TITLE: How to prove that $2+2 = 2*2 = 2^2 \cdots= 4$
QUESTION [5 upvotes]: One day, I noticed that
$2+2 = 2*2 = 4$.
Later, I learned that 
$2+2 = 2*2 = 2^2 = 4$
Multiplication is an abstraction of a lot of additions , exponential is an abstraction of a lot of multiplications...  I'm sure there is always an abstraction of the previous operation. 
But my questions is : if I use any abstraction with the numbers $2$ and $2$ does it always result $4$ ?

REPLY [10 votes]: I think the "deep" reason for this is that these are all binary operations.
Given a binary operation $\ast$ on integers at least $2$, define $\ast'$ by $$m\ast' n = \overbrace{m\ast m\ast \cdots \ast m}^{n\text{ times}}.$$
Always associate to the right (for concreteness), so that $a\ast b\ast c = a\ast(b\ast c)$ and so on.
Now starting with any binary operation $\ast$, define $\ast_1=\ast$ and $\ast_{n+1} = \ast_n'$. Then for all $n$ we clearly have $2\ast_n 2 = 2\ast_{n-1} 2 = \cdots = 2\ast 2$.
The sequence you consider is given by taking $\ast$ to be $+$.