# Embeddings into outer models

## TL;DR

This paper investigates elementary embeddings between models of ZFC, constructing systems of such embeddings that reflect complex order structures like the real line, advancing understanding of model interactions in set theory.

## Contribution

It introduces methods to construct commuting systems of elementary embeddings between models of ZFC that mirror various canonical orders, including the real line.

## Key findings

- Constructed commuting systems of embeddings between models of ZFC.
- Embedded complex order structures such as the real line into model systems.
- Demonstrated the versatility of embeddings in representing various linear and partial orders.

## 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 $\mathbb R$.