# Condensable models of set theory

**Authors:** Ali Enayat  
**arXiv:** [1910.04029](https://arxiv.org/abs/1910.04029)  
**Date:** 2019-10-09  
**Summary page:** https://www.summarizepaper.com/en/paper/1910.04029v4/

## Key points

- Condensability in models of set theory is explored by Ali Enayat
- A model M is condensable if there exists an ordinal v within M that satisfies certain conditions
- The author establishes the existence of condensable models of ZFC under a modest set-theoretic hypothesis
- Every definable element in these models belongs to their well-founded part
- Various characterizations of countable condensable models of ZF are provided in the paper
- These characterizations offer insights into the nature and behavior of countable models with this property
- Enayat's work sheds light on condensability in set theory models and offers new perspectives on their structure and definability
- The findings contribute to our understanding of foundational aspects in mathematics and provide valuable tools for further research.

## Summary

In the paper "Condensable models of set theory," Ali Enayat explores a concept called condensability in models of set theory. A model M is considered condensable if there exists an ordinal v within M such that the rank initial segment of M determined by v is both isomorphic to M and serves as an elementary submodel of M for infinitary formulae in the well-founded part of M. 

The author establishes, under a modest set-theoretic hypothesis, the existence of condensable models M of ZFC (Zermelo-Fraenkel set theory with the axiom of Choice) where every definable element in M belongs to its well-founded part. This result has significant implications for understanding the structure and properties of these models. Additionally, the paper provides various characterizations of countable condensable models of ZF (Zermelo-Fraenkel set theory without the axiom of Choice). These characterizations offer insights into the nature and behavior of countable models that possess this unique property. Overall, Enayat's work sheds light on condensability in set theory models and offers new perspectives on their structure and definability. The findings contribute to our understanding of foundational aspects in mathematics and provide valuable tools for further research in this area.