Just as a general comment, I think it's important to reiterate the
actual scope of the paper.

One important assumption I assume is that a complete description of
reality is independent of humans and anything carrying human
"baggage."  Admittedly, a mathematical structure as defined within the
context of model theory is not necessarily the only sort of thing
devoid of human baggage.  The wider consequence is that what I provide
later is certainly not the only possible complete description of
reality, if it even is one at all.  Another assumption is that the
structure U having the property that all structures are elementarily
embeddable within it would have to be a structure that a complete
description of reality describes.

The scope is further limited in that absolutely no other properties of
U are investigated and the paper is not meant to be a complete
investigation of U in its current form.  I would like to examine
further properties of U and that would involve a sequel-ish type
paper.  As such, many important issues are not investigated.
These remarks will have to be added such as in a conclusion section
and thanks to you, Bruno for providing further discussion.


>
> But your mathematical structures are sets, this is why you work in a
> set theory (NF). They are models of first order theory, and the
> function and relation symbols are interpreted in term of sets and
> subsets. OK?

Yes.


>
> But you have to know if you are using real as cut or real as Cauchy
> limit if you do constructive analysis.
> I know only the partial computable functions as something really
> machine or language independent. Theories are never universal, except
> *perhaps* with respect to injection (inclusion, embedding). All first
> order recursively enumerable set theory only scratch the arithmetical
> truth.

Maybe a better example would be a construction of the integers given
in the wiki article:

http://en.wikipedia.org/wiki/Integer#Construction

Once the integers have been constructed so, it is not necessary to
refer to how they were constructed.

Perhaps I am wrong and the method I use to construct U will matter
more than I think.  Well, it's more of an existence proof than a
construction anyway.

>
> What is really 'baggage-free' in first order theories is the relation
> of consequences, between premise and conclusion. But all premise are
> already baggages.

Yes but the structures themselves are baggage-free.  The baggage is in
the premise that any structures are relevant concerning a complete
description of reality.

>
> Are you not confusing mathematical object with first order theories
> describing those mathematical objects, or with the models of those
> theories.
> What about incompleteness?
> You say "she can know", but how do you define "knowing" for a
> mathematical structure.

Given the property of U that *all* structures are elementarily
embeddable within it establishes this without needing to get into what
knowing means for structures.  It certainly would be cute to provide
such a definition of knowing but that would likely involve baggage in
the introduction of even more premises.

What about incompleteness, indeed?

If one looks at the set of formulas satisfied by U, called the theory
of U, that set is automatically complete and consistent as are all
"theories of".  I'm not sure if this answers your question.

>
> Let me drop my pen. How do you predict what will happen in your
> theory, and how do you relate this to possible reality. mechanism
> shows that such question are not trivial, and that physics (as
> conceived today) just miss the second part of the question, by
> assuming a mind-brain identity thesis which is not correct. A lot of
> discussion have been done on just this.

This is as I mentioned in the beginning of this post beyond the scope
of the paper.  I would add that this is not intended to be a science
paper but a math paper, or perhaps a theoretical physics paper whose
sole intention is to answer Tegmark's question about which structure
is isomorphic to reality.  So no testable predictions are involved as
it is not intended to be a science paper.

>
> The problem is that I am not really a set theoretical realist. So I am
> not even sure those question are meaningful.

The basic underlying view of the role of mathematics in a complete
description of reality will dictate, at times, whether or not the
paper is to be rejected outright because of its premises.

>
> And of consciousness. I'm afraid too.

Yes but again, that is not an intended issue for discussion.


>
> You can save the theory by supposing some strong anti-comp axioms. but
> most usual weakening of comp, like with supermachine, or machines with
> oracles, will not do. All this follows from what is argued in the
> SANE04 paper (or my thesis, alas in french).
> My work refute Tegmark approach (even if much older!). Level 1 and 2
> appears to concerned geographical question, and the laws of physics
> are given by what is universally observable by universal machine,
> blurring level 3 and 4. I should probably write a paper, but this
> follows easily from the Universal Dovetailer Argument, once you get
> the non triviality of Church Thesis.

In my opinion, assuming even more things would necessarily weaken the
theory and make it even less non-controversial.


> You still beg the question. (First to be sure I do agree with you on
> the conclusion). A physicalist will just tell you that, if observation
> needs tool, their work consists from abstracting from those tools to
> get the real, perhaps still unknown, thing. And that thing will be
> conceive as substantial, primitively material or physical. They will
> say that anything mathematical is a product of the human mind, and
> that it simplifies the physical reality, and that we should never
> equated both.
> Now, what I show, is that if we assume that we are digitalisable
> machine, then physical reality is given by a very special mathematical
> 'structure' (which by the way are not elementary embaddable in first
> order theories, already (the model of the modal logic G (and S4Grz)
> are not definable by first order relations, nor are the arithmetical
> quantum logic. This is of course coherent with the fact that your
> theory need an anti-computationalist foundation.

Indeed.  The assumptions are necessary to make the U have any
significance and, while U is baggage-free, the assumptions are not.

>
> And what do you mean by 'complete description' of reality? Sometimes I
> call that a realm. By incompleteness there is no complete theory of
> the arithmetical realm, but (assuming mechanism) the mathematical
> structure (N, +, x) is already reality-complete, even if non
> effectively axiomatisable.
>
Excellent question.  I was just about to say that the words complete
and reality have human baggage.  I would say that a complete
description of reality would have to provide a "dictionary" or one-to-
one correspondence between that description and everything that is
real.  But I still haven't really defined what "real" means.  I could
say that reality is the totality of all that exists but then what do I
mean by totality and exists?

I believe that an analysis of language would entail that *nothing* has
a definition free of "atomic concepts".  Thus I would say that all
words in language have definitions that carry human baggage.  However,
U has no human baggage; it can be defined utterly formally and I would
think in a machine language.

Indeed, I will have to add what I think a complete description of
reality is to the paper.


>
> But models of G are not first oder theoretical, even for just the
> propositional part. You can represent them IN term of first-order
> theoretically describable objects (like set), but the Kripke structure
> of G needs second order axiom to be handled directly. Going through
> *some* theory of sets, in my opinion, introduced baggages.

G?

If baggage includes what the axioms are, then yes there is baggage in
axiomatic systems.  I would say that (N, 0, S, <) ie "the arithmetic
structure" is free of baggage for one could, say, take the equivalence
class of all of its descriptions and call *that* the structure.

>
> Yes. With mechanism, elementary (first order) arithmetic, even without
> the induction axioms, is already the 'theory of everything'.
> I can explain how to derive the whole of physics from it, including
> both quanta and qualia.

Sounds appealing.  What do you mean by "theory of everything"?

> To be sure *any* first order logic Sigma_1 complete theory will do.
> The Pi_i comes from inside, and already belongs to the mind of the
> numbers or finite thing. I could use the hereditarily finite sets.
> Even equational theory without logic (like a Diophantine equation, or
> the SK combinators) can do the job.
> If we are machine, it is just absolutely undecidable to know if there
> is more at the basic existence level. From inside, by reflexive
> incompleteness, there is *much* more. The intuitive reason is that
> machine cannot distinguish realities from dreams (computations).
>

My reasoning for U and its significance is more of a top-down approach
than a bottom-up approach, I would say.  It would be interesting to
see if they ultimately are equivalent, depending on what "theory of
everything" means.  Tegmark defines that as a complete description of
reality and I'm unsure how you're defining it.

>
> yes, p is arbitrary (sigma_1) arithmetical sentence, like Ex(x+s(s(0))
> = s(s(s(s(s(0))))))
>
> > and B a
> > predicate something along the lines of "believes"?
>
> It is Gödel provability predicate (Beweisbar). Bp, for example B('Ex(x
> +s(s(0)) = s(s(s(s(s(0))))))') is an arithmetical sentence, with 'Ex(x
> +s(s(0)) = s(s(s(s(s(0))))))' the Gödel number of the string "Ex(x
> +s(s(0)) = s(s(s(s(s(0))))))".
>
> > If that machine
> > can prove p->Bp I take that to mean that it can prove that if a
> > statement is true then it believes that statement.
>
> You would have to quantify on "p". That is interesting, but I use only
> that: for all Sigma-1 p the machine prove p -> Bp. It is the difficult
> part in Hilbert Bernays proof of Gödel's second incompleteness
> theorem. See Boolos 1993 for a readable proof. Now, Bp is itself
> Sigma_1, so that such machine verify that for all arithmetical p, Bp -
>  > BBp, and this makes such machine Löbian (we get Löb's formula: B(Bp
> -> p) -> Bp). The main axiom of G (and G*).
> Note that you need induction for this. The believer is a richer theory
> than the theory of everything!

That would be a parallel to the U in my paper in that things such as
consciousness would be richer theory than the TOE.


>
> Which reality?  Assuming mechanism there are only numbers' dream, and
> some can cohere to give the 'illusion' of physical reality to
> population of numbers or machines.
> Eventually I think the big whole is not nameable by its internal
> creature. I do like the idea that there is no universal set (in ZF,
> VNBG, ...). From outside we might say that arithmetical truth is a
> name for that big whole, in the 'theology' of the simple Löbian
> arithmetic (Robinson arithmetic + induction). By Tarski theorem, such
> a notion of truth is not describable by the machine itself, and this
> makes it possible for arithmetical truth to play the role of the ONE
> in Plotinus in the theology of such a simple machine/theory. Mechanism
> leads to a "simple" reinterpretation of the basic of Plato-Plotinus,
> and shows that mechanism is incoherent with Aristotelianism (with a
> basic physical universe with creature in there).

I am taking the word reality to be an ultimate.  There is only one
such ultimate perhaps with differing descriptions.

>
> > But does mechanism imply that physical and mental --can not-- be
> > isomorphic to a mathematical structure, even if it implies they are
> > not literally mathematical structures?
>
> It is an open problem, with the mechanist assumption. My feeling is
> that consciousness is not a mathematical structure. It is more like a
> mathematical 'phenomenon'. A tendency of universal number to infer
> coherent universal extension to their dreams. To be short.
>
You mentioned earlier that
[quote]
With mechanism, elementary (first order) arithmetic, even without
the induction axioms, is already the 'theory of everything'.
I can explain how to derive the whole of physics from it, including
both quanta and qualia.[end quote]

Consciousness is something and as such included within 'everything'
and therefore, especially since you can derive qualia, it would appear
that consciousness is a mathematical structure (as yet perhaps
undiscovered).  I've seen attempts to formalize consciousness but I
believe that since consciousness does not have a controversy-free
definition that it adds to the dilemma of whether or not consciousness
is isomorphic to a structure.  How are you defining consciousness?



>
> Good luck. Your paper is very clear, and interesting (despite it
> misses somehow both mind and matter!). I hope it is original enough to
> be published. Don't hesitate to submit to some logical journal,
> perhaps the bulletin of the society for symbolic logic:
>
> http://www.aslonline.org/journals-bulletin.html

Thanks, I will someday try to but I certainly will be adding to it in
light of your insights.
The very worst case scenario is that I have to drop the entire physics
level 4 angle and just propose it's an interesting structure but that
it is interesting remains to be seen in that I have not investigated
the nature of U.  Yet.

>
> Of course it would even be better if you succeed in taking into
> account the mechanist constraints, or at least make clear that you
> need some non-comp axioms. Even in that case you will have to define
> your notion of 'intersection of structures' in a way which take into
> account the infinitely many first order theories describing your
> observer's state.

I actually don't think it will be hard to define the notion of the
intersection of structures.  I wager I shouldn't try to too hastily.
For one thing, that notion might already be present in model theory.
My general idea would be to define an isomorphic copy of the
intersection that belongs to U by sending the structures we're
intersecting into U via the elementary embedding function and
intersecting them within U.  But this is just my first thought.

Indeed category theory is not mentioned in the paper when perhaps it
should be.  I need to look at how a structure in model theory is like
a category, and unlike.

Thanks again.

Brian

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