24/Aug/2012: Joel David Hamkins, and Lajos Soukup

11:00–12:00
Fields institute, Room 210

Speaker:  Joel David Hamkins (The City University of New York)

Title: Every countable model of set theory embeds into its own constructible universe.

Abstract:  I shall give an account of my recent theorem showing that every countable model of set theory $M$, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. 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 Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $\text{Ord}^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other.  Indeed, the bi-embedability classes form a well-ordered chain of length $\omega_1+1$.  Specifically, the countable well-founded models are ordered by embedability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $\text{Ord}^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory—in particular, every model of ZFC—is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.

Questions and commentary concerning Hamkins' talk can be made in here.

13:30–15:00
Fields institute, Room 230

Speaker: Lajos Soukup (Alfréd Rényi Institute of Mathematics)

Title: On properties of ladder systems on $\omega_1$.