Philosophy
Second-Order Logic and the Notion of Mathematical Structure
Quick fact
Second-order logic can uniquely characterize the natural numbers up to isomorphism, something first-order logic cannot do. This categoricity comes, however, at the cost of sacrificing completeness and compactness—two properties that make first-order logic so tractable.
Why this is interesting
You've learned that logic can talk about objects like numbers. But what if logic could also talk about all properties and relations—and that extra power suddenly revealed the true structure of the mathematical universe?