Claim. If $x,y \in \mathbb N$, then at least one of these is true: (a) $x>y$; (b) $x=y$; or (c) $x<y$.
It seems that the usual proof of this claim uses induction. Is it possible to prove this without induction? (Or is perhaps this claim false without the axiom of induction?)
If we drop the induction axiom (but keep the other four Peano axioms), then we could have $$\mathbb N = \{0,\bigstar_0,1,\bigstar_1,2,\bigstar_2,\dots \}.$$
And now $0 \nless \bigstar_0$ , $0\neq \bigstar_0$, and $0 \ngtr \bigstar_0$.