We gave a formal definition of *ordered pair, *which* *allows us to discuss relations and functions in set theory. We presented the intuitive picture of the universe of sets in terms of the *cumulative hierarchy*, that we will be fleshing out in subsequent lectures.

We proved **Cantor’s theorem** stating that for any set and the **Knaster-Tarski theorem** stating that any order-preserving function from a complete lattice to itself has a fixed point, as a corollary of which we obtained the **Schröder-Bernstein theorem** stating that if then in fact .

We briefly mentioned Cantor’s work on the theory of *sets of uniqueness for trigonometric series*, that led to his discovery of the ordinal numbers; Kechris’s very nice expository paper on the subject is here, the result that countable closed sets are of uniqueness is discussed in pages 3-13, and I highly recommend it.

**Remark:** Cantor’s original proof of the Schröder-Bernstein theorem used the axiom of choice and will be discussed in a future lecture, together with a more “combinatorial” proof.