Every other branch of maths uses proof. Mathematical logic turns the floodlight around and studies proof itself — reasoning, truth and the very machinery of mathematics, treated as objects to be examined. What is a valid argument? What does it mean for a statement to be true? Can every true thing be proved? Is there a mechanical procedure to settle any question at all?
The answers, worked out in a single astonishing century, are among the deepest and most
unsettling in all of thought. Logic is where mathematics discovered its own limits — and, in the
same breath, laid the theoretical foundation for the
One distinction runs through everything here, and it is subtle enough to have taken geniuses to make it sharp. There is syntax — what you can derive by pushing symbols around according to rules (proof) — and there is semantics — what is actually true in the worlds a statement describes (meaning). The whole drama of logic is the relationship between these two: when they agree (soundness, completeness), and the shocking places where they must forever come apart (Gödel). Keep the pair — provable versus true — in mind, and the course tells a single story.
This course climbs in three stages, from symbols to their limits.
We begin with the simplest reasoning of all — statements that are just true or false, and the tables that combine them. It looks like a game of Xs and Os. By the end of the course, that same game will have told us the limits of mathematics itself.