I have the following proof but it seems suspiciously simple and wonder if I did anything wrong?
Suppose addition in the natural numbers defined as $+:\def\N{\mathbb N} \N \times \N \to \N$, and two axioms are given:
- For all $n$ in $\N$, $1+n=f(n)$ where $f$ is bijective.
- For all $m,n$ in $\N$, $f(m)+n=f(m+n)$.
Then I am attempting to prove that $\forall m, n\in N$, $m+f(n)=f(m+n)$
Inductive step (proving $P(n+1)$）:
suppose $P(n) =m+f(n)=f(m+n)$ is true and holding $n$ fixed, start with $(m+1)+f(n)$, by inductive hypothesis, $(m+1) + f(n) = f(m+1+n)$, hence $P(n+1)$ is true.
Why is this so short? Am I tripping?
I suspect you don't have an axiom directly stating $+$ is commutative. This means that even though you indeed can conclude $f((m+1)+n)=(m+1)+f(n)$, you cannot equate the LHS with $f(m+(n+1))$ (which is what you need to prove $n+1$). Notice I have also added extra parentheses since not even associativity is given.