# Blog Archives

# Topic Archive: homogeneity

# 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

This will be a talk 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 $langletext{HF}^M,{in^M}rangle$ the corresponding nonstandard hereditary finite sets of $M$, then for any consistent computably axiomatized theory $T$ in the language of set theory, with $Tsupset ZF$, there is a submodel $Nsubsetlangletext{HF}^M,{in^M}rangle$ such that $Nmodels T$. 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 $langletext{HF}^M,{in^M}rangle$.

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 $langle M,in^Mrangle$. 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 $M$.

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