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Topic: Why can no one in sci.math understand my simple point?
Replies: 660   Last Post: Dec 7, 2017 8:06 AM

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 Peter Webb Posts: 945 Registered: 12/6/04
Re: Why can no one in sci.math understand my simple point?
Posted: Jun 20, 2010 9:38 PM

"Tim Little" <tim@little-possums.net> wrote in message
news:slrni1td72.jrj.tim@soprano.little-possums.net...
> On 2010-06-20, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote:
>> "Tim Little" <tim@little-possums.net> wrote in message
>>> However, there is no way that you can then prove the existence of a
>>> finite algorithm accepting only the *finite* input n and producing the
>>> n'th antidiagonal digit.

>>
>> Of course I can prove the existence of a finite algorithm. I can produce
>> it.
>>
>> To produce the nth decimal of the anti-diagonal, look at the first n
>> items on the purported list of all computable Reals.

>
> In what way is that "a finite algorithm accepting only the *finite*
> input n"?

Its finite, and its an algorithm.

>It either must embed the list within the algorithm (thus
> rendering it not finite),

No, it embeds the first n digits of the first n terms to calculate the first
n decimal places, and does for all n.

That a finite process can be used to determine a number to any required
degree of accuracy makes it computable.

> or accept the list as additional ainput
> (thus violating the condition that it accept only the finite input n).
>
>

>> Well, I think I am making a pretty good hand of my case. But yes, I
>> know what I am saying is not accepted. I am genuinely confused by
>> this - the fact that my opinions are unique, and I would expect
>> therefore to be almost certainly wrong (with probability 1). On the
>> other hand my argument seems pretty solid to me.

>
> Read through very carefully the definition of a computable real,
> perhaps also reading some of the other known results on computable
> reals and following their proofs. At some point you will hopefully
> see why the "construction" in Cantor's diagonal proof does not qualify
> as a finite algorithm in the sense of the definition of computable
> real.
>

Its an explicit construction. I can even do it by hand to tell you the
anti-diagonal to any required degree of accuracy in a finite number of
steps. That makes it computable.

>
> - Tim

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