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A soft consequence of ${\bf \Delta}^1_2$-determinacy is that the theory of $(\text{HOD}^{L[x]},\omega_2^{L[x]})$ is stable on a Turing cone. Woodin guessed that this has the following inner model theoretic explanation: $ \text{HOD}^{L[x]}$ is an iterated ultrapower (via an iteration tree) of the minimal canonical inner model $M_1$ with a Woodin cardinal, and $\omega^{L[x]}_2$ is the image of $M_1$'s Woodin cardinal.

A soft consequence of ${\bf \Delta}^1_2$-determinacy is that the theory of $(\text{HOD}^{L[x]},\omega_2^{L[x]})$ is stable on a Turing cone. Woodin guessed that this has the following inner model theoretic explanation: $ \text{HOD}^{L[x]}$ is an iterated ultrapower (via an iteration tree) of the minimal canonical inner model $M_1$ with a Woodin cardinal, and $\omega^{L[x]}_2$ is the image of $M_1$'s Woodin cardinal. (EDIT: We add that a lot of interesting descriptive set theory, not all due to Woodin, went into this guess: for example, it was conjectured by Kechris-Martin-Solovay in Introduction to Q-Theory that the set $$Q_3 = \{x\in \omega^\omega : x\text{ is }\Delta^1_3\text{ in a countable ordinal}\}$$ is the set of reals in the ultimate inner model with a $\Delta^1_3$ wellorder of the reals, which at that time was believed to have large cardinals at the level of $I_3$ (!). The same paper includes a proof of the following theorem of Martin: assuming ${\bf \Delta}^1_2$-determinacy, for a Turing cone of $x$, $Q_3 = \omega^\omega \cap \text{HOD}^{L[x]}$. Then Woodin, in his work with Shelah, realized that the large cardinal level of this ultimate inner model should actually be exactly one Woodin cardinal, since any significantly stronger hypothesis implies there is no $\Delta^1_3$ wellorder of the reals.)

There are many applications of core model theory at the level of Woodin cardinals in determinacy theory. One is Steel's theorem that PFA implies $\text{AD}^{L(\mathbb R)}$ (see his paper PFA implies $\text{AD}^{L(\mathbb R)}$). This is proved by the core model induction, another discovery of Woodin's. Another example is Sargsyan's result (see his thesis A Tale of Hybrid Mice) that if there is a Woodin limit of Woodin cardinals, then there is a pointclass $\Gamma\subseteq P(\mathbb R)$ such that $L(\mathbb R,\Gamma)$ satisfies $\text{AD}_\mathbb R + \Theta\text{ is regular}$. Before Sargsyan's work, this theory was conjectured to be extremely strong, at least at the level of a supercompact cardinal. There's also Steel's theorem that assuming AD and $V = L(\mathbb R)$, every regular cardinal below $\Theta$ is measurable and $\text{HOD}$ is a model of GCH. The proofs involve analyzing HOD as a fine structure model; $\text{HOD}$ is closely related to (but distinct from) the minimal inner model $M_\omega$ with $\omega$ Woodin cardinals, for example, $V_\Theta^\text{HOD}$ is an iterate of $M_\omega|\delta$ where $\delta$ is the least Woodin of $M_\omega$. Again see HOD as a Core Model.

There are many applications of core model theory at the level of Woodin cardinals in determinacy theory. One is Steel's theorem that PFA implies $\text{AD}^{L(\mathbb R)}$ (see his paper PFA implies $\text{AD}^{L(\mathbb R)}$). This is proved by the core model induction, another discovery of Woodin's. Another example is Sargsyan's result (see his thesis A Tale of Hybrid Mice) that if there is a Woodin limit of Woodin cardinals, then there is a pointclass $\Gamma\subseteq P(\mathbb R)$ such that $L(\mathbb R,\Gamma)$ satisfies $\text{AD}_\mathbb R + \Theta\text{ is regular}$. Before Sargsyan's work, this theory was conjectured to be extremely strong, at least at the level of a supercompact cardinal. There are also Steel's theorems that assuming AD and $V = L(\mathbb R)$, (1) every regular cardinal below $\Theta$ is measurable and (2) $\text{HOD}$ is a model of GCH. The proofs of both theorems involve analyzing HOD as a fine structure model; $\text{HOD}$ is closely related to (but distinct from) the minimal inner model $M_\omega$ with $\omega$ Woodin cardinals, for example, $V_\Theta^\text{HOD}$ is an iterate of $M_\omega|\delta$ where $\delta$ is the least Woodin of $M_\omega$. Again see HOD as a Core Model.

There are many applications of core model theory at the level of Woodin cardinals in determinacy theory. (At this point the subjects of inner model theory and higher descriptive set theory are so interconnected that it is impossible to say which subject is applied to which.) One is Steel's theorem that PFA implies $\text{AD}^{L(\mathbb R)}$ (see his paper PFA implies $\text{AD}^{L(\mathbb R)}$). This is proved by the core model induction, another discovery of Woodin's. Another example is Sargsyan's result (see his thesis A Tale of Hybrid Mice) that if there is a Woodin limit of Woodin cardinals, then there is a pointclass $\Gamma\subseteq P(\mathbb R)$ such that $L(\mathbb R,\Gamma)$ satisfies $\text{AD}_\mathbb R + \Theta\text{ is regular}$. Before Sargsyan's work, this theory was conjectured to be extremely strong, at least at the level of a supercompact cardinal. There's also Steel's theorem that assuming AD and $V = L(\mathbb R)$, every regular cardinal below $\Theta$ is measurable and $\text{HOD}$ is a model of GCH. The proofs involve analyzing HOD as a fine structure model; $\text{HOD}$ is closely related to (but distinct from) the minimal inner model $M_\omega$ with $\omega$ Woodin cardinals, for example, $V_\Theta^\text{HOD}$ is an iterate of $M_\omega|\delta$ where $\delta$ is the least Woodin of $M_\omega$. Again see HOD as a Core Model.

There are many applications of core model theory at the level of Woodin cardinals in determinacy theory. One is Steel's theorem that PFA implies $\text{AD}^{L(\mathbb R)}$ (see his paper PFA implies $\text{AD}^{L(\mathbb R)}$). This is proved by the core model induction, another discovery of Woodin's. Another example is Sargsyan's result (see his thesis A Tale of Hybrid Mice) that if there is a Woodin limit of Woodin cardinals, then there is a pointclass $\Gamma\subseteq P(\mathbb R)$ such that $L(\mathbb R,\Gamma)$ satisfies $\text{AD}_\mathbb R + \Theta\text{ is regular}$. Before Sargsyan's work, this theory was conjectured to be extremely strong, at least at the level of a supercompact cardinal. There's also Steel's theorem that assuming AD and $V = L(\mathbb R)$, every regular cardinal below $\Theta$ is measurable and $\text{HOD}$ is a model of GCH. The proofs involve analyzing HOD as a fine structure model; $\text{HOD}$ is closely related to (but distinct from) the minimal inner model $M_\omega$ with $\omega$ Woodin cardinals, for example, $V_\Theta^\text{HOD}$ is an iterate of $M_\omega|\delta$ where $\delta$ is the least Woodin of $M_\omega$. Again see HOD as a Core Model.