h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi, a few things I'd change in this to make things a little clearer/more precise:

It might be an idea to define real numbers when you first mention them. Something about an infinite decimal number.

When you say the integers are countable, you go on to say that you can 'count' them, footnoted 'given infinite time'. Seems circular to me - which infinite time? I think maybe would be better to talk about naming the first, then the second, and going on through them one by one, and given any of the things in the set, you get to them sometime.

I think your way of intuitively saying that the real numbers are not countable is not helpful - the same argument works for the rationals, which are of course countable. You can't count them up in their natural order but you can in some order. The reason that the reals are uncountable is nothing to do with the intuitive reason you give.

The bit about the cardinality of [0,1] being the same as that of the real line - if all you want to show is that there are more real numbers than natural numbers, then showing that there are more real numbers in [0,1] is enough?

The proof itself looks good, though you'll still get people not get it - thats the nature of this subject

I think you are ok avoiding the problem with 0.123999999999.... = 0.1240000000... because your rule changes things to 1s and 2s, but it is a technical point that might be worth mentioning in a footnote or something.

Incidentally, you might want to link to my entry on 'Bigger and Bigger Infinities', http://www.bbc.co.uk/h2g2/guide/A593552 which uses an abstracted version of Cantor's diagonal argument.

Hi!

Excellent work - articulate and understandable. Though perhaps a little more could be put into it... (not a criticism!!!) I'd like to know if (perhaps!) it could be mentioned that Turing used a modified version of this proof in his argument concerning the halting problem?

Yours,

Jordan

This is so horribly wrong!

It's not necessary to speak of "infinite number of decimals" - infinity is not a number.

We only need to be able to have arbitrarily large (but finite) decimals.

1) It isn't necessary to say this is a proof by contradiction - Cantor did not present it that way. He simply showed that it is always possible to *construct* a number which is not on any putative complete list.

2) The issue of represention of numbers in two ways is ignored; for example
0.2 = 0.199999... (a failing of the decimal system).

3) What's with the finite example?
It's bound to fail since we know that there are an infinitude of real numbers - in fact the proof shows there is more than a countable infinity of them.