On 29 Dec 2010, at 13:50, Brian Tenneson wrote:

If a complete description of arithmetical truth is not possible, what
exactly are we talking about?

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.

But if arithmetical truth can not be described then what are we
talking about?

About God, or Everything, or the ONE, or the one-who-has-no- description, etc. The big thing, that is, or the big nothing ...



What our intuition tells us?  I'm still unclear as to
why a complete description of reality is not possible because we
apparently can know what we're talking about when we talk about
arithmetical truth, despite arithmetical truth not being recursively
enumerable.

The proof that we cannot *know* what is arithmetical truth is related to the fact that we cannot know if mechanism is true. So we can already not know if some arithmetical relations make people or zombie. We can only know that IF mechanism is true THEN we cannot know which number(s)/machine(s) we are, etc. We can have a description of reality, but not a complete theory answering all the questions that we can formulate in the theory.



 How does one even prove that "arithmetical truth is not
recursively enumerable" if one does not have a complete description of
"arithmetical truth."

Because we can have the intuition that any closed arithmetical formula is either true or false, so that the function from the set of arithmetical formula to {true, false} is well defined. But it is not a computable function. Most functions are not computable. Most arithmetical problems are not solvable. That is proved by diagonalization. The shorter way to see this is that the set of all functions from N to N is non enumerable, but the set of total computable function is enumerable (although not recursively enumerable). The set of true arithmetical propositions is well defined in classical mathematics, but the view from inside is not (intuitionistic arithmetics/analysis, constructive arithmetic, self-extending autonomous structure, etc.). Recursion theory, a branch of math, studies the degrees of insolubility. Church thesis makes such study universal in a strong sense which permeates the consequence of the mechanist hypothesis.





Take whatever level of description of arithmetical truth necessary to
prove that statement and analogously take "reality" to be described to
that level, then it no longer seems your implausibility argument based
on arithmetical truth not being recursively enumerable is applicable.
What I mean is that couldn't we make statements about a complete
description of reality such as it is not recursively enumerable?

Accepting some hypothesis, we can make statement on 'reality'.
With the hypothesis "digital mechanism", we can state that reality is not recursively enumerable, neither pi_i or sigma_i for any positive integer i, etc. In particular we can know, relatively to that hypothesis, that reality does not admit any finite complete description.



How
does that mean a complete description of reality does not exist, as
you said?

It means that the "theory of everything" cannot answer all question written in its language. It means that many surprises will ever attend us, and that some mystery will remain forever. But we can account epistemologically for those surprises and mysteries.



There is a complete description of reality, though the most evident
one is infinite in length, and not recursively enumerable (since
arithmetical truth isn't).

OK. But that is not an effective theory, nor a theory at all with the definition given. You talk about the model, which is what we talk about: the intended model of a theory of everything (including matter, consciousness, taxes and death, ...).



What I'm after is analogous to a finite basis for a vector space over
an infinite field.  Can any infinite TOE be abridged into a finite
document that, when provided with how to expand the finite document
(like the span of a basis in my analogy), is the original infinite
document?

Yes, for the outside view of the reality (with some hypothesis). A physicalist can take a Hilbert space and a universal wave function or unitary rotation. A mechanist has to choose anything equivalent to a turing universal system. But this will be incomplete for all inside relative views obtained by the inner universal subsystems. The coherence of the dreams pertain on the whole ontology, which is 3- expandable, but without purpose, given that the 3-expansion is useless, except for "real ultimate prediction", or extracting the physical laws by extracting the measure that the universal number can deduce or infer from inside. The key in all this consists in the distinction between the 3-outside universal view (given by the intended model of the TOE), and the epistemology of the observers "emulated" by what exists (in that theory).



If so, that would entail that reality is somehow self-similar or, in a
sense, holographic in nature.

Indeed, as it is the case for the UD* (the expansion of the UD). I suspect the Mandelbrot set to be a compact projection of such an expansion. That would be the case if the rational Mandelbrot set is Turing universal.

A faithfuth correct view of the 3-everything would be given by the Mandelbrot set. The minibrots would named constructive transfinite ordinals.
http://www.youtube.com/watch?v=9G6uO7ZHtK8
http://www.youtube.com/watch?v=kjW0a6WIAxw
A measure (notably physics) would be some integral on the border of the M set.



Or if there is an infinite TOE whose number of symbols is of a certain
cardinality, what is the smallest cardinal that that TOE can be
compressed into?

I thought we agree that a theory has to be finite?




 IF that is provably finite then I think that would
be interesting.

With mechanism: what exist basically (the true relation between numbers) is conceptually very simple, and is enough to understand that the "every appearances" is infinitely complex, but highly structured.




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.

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 am really not seeing how one can have an outside view of the
ultimate reality based on what you've said.  Can you explain it in
more simple terms?  If there were an outside to ultimate, then it
wasn't ultimate to start with.

You are the one invoking NF so that the universe U of sets is a set! You should be less annoyed than anyone else with the idea of an ultimate reality being an object by itself.
Such U belongs to U.
So that U can serve itself as an outside view of itself!
But I don't need NF, because with mechanism just a diophantine degree 4 polynomial can do the trick, or any other universal relation between numbers. Those are extensionally equivalent with the expanding of the UD, which contains/emulates infinitely many versions of itself.
UD emulates UD.
For computation, we have the Church's thesis making a notion of computation truly universal one. We don't have this for structure, theories, proofs, etc.

Even a physicalist can suggest a 3-view from outside like the quantum vaccum (which is really just a quantum universal dovetailer) or something like Everett's Universal (Schroedinger) Waves, or a universal (Heisenberg) matrix. But this does not gives the inside view per se. You need a theory of mind (first person, qualia) and a way to unify it with the observation and the 3-big-reality. Digital mechanism gives all this on a plate at once. It justifies, by the self-reference logics and their intensional variants how the consciousness/realities or qualia/quanta couplings emerge from the number relations, from the points of view of those universal (Löbian) numbers-relations.

Bruno

http://iridia.ulb.ac.be/~marchal/



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