Question about Cantor's Diagonalization Proof My discrete class acquainted me with me Cantor's proof that the real numbers between 0 and 1 are uncountable. I understand it in broad strokes - Cantor was able to show that in a list of all real numbers between 0 and 1, if you look at the list diagonally you find real numbers that are not included on the list- but are clearly in between 0 and 1. Thus that particular set is uncountable (and I assume uncountably infinite? Is that a thing?).
My question is this - how is the list constructed? Or is that beside the point, as no list could ever entail every single real number anyway, and looking at any such list diagonally would yield new real numbers 0 < x < 1? 

The list isn't constructed explicitly.  What you're doing is starting out by saying "Suppose for contradiction there exists a bijection between $\mathbb{N}$ and $[0, 1]$".  If this were the case, then we could write down an ordered list: let $x_1$ be the real number associated with $1$, let $x_2$ be the real number associated with $2$, etc.
From here, one arrives at the contradiction by demonstrating that this "bijection" couldn't have actually associated a natural number to every real number in $[0, 1]$.  In particular, if you let $x$ be the real number such that the first digit is the opposite of the first digit of $x_1$, the second digit is the opposite of the second digit of $x_2$, and so forth, it's clear that $x$ is distinct from every real number in the list.

The real numbers are written in their binary expansion; by "opposite", I just mean $\Big( \text{Whatever the digit is} \Big) + 1 \pmod{2}$.
