# Proof 2+2=4

First define what we mean by 2, 4, etc, using the successor function $$s(n)$$ that maps each number $$n$$ onto the number immediately after it, so:

\begin{align} s(1) & = 2 \label{a}\tag{Eq.1} \\ s(2) & = 3 \label{b}\tag{Eq.2} \\ s(3) & = 4 \label{c}\tag{Eq.3} \\ \end{align}

And so on. Secondly define what we mean by addition:

\begin{align} n + 1 & = s(n) \label{d}\tag{Eq.4} \\ n + s(k) & = s(n + k) \label{e}\tag{Eq.5} \end{align}

Take $$\ref{d}$$ and substitute $$n = 2$$:

$2 + 1 = s(2)$

From $$\ref{b}$$ we can substitute $$s(2)$$ for $$3$$ to get

$2 + 1 = 3 \label{f}\tag{Eq.6}$

Take $$\ref{e}$$ and substitute $$n = 2$$, $$k = 1$$:

$2 + s(1) = s(2 + 1)$

From $$\ref{a}$$ we can substitute $$s(1)$$ for $$2$$, and from $$\ref{f}$$ we can substitute $$2 + 1$$ for $$3$$, therefore:

$2 + 2 = s(3)$

From $$\ref{c}$$ we can substitute $$s(3)$$ for $$4$$, therefore:

$2 + 2 = 4$