On Sun, 11 Feb 2007 21:27:06 +0900, toshiaki wrote:
> "Dave Seaman" <dseaman@no.such.host> wrote in message > news:eqkmb4$i3v$1@mailhub227.itcs.purdue.edu... >> What do you mean by "symbol"? Is an infinite digit string a "symbol"?
> It's also symbol. I mean by symbol a row of symbols. anothers are: > square root2 > pai
You snipped the question that prompted this, leaving no context.
> .................................. > Pai cannot be represented precisely by decimal, so that we use pai.
I take it you mean "pi". But you're wrong: pi can be represented precisely by an infinite decimal.
> My explanation may have been insufficient. I shall make effert to > make clear my argument as far as possible. > In proof of Cantor: > A list is assumed to include all reals, but it in fact includs decimal > representation of reals, which is build by countable digits.
The proof merely assumes that f: N -> R is a mapping (a "list") of the natural numbers to the reals. We don't make any assumption as to whether f is a surjection or not. In fact, the proof shows that f is *not* a surjection.
The codomain of f is R, not "representations of R". That is, the members of the list are actual real numbers, not decimal digit strings.
> In this stage, > a set of all countable decimals is assumed, but it is not correct.
The proof makes no assumption other than that f: N -> R is a mapping of the naturals to the reals. That is, for each natural number n, we are given a real number f(n).
> By making diagonal number, we can only add new decimal which is not > on a list. If we assume infinite list, it means that this operation of > making new number doesn't complete.
The word "complete" is not a part of the proof. What are you talking about? If we know the n-th digit of x for each n, then we know x.
Given f: N -> R, the proof shows that there is an x_f in R\ran(f). Hence, no such f is a surjection.
> Certainly reals are more than rationals. But it is valid only in finite > case.
What is the "finite case" of |N| < |R|?
-- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
