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.

(*) <x-tad-bigger>The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy</x-tad-bigger><x-tad-bigger> by </x-tad-bigger><x-tad-bigger>Kenneth Sylvan Guthrie
</x-tad-bigger>

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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