can be finitely describable.
This is simply wrong.
That's good!
The argument is made that a TOE can be in the form of a logical
structure which is a tuple consisting of an underlying set, a set of
distinguished constants (like zero), functions (like successor), and
relations (like less than) on this underlying set. Making the
additional assumption that if there is a structure such that *all*
logical structures can be "embedded" within it, then this type of
universality endows such a structure with the same structure as
reality. Thus this sort of ultimate structure would be in an intuitive
sense like ultimate reality. Thus a description of this ultimate
structure would be a description of reality.
You have to distinguish the outside and inside view of the ultimate
reality.
What is the outside view of the ultimate reality?
Assuming mechanism, arithmetical truth is enough. It is infinite and
non algorithmically generable.
It is just absolutely undecidable that there is anything more needed
for the outside view. So by Occam razor ...
If we are "machine" (survive at some substitution for some level, or
"are fully turing emulable" with "fully" meaning that consciousness is
preserved) then the inside "knowledge" view of arithmetical truth will
be FAR BIGGER than arithmetical truth. Indeed, it corresponds
plausibly to George Levy first person plenitude. That things is much
less definable (if I can say) than arithmetical truth. There is here
an analog of Skolem paradox, with the existence of a structure which
looks small from outside and is immensely big from inside.
I don't understand how "a view of arithemetical truth" can be bigger
than arithmetical truth - unless the view includes some arithmetical
untruths.
Brent
How is it possible
to have an outside view of the ultimate reality?
By assuming mechanism. If mechanism fails the UDA test, we will know
that arithmetical truth is too big or to small. Plausibly too small
(for the reason that we can already conceive, apparently, arithmetical
truth (but an ultrafinitist could contest this).
Church thesis rehabilitates the Pythagorean Neoplatonist theology. The
relation between the numbers (the true arithmetical relation or
propositions) defines, by the first person indeterminacy, a
differentiating flux of consciousness, which, by the non awareness of
the basic UD execution time (defined by those relations), filters *in
the limit* the possible and sharable infinite consistent histories.
The reason why you are conscious here and now is rather easy. It
depends only on the truth of one sigma_1 arithmetical sentence
(asserting the existence of your computational state relatively to
some universal number). But there are infinitely many equivalent
sentences, both with the same universal numbers and with different
universal numbers. Most relations involve pretty big numbers.
What the first person indeterminacy makes complex is the relation
between 'here and now', and 'there and then'. This, unfortunately
depends on that infinity of universal (among others) numbers, and all
those sigma_1 propositions, and actually all their proofs, which
define the relative measure.
Below your substitution level, an infinity of universal number
compete, for proving your existence, together with all random oracles.
Fortunately, the constraints of self-referential correctness for the
Löbian observers are enough to shed some light on this, and to make
quite plausible that the bottom physical reality will appear linear
and extremely symmetrical, then self-observation break the symmetries
from the point of view of the self-observing machines.
You can see this as Everett spirit applied to arithmetic. Everett
embeds the physicist in the quantum reality. DM embeds the
mathematician in the arithmetical reality.
I choose the numbers, but any finite combinatorial objects would do.
Physics, and none of the internal views depends on the initial choice,
except perhaps(*) some diagonal systems build to lead to anomalous
measure.
Bruno
(*) a conjecture by Jacques Bailhache on this list.
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