Are real numbers uncountable?(knol.google.com)
knol.google.com
Are real numbers uncountable?
http://knol.google.com/k/are-real-numbers-uncountable#
4 comments
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.
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.
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.
It's a simple question which he should be able to answer with a finite string of digits. No words necessary.
"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.