Hi Max,
I will first comment what you say about Gödel's theorem.
You say (pp 19, 20) that Gödel's second incompleteness theorem implies
that we can never be 100% sure that "this" (Peano Arithmetic, real
numbers, ...) is consistent, and that this would leave open the
possibility that a finite length proof of "0 = 1".
This is a very common misconception of Godel's incompleteness,
sometimes advocated by "relativists".
By "common" I mean that most good popularizations of Godel's results
address correctly this misconception. I am mainly thinking about
Smullyan's many books on this subject, or the more recent, quite
excellent, book by Torkel Franzčn "Gödel's Theorem An incomplete Guide
to its Use and Abuse". I certainly recommand it to anyone interested
in this list subject. Franzčn is a little weak on the *Use* of Gödel's
theorem, but quite excellent on the so widespread *Misuses* and
*Abuses*.
It is hard for me to believe you are serious on Gödel. Even if we grant
some possibility of doubting the consistency of Peano Arithmetic PA
(say) I don't see how you derive from Gödel's second theorem the
possibility of a finite proof of 0=1. Gödel's theorem is itself
provable in PA, so your doubt would have a circular origin. Would PA
proves its consistency, this could be doubtful too: after all, all
inconsistent theories do proof their own consistency. Then it is easy
to provide everyday informal quite convincing proof of the consistency
of PA by using the fact that the axioms of PA are satisfied by the
model (N, +, *), and the inference rule of PA are truth preserving).
Formally, the consistency of PA can be proved in weak fragment of ZF
(Zermelo Fraenkel set theory) by transfinite induction up to the little
constructive ordinal epsilon zero (Gentzen theorem).
Now, what is curious and amazing, is the following consequence of the
second incompleteness theorem: given that PA is consistent, but cannot
prove its consistency, it follows that the theory PA + [PA is
inconsistent], that is PA with the addition of the axiom Bf (beweisbar
false = false is provable) has to be consistent too! (why? because if
you can derive a contradiction in PA from Bf, you would prove in PA
that Bf -> f, that is ~Bf = PA's consistency, contradicting the second
incompleteness theorem. NOW, by Godel's COMPLETENESS (not
INcompleteness) theorem, all first order theory is consistent if and
only if the theory has a model (in the logician sense, that is a model
is a mathematical structure satisfying the axioms. I think your
misconception could come from this fact. Indeed the completeness
theorem entails that the theory PA+Bf , being con,sistent by Godel II,
has a model! So there is a mathematical structure which satisfies the
axiom of PA + there is a proof of a falsity. But PA can prove (like
weaker theories) that 0 is not (a godel number coding) a proof of f,
and that 1 is not a proof of f, and that 2 is not a proof of f, etc.
That is, for each natural number n, PA can prove that n is not the
godel number of a falsity f. Thus, in the model of PA+Bf, the object
corresponding to a proof of a falsity has to be different from any
natural number. logician describes such object has an infinite non
standard numbers, and it can't correspond to anything looking like a
finite proof of f, or 0=1.
By the way, this list mixes people with diploma and without, you could
have asked or participate, but then this is what you are doing now,
isn't it? I have a phd in logic and computer science, although my
motivation has always been biology and/or theology, I mean fundamental
questioning. People without diploma are often better on new or very old
questions because they are less prejudiced by granting less theories.
It is also why I like to interview directly universal machines.
What is much more annoying in your paper, and shows that you have never
really consulted this mailing list, is that you are still burying
under the rug the mind body problem, or the first person/third person
relation problem. Your use of the frog/bird distinction illustrates
that you are using implicitly, despite your mathematicalism which I
appreciate, some "mind-matter"-like identity theory capable of giving
sense to the notion of a physical structure and of an observer
belonging to it. This *can* make sense, but, especially with the
computationalist hypothesis (= I am turing emulable), such a thing has
to be justified. This follows from the Universal Dovetailer Argument +
the Movie-graph Argument. I have already show that the
computationalist hypothesis (roughly: there is a level of description
where I am Turing emulable) entails the falsity of the computational
universe thesis. Physicalness, with comp, is a global internal feature
of arithmetical reality emerging from "machine's dream gluing", to be
short.
Another problem, is that, although I agree with mathematicalism, I have
no clue of what could be "All Mathematics". But with the Church Thesis,
or Church-Turing thesis, it can be argued that arithmetical reality is
enough (even for set-theoretical talking machine). Analysis and
everyday informal mathematics can be justified from inside too. See my
other posts or my work (hmm...I should update my webpage with my last
papers). About some of your point on mathematics, I think category
theory could help you, but note that it will not help for defining the
whole of math.
I finish by saying that I am ok with your ERH (there exists an external
physical reality completely independent of us human).
But I do not believe there could exist a corresponding external
physical reality completely independent of us lobian entity, once we
assume the comp hyp, or even very weak version of the comp hyp). CF the
UDA reasoning.
About the MUH (Our external physical reality is a mathematical
structure), I find this very vague and ambiguous and have to think more
about how you derive it. With comp the physical has to emerge from a
notion of first person plural sharable experiences. Of course I am not
saying that comp is true, but I have made that first person sharable
experience enough precise so that it can be empirically tested, and I
can already show that many feature of quantum mechanics are
consequences of it. I hope to derive the local exploitability of
universal quantum machine in the neighborhood of (almost all) classical
universal, in the years to come, or to refute it, and thus refute the
comp hyp.
I show also that Godel's results (and Löb Solovay generalizations, ref
in my Lille thesis) provide a transparent arithmetical interpretation
of Plotinus theology, including his platonist theory of matter. The
UDA, then, can relate Plotinus with the comp hyp, and does illustrate
new relationships between Pythagorean and Platonist "theologies" (in
the greek sense, not necessarily in the Christian sense). See some
"theological" threads in the archive.
Bruno
Le 11-avr.-07, ŕ 17:25, Max a écrit :
>
> Hi Folks,
>
> After a decade of procrastination, I've finally finished writing up a
> sequel to that paper that I wrote back in 1996 (Is "the theory of
> everything'' merely the ultimate ensemble theory?) that's been the
> subject of so much interesting discussion in this group.
> It's entitled "The Mathematical Universe", and you'll find it at
> http://arxiv.org/pdf/0704.0646 and
> http://space.mit.edu/home/tegmark/toe.html
> - I'd very much appreciate any comments that you may have.
>
> The purpose of this paper is both to clarify what I mean by the Level
> IV Multiverse and to further explore various implications, so it has
> lots of discussion of stuff like the simulation argument, the relation
> to Schmidhuber's ideas, Gödel incompleteness and Church-Turing
> incomputability. Please let me apologize in advance for the fact that
> Sections III, IV and the appendix of this paper are quite technical,
> so if you're among the 99.99% who don't have a Ph.D. in theoretical
> physics, perhaps skip those sections. I've added links to more
> accessible papers touching on some of these issues at
> http://space.mit.edu/home/tegmark/toe.html, and I'll try to write
> something less obtuse soon.
>
> Finally, if you discover a good time stretching device, please let me
> know! Although I'm embarrassed that I haven't found the time to follow
> and participate in the fascinating discussions in this group, the fact
> that there's such interest has inspired and motivated me to continue
> pursuing these ideas despite the discouragement from mainstream
> academia. So thanks for the encouragement!
>
> Max
> ;-)
>
>
> >
>
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Apr 18 2007 - 06:29:17 PDT