"Tim Little" <email@example.com> wrote in message news:firstname.lastname@example.org... > On 2010-06-19, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: >> I see that you have stopped responding to my arguments, and have >> employed the rather lame tactic of suggesting I look at definitions >> on the web. > > I don't care where you get your definitions from, so long as they are > not from your own orifices. You are using mathematical terminology in > a completely incorrect manner, and further discussion is unlikely to > be fruitful until you learn what the words you are using mean. > > >> In the mean time, how about answering two questions for me: >> >> 1. Do you believe it is possible to create a list of all computable >> Reals? > > What do you mean by "create"? Such an list can be proven to exist, > and I even provided a well-defined mapping from N to the set of > computable reals earlier in this thread. >
Hmmm. But you cannot provide me with a list of all Computable numbers, can you? Which is what Cantor's diagonal proof requires.
Of course there exists a mapping from N to computable numbers. But Cantor's proof requires more than that; it requires the mapping to be recursively enumerable such that we can also explicitly list them.
That a set cannot be listed is not the same as it is uncountable, as the set of Computable numbers clearly illustrates. You cannot give me a list of all Computable numbers, because I can use a diagonal construction to form a computable Real not on the list. Yet the set of computable Reals in countable. Cantor's proof applied to Reals does *not* prove they are uncountable; it merely proves what Cantor said it proved, that they cannot be formed into a list. This is not neccesarily the same as being uncountable, witness the set of computable Reals which also cannot be listed but is in fact countable.
> If that doesn't answer your question, you'll have to clarify what you > mean by it. >
What I mean is that Cantor proved you cannot provide a list of all Reals. This does not mean that the Reals are uncountable. Exactly the same form of proof can be applied to all Computable Reals, and be used to prove that you cannot provide a list of all Computable Reals, either. Yet these are countable.
Clearly the predicate "cannot be listed" is *not* the same as "is uncountable", because computable Reals have the first property but not the second.
And to the extent that Cantor's diagonal proof merely proves that the Reals "cannot be listed", this does not automatically prove "they are uncountable".
> >> 2. Do you believe the computable Reals are countable? > > Obviously. > > > - Tim