Le 05-juin-06, à 18:37, Tom Caylor a écrit :
> Not to try to answer the Puzzle, but just some thoughts for the
> conversation:
That's the right spirit!
>
> At one glance, it seems that the argument is trying to transcend
> Godel's Incompleteness theorems. The Universal Language is trying to
> be both universal (complete) and consistent.
The Universal Language is trying to be universal, and actually will
succeed, if we accept Church thesis. That is, concrete languages like
fortran, python, java, c++ .... , which can all be proved equivalent
with respect to the ability to define *computable functions*, *are*
universal once we accept Church thesis.
But now let me say something quite important. The term "consistent" is
not genuinely used here. Consistency is said about a THEORY, and until
now we are just talking about "programming language" or machine. A
theory is inconsistent if it proves a falsity, but a programming
language just prove nothing. There is no axioms, nor any inference
rules. There is no theory (not yet!); just primitive instructions which
can be used to explain how to compute functions and which can be
interpreted by some entities/machines.
I am not saying your intuition fails you completely because we are
indeed very near a proof of incompleteness theorem for ALL *theories*,
but this will be a consequence (here) of accepting Church thesis, and
thus understanding how fortran (say) can escape the use of
diagonalization for going outside the sequence of functions defined in
the language. That is, at some point you will see that incompleteness
of theories, i.e. with respect to provability, is a consequence of
completeness of universal machine/language, that is with respect to
computability.
Of course you make me anticipating a little bit, but once you will see
that Church thesis CT can be consistently assumed despite the
diagonalization, you will see that (with CT):
1) the notion of COMPUTABILITY is ABSOLUTE, and will not depend on any
choice of (universal) language, formal system or machine: there exist
universal languages (or machines), and they are all equivalent with
respect of the definable, codable, describable, computable functions.
but then as a consequence (and that's a part of the price):
2) the notion of PROVABILITY is necessarily RELATIVE, and will always
depend on the choice of a particular formal system or machines. In
particular there will never be any universal theory, even if the
discourse domain is restricted just to positive integers or natural
numbers.
Ah. You see computer science is nice: it assesses both the relativist,
at the level of theories and proof, and the absolutist, at the level of
programming languages and computations.
A good understanding of this will help you later to get a better
appreciation of G and G*, which show that, although provability is a
relative notion, there are universal feature of provability which can
be captured by some modal logics.
>
> This Puzzle seems to corresponding to part of Step 7 in your Universal
> Dovetailer Argument (UDA). In that Step, you perhaps answer the Puzzle
> so I don't want to simply quote the answer, which might short-circuit
> my/our understanding.
I'm not sure. Step seven introduces the UD and thus relies on Church
thesis and I suppose people already knows why the UD and CT is possible
despite diagonalization.
> But I have a problem with answering that the
> programs which conclude that 0 = 1 simply run forever.
> Couldn't you
> build any "complete" system or theories by simply letting programs run
> forever.
You seem to be close to the right idea here ...
> This seems to be an artificial/arbitrary path to the truth.
> Couldn't you conclude whatever you want with this method? Perhaps I'm
> just proving your point.
... but a little bit less here :)
>> PS Rereading some recent mails I wrote, I am ashamed of my style (when
>> I complete a sentences!) and by my enduring mishandling of the
>> singular/plural (the "s" problem). Please accept my apologies, and
>> don't hesitate to correct me or to ask questions in front of
>> ambiguities. Thanks for the interest anyway.
>>
>
> It is probably mainly because of English being not your native
> language.
Alas this is only partially true. Well, perhaps it is due to my use of
both english and french all the time, but I have a tendency to mess up
the "s" in french too. The "s" rule are 90% opposed in french and
english.
A deeper explanation could perhaps be related to .... the
reason-&-person thread! Once you allow, like Parfit and some people in
this list to do thought experiments in which amnesia is accepted, then,
as I have already try to explain to Lee Corbin some month ago, you will
converge toward the idea that there is only one person possible. For
example, if you think that after a duplication Washington/Moscow both
of "you" continue to be you *at the first person*, then you should
already accept that all the descendants of the amoeba, that is all of
us, are in reality the same person. Put in another way, personal memory
is capital for personal identity. (I have discovered recently in a book
by Guthrie that the Pytahgorean already insisted on that(*)). This
should be true for individuals, nations, community, etc. This is
obvious with my UDA definition of first person discourse which is just
the content of a personal diary/memory. But this will appear as being
less obvious once we interview the lobian machine, so that, although
the UDA definition is enough to get the "reversal", incompleteness will
utimately justified that a non trivial notion of first person will
emerge from incompleteness, I mean a notion of first person which
relies not exclusively on personal memory.
(*) The Pythagorean Sourcebook and Library: An Anthology of Ancient
Writings Which Relate to Pythagoras and Pythagorean Philosophy by
Kenneth Sylvan Guthrie
http://www.amazon.com/gp/product/0933999518/ref=pd_sim_b_1/103-1630254
-7840640?%5Fencoding=UTF8&v=glance&n=283155
> My familiarity with a few non-English Latin-based languages
> helps in my understanding, I hope.
Thanks for trying anyway.
> If only there was a Universal
> Language.
Church thesis *is* the statement that there is a universal language,
for computation, and I hope you are not taking my "refutation" of
Church thesis too much seriously. The "refutation" is wrong, I have
really make an error (on purpose). The question is which one?
You did thought at some moment that I was using the axiom of choice and
that I was leaving the finite area and I told you I didn't. Indeed I
did not even use the excluded middle principle and the transfinite
extensions were all definable intuitionistically, that is
constructively (indeed all the growing functions where programmable).
Now, to solve *the puzzle*, there is indeed a need to leave the
constructive (intuitionist) area. Mathematicians do that all the time,
even if not always consciously. Before proceeding let me ask you if you
accept the following reasoning.
Oh, I'm interrupted, and I will do it in another post asap. Still today
I hope.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Tue Jun 06 2006 - 09:06:40 PDT