# The language of Stratified Sets is confluent and strongly normalising

## TL;DR

This paper proves that the language of Stratified Sets, used in various set theories, is confluent and strongly normalising, ensuring consistent and predictable rewriting behavior.

## Contribution

It establishes that stratification conditions lead to confluence and strong normalisation in the syntax of Stratified Sets, connecting logic with rewriting properties.

## Key findings

- Syntax forms a nominal algebra for substitution
- Stratification implies confluence
- Stratifiability implies strong normalisation

## Abstract

We study the properties of the language of Stratified Sets (first-order logic with $\in$ and a stratification condition) as used in TST, TZT, and (with stratifiability instead of stratification) in Quine's NF. We find that the syntax forms a nominal algebra for substitution and that stratification and stratifiability imply confluence and strong normalisation under rewrites corresponding naturally to $\beta$-conversion.