|
|
Re: The WHO CARES proof of anti-anti-diagonalisation
Posted:
Jun 9, 2005 7:52 AM
|
|
Stephen Montgomery-Smith wrote:
> HERC777 wrote:
>
>>>> x(i) = not F(i,i)
>>>> therefore F is incomplete .. is used in most every proof of
>>>> incompleteness in all theory.
>>>
>>>
>>>
>>> Yes, this is what is used. So let's go with the computable function
>>> example (which I presume that you are hinting at with A.I.) One way to
>>> see the Turing halting theorem is this. If you give me a computer
>>> program which you claim can check if programs halt or not, I can create
>>> a program for which your program will fail. It is a highly constructive
>>> proof.
>>
>>
>>
>> That doesn't prove F is incomplete.
>
>
> I have to admit that your thoughts are too fast for me to follow
> completely. When you use the word "incomplete" I presume that you are
> refering to the notion that there are statements that can be neither
> proven or disproven in certain theories.
>
>
>>
>> You extend the proof of nonexistence from A FUNCTION to THE COMPLETE
>> MATHEMATICAL SET.
>>
>> What function, if any, did the halting proof disprove?
>> What real, if any, did Cantors proof disprove?
>>
>> Sure you can disprove certain specific defined functions don't exist.
>> the function that calculates halt values for every program, the
>> function that sums infinite objects in a microsecond, the function that
>> tells me the weather 10 days from now, the function that tells me the
>> maximum output of a program of size n, the function that does this...
>> NO PROBLEM.
>
>
> Thanks. I feel that we are making progress. Here is one place where
> you have accepted that the diagonal style argument works.
>
>
>> But when you make a proof about reals, AND YOU SAY ITS VALID BECAUSE IT
>> WORKED WITH THE FUNCTIONS.
>>
>> THEN YOU BETTER ALSO HAVE A SPECIFIC, DEFINABLE REAL THAT YOU ARE
>> MAKING PROOFS ABOUT.
>> YOU DON'T.
>>
>> What you've done is pegged a lease on a bit of beach front property,
>> everybody agreed, then you claimed ownership on all the sand in the
>> world. WORKED FOR THE SPECIFIC CASE! typical americans.
>
>
> I am having a hard time understanding you completely here, but I think I
> can see what you are getting at.
>
> There is something profoundly counterintuitive about the proof that the
> real numbers are uncountable. It starts with a puported list of real
> numbers, and then creates just ONE more. How is creating just ONE more
> real number showing that it is uncountable.
>
> The problem is, you are thinking about it in the wrong way. Rather,
> think of it as similar to the proof that the halting problem is
> uncomputable. The proof is really stated as a challenge - you give me a
> list of reals that you claim contains all of them, and I in turn will
> respond with a real number that is not in your list.
He doesn't accept that proof either.
--
Will Twentyman
email: wtwentyman at copper dot net
|
|
