# How do we know that the number of points on a line is uncountable?

The question is simple one, how do we know that there is a bijection between the points in a line and an uncountable set (such as the power set of natural numbers)? Why don’t countably infinite sets that are dense (such as the rational numbers) correspond 1-1 with all the points on a line? The simplest possible explanation that assumes minimal background knowledge of the inquirer would be greatly appreciated.

-Thank you

• Do you know why the set of real numbers is uncountable? Commented Aug 10 at 5:14
• @ThomasAndrews Yes, it is because it contains more elements than can be corresponded to by the set of natural numbers, this is shown by Cantor’s diagonal argument. Commented Aug 10 at 5:18
• Without essentially knowing the line is in one-to-one correspondence with the real numbers, you actually can't show the line is uncountable. Euclid's postulates hve a model with a countable line, and a countable plane. Commented Aug 10 at 5:18
• I mean, there exist rather simple models of plane geometry - for example, the set of pairs of points $(x,y)$ with $x,y$ algebraic numbers satisfies Euclid's postulates. Commented Aug 10 at 5:24
• But to prove it, you have to define geometry in set theory. There is no such thing as the functions, the natural numbers, etc in formal geometry. Commented Aug 10 at 5:27