Philosophy
The Logic of Second-Order Logic and Expressive Power
Quick fact
In second-order logic, the natural numbers can be characterized up to isomorphism—meaning all models of the Peano axioms are identical. This is impossible in first-order logic due to the Löwenheim–Skolem theorems, which force the existence of nonstandard models.
Why this is interesting
You know how we can say 'for all numbers' in logic? What if we could also say 'for all properties of numbers'? That simple change leads to a logic so powerful it can pin down the natural numbers exactly—but also breaks some of the most beloved properties of ordinary logic.