This will be a short lecture series given at the conclusion of the graduate logic class in the Mathematical Logic group at Fudan University in Shanghai, June 13, 18 (or 20), 2013.

I will present an elementary introduction to the theory of universal orders and relations and saturated structures. We’ll start with the classical fact, proved by Cantor, that the rational line is the universal countable linear order. But what about universal partial orders, universal graphs and other mathematical structures? Is there a computable universal partial order? What is the countable random graph? Which orders embed into the power set of the natural numbers under the subset relation $\langle P(\mathbb{N}),\subset\rangle$? Proceeding to larger and larger universal orders, we’ll eventually arrive at the surreal numbers and the hypnagogic digraph.