Every Countable Model of Set Theory Embeds into Its Own Constructible Universe

The main theorem of this article is that every countable model of set theory 〈M,∈ 〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈L ,∈ 〉. In other words, there is an embedding j : M → L that is elementary for quantifier-free assertions. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory 〈M,∈ 〉 and 〈N,∈ 〉, either M is isomorphic to a submodel of N or conversely. Indeed, they are pre-well-ordered by embeddability in order-type exactly ω1+1. Specifically, the countable well-founded models are ordered under embeddability exactly 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 Ord ; 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 plus large cardinals—is isomorphic to a submodel of the hereditarily finite sets 〈HF ,∈ 〉 of M . Indeed, 〈HF ,∈ 〉 is universal for all countable acyclic binary relations.