Philosophy
Second-Order Logic and the Expressive Power of Quantification
Quick fact
In second-order logic, you can write a single sentence that fully describes the natural number structure: any two models satisfying it are isomorphic. First-order logic can never do this—by the Löwenheim-Skolem theorem, any first-order theory with an infinite model has models of every infinite size.
Why this is interesting
What if, in addition to saying 'every frog is green,' you could also say 'there is a property that every green thing shares'? That leap—from talking about things to talking about their properties—explodes the limits of logical expression.