It's a well-known fact of logic that if from $$P$$ you can get $$Q$$, then from $$P$$ you can also get $$Q$$ and $$Q$$.* So since you can get two dimes and a nickel from a quarter, you can get two dimes and a nickel and two dimes and a nickel from a single quarter. Come to learn about linear logic, which is a version of logic which doesn't claim that you can get infinite amounts of money from a quarter. *For example, since $$n = 2$$ implies that $$n = 1 + 1$$ then, $$n = 2$$ implies that $$n = 1 + 1$$ and also $$n = 1 + 1$$.

Löb's theorem is a beautiful theorem with a deceptively short proof. It states that $$\square (\square P \to P) \to \square P$$ for all $$P$$---that if you can show that proving $$P$$ is sufficient to make $$P$$ true, then you can prove $$P$$. Löb's theorem has a variety of applications, from enabling robust cooperation in the prisoner's dilemma, to curing social anxiety, from proving Gödel's incompleteness theorem, to proving that the halting problem is undecidable. I will present a few proofs of Löb's theorem, all of which are twisty in subtly different ways. We will spend the rest of the time working on wrapping our minds around these proofs, and discussing related topics.