"Tim Little" <email@example.com> wrote in message news:firstname.lastname@example.org... > On 2010-06-19, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: >> Umm ... they can be listed, but the list is not computable? > > Correct. Is that difficult for you to understand? > > >> How are you going to give me a list of all computable Reals if you >> cannot compute what the first, second, third etc items in the list >> are? > > There is no single *algorithm* that computes them all. There is > however a clear mathematical definition of such a list. > > >> In Cantor's diagonal proof, the list of Reals is provided in >> advance, such that the nth digit of the nth item is known. > > Where is the phrase "provided in advance" used in the proof?
Because Cantor's proof requires us to know the nth digit of the nth item on the list.
Indeed Cantor's proof starts with a list, and then proves there is a Real not on the list.
If you don't what is on the list already, you cannot possibly prove that some Real is missing.
My proof is *exactly* the same, except for inserting the word "computable" in front of Real.
> You > appear to be mistaking a simple form of conditional introduction in > the proof for some completely unfounded computability assumption. > > Suppose I say "suppose x is a real number. Therefore (x has some > property)". Do you take this to mean that x must be a *computable* > number, since it is "provided in advance"? >
> > >> All I am asking for in my proof that the computable reals cannot be >> listed is exactly the same thing as Canor's proof requires - a list >> of (computable) Reals provided in advance, such that the nth digit >> of the nth item is known. > > Known? To whom? All is required is that it exists. That is, that > the list is a function from N to R and not some other type of > mathematical object. >
No. Cantor's proof requires that the nth digit of the nth term is known.
Otherwise the diagonal number cannot be constructed.
> >> OK, give me a list of all computable Reals. In exactly the same form >> as Cantor requests that a list of Reals be produced, such that I can >> identify the nth digit of the nth term. Exactly as per Cantor's >> proof that the Reals cannot be listed. > > You are reading a great deal of things into Cantor's proof that simply > are not there. I have now three times provided an explicitly defined > mapping from N to the set of computable reals. That is entirely > sufficient for Cantor's proof.
No you haven't.
If you have an explicit list, you could post it. In fact, just like Cantor, I only need to know the nth digit of the nth term for all n, so you can just post that if you like.
> > However, even that much is unnecessary! Cantor's proof starts with a > conditional introduction and then discharges that conditional later in > the proof, so that *nothing* need be provided "in advance". >
Incorrect. He says "if you give me any purported list of all Reals, I will produce a Real not on the list".
Exactly as I do for computable numbers, in fact.
> >> Given the list, its trivially easy to explicitly compute the >> anti-diagonal. Cantor provides an explicit construction. > > Given the digits of Chaitin's Omega, it's trivially easy to explicitly > compute Omega+1. That does not make Omega+1 computable. >
But you can't give me all the digits of Chaitan's Omega, can you?
Its not computable.
> >> You cannot give me a list of all computable numbers. >> >> Try, if you like. > > Already done 3 times now. You have completely ignored it each time. >
Give me a list of all computable numbers in the same form as Cantor's proof requires a purported list of all Reals, ie a list where the nth digit of the nth term is known.
Surely this cannot be too hard? After all, a definition of a "computable Real" is that it can be determined to any required degree of accuracy, which is equivalent to the statement its decimal expansion can be determined. So by definition if the list contains only computable Reals, we know the nth digit of each term.