# The sharp for the Chang model is small

## TL;DR

This paper constructs a sharp for the Chang model under weaker large cardinal assumptions, specifically using a cardinal with an extender of a certain length, extending Woodin's results.

## Contribution

It demonstrates the existence of a sharp for the Chang model assuming only a cardinal with a long extender, weakening previous large cardinal requirements.

## Key findings

- Constructs a sharp for the Chang model with weaker assumptions
- Uses a cardinal with an extender of length κ^{+ω_1}
- Extends Woodin's results to weaker hypotheses

## Abstract

Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal $\kappa$ having an extender of length $\kappa^{+\omega_1}$.