Here’s the goal for the time that is remaining: see how ordinals lead to cardinals (Section 2.5), and then see some of the result of Ramsey theory for cardinalities beyond countable, in particular the negative result Proposition 2.36, and the slight modification that turns the negative result positive (Theorem 2.38, the Erdos-Rado theorem). Given the time that we have, the treatment will necessarily be a little sketchy, but hopefully the key ideas will be clear.

Last time we saw the definition of a well-ordered set, and saw that being well-ordered is equivalent to having no infinite descending chains. We ended with the proof that every subset of the ordinals is well-ordered.

After I quickly recap that proof, **Ted** will present the argument that every set can be well-ordered (by a suitably chosen ordering), as long as we assume that axiom of choice, which **Ted** will also introduce. Everyone should think about exercise 2.18 (the axiom of choice is equivalent to the statement that every set can be well-ordered), and we’ll discuss that. I’ll also make some comments about axiom of choice.

Then we’ll move on to cardinals (Section 2.5). Everyone should read through Section 2.5, and attempt as many of the exercises as time allows.

We’ll go quickly through exercise 2.19 (the exactly formula is really an unnecessary red herring here; what’s important is the picture that goes with the formula). **Luca** can present exercise 2.21 (\({\mathbb R}\) and \([0,1]\) have the same cardinality). We’ll talk about exercise 2.22 (cardinality is an equivalence relation) as a group.

**Greyson** can present exercise 2.23 (\(\omega^\omega\) and \(\omega\) have the same cardinality).

We’ll then introduce the definition of cardinality, and **Bailee** can present Cantor’s famous theorem that there is no largest cardinality.