On 28 Dec 2010, at 22:43, Brent Meeker wrote:

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!


On Dec 27, 8:36 am, Bruno Marchal <marc...@ulb.ac.be> 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

Of course, this is a matter of definition.

I am taking 'TOE' to be, as a working definition, a complete
description of reality or a complete description of everything that

That does not exist. Arithmetical truth is already not recursively

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

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.

The inside view might include some arithmetical untruth, but in that case it can be shown that the machine will lose its Löbianity, so this is not the interesting case. (think of PA + beweisbar('false')). It is more related to the fact that the inside (first person) view includes some non arithmetical truth. This is simpler to explain with the definition of the first person view in AUDA. The first person is the knower and it is defined by Bp & p, that is: the conjunction of provability and truth. But truth is already non arithmetical, making the knower already not definable by the first person, that is by itself. In that sense the knower feels as being something different and bigger from any definition it can handle, including "Bp". That is actually why Lucas and Penrose infer from Gödel that we are not machine, but this is incorrect, because if that was correct we would conclude that machine could prove correctly that they are not machine.

Another way to illustrate this is by using Skolem paradox in set theory. It is not a contradiction: just an amazing fact.

Consider the set theory ZF. It is a first order logical theory, so that we can apply the Löwenhein-Skolem theorem saying that there is enumerable model of ZF. A model is just an informal set satisfying all the axioms of ZF. Furthermore it can be shown that there is transitive enumerable model M, and this means that if A belongs to M, then A is included in M: all elements of A are element of the model M. But ZF proves the existence of a non countable set A. So it looks like if there is a non countable set included in a countable set! The solution of that paradox is that the non countable set is just non countable when seen in the model M. It means that the bijection between A and M is not itself a set in M. The bijection go outside of the universe M. This shows also that the notion of cardinality is relative to the model in which we are working. M is countable, but as seen in the model M it is bigger than the uncountable.

For a mechanist, this should not be so much more astonishing than the fact that a finite machine can conceive an infinite set, or that a little brain can conceive a vast, even infinite, universe.

It is hard to explain more without getting more technical. I suggest you search on "Skolem Paradox". It illustrates that phenomenon very well. But the "real" explanation is that the first person is directly defined by an evocation to the non arithmetical notion of truth. The inside first person view of arithmetic is not arithmetic. In a sense the first person associated to a machine is not a machine, from its personal point of view. It explains also why it is necessary to make a leap of faith when saying "yes" to a digitalist surgeon, and why mechanism is intuitively unbelievable by machines.


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.


(*) a conjecture by Jacques Bailhache on this list.

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