I've been given the following explanation of why the Cantor set is uncountable using base-3 (shortened):
In base-3 we write some arbitrary number $x \in [0,1]$ as $$x = 0.b_1b_2b_3..._3 = \frac{b_1}{3}+\frac{b_2}{3^2}+\frac{b_3}{3^3} +\enspace ...$$ with $b_i \in \left\{0,1,2\right\}$. So we can write $x = \sum_{i=1}^{\infty}b_i3^{-i}$. In this format, the Cantor set has some properties. We divide $[0,1]$ into three equal pieces. The first digit, $b_1$, tells us whether $x$ is in the left, middle or right of a given line segment and so on. Assuming that $\exists$ some $k \in \mathbb{N}$ such that $b_k = 1$ then $x$ will be in some middle third of a given interval so $x \notin \mathcal{C}$. Conversely, by the definition of the base-3 expansion, if $b_n \in \left\{0,2\right\}$ for all $n \in \mathbb{N}$, $x$ will never be in the middle third of any interval, so $x \in \mathcal{C}$
With this formulation of the Cantor set we can easily show that it is uncountable. Suppose instead the opposite, that $\mathcal{C}$ is countable, so $\mathcal{C} = \left\{x^1, x^2, x^3, ...\right\}$ where each element of $\mathcal{C}$ is written like: $$x^1 = 0.b_1^1b_2^1b_3^1..._3$$ $$x^2 = 0.b_1^2b_2^2b_3^2..._3$$ $$x^3 = 0.b_1^3b_2^3b_3^3..._3$$ $$\vdots$$
Here is where I get lost:
Let $\left(b_1,b_2,b_3,...\right)$ be the sequence that differs from the diagonal sequence $\left(b_1^1,b_2^2,b_3^3,...\right)$. $\bf\text{In other words if $b_i^i = 2$ then $b_i = 0$.}$ As a result, $x = 0.b_1b_2b_3..._3$ never appears in $\mathcal{C}$!
Does this mean that because the element composed by this sequence differs at the diagonal every time that it will never equal to any element of $\mathcal{C}$? So we can't write every element of $\mathcal{C}$? Could we also say "if $b_i^i = 0$, then $b_i = 2$?".
Yes, yes, yes. This is the Cantor's diagonal argument