
Are real numbers uncountable? - fogus
http://knol.google.com/k/are-real-numbers-uncountable#
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chancho
This is a goldmine.

 _"Assuming any real number can be represented using base 10 (this is of
course false, since it is not possible to represent all real numbers in any
given radix system)"_ So that bit in parens pretty much refutes his whole
post.

And the tree: "One can traverse it sequentially..." Go depth-first or in-order
and you'll never get off the left spine. Go breadth-first and you'll never get
past the first level because _the root has (countably) infinitely many
children._ This is actually a very intuitive argument for why the reals (oops
I mean decimals) _are not_ countable.

And the "refutation" of Cantor's argument of the reals is fun too. I've never
read the original argument, but I always figured that you're supposed to jump
over the entries i/j for which i and j are not relatively prime. They're easy
enough to spot. I can't be 100% sure my logic is sound, but I'm sure this
guy's isn't.

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cchooper
I upvoted this because it's a wonderful example of mathematical crankiness.
I've always wondered why cranks are attracted to the same things: Cantor's
diagonalization proof, squaring the circle, 1=0.999.. Perhaps it's because the
problems are easy to understand, but the proofs aren't.

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tlholaday
Yes, Real numbers are uncountable. Every real number can be expressed as the
sum of decimal fractions, but not as a finite or regular sum of decimal
fractions.

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fexl
I asked the author which natural number N corresponds to the real number 1/3.

It's a simple question which he should be able to answer with a finite string
of digits. No words necessary.

