Lecture Notes 11/14 – 11/30 (the dynamical systems approach to Ramsey theory) are available.

Lecture Notes 11/04 – 11/09 (the Paris-Harrington Theorem) are available.

Homework 11, the final homework set, is now available and is due 12/05.

Solutions to HW 10 are also available.

Lecture Notes 11/02-11/09 (Basic facts about first-order logic) are available.

**Problem 2**

As we saw in class, you can define the relation using the formula . Furthermore, given any number , we can represent it as , where we apply -times.

Now show that .

**Problem 3**

For any *specific* finite number , you can find a formula that says: “There exist *exactly* distinct elements”.

**Problem 4**

One direction is trivial. For the other direction, use the completeness theorem. Argue that a theory is consistent if and only if it has a model. Then use the fact that proofs are finite, in particular can use at most finitely many formulas.

**Problem 5**

Extend the language of arithmetic by a new constant symbol . Add sentences of the form , , , … to . Apply the compactness theorem.

Homework 10 (due 11/14) is available.