Is the set of reals, R, a countable set?

Afraid not; Cantor's famous diagonal proof shows otherwise. you can read about it in Wikipedia, at this link

Thanks!
Thanks alot! Cantor's proof by contradiction makes perfect sense.

A good description of Cantor's diagonal proof is here:

http://www.mathpages.com/home/kmath371.htm

Thanks Thomas!

The mapping of rational number to integer, and to squares etc, cannot, as a mapping, be restricted to following the sequence in which integers, etc are presented (1, 2, 3, 4, etc). There is no reason why a mapping of a rational number can't be mapped to the same integer twice, etc.

So, we need to make a rule about our mapping procedure. We must state that each mapping must result in a unique pair, with unique individual elements in that pair. But if we do that then we no longer have a mapping. So we need to admit that we are not making a mapping but constructing a relationship. We, and Cantor, must state what that relationship is.

In examining if something is countable, you don't map rational numbers to integer -- you try to map integer numbers to whatever set's countability you intend to prove. So to prove that the rational numbers are countable, you try to find a way to go through the set of integers, and make sure that for each integer, there is exactly one rational number it maps to. There are ways to make sure that each rational number gets an integer mapped onto it (or in other words, "gets counted"). Some rational numbers get counted several times. (As in 1/1, 2/2, 3/3, ...), but that doesn't matter, because it only proves that there are still integers left over after each distinct rational number got counted.

The same issue arises when you prove that the real numbers are not countable. Some decimal representations stand for the same real number (1=0.9999999...), and this raises possibility that the real numbers only appear uncountable because Cantor's proof "wastes" integers. The article I pointed to acknowledges this.

But a popular treatment of a moderately difficult mathematical point will only bear so many technical complications.

Interesting points indeed guys.

Re: Is the set of reals, R, a countable set?
The injunction to '...' or 'carry on calculating indefinitely', does not appear to be a mathematical device, or a rule that can be accomodated by mathematics. Only with an outcome already in mind can the rule '...' be realised in mathematics.
We can use the example of a pattern to show this. Mathematics is used to show us many ways of presenting and recognising the same pattern, but if the pattern has no limits then we don't have a pattern at all.
The injunction to '...' negates any differences between fractions, integers, etc. The fact that we map the symbols associated with fractions to the symbols associated with integers, for example, does not mean that we are mapping fraction to integer. The injunction to '...' removes the identity of a symbol. Again, we need not rush to consider these things in the jargon commonly used.