On 31 Dec 2010, at 13:31, Brian Tenneson wrote:

So if a TOE is not recursively enumerable, it still might be captured
in some finite set of statements like how arithmetic can be captured
in the Peano axioms.  That arithmetical truth is not recursively
enumerable can somehow be derived from these first principles, right?
Perhaps the same is true of a TOE.  That's what I'm wondering.


Let us distinguish carefully the "everything" or ultimate reality and the "theory of everything" which is supposed to be finitely presentable, and having a recursively enumerable set of theorems (because we want proofs in the theory being checkable). And then we have to distinguish the ontology (what exists primitively), and the epistemology (in the large and different senses of what can be known, believed, guessed, felt, observed).

Once we aim at a theory of everything, the same theory has to handle both the ontology and the epistemology, and their relations.

Then, what I am saying, is this: IF my consciousness is invariant for a local digital substitution of my brain (or generalized brain) then the theory of everything is ROBINSON ARITHMETIC (RA). That is a very tiny fragment of arithmetic, in which you can not even prove that for all number x you have x + 0 = x. It is a very tiny fragment of arithmetic, much smaller than PEANO ARITHMETIC (PA). PA is RA + the induction axioms.

This is not an arbitrary suggestion: it is really the end result of UDA +AUDA. It makes DM testable.

The theory has a recursively enumerable ontology: indeed it is the set {0, 1, 2, 3, ...}, naturally described in RA or PA by 0, s(0), s(s(0)), s(s(s(0))), ... That ontology is highly structured, by addition and multiplication, and has a standard model, or a standard notion of truth: we say that a sentence like AxEyP(x,y) is true if it is the case that for all numbers n it exists a number m such that it is the case that P(n, m).

That standard model can be represented by the set of arithmetical assertions which are true, or even by the set of the Gödel numbers of those true statements. That is what I call arithmetical truth, and that is *not* recursively enumerable: indeed it is at the top of the arithmetical insolubility hierarchy (but at the bottom of the analytical hierarchy, note).

So, the 3-complete description of the ontological structure, which is (N, +, *) is not at all captured by the "theory of everything", RA. And things will be much worse for the epistemology. RA is not, unlike PA, löbian.

So, in what sense RA is the theory of everything?

Because RA although not Löbian, is already universal. It is the simple first order specification of a Turing universal machinery. RA is sigma_1 complete, and it proves all true statements having the shape ExP(x,y), and this makes RA equivalent with a Universal Dovetailer. In particular RA emulates, in all possible ways (by many proofs) all correct Löbian entities, including their interaction with all initial segment of all oracles.

And all the rest is provided by the epistemology, which is given by the internal views of all those Löbian entities, emulated by RA. (the arithmetical hypostases in AUDA).

And *that* is very big. Not only it extends vastly arithmetical truth, but it extends analytical truth, set theoretical truth, Category theoretical truth, etc.

Note that this is trivial. Löbian machines have much stronger cognitive abilities than RA. They are universal, like RA, but they know that their are universal, and their first person views are distributed on the infinite border of the sigma_1 truth: they have to develop richer mathematics, like they have to develop themselves. But the more they develop, the bigger will be their relative ignorance. G and G* remains invariant in intension, but grows in extension for any self-observing universal machine, and G* grows more quickly than G.






How can we know that?  "Reality is the totality of all that exists" is
a finite complete description.

Well, that is my favorite definition of reality. But it is not a theory: you don't say what exist. RA says what exist. It says that 1 exists (Ex(x = s(0)), it says that you current computational states exists.



Now I'm assuming, for the sake of argument, that "totality," "all",
and "exists" have finite complete descriptions.  And if these words
don't, then no words do and we might as well be talking about
asdjedwjef.

Well, as you know, if we limit ourself to first order language, we can talk in a metaphysics-free language. With mechanism this works well for the ontology, but for the epistemology we have to give meaning to the standard model, and eventually to the full analytical truth, and beyond. To explain mechanism, we need people having a notion of consciousness, enough for doubting they can survive a digital substitution.




What I'm after is what else we can say about a TOE.  Given what a TOE
is, it does answer all questions written in its language.

Are you not confusing the theory, and the subject of the theory.

We might need to introduce an intermediate notion. Let us call that the realm. With mechanism the numbers, structured by addition and multiplication is the Realm of Everything: ROE. But that is what I called above the ontology. For a (naïve) physicalist the Realm is given by particles structured by their force (fermions and bosons, the quantum law).

The epistemology will be given by higher order substructures of the ontological structure. Physical brains for the physicalist, Löbian numbers for the digital mechanist. For a mechanist, a physical brain appearance for numbers has to be justified by the numbers' structure.



One way to describe something, a real basic way to describe something,
is to form an aggregate of all things that meet that description.
There may be no effective procedure for deciding whether or not A is
in that aggregate, whatever.  The point is that that is one way to
describe something.
Thus reality basically describes itself.
Reality is an aggregate and as such is a TOE, a complete description
of reality.

But that is the trivial "TOE". You are saying "take the territory for map". We search for "first principle". Physicalist believes that it is some particles and very few laws, or even just the quantum vacuum. What I say is that if DM is true then any first order sigma_1 complete theory will do for the ontology, and the rest is in the epistemology (entailing that the physical laws are *really laws*: they are invariant for all Löbian entities, and their are extractible from self- reference). With mechanism, we have that physics, psychology, theology (the whole epistemology in a large sense) is independent of the basic realm, which need only to be universal (sigma_1 complete).
But then, the internal epistemology has an 'unboundable' complexity.
It is open if the number's physics alone has a boundable or unboundable complexity, but intuitively it would be weird that it has a boundable complexity).





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 is the nature of a TOE, though?  I know actually finding a TOE
might be difficult to say the least but we can at least say some
things about the nature of a TOE.

It is what unifies what we know, believe, feel, observe, guess .. into a coherent picture.




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

Axiom schemata in ZF are infinitely many statements, aren't they?

Oh! Take Von Neumann Bernays Gödel instead. It is more powerful than ZF, and it admits finite presentation. ZF is OK. I really meant finite or recursively enumerable theories, or theories having a finite presentation. They are (abstract ) machine or programs.





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.

Well, that's what I've been trying to prove.  I guess I can stop now
=)

Sorry :-/
The facts is that if mechanism is true, I really don't see how we can escape the conclusion that elementary arithmetic is enough, and the internal epistemology of arithmetic "explains" all the dreams, why their are stable, why some are sharable, etc. Eventually, interpreting the ONE of Plotinus by Arithmetical truth gives back a transparent interpretation of a neopythagorean neoplatonist Theology. And my point is only that the physics of machine is already enough precise to be confronted with the observation, and the gift is big given that we have a way to explain qualia and quanta at once.
BTW, a lot of works remains ...
I doubt you will find something as universal as a something like Post- Church-Turing universal. IMO, the real bomb of the 20th century is Church thesis and the discovery of the universal 'numbers'. The embedding relation you are searching is, I think, partial or total turing emulation. The UD builds them all. The UD* is just a splashed universal machine. The UD is an avatar of the universal machine. The keyword is 'universal' and it make sense by Church's thesis (Post's law, Turing's thesis). Perhaps the object you search is some initial object in a category with Turing-morphism, or something ...


It is the overall, basic structure of what exists.  The reduced
product of all structures is a key to this.

Well, with mechanism, what exists is not what is important. I chose the numbers, but I could prefer the combinators, or the hereditarily finite n-categories, etc. What is important is the epistemology, that is mind and matter, person and personal experiences, the stability of dreams, laws and contingencies ..., the relation between universal numbers, etc. Matter is part of epistemology (that is what the UDA, including the Movie-Graph-Argument is all about).




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 ...


But U is not outside U.

U is belonged by U.



read more »

Sorry, it seems to have eaten your post here...hmm.


I cut and paste the last paragraphs, hoping you get it right:


<<

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 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 (1-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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