How to prove that $2+2 = 2*2 = 2^2 \cdots= 4$

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 $4$ ?

Solutions Collecting From Web of "How to prove that $2+2 = 2*2 = 2^2 \cdots= 4$"

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 $+$.