Bumping to mention/advertise a recent development:

Today, William Chan and Stephen Jackson posted this paper to the arxiv. They proved that a broad class of forcings (under reasonable hypotheses) always kill determinacy. Specifically, they showed

Theorem 3.2: ZF + AD proves that every well-orderable forcing of cardinality $<\Theta$ forces $\neg$AD

and

Theorem 5.6: ZF + AD + "$\Theta$ is regular" proves that every nontrivial forcing which is a surjective image of $\mathbb{R}$ forces $\neg$AD, as does ZF + AD$^+$ + $\neg$AD$_\mathbb{R}$ + V=L$(\mathcal{P}(\mathbb{R}))$.

Below I'll give a tissue-thin, hopefully-not-too-inaccurate description of their paper.

The arguments in the paper generalize the existing proofs of "determinacy-destruction" for a wide class of forcings under various assumptions (specifically: Cohen forcing, $Col(\omega_1,\omega_2)$, and $Col(\omega,\omega_1)$) from prior analysis by Ikegami and Trang:

Ikegami and Trang initiated the study of the preservation of AD under forcing. They showed that many forcings, such as Cohen forcing, can never preserve AD. They also showed that if one is working with natural models of AD, i.e. models satisfying ZF + AD+ + V = L$(\mathcal{P}(\mathbb{R}))$, then any forcing which preserve AD must preserve $\Theta$, where $\Theta$ is the supremum of the ordinals which are surjective images of $\mathbb{R}$. They also showed that the consistency of ZF + AD+ + $\Theta>\Theta_0$ implies the consistency of ZF + AD and there is a forcing which preserve AD and increases $\Theta$. Thus necessarily this forcing must disturb $\mathcal{P}(\mathbb{R})$ by adding a new set of reals

(Unfortunately, it looks like this work hasn't appeared yet, and Ikegami and Trang don't appear in the bibliography. My quick impression, though, is that the Chan/Jackson paper subsumes their results.)

Chan and Jackson isolate a common combinatorial aspect of these examples: the ground club property at $\kappa$ (Definition 2.9), that any new club in $\kappa$ contains a ground club in $\kappa$. The ground club property provides a connection between Ramsey properties and reals (slightly rephrased):

Lemma 2.10: If $V[G]$ is a generic extension via a forcing with the ground club property at $\kappa$ and $V[G]\models\kappa\rightarrow(\kappa)^\omega_2$, then $\mathbb{R}^V=\mathbb{R}^{V[G]}$.

From this they quickly deduce Theorem 3.2 above (and a different argument, joint with Goldberg, improves the theorem by replacing "cardinality $<\Theta$" with "adds a real" - Corollary 3.5).

This lemma is also fundamental to the proof of Theorem 5.6, but the argument there is much more complicated, and relies on an analysis of the behavior of $\Theta$ after forcing, building off of a previous result by Ikegami and Trang. In particular, they show that (in ZF + AD) any nontrivial forcing which is a surjective image of $\mathbb{R}$ and preserves AD must add a real (Fact 4.4), and so must preserve $\Theta$ (Lemma 4.3 - which surprised me quite a bit).

I hope I haven't misrepresented anything here; regardless of my understanding, this is certainly a very relevant paper.

Bumping to mention/advertise a recent development:

Today, William Chan and Stephen Jackson posted this paper to the arxiv. They proved that a broad class of forcings (under reasonable hypotheses) always kill determinacy. Specifically, they showed

Theorem 3.2: ZF + AD proves that every well-orderable forcing of cardinality $<\Theta$ forces $\neg$AD

and

Theorem 5.6: ZF + AD + "$\Theta$ is regular" proves that every nontrivial forcing which is a surjective image of $\mathbb{R}$ forces $\neg$AD, as does ZF + AD$^+$ + $\neg$AD$_\mathbb{R}$ + V=L$(\mathcal{P}(\mathbb{R}))$.

• Note that we generally don't have $L(\mathbb{R})^V[g]=L(\mathbb{R})^{V[g]}$ (which would of course contradict these results).

Below I'll give a tissue-thin, hopefully-not-too-inaccurate description of their paper.

The arguments in the paper generalize the existing proofs of "determinacy-destruction" for a wide class of forcings under various assumptions (specifically: Cohen forcing, $Col(\omega_1,\omega_2)$, and $Col(\omega,\omega_1)$) from prior analysis by Ikegami and Trang:

Ikegami and Trang initiated the study of the preservation of AD under forcing. They showed that many forcings, such as Cohen forcing, can never preserve AD. They also showed that if one is working with natural models of AD, i.e. models satisfying ZF + AD+ + V = L$(\mathcal{P}(\mathbb{R}))$, then any forcing which preserve AD must preserve $\Theta$, where $\Theta$ is the supremum of the ordinals which are surjective images of $\mathbb{R}$. They also showed that the consistency of ZF + AD+ + $\Theta>\Theta_0$ implies the consistency of ZF + AD and there is a forcing which preserve AD and increases $\Theta$. Thus necessarily this forcing must disturb $\mathcal{P}(\mathbb{R})$ by adding a new set of reals

(Unfortunately, it looks like this work hasn't appeared yet, and Ikegami and Trang don't appear in the bibliography. My quick impression, though, is that the Chan/Jackson paper subsumes their results.)

Chan and Jackson isolate a common combinatorial aspect of these examples: the ground club property at $\kappa$ (Definition 2.9), that any new club in $\kappa$ contains a ground club in $\kappa$. The ground club property provides a connection between Ramsey properties and reals (slightly rephrased):

Lemma 2.10: If $V[G]$ is a generic extension via a forcing with the ground club property at $\kappa$ and $V[G]\models\kappa\rightarrow(\kappa)^\omega_2$, then $\mathbb{R}^V=\mathbb{R}^{V[G]}$.

From this they quickly deduce Theorem 3.2 above. (A different argument, joint with Goldberg, improves the theorem by replacing "cardinality $<\Theta$" with "adds a real" - Corollary 3.5.)

This lemma is also fundamental to the proof of Theorem 5.6, but the argument there is much more complicated, and relies on an analysis of the behavior of $\Theta$ after forcing, building off of a previous result by Ikegami and Trang. In particular, they show that (in ZF + AD) any nontrivial forcing which is a surjective image of $\mathbb{R}$ and preserves AD must add a real (Fact 4.4), and so must preserve $\Theta$ (Lemma 4.3 - which surprised me quite a bit).

I hope I haven't misrepresented anything here; regardless of my understanding, this is certainly a very relevant paper.

Bumping to mention a recent development:

Today, William Chan and Stephen Jackson posted this paper to the arxiv. They proved that a broad class of forcings (under reasonable hypotheses) always kill determinacy. Specifically, they showed

Theorem 3.2: ZF + AD proves that every well-orderable forcing of cardinality $<\Theta$ forces $\neg$AD

and

Theorem 5.6: ZF + AD + "$\Theta$ is regular" proves that every nontrivial forcing which is a surjective image of $\mathbb{R}$ forces $\neg$AD, as does ZF + AD$^+$ + $\neg$AD$_\mathbb{R}$ + V=L$(\mathcal{P}(\mathbb{R}))$.

Below I'll give a tissue-thin, hopefully-not-too-inaccurate description of their paper.

The arguments in the paper generalize the existing proofs of "determinacy-destruction" for a wide class of forcings under various assumptions (specifically: Cohen forcing, $Col(\omega_1,\omega_2)$, and $Col(\omega,\omega_1)$) from prior analysis by Ikegami and Trang:

Ikegami and Trang initiated the study of the preservation of AD under forcing. They showed that many forcings, such as Cohen forcing, can never preserve AD. They also showed that if one is working with natural models of AD, i.e. models satisfying ZF + AD+ + V = L$(\mathcal{P}(\mathbb{R}))$, then any forcing which preserve AD must preserve $\Theta$, where $\Theta$ is the supremum of the ordinals which are surjective images of $\mathbb{R}$. They also showed that the consistency of ZF + AD+ + $\Theta>\Theta_0$ implies the consistency of ZF + AD and there is a forcing which preserve AD and increases $\Theta$. Thus necessarily this forcing must disturb $\mathcal{P}(\mathbb{R})$ by adding a new set of reals

(Unfortunately, it looks like this work hasn't appeared yet, and Ikegami and Trang don't appear in the bibliography. My quick impression, though, is that the Chan/Jackson paper subsumes their results.)

Chan and Jackson isolate a common combinatorial aspect of these examples: the ground club property at $\kappa$ (Definition 2.9), that any new club in $\kappa$ contains a ground club in $\kappa$. The ground club property provides a connection between Ramsey properties and reals (slightly rephrased):

Lemma 2.10: If $V[G]$ is a generic extension via a forcing with the ground club property at $\kappa$ and $V[G]\models\kappa\rightarrow(\kappa)^\omega_2$, then $\mathbb{R}^V=\mathbb{R}^{V[G]}$.

From this they quickly deduce Theorem 3.2 above (and a different argument, joint with Goldberg, improves the theorem by replacing "cardinality $<\Theta$" with "adds a real" - Corollary 3.5).

This lemma is also fundamental to the proof of Theorem 5.6, but the argument there is much more complicated, and relies on an analysis of the behavior of $\Theta$ after forcing, building off of a previous result by Ikegami and Trang. In particular, they show that (in ZF + AD) any nontrivial forcing which is a surjective image of $\mathbb{R}$ and preserves AD must add a real (Fact 4.4), and so must preserve $\Theta$ (Lemma 4.3 - which surprised me quite a bit).

I hope I haven't misrepresented anything here; regardless of my understanding, this is certainly a very relevant paper.

Bumping to mention/advertise a recent development:

Today, William Chan and Stephen Jackson posted this paper to the arxiv. They proved that a broad class of forcings (under reasonable hypotheses) always kill determinacy. Specifically, they showed

Theorem 3.2: ZF + AD proves that every well-orderable forcing of cardinality $<\Theta$ forces $\neg$AD

and

Theorem 5.6: ZF + AD + "$\Theta$ is regular" proves that every nontrivial forcing which is a surjective image of $\mathbb{R}$ forces $\neg$AD, as does ZF + AD$^+$ + $\neg$AD$_\mathbb{R}$ + V=L$(\mathcal{P}(\mathbb{R}))$.

Below I'll give a tissue-thin, hopefully-not-too-inaccurate description of their paper.

The arguments in the paper generalize the existing proofs of "determinacy-destruction" for a wide class of forcings under various assumptions (specifically: Cohen forcing, $Col(\omega_1,\omega_2)$, and $Col(\omega,\omega_1)$) from prior analysis by Ikegami and Trang:

Ikegami and Trang initiated the study of the preservation of AD under forcing. They showed that many forcings, such as Cohen forcing, can never preserve AD. They also showed that if one is working with natural models of AD, i.e. models satisfying ZF + AD+ + V = L$(\mathcal{P}(\mathbb{R}))$, then any forcing which preserve AD must preserve $\Theta$, where $\Theta$ is the supremum of the ordinals which are surjective images of $\mathbb{R}$. They also showed that the consistency of ZF + AD+ + $\Theta>\Theta_0$ implies the consistency of ZF + AD and there is a forcing which preserve AD and increases $\Theta$. Thus necessarily this forcing must disturb $\mathcal{P}(\mathbb{R})$ by adding a new set of reals

(Unfortunately, it looks like this work hasn't appeared yet, and Ikegami and Trang don't appear in the bibliography. My quick impression, though, is that the Chan/Jackson paper subsumes their results.)

Chan and Jackson isolate a common combinatorial aspect of these examples: the ground club property at $\kappa$ (Definition 2.9), that any new club in $\kappa$ contains a ground club in $\kappa$. The ground club property provides a connection between Ramsey properties and reals (slightly rephrased):

Lemma 2.10: If $V[G]$ is a generic extension via a forcing with the ground club property at $\kappa$ and $V[G]\models\kappa\rightarrow(\kappa)^\omega_2$, then $\mathbb{R}^V=\mathbb{R}^{V[G]}$.

From this they quickly deduce Theorem 3.2 above (and a different argument, joint with Goldberg, improves the theorem by replacing "cardinality $<\Theta$" with "adds a real" - Corollary 3.5).

This lemma is also fundamental to the proof of Theorem 5.6, but the argument there is much more complicated, and relies on an analysis of the behavior of $\Theta$ after forcing, building off of a previous result by Ikegami and Trang. In particular, they show that (in ZF + AD) any nontrivial forcing which is a surjective image of $\mathbb{R}$ and preserves AD must add a real (Fact 4.4), and so must preserve $\Theta$ (Lemma 4.3 - which surprised me quite a bit).

I hope I haven't misrepresented anything here; regardless of my understanding, this is certainly a very relevant paper.