The main idea is the following. We have a set $I$ of ``codes'', and a set, e.g., N of ``inputs''. Very often, if the goal were not true, then there would be a function $F : I \times N \rightarrow \{0,1\}$ that ``efficiently'' encodes some property about the pair $(i,n) \in I \times N$. One can then build from the diagonal a new code $K \in I$ such that, for all $n \in N$, $F(K,n) = 1 - F(n,n)$. Very often, every element $i \in I$ can be encoded as an element of $N$. The contradiction is obtained with $F(K,K) = 1-F(K,K)$.
Let's see some concrete examples.
#1. Real numbers are uncountable
Let $I = N = \mathbb{N}$. Assume there exists a function $F : I \times N \rightarrow \{0,1\}$ such that for every real number $x \in [0,1]$, there exists a code $i \in I$ such that $F(i,n)$ is the $n$-th bit of $x$ in its binary expansion.
We now build a member $K \in I$ from the diagonal of $F$ by $F(K,n) = 1-F(n,n)$, i.e., the real number encoded by K is the one whose $n$-th bit is $1-F(n,n)$. But then the $K$-th bit of $K$ is $F(K,K) = 1 - F(K,K)$. Whence a contradiction.
#2. The Halting problem
Let $I$ be the set of codes of Turing machines. Assume that there exists a machine $F$ which solves the Halting problem, i.e., for any $(i,n)$, $F$ terminates on $(i,n)$, and $F(i,n) = 1$ if the $i$-th machine terminates on input $n$, and $0$ otherwise. Again, we can think of $F$ as an array with rows indexed by Turing machines, and columns indexed by all possible inputs.
We now build a machine $K \in I$ from the diagonal of $F$. On the input $n$ (a code for a Turing machine), the machine $K$ returns (any value is ok) if $F(n,n) = 0$ and runs forever otherwise. Since $K \in I$, the value $F(K,K)$ is defined. But, $K$ does not terminate on input $K$ iff $F(K,K) = 0$ iff $K$ terminates on input $K$. Whence a contradiction. Note that here the expression ``$F(K,n) = 1 - F(n,n)$'' from the introduction is to be understood as: $K$ does not terminate on input $n$ if $F(n,n)$ returns $1$.
#3. The Time Hierarachy theorem
Let $I$ be the set of (codes of) Turing machines. I am merely restating the proof on wikipedia. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a time constructible function. Consider the following problem
$$
H = \{ (i,n),~\text{the machine } i\in I \text{ accepts } n \text{ within } f(|n|) \text{ steps}\}
$$
Here, ``accept'' means returning $1$. It is not difficult to see that $H \in \mathsf{DTIME}(f(m)^3)$ where $m$ denotes the size of $(i,n)$. The goal is to prove that $H$ does not belong to $\mathsf{DTIME}(f(m/2))$. Assume it is the case. Then, there exists a machine $F$ which terminates on each input $(i,n)$ within $f(m/2)$ steps such that $F(i,n) = 1$ if $(i,n) \in H$ or $0$ otherwise.
We now build a machine $K$ from the diagonal of $F$. The machine $K$ accepts the input $i$ if $F$ rejects $(i,i)$, and rejects it otherwise. Let $s$ be the size of $K$, and $m \simeq 2s+1$ be the size of $(K,K)$. The function $F$ takes about $f(m/2) \simeq f(s)$ steps to compute $F(K,K)$. In particular, $K$ terminates within $f(s)$ steps on input $K$. Now, $K$ accepts input $K$ (which it does within $f(s)$ steps) iff $F$ rejects input $(K,K)$ iff $K$ does not accept $K$ within $f(s)$ steps iff $K$ rejects $K$ (which it does within $f(s)$ steps). Whence a contradiction. Here, the expression ``$F(K,n) = 1 - F(n,n)$'' is to be understood as: $K$ rejects (accepts) $n$ if $F$ accepts (rejects) $(n,n)$.
pb
