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Re: Grrrr! (was): infinite dimensional vector space
Posted:
Jun 25, 2003 8:40 PM
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David Petry wrote:
> David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<68bjfv0qpm20kt7fgnup431sch6ii38hus@4ax.com>...
>
>
>>(Or maybe I'm wrong - how _does_ one give a proof that there
>>exist irrational numbers by "diagonalization"?
>
>
> Start with a list of rational numbers between 0 and 1, and
> diagonalize over the decimal expansions of those numbers. It
> works just fine. The technical detail of what to do with
> numbers that end in either all 9's or all 0's is easily
> handled, and in the end you get an honest-to-god irrational
> number.
>
> Comment: if the above proof were more widely known, there
> would be much less confusion about Cantor's nonsense.
How do you know that you can't select alternately 4 and 5
as the digit that does not match the n-th digit of the
n-th number. If you can do this, you get
.4545454545.... which is rational (45/99 = 5/11).
Martin Cohen
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