Re: There are uncountably many irrationals

From: John Savard (jsavard_at_excxn.aNOSPAMb.cdn.invalid)
Date: 01/12/05 

Date: Wed, 12 Jan 2005 14:45:21 GMT

On 11 Jan 2005 10:38:54 -0800, mueckenh@rz.fh-augsburg.de wrote, in
part:
>Robin Chapman wrote:

>> No. There is no proof that the rationals are uncountable.

>Again: Look it up, try to understand it, then defend it, if you can.

I cannot defend any "proof" advanced that the rationals are uncountable.

But I can prove that the rationals are countable.

1/1 -> 2/1 3/1
        | |
        v v
1/2 <- 2/2 3/2
               |
               v
1/3 <- 2/3 <- 3/3

Of course, there is a proof that the *real* numbers are uncountable,
because, in addition to rationals, they include completed infinities of
sequences of random digits.

The square root of two has been *proven* not to be a rational number.
This as long ago as the ancient Greeks. To reject irrationals is to say
that the diagonal of a perfect mathematically abstract square doesn't
really have a length, and that doesn't lead to a very sensible and
useful definition of length.

If you have the countable sequence of computable numbers, or even the
countable sequence of rational numbers, you do have the strange-seeming
condition that you can find partial matches to any irrational number in
that sequence _to any precision whatever_ on that list, and yet no one
element need be an exact match - because there are vastly more
irrational numbers than could all fit on such a list.

Of course that is bizarre... and since nothing in the physical world is
measured to a precision of one part in 10 to the power
4,000,000,000,000,000, let alone to infinite precision, it is hard to
see how the rational numbers can be called inadequate.

Even if one accepts irrational numbers, the ones that are of any
mathematical use, like pi and the square root of two, also form a
countable set. The irrationals only become uncountable thanks to
infinite sequences which are off the beaten path because their general
plan, as they go to infinity, differs from nameable numbers... except
they have no general plan that anyone can refer to!

You can perhaps find a way to say, therefore, that the irrational
numbers with which you are uncomfortable are useless or uninteresting to
you, without incorrectly claiming that mathematicians have made a
mistake. They are not wrong - but the real number system is a
mathematical abstraction of a high order, and is not very "real", if by
that you mean physical.

John Savard
http://home.ecn.ab.ca/~jsavard/index.html

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