# Successive failures of approachability

## TL;DR

This paper constructs a model demonstrating the failure of the approachability property across a wide range of regular cardinals, impacting the existence of special Aronszajn trees in set theory.

## Contribution

It shows, within ZFC, that the approachability property can fail at all regular cardinals in a specified interval, affecting the existence of special Aronszajn trees.

## Key findings

- Approachability property fails at all regular cardinals in the interval [ℵ₂, ℵ_{ω^2+3}]
- No special Aronszajn trees exist in the constructed model
- ℵ_{ω^2} is a strong limit in this model

## Abstract

Motivated by showing that in ZFC we cannot construct a special Aronszajn tree on some cardinal greater than $\aleph_1$, we produce a model in which the approachability property fails (hence there are no special Aronszajn trees) at all regular cardinals in the interval $[\aleph_2, \aleph_{\omega^2+3}]$ and $\aleph_{\omega^2}$ is strong limit.