Every countable model of set theory is isomorphic to a submodel of its own constructible universe, Barcelona, December, 2012

This will be a talk for a set theory workshop at the University of Barcelona on December 15, 2012, organized by Joan Bagaria.

Vestíbul Universitat de Barcelona

Abstract. 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-embeddability classes form a well-ordered chain of length $\omega_1+1$.  Specifically, the countable well-founded models are ordered by embeddability 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.

Article | Barcelona research group in set theory

Victoria Gitman

Victoria Gitman earned her Ph.D. under my supervision at the CUNY Graduate Center in June, 2007.  For her dissertation work, Victoria had chosen a very difficult problem, the 1962 question of Dana Scott to characterize the standard systems of models of Peano Arithmetic, a question in the field of models of arithmetic that had been open for over forty years. Victoria was able to make progress, now published in several papers, by using an inter-disciplinary approach, applying set-theoretic ideas—including a use of the proper forcing axiom PFA—to the problem in the area of models of arithmetic, where such methods hadn’t often yet arisen.  Ultimately, she showed under PFA that every arithmetically closed proper Scott set is the standard system of a model of PA.  This result extends the classical result to a large new family of Scott sets, providing for these sets an affirmative solution to Scott’s problem.  In other dissertation work, Victoria untangled the confusing mass of ideas surrounding various Ramsey-like large cardinal concepts, ultimately separating them into a beautiful hierarchy, a neighborhood of the vast large cardinal hierarchy intensely studied by set theorists.  (Please see the diagram in her dissertation.)  Victoria holds a tenure-track position at the New York City College of Technology of CUNY.

Victoria Gitman

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Victoria Gitman, “Applications of the Proper Forcing Axiom to Models of Peano Arithmetic,”  Ph.D. dissertation for the Graduate Center of the City University of New York, June 2007.

Abstract. In Chapter 1, new results are presented on Scott’s Problem in the subject of models of Peano Arithmetic. Some forty years ago, Dana Scott showed that countable Scott sets are exactly the countable standard systems of models of PA, and two decades later, Knight and Nadel extended his result to Scott sets of size $\omega_1$. Here it is shown that assuming the Proper Forcing Axiom, every arithmetically closed proper Scott set is the standard system of a model of PA. In Chapter 2, new large cardinal axioms, based on Ramsey-like embedding properties, are introduced and placed within the large cardinal hierarchy. These notions generalize the seldom encountered embedding characterization of Ramsey cardinals. I also show how these large cardinals can be used to obtain indestructibility results for Ramsey cardinals.

Every countable model of set theory embeds into its own constructible universe, Fields Institute, Toronto, August 2012

This will be a talk for the  Toronto set theory seminar at the Fields Institute, University of Toronto, on August 24, 2012.

Abstract.  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 Image: Igougo.comwith 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-embeddability classes form a well-ordered chain of length $\omega_1+1$.  Specifically, the countable well-founded models are ordered by embeddability 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.

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