Re: (Not quite) Cantor's diagonal proof

From: |-|erc (spam_at_fodder.abc)
Date: 10/20/04 

Date: Wed, 20 Oct 2004 05:36:53 GMT

"Saint Cad" <saintcad@emailblackhole.com> wrote in > >> >>
> >>
> >> >>
> >> >> Here's another question. Suppose I list all of the repeating and
> >> >> terminating decimals (i.e. rationals). What in Cantor's proof
> >> >> prevents
> >> >> me
> >> >> from finding some sort of order for them so that I can use the
> >> >> diagonal
> >> >> method to create a repeating decimal not in the list, thereby
> >> >> "proving"
> >> >> that
> >> >> the rationals are uncountable? I know that I can't actually do this
> >> >> (the
> >> >> rationals are countable), but it would take someone else to explain
> >> >> why
> >> >> in
> >> >> order to dismiss this "counter" to his method. Again, Cantor's method
> >> >> may
> >> >> be shown to not be invalid, but it take more than just his method by
> >> >> itself.
> >> >
> >> > Since that are several well published surjective functions from the
> >> > naturals to the rationals, a proof that the rationals are uncountable
> >> > would, one hopes, be impossible, and would devastate mathematics if one
> >> > were ever found.
> >>
> >> Exactly! But my question is can Cantor's proof by itself be shown to not
> >> suffer this flaw?
> >
> > Please read the replies to you made by Tim Peters, Dave Seaman, Daniel
> > W. Johnson, and Robert Low (all of which were made after you wrote
> > this).
> > Especially Tim Peters has pointed out your flaws in not understanding
> > Cantor's Diagonal Argument with the clarity of a thousand suns.
> >
> > -Leonard Blackburn
>
> Let me start by saying that I don't believe Cantor's Method is in fact
> invalid. I know from when I got my BA in math that the proof as presented
> in class left a lot to be desired. I guess the issue I was trying to raise
> is: are there arguments against Cantor's Method that need results other that
> the method by itself to dismiss?
>
> The first "disproof" of Cantor was one presented by another student (not
> me!) at the time. I never liked that argument either. It struck me as
> similar to the argument to "the number of naturals is finite since listing 1
> to N is a finite cardnality and adding one more (i.e. 1 to N+1) is finite.
> QED." I do have to admit though that the idea a couple of posters presented
> (including Tim) that the original list of reals is arbitrary is a good
> point, one the professor didn't make when I originally learned it.
>
> The second "disproof" was in honor of James Harris. I was thinking of his
> idea of a counterexample to Wiles proof to see where it broke down. I had
> tried it myself and I kept running into the same problem - I could never
> show that the number created by the diagonal was repeating. I posted it in
> hopes that someone could show me why I kept running into the same problem
> over and over without the circular argument of "It fails because the
> rational are countable" and could instead point out where exactly Cantor's
> Method fails on rational numbers.
>

I've shown this... 3 times now. Yes it fails because the rationals are countable, so what does
'they are countable' mean?

1 Given a countable list containing all rationals
2 Given any *new number* that has a repeating decimal
3 Given any number with repeating decimal, we can calculate the integer numerator and denominator.
4 Given any numerator and denominator pair, we can calculate its position in the list.

   numerator 21111
--------------------------->
| / / / / /
| / / / /
| / / /
| / /
| /
|
| 10,000 *
v
denominator

5 Given it has a fixed position in the countable list, it cannot be new. Contradiction to 1&2.
6 Corollary : Every complete list of rationals has an irrational diagonal

HTH
Herc

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