On 13 Sep 2012, at 12:40, benjayk wrote:
Bruno Marchal wrote:
Some embeddings that could be represented by this number relations
could
"prove" utter nonsense. For example, if you interpret 166568 to mean
"!=" or
"^6" instead of "=>", the whole proof is nonsense.
Sure, and if I interpret the soap for a pope, I can be in trouble.
Right, but that's exactly what Gödel is doing. 11132 does not mean "="
anymore than "soap" means "pope", except if artificially defined.
Nor do "=" itself, nor "nor".
But even
than the meaning/proof is in the decoding not in 11132 or "soap".
No, it is not. It is in the rule governing "=" on which you agree on.
If not you beg the question and take the proving machine for a zombie.
When I ask my computer to send a mail, he understands very well
(usually), despite the ambiguity and arbitrariness of the coding.
Today, it does not observes itself in practice, so the sense is still
distributed on a much larger spectrum than the specific task that it
implements, but that's is contingent.
If we just take Gödel to make a statement about what encodings
together with
decoding can express, he is right, we can encope "pope" with "soap"
as well,
but this shows something about our encodings, not about what we use
to do
it.
Indeed.
Bruno Marchal wrote:
That is why we fix a non ambiguous embedding once and for all.
How using only arithmetics?
Like a german grammar written in german. PA talks arithmetic, so we
have to translate in arithmetic. Arithmetic is Turing universal, so we
can do that without any trouble. We can even so it without classical
logic, only the usual axioms for "=", and diophantine polynomial of
degree less than 4. It took 70 years to prove this: it is not obvious
at all, and I find this quite surprising.
Bruno Marchal wrote:
Thus Gödel's proof necessarily needs a meta-level,
Yes. the point is that the metalevel can be embedded non ambiguously
in a faithfull manner in arithmetic.
It is the heart of theoretical computer science. You really should
study the subject.
You should stop studying and start to actually start to question the
validity of what you are studying ;)
Studying implies questioning the validity all along ;)
Sorry, I just had to say that, now that you made that remark
numerous times.
It is like saying "You should really study the bible to understand why
christianity is right.".
You seem to talk about Gödel's work, with weird assertion like "I
don't need to study it to say ...", where a simple study of Gödel
would help you to see that you miss something.
Studying the bible in detail will not reveal the flaw unless you are
willing
to question it (and then studying it becomes relatively superfluous).
LOL (we don't have to study no more).
Bruno Marchal wrote:
It
seems to be very fundamental to the idea of the proof itself, not
the proof
as such. Maybe you can explain how to solve it?
But please don't say that we can embed the process of assigning
Gödel
numbers in arithmetic itself.
?
a number like s(s(0))) can have its description, be 2^'s' * 3^(... ,
which will give a very big number, s(s(s(s(s(s(s(s(s(s(s(s...
(s(s(s(0))))))))))))...))). That correspondence will be defined in
term of addition, multiplication and logical symbols, equality.
I don't see what your reply has to do with my remark. In fact, it just
demonstrates that you ignore it. How to do this embedding without a
meta-language (like you just used by saying 'have its description' -
there
is no such axiom in arithmetic).
There is no problem at all given that arithmetic contains its
metalanguage. So we do use a metalanguage, which is just arithmetic
itself. The second incompleteness theorem
not-provable('0=1') implies not-provable('not-provable('0=1')')
*is* a theorem of arithmetic.
(this is reflected by the fact that G proves ~[]f -> ~[](~[]f)).
You do need a part non accessible to the arithmetic-machine to say
that ~[]f is true, but the machine can guess that.
It is not entirely obvious that we can define "provable" entirely in
arithmetic, or in any programming language, but we can, like we can
define it entirely in German, or in Fortran.
It is no more ambiguous that we can ask a universal to generate the
prime numbers, or send mails.
Bruno Marchal wrote:
This would need another non-unique embedding
of syntax, hence leading to the same problem (just worse).
Not at all. You confuse the embedding and its description of the
embedding, and the description of the description, but you get this
trivially by using the Gödel number of a Gödel number.
Maybe actually show how I am wrong rather than just saying that I
confuse
everything?
I can't open a parenthesis and provide in one simple sentence the
basic of mathematical logic.
It takes time for anyone to understand that metamathematics can be
arithmetized.
There are many good books on that subject.
I would say that the second part of sane2004 introduces concisely to
the subject, may be we can start from that.
Bruno Marchal wrote:
For more detail and further points about Gödel you may take a look
at this
website: http://jamesrmeyer.com/godel_flaw.html
And now you refer to a site pretending having found a flaw in Gödel's
proof. (sigh).
You could tell me at the start that you believe Gödel was wrong.
I tried to be fair and admit that Gödel did prove something (about
what
numbers can express together with a meta-level).
If you believe that Gödel proved something about arithmetics as
seperate
axiomatic systems, then the site clearly shows numerous cricitical
flaws.
Define "separate" axiom system.
It is just non ambiguous relatively to any universal system of your
choice.
I think it is more easy to grasp in term of machine than in term of
axiomatizable theories.
It
is not pretending anything. It is clearly pointing out where the
flaws lie
(and similar flaws in other related proofs). I haven't even see any
real
attempt to show how he is wrong.
He attributes meaning in the proof that is not there.
I have given many times a simpler proof than Gödel one, based directly
on Church thesis. I will do it again on the FOAR list, soon or later.
I will tell you.
All responses amount to little more than
denial or authoritative argument or obfuscaction.
The main reason that people don't see the flaw is because they
abstract so
much that they abstract away the error (but also the meaning of the
proof)
and because they are dogmatic about authorities being right.
No, it is "just" math. And Gödel's theorem is a theorem in arithmetic,
re-obtained many times by humans and computers alike (like for testing
logical programming: The theorem has been "found" by the Boyer-Moore
automated prover machine, for example.
Meyer might just make a sort of vague philosophical point unrelated to
the theorem and its proof. There are tuns of "refutation" of Gödel,
like there are tuns of proof that PI is rational. It would be a full
time job to refute them, especially when you see that the style is
vague and opaque, and not open to being refuted.
That's why studying will not help much. It just creates more
abstraction,
further hiding the error.
This is self-defeating. And looks like non sense when talking
factually about machines and/or formal theories.
Bruno
http://iridia.ulb.ac.be/~marchal/
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