# The countable models of ZFC, up to isomorphism, are linearly pre-ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic to a submodel of its own $L$, New York, 2012

This will be a talk on May 18, 2012 for the CUNY Logic Workshop on some extremely new work. The proof uses finitary digraph combinatorics, including the countable random digraph and higher analogues involving uncountable Fraisse limits, the surreal numbers and the hypnagogic digraph.

The story begins with Ressayre’s remarkable 1983 result that if $M$ is any nonstandard model of PA, with $\langle\text{HF}^M,{\in^M}\rangle$ the corresponding nonstandard hereditary finite sets of $M$, then for any consistent computably axiomatized theory $T$ in the language of set theory, with $T\supset ZF$, there is a submodel $N\subset\langle\text{HF}^M,{\in^M}\rangle$ such that $N\models T$. In particular, one may find models of ZFC or even ZFC + large cardinals as submodels of $\text{HF}^M$, a land where everything is thought to be finite. Incredible! Ressayre’s proof uses partial saturation and resplendency to prove that one can find the submodel of the desired theory $T$.

My new theorem strengthens Ressayre’s theorem, while simplifying the proof, by removing the theory $T$. We need not assume $T$ is computable, and we don’t just get one model of $T$, but rather all models—the fact is that the nonstandard models of set theory are universal for all countable acyclic binary relations. So every model of set theory is a submodel of $\langle\text{HF}^M,{\in^M}\rangle$.

Theorem.(JDH) Every countable model of set theory is isomorphic to a submodel of any nonstandard model of finite set theory. Indeed, every nonstandard model of finite set theory is universal for all countable acyclic binary relations.

The proof involves the construction of what I call the countable random $\mathbb{Q}$-graded digraph, a countable homogeneous acyclic digraph that is universal for all countable acyclic digraphs, and proving that it is realized as a submodel of the nonstandard model $\langle M,\in^M\rangle$. Having then realized a universal object as a submodel, it follows that every countable structure with an acyclic binary relation, including every countable model of ZFC, is realized as a submodel of $M$.

The proof, in brief:  for every countable acyclic digraph, consider the partial order induced by the edge relation, and extend this order to a total order, which may be embedded in the rational order $\mathbb{Q}$.  Thus, every countable acyclic digraph admits a $\mathbb{Q}$-grading, an assignmment of rational numbers to nodes such that all edges point upwards. Next, one can build a countable homogeneous, universal, existentially closed $\mathbb{Q}$-graded digraph, simply by starting with nothing, and then adding finitely many nodes at each stage, so as to realize the finite pattern property. The result is a computable presentation of what I call the countable random $\mathbb{Q}$-graded digraph $\Gamma$.  If $M$ is any nonstandard model of finite set theory, then we may run this computable construction inside $M$ for a nonstandard number of steps.  The standard part of this nonstandard finite graph includes a copy of $\Gamma$.  Furthermore, since $M$ thinks it is finite and acyclic, it can perform a modified Mostowski collapse to realize the graph in the hereditary finite sets of $M$.  By looking at the sets corresponding to the nodes in the copy of $\Gamma$, we find a submodel of $M$ that is isomorphic to $\Gamma$, which is universal for all countable acyclic binary relations. So every model of ZFC isomorphic to a submodel of $M$.

The proof idea adapts, with complications, to the case of well-founded models, via the countable random $\lambda$-graded digraph, as well as the internal construction of what I call the hypnagogic digraph, a proper class homogeneous surreal-numbers-graded digraph, which is universal for all class acyclic digraphs.

Theorem.(JDH) Every countable model $\langle M,\in^M\rangle$ of ZFC, including the transitive models, is isomorphic to a submodel of its own constructible universe $\langle L^M,\in^M\rangle$. In other words, there is an embedding $j:M\to L^M$ that is quantifier-free-elementary.

The proof is guided by the idea of finding a universal submodel inside $L^M$. The embedding $j$ is constructed completely externally to $M$.

Corollary.(JDH) The countable models of ZFC are linearly ordered and even well-ordered, up to isomorphism, by the submodel relation. Namely, any two countable models of ZFC with the same well-founded height are bi-embeddable as submodels of each other, and all models embed into any nonstandard model.

The work opens up numerous questions on the extent to which we may expect in ZFC that $V$ might be isomorphic to a subclass of $L$. To what extent can we expect to have or to refute embeddings $j:V\to L$, elementary for quantifier-free assertions?

## 4 thoughts on “The countable models of ZFC, up to isomorphism, are linearly pre-ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic to a submodel of its own $L$, New York, 2012”

1. Victoria Gitman on said:

Wow! This is an amazing theorem! I also never thought I would see surreal numbers used in a set theoretic paper, it makes me want to learn all about them. I can’t wait to hear the details. Hopefully, I can make arrangements to come to the city.

• Joel David Hamkins on said:

Thanks for the vote of confidence, Vika. It means a lot to me.

2. Thomas Benjamin on said:

Very, Very Interesting! How does this apply to the countable models which are forcing extensions? How does this relate (as countable models are deemed ‘toy models’) to the notion of the set-theoretic multiverse?

• Joel David Hamkins on said:

I’m glad you like it. I’m working furiously on the article, which should be ready fairly soon, and I’ll make a post when it is. The theorem shows that all forcing extensions of a countable model of set theory are isomorphic to a submodel of the original model. This submodel, of course, will not be an amenable class in the original model, and the isomorphism is visible only from an external perspective to the model. Nevertheless, it is not currently clear to what extent we may hope to prove or refute the hypothesis that there is a class embedding of V into L that is elementary for quantifier-free assertions.