Embeddability amongst the countable models of set theory, plenary talk for ASL / Joint Math Meetings in Baltimore, January 2014

A one-hour plenary talk for the ASL at the Joint Math Meetings, January 15-18, 2014 in Baltimore, MD.

Abstract. A surprisingly vigorous embeddability phenomenon has recently been uncovered amongst the countable models of set theory.  In particular, embeddability is linear:  for any two countable models of set theory, one of them is isomorphic to a submodel of the other.  In particular, every countable model of set theory, including every well-founded model, is isomorphic to a submodel of its own constructible universe, so that there is an embedding $j:M\to L^M$ for which $x\in y\iff j(x)\in j(y)$. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues arising as uncountable Fraïssé limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers.