**NOTE**: this is probably too much for one meeting, so some of this may/will get put off until Thursday/Friday.

At the end of last time, I presented a simple example of *compactness argument* — deducing a finite statement from an infinite one. We’ll start on Tuesday with a more non-trivial example of compactness — deducing finite Ramsey’s theorem from the infinite Ramsey theorem. **Colin** and **Henry** can present this. Everyone should also read this (it is Section 2.2).

If you’ve seen some topology (or, even if you haven’t…), you can read Section 2.3, to learn why applications of Konig’s tree lemma are referred to as “compactness” arguments, but I don’t plan to spend any time talking about this. Instead, we will spend the next while looking at some infinite Ramsey theory, beyond just countable infinity.

Everyone should read Section 2.4, that introduces ordinals, ordinal arithmetic (addition, multiplication, exponentiation), transfinite induction, and the axiom of choice, and do all the exercises (including verifying the various assertions made throughout to illustrate the definitions). For presentations:

**Casey**and**Alex**can give us the definition of ordinals, explain successor and limit ordinals, define ordinal addition, and show that it is not commutative**Greyson**and**Nicholas**can define ordinal multiplication and exponentiation, and say what \(\varepsilon_0\) is**Bailee**and**Joe**can introduce*well-ordering*(Definition 2.14), and show that well ordering is equivalent to having no infinite descending chains (Proposition 2.15).- Back to
**Greyson**and**Nicholas**, who can show that the collection of ordinals is well-ordered (Proposition 2.26) **Ted**and**Ryan**can introduce the axiom of choice, and present the argument (beginning the middle of page 62) that, assuming axiom of choice, every set can be well ordered.

Everyone should think about Exercise 2.8 (that the implication goes the other way — so the axiom of choice is *equivalent* to the statement that every set can be well ordered), and someone can volunteer a proof.