Notatebility of Uncountable Sets I have noticed a pattern. The set of integers is infinite. Therefore one at first would think it impossible to come up with a notation allowing the representation of all integers. This though becomes easy actually to get around. Simply allow larger integers to take up larger notation. Now look at the rational numbers. They are dense, so there are an infinite number of rational numbers between 0 and 1. Seemingly impossible, but not. By dividing two integers, you can come up with any rational numbers. Algebraic, and even all the computable numbers can be notated. Yet when you get to the real numbers, the great uncountable set, there is no finite notation to represent all the real numbers. Why is it that countable sets can be notated, but it seems that uncountable sets can not? Has this been studied before?

Other people have answered why uncountable sets cannot be notated.
I'll just add that some countable sets cannot be "usefully" notated. So it depends on your definition of notation.
For example, the set of all Turing machines is countable. The set of Turing machines that do not halt is a subset, so is also countable. But there is no practical notation - that is, no programatic way - which allows for the enumeration of the set of non-halting Turing machines. A notation system is only as useful as it is computable, and there is no computable enumeration of this set.
