On 12 Sep 2012, at 14:05, benjayk wrote:
Bruno Marchal wrote:
On 11 Sep 2012, at 12:39, benjayk wrote:
Our discussion is going nowhere. You don't see my points and assume
I want to
attack you (and thus are defensive and not open to my criticism),
and I am
obviously frustrated by that, which is not conducive to a good
discussion.
We are not opertaing on the same level. You argue using rational,
"precise"
arguments, while I am precisely showing how these don't settle or
even
adress the issue.
Like with Gödel, sure we can embed all the meta in arithmetic, but
then we
still need a super-meta (etc...).
I don't think so. We need the understanding of elementary arithmetic,
no need of meta for that.
You might confuse the simple truth "1+1=2", and the complex truth
"Paul understood that 1+1=2". Those are very different, but with
comp,
both can be explained *entirely* in arithmetic. You have the right to
be astonished, as this is not obvious at all, and rather counter-
intuitive.
There is no proof that can change this,
and thus it is pointless to study proofs regarding this issue (as
they just
introduce new metas because their proof is not written in
arithmetic).
But they are. I think sincerely that you miss Gödel's proof. There
will be opportunity I say more on this, here, or on the FOAR list. It
is hard to sum up on few lines. May just buy the book by Davis (now
print by Dover) "The undecidable", it contains all original papers by
Gödel, Post, Turing, Church, Kleene, and Rosser.
Sorry, but this shows that you miss my point. It is not about some
subtle
aspect of Gödel's proof, but about the main idea. And I think I
understand
the main idea quite well.
If Gödels proof was written purely in arithmetic, than it could not be
unambigous, and thus not really a proof.
What? this is nonsense.
The embedding is not unique, and
thus by looking at the arithmetic alone you can't have a unambigous
proof.
This does not follow either. *Many* embeddings do not prevent non
ambiguous embedding.
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.
That is why we fix a non ambiguous embedding once and for all. What
will be proved will be shown independent of the choice of the
embeddings.
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.
or alternatively a
level-transcendent intelligence (I forgot that in my prior post) to
be true,
because only then can we fix the meaning of the Gödel numbers.
Gödel could have used it, like in Tarski theorem, but Gödel ingenuosly
don't use meaning or semantic in he proof. It is a very constructive
proof, which examplifies the mechanisability of its main
diagonalization procedure. This has lead to a very great amount of
results, the most cool being Solovay arithmetical completeness theorem
for the logic of self-reference.
You can, of course *believe* that the numbers really exists beyond
their
axioms and posses this transcendent intelligence,
What do you mean by exists beyond the axiom.?
What transcendent intelligence is doing here?
so that they somehow
magically "know" what they are "really" representing. But this is
just a
belief and you can't show that this is true, nor take it to be
granted that
others share this assumption.
No need of that belief. Machine's belief are just supposed to be made
of the axioms and the rules generating them, which can include inputs,
and other possible machines. It is model by Gödel's provability
predicate for "rich" machines.
I don't see how any explanation of Gödel could even adress the
problem.
You created a problem which is not there.
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.
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.
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.
Gödel is often misused, but it is a type of result that you can prove
in many different ways, like Pythagoras theorem, it is also
constructive, so it looks like "Pi is rational because everyone know
that Pi = 3,14".
Bruno
http://iridia.ulb.ac.be/~marchal/
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