Prove that $2+2=4$. Before you might chastise this quesion, I understand that we all know $2+2=4$. But a while ago I just stumbled across this paper which formally proves that $2+2=4$: http://www.cs.yale.edu/homes/as2446/224.pdf
However, seeing that I had never taken any courses in number theory, I had no exposure to congruence relations, so I have a hard time understanding those areas of the proof. Also, at 1.5 pages, I personally find the proof in the paper to be rather long.
I would write my own proof here (e.g. maybe show that $2+2 \le 4$ and $2+2 \ge 4$ in order to achieve equality?), but I fear that it will be extremely faulty, so I had been wondering for a while: 

Are more concise ways to write a formal proof which shows that $2+2=4$, preferably without the use of any congruence relations? I would like to see different ways if possible.


I don't know about more concise, but I will sketch the outline of a simpler proof starting with Peano's Axioms for the natural numbers $(N,0,S)$.
We will construct the function $add: N^2\to N$ such that 
$\forall a\in N: add(a,0)=a$
$\forall a,b\in N: add(a,S(b))=S(add(a,b))$
Begin by constructing the set $add$ such that
$\forall a,b,c:[(a,b,c)\in add \iff (a,b,c)\in N^3$
$\land \forall d:[\forall e,f,g: [(e,f,g)\in d\implies (e,f,g)\in N^3]$
$\land \forall e\in N: (e,0,e)\in d$
$\land \forall e,f,g:[(e,f,g)\in d \implies (e,S(f),S(g))\in d]$
$\implies (a,b,c)\in d]]$
Then prove that $add$ is the required function (see full formal proof in DC Proof format, 728 lines).
Then define $1=S(0), 2=S(1), 3=S(2), 4=S(3).$
Then prove, in turn, that $add(2,0)=2, add(2,1)=3, add(2,2)=4$ as required.
