# The Destruction of the Axiom of Determinacy by Forcings on $\mathbb{R}$ when $\Theta$ is Regular

## TL;DR

This paper demonstrates that forcing extensions on the real numbers under certain conditions destroy the Axiom of Determinacy, especially when the ordinal $ heta$ is regular, highlighting limitations of AD in these models.

## Contribution

It proves that forcing on sets of reals in models with AD and regular $ heta$ necessarily negates AD, extending understanding of the fragility of AD under forcing.

## Key findings

- Forcing on wellorderable sets of size less than $ heta$ destroys AD.
- Forcing on surjective images of $
eals$ also negates AD when $ heta$ is regular.
- In $L(
eals)$ models, nontrivial forcing always destroys AD.

## Abstract

$\mathsf{ZF + AD}$ proves that for all nontrivial forcings $\mathbb{P}$ on a wellorderable set of cardinality less than $\Theta$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$. $\mathsf{ZF + AD} + \Theta$ is regular proves that for all nontrivial forcing $\mathbb{P}$ which is a surjective image of $\mathbb{R}$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$. In particular, $\mathsf{ZF + AD + V = L(\mathbb{R})}$ proves that for every nontrivial forcing $\mathbb{P} \in L_\Theta(\mathbb{R})$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$.