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? 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.
> 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.
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".
> 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.
> 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.
