For one of my homework assignments I was given the following complaints about his argument:
Every rational number has a decimal expansion so we could apply the Cantor Diagonalization Argument to show that the set of rational numbers between 0 and 1 is also uncountable.However, because we know that any subset of Q must be countable, there must be a flaw in Cantor’s Diagonalization Argument.
My idea is that the flaws in this argument are the following: Firstly, the set of numbers between 0 and 1 does not only consist of rationals - there are irrational numbers as well. Therefore, the argument that the set of numbers between 0 and 1 is rational and therefore countable is incorrect. However, I might be misunderstanding the problem because it is talking about the set of rationals between 0 and 1 specifically, so I'm just generally confused.
Any hints or suggestions would be greatly appreciated, thank you!
Edit: After reading the comments, I understand my initial thoughts above were incorrect. So, since this is looking at the rationals in the set (0,1), would I be correct in saying that if we used Cantor's method, there is no way of ensuring that the diagonal taken would also be a rational (since a rational can be written as a repeating decimal expansion). Since the diagonal might not ever repeat (because in theory, each added number in the diagonal can continue infinitely and never repeat). Would this argument be correct?