Especially Cantor's Diagonal Argument
and the Continuum Hypothesis.
Let's start with the diagonal argument, as found at the Wikipedia page.
A finite context I said. So what would happen if we simply leave out all of the ellipsis
in the first picture
and proceed without them? What can be done analogously is the following.
- Generate all binary fractions in $[0,1]$ with $N$ bits
- Shuffle these $2^N$ rational numbers in random order
- Apply the diagonal argument to the first $N$ of them
The idea is that the first $N$ fractions are in one-to-one correspondence with a finite initial segment of the naturals.
Here is an example for $N=4$ with natural numbers mapping (= index) on the left and the numbers themselves on the right:
\begin{matrix}
0 & 0.\color{red}{1}010 \\
1 & 0.0\color{red}{0}10 \\
2 & 0.01\color{red}{1}1 \\
3 & 0.110\color{red}{0} \\
4 & 0.1101 \\
5 & 0.1111 \\
6 & 0.0001 \\
7 & 0.\color{red}{0101} \\
8 & 0.1001 \\
9 & 0.0110 \\
10 & 0.0011 \\
11 & 0.1000 \\
12 & 0.1110 \\
13 & 0.1011 \\
14 & 0.0100 \\
15 & 0.0000 \end{matrix}
$\color{red}{\mbox{Red font}}$ is used for the diagonal and the outcome of the argument, which is a number that is clearly not in the index range $[0,N-1]$.
But all fractions with a finite number of bits $N$ are in the index range $[0,2^N-1]$, so the number that is diagonalized out must be in the index range $[N,2^N-1]$.
As a physicist by education, accustomed to limits and calculus, I would say that if $N \to \aleph_0$
then what is the cardinality of the whole set, including the elements that have been diagonalized out? I would say that it is: $2^N \to 2^{\aleph_0}$.
Isn't the above a decent heuristics that the Continuum Hypothesis is true?