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

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 light, a bat using sonar, a blind
person with keener touch and hearing, or the latest overpriced
robotic vacuum cleaner, all agree on whether the door is open.
The key current challenge in physics is deriving this semiclas-
sical consensus view from the fundamental equations specifying
the bird perspective. In my opinion, this means that although
understanding the detailed nature of human consciousness is an
important challenge in its own right, it is not necessary for a
fundamental theory of physics."

This assumption is not sustainable if you assume that the brain works like a universal system, or more weakly that your generalized brain (can include the physical universe) works like a universal system. The generalized brain is any portion of physical reality that you have to emulate digitally (and physically, a priori) to survive a functional substitution, or just to feel nothing change from your perspective in such a substitution. All that makes sense assuming digital mechanism. This entails that a theory of everything is given by *any* first order specification of a universal (in the sense of Turing) system (language, theory, machine, number).

But consciousness, or the most basic 'frog view' (the first person) cannot be attached to any particular third person describable frog embedded in a structure or a computation, but in an infinity of them, related by different type of accessibility relations, or neighborhood relations, depending of the points of view defined.

Mechanism makes it possible to reduce the mind-body problem to a reduction of physics to number self-reference theory. The splitting of such theory into the deductive and the inductive part of those logics makes it possible to derive a notion of both quanta and qualia.

Mechanism makes the inference of consistency (a part of consciousness) a key ingredient in the making of the physical realities, which appear to be first person plural sharable computations.

Physical reality is not a human mental object, but still a lobian mental object.

Mechanism intertwined completely the level 3 and 4, in an highly mathematically structured way. This answers a criticism by Deutch on that kind of everything theory, because mechanism makes physics the mathematical non trivial border of the universal (lobian) person. It makes mechanism testable. And indeed indeterminacy, non locality and non)-clonability are 'easily' derivable. The person, alias any universal system, becomes Löbian when it can prove its own Sigma_1 completeness: it proves p -> Bp for all p Sigma_1, so that Bp -> BBp, B(BB->p)->Bp, etc. Such machine are aware of their incompleteness. I see a reference on incompleteness in your reference, but none in your text.

Your paper is a nice little paper. I am not a NF expert, so I cannot judge the originality, but I took pleasure reading it, and you have probably reawaken my taste for NF. Nice presentation of first order logic, too. You might consult the archive of this list, or my url, but assuming mechanism, and even strong weakening of mechanism, entails that you cannot make the physical, nor the mental, a purely mathematical thing, except in a necessarily informal way (mentioning the logician notion of standard model of arithmetic, for example). Both the mental and physical, or the coupling first person/its third person possible computations, emerges from the purely (first order) arithmetical relation existing among numbers, or combinators, or lisp programs. But it only "emerges" from inside, and that inside, including the first person", can never completely self-reflect itself in it completely (which justifies a tree of transfinite progression and diversification, some very deep one like most probably ours).

Note that a universal dovetailer, or a Sigma_1-complete theory (seen in extension) are universal object, and can simulate themselves. They play the role of you universal object, with simulation playing the role of (not always elementary) embeddings (from inside, or from the first person views the embeddings are not elementary, nor the accessibility relations.

You might improve your theory by addressing concrete problems, like why physics has this shape? why qualia? why quanta? are there really constant in physics? why superposition?, why complex numbers? why dream, why pleasure, why symmetry, why irreversibility (if any?), why suffering, etc.



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