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. The embedding is not unique, and
thus by looking at the arithmetic alone you can't have a unambigous proof.
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.

Thus Gödel's proof necessarily needs a meta-level, 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.
You can, of course *believe* that the numbers really exists beyond their
axioms and posses this transcendent intelligence, 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.

I don't see how any explanation of Gödel could even adress the problem. 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. This would need another non-unique embedding
of syntax, hence leading to the same problem (just worse).

For more detail and further points about Gödel you may take a look at this

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