"Tim Little" <email@example.com> wrote in message news:firstname.lastname@example.org... > On 2010-06-20, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: >> "Tim Little" <email@example.com> wrote in message >>> 1) Recursive enumerability has nothing to do with it. >> >> So you assert. Personally, I think that the fact that the computable >> Reals >> are not recursively enumerable is intimately related, as a set being >> recursively enumerable in this context basically means the same thing as >> there being a mapping from N to exactly that set, ie a list of elements. > > Look at the first theorem on http://en.wikipedia.org/wiki/Countable_set . > I use Wikipedia since if I quoted from a textbook you might not be > able to verify it. > > What you have just said as the definition for "recursively enumerable" > is *exactly* equivalent to being a countable set! >
Do you believe the set of all computable numbers is recursively enumerable?
Because Wikipedia only claims that the computable numbers are recursive, not that they are recursively enumerable.
If you nthink that they are recursively enumerable, you have to produce a partial recursive function whose domain is exactly the uncomputable numbers, and I dare say you can't do that.
> > I suggest you look up the definition for a recursively enumerable set, > and see how it differs from the definition of a countable set. > > >>> 2) The word you should be using instead of "recursively enumerable" >>> is "surjective". >> >> Well, recursively enumerable is a property of sets, and surjective is a >> property of mappings, so dunno what you are talking about. > > Did you forget what you wrote? Look again at what you claimed: > > "What Cantor proved (in more modern parlance) is that there is no > recursively enumerable function from N -> R". > > You were the one applying the term "recursively enumerable" to a > function. I was merely correcting your mistake. >
Whoops. I meant a recursively enumerable set.
> >> Well, that's very rude of you. > > Yes, it was. Polite correction of your errors wasn't getting > anywhere, and you did not appear to be aware of how bad it looks to be > declaiming "Cantor didn't prove that the reals were uncountable!" > while obviously not grasping some of the most basic concepts in set > theory, proof, and computability. >