# Initial self-embeddings of models of set theory

## TL;DR

This paper explores the properties of initial self-embeddings in models of set theory, extending classical results about rank-initial self-embeddings to broader classes of initial-embeddings with transitive images.

## Contribution

It generalizes Friedman’s classical theorem by investigating proper initial-embeddings with transitive images in models of set theory fragments.

## Key findings

- Established conditions for initial-embeddings with transitive images.
- Extended classical results to broader classes of embeddings.
- Provided new insights into the structure of models of set theory.

## Abstract

By a classical theorem of Harvey Friedman (1973), every countable nonstandard model $\mathcal{M}$ of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding $j$, i.e., $j$ is a self-embedding of $\mathcal{M}$ such that $j[\mathcal{M}]\subsetneq\mathcal{M}$, and the ordinal rank of each member of $j[\mathcal{M}]$ is less than the ordinal rank of each element of $\mathcal{M}\setminus j[\mathcal{M}]$. Here we investigate the larger family of proper initial-embeddings $j$ of models $\mathcal{M}$ of fragments of set theory, where the image of $j$ is a transitive submodel of $\mathcal{M}$.