# All natural numbers are equal.

I saw the following “theorem” and its “proof”.

I can’t explain well why the argument is wrong. Could you give me clear explanation so that kids can understand.

Theorem: All natural numbers are equal.
Let $a, b \in \mathbb{N}$, then a=b.

Proof by induction.
Let $m=\max\{a, b\}$. We will prove that the theorem holds for all $m\in \mathbb{N}$.
If $m=1$, then $\max\{a,b\}=1$, so $a=b=1$.

Now assume that it holds for $m=k$ for some number $k$.
Now let $\max\{a, b\}=k+1$. Then $\max\{a-1, b-1\}=k$ and thus by assumption $a-1=b-1$, so $a=b$.

Therefore, the proof is complete.

The error lies in the fact that when decreasing $1$ you may get out of the set $\mathbb{N}$ – and indeed, $\max\{1,2\}=\max\{0,1\}+1$, but $0 \ne 1$, and $0\notin \mathbb{N}$ as you defined it.