scott <sco### [at] scottcom> wrote:
> >> Genuine question then, why isn't the set of real numbers countable,
> >> given that you could represent each one with two integers (one for the
> >> digits before the decimal point, one for the digits after the decimal
> >> point)?
> >
> > The set of all strings that contain all possible combinations of two
> > or more digits (in other words, the set of all possible decimal
> > representations) is uncountable. In other words, it's not possible to
> > enumerate them all. This might be a bit surprising (it was to me...)

> This is surprising to me too, I would have thought that because every 
> string of digits (without a decimal place) represents an integer then it 
> would be countable.

Note that an infinite string of digits does not represent an integer
(because no integer is infinite.) That's probably what causes the
confusion.

("No real number is infinite either." That's correct. But the decimal
representation of a real number can be.)

That's not to say that no set containing infinite decimal representations
is countable. For example the set of rational numbers contains infinitely
many values which decimal representation is infinite, yet this set is
countable. (It just that, obviously, the set of rationals does not
contain *all* possible infinite decimal representations.)

"Decimal representation" in itself might be a problematic thing to rely
on when dealing with infinite sets, because it's not very intuitive.

> > However, we can construct such a string: Take the first digit of the first
> > string and invert it, the second digit of the second string and invert it,
> > and so on. In other words, with the example above the string would be
> > 0110... etc.

> I understand that, but don't understand why that proves it is 
> uncountable.

Because there's no way to enumerate them. (More technically, there exists
no bijective function between the two sets.)

> Because you can do a similar thing with a list of all 
> natural numbers, ie calculate the sum and then you get a new natural 
> number that wasn;t in the list to start with.

You cannot sum all the natural numbers because there's an infinite
amount of them.

Any sum of a finite amount of natural numbers gives a number that was
already in the set.

-- 
                                                          - Warp

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