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

摘要

The main theorem of this article is that every countable model of set theory 〈M, ε~M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈L~M, ε~M〉 by means of an embedding j: M → L~M. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ε~M〉 and 〈N, εN〉 are countable models of set theory, then either M is isomorphic to a submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω_1 + 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 OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the 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~M, ε~M〉 of M. Indeed, 〈HF ~M, ε~M〉 is universal for all countable acyclic binary relations.