Embeddings into outer models

Abstract

We explore the possibilities for elementary embeddings $j : M \to N$, where $M$ and $N$ are models of ZFC with the same ordinals, $M \subseteq N$, and $N$ has access to large pieces of $j$. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line $\mathbb R$.