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

Saturday January 18, 2014, 2:00 p.m.-2:50 p.m, Room 319 BCC

**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.

Has a paper with proofs appeared on your website?

The paper has appeared in the current issue of the JML, and you can find links at http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe/; also http://www.worldscientific.com/doi/abs/10.1142/S0219061313500062. (Note that the preprint version on the arxiv has a theorem on uncountable Fraisse limits that is not correct, and which does not appear in the published version.)

Why not update the arXiv version to take out the incorrect statement?

Oh, indeed I updated the arxiv article in February. Version v3 makes the correction.

Oh good, I was a bit mystified in thinking you wouldn’t.