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An unpublished theorem due to Woodin (which appears without proof as Theorem 7.35 of "In search of Ultimate L") states that if the $\Omega$ conjecture holds and there is a proper class of Woodin cardinals, then there is a partially ordered set $\mathbb P$ such that if $G\subseteq \mathbb P$ is a $V$-generic filter and $\mathbb R^* = (\mathbb R)^{V[G]}$, then $V(\mathbb R^*)$ is a model of AD. The hypotheses of this theorem are known to be consistent: in fact they hold in any model with a proper class of Woodins that satisfies Woodin's weakly homogeneous iteration hypothesis (WHIH), and this includes many of the current fine structure models by Steel's theorem that such models satisfy UBH; see Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Definition 10.4 and Theorem 10.151. The model $V(\mathbb R^*)$ will in this context satisfy AD plus a proper class of Woodin cardinals. If there is a proper class of cardinals with some stronger large cardinal property in $V$, say a proper class of extendible cardinals for concreteness, then $V(\mathbb R^*)$ will have a proper class of extendible cardinals by standard lifting arguments; but the $\Omega$ conjecture is not known to be consistent with the existence of a proper class of extendible cardinals (or even a proper class of measurable Woodin cardinals).

In summary, if ZFC plus the $\Omega$ conjecture is consistent with a proper class of $X$ cardinals, then the Axiom of Determinacy is consistent with a proper class of $X$ cardinals, for any $X\in \{\text{Woodin, Extendible, Superduperhuge, etc}\}$. Moreover the $\Omega$ conjecture is consistent with a proper class of Woodin cardinals assuming the existence of a proper class of Woodin cardinals is consistent.

An unpublished theorem due to Woodin (which appears without proof as Theorem 7.35 of "In search of Ultimate L") states that if the $\Omega$ conjecture holds and there is a proper class of Woodin cardinals, then there is a partially ordered set $\mathbb P$ such that if $G\subseteq \mathbb P$ is a $V$-generic filter and $\mathbb R^* = (\mathbb R)^{V[G]}$, then $V(\mathbb R^*)$ is a model of AD. The hypotheses of this theorem are known to be consistent: in fact they hold in any model with a proper class of Woodins that satisfies Woodin's weakly homogeneous iteration hypothesis (WHIH), and this includes many of the current fine structure models by Steel's theorem that such models satisfy UBH; see Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Definition 10.4 and Theorem 10.151. The model $V(\mathbb R^*)$ will in this context satisfy ZF + DC + AD plus a proper class of Woodin cardinals. If there is a proper class of cardinals with some stronger large cardinal property in $V$, say a proper class of extendible cardinals for concreteness, then $V(\mathbb R^*)$ will have a proper class of extendible cardinals by standard lifting arguments; but the $\Omega$ conjecture is not known to be consistent with the existence of a proper class of extendible cardinals (or even a proper class of measurable Woodin cardinals).

In summary, if ZFC plus the $\Omega$ conjecture is consistent with a proper class of $X$ cardinals, then the Axiom of Determinacy is consistent with a proper class of $X$ cardinals, for any $X\in \{\text{Woodin, Extendible, Superduperhuge, etc}\}$. Moreover the $\Omega$ conjecture is consistent with a proper class of Woodin cardinals assuming the existence of a proper class of Woodin cardinals is consistent.

An unpublished theorem of Woodin (which appears without proof as Theorem 7.35 in "In search of Ultimate L") states that if the $\Omega$ conjecture holds and there is a proper class of Woodin cardinals, then there is a partially ordered set $\mathbb P$ such that if $G\subseteq \mathbb P$ is a $V$-generic filter and $\mathbb R^* = (\mathbb R)^{V[G]}$, then $V(\mathbb R^*)$ is a model of AD. The hypotheses of this theorem are known to be consistent: in fact they hold in any model with a proper class of Woodins that satisfies Woodin's weakly homogeneous iteration hypothesis (WHIH), and this includes many of the current fine structure models by Steel's theorem that such models satisfy UBH; see Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Definition 10.4 and Theorem 10.151. The model $V(\mathbb R^*)$ will in this context satisfy AD plus a proper class of Woodin cardinals. If there is a proper class of cardinals with some stronger large cardinal property in $V$, say a proper class of extendible cardinals for concreteness, then $V(\mathbb R^*)$ will have a proper class of extendible cardinals by standard lifting arguments; but the $\Omega$ conjecture is not known to be consistent with the existence of a proper class of extendible cardinals (or even a proper class of measurable Woodin cardinals).

In summary, if ZFC plus the $\Omega$ conjecture is consistent with a proper class of $X$ cardinals, then the Axiom of Determinacy is consistent with a proper class of $X$ cardinals, for any $X\in \{\text{Woodin, Extendible, Superduperhuge, etc}\}$. Moreover the $\Omega$ conjecture is consistent with a proper class of Woodin cardinals assuming the existence of a proper class of Woodin cardinals is consistent.

An unpublished theorem due to Woodin (which appears without proof as Theorem 7.35 of "In search of Ultimate L") states that if the $\Omega$ conjecture holds and there is a proper class of Woodin cardinals, then there is a partially ordered set $\mathbb P$ such that if $G\subseteq \mathbb P$ is a $V$-generic filter and $\mathbb R^* = (\mathbb R)^{V[G]}$, then $V(\mathbb R^*)$ is a model of AD. The hypotheses of this theorem are known to be consistent: in fact they hold in any model with a proper class of Woodins that satisfies Woodin's weakly homogeneous iteration hypothesis (WHIH), and this includes many of the current fine structure models by Steel's theorem that such models satisfy UBH; see Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Definition 10.4 and Theorem 10.151. The model $V(\mathbb R^*)$ will in this context satisfy AD plus a proper class of Woodin cardinals. If there is a proper class of cardinals with some stronger large cardinal property in $V$, say a proper class of extendible cardinals for concreteness, then $V(\mathbb R^*)$ will have a proper class of extendible cardinals by standard lifting arguments; but the $\Omega$ conjecture is not known to be consistent with the existence of a proper class of extendible cardinals (or even a proper class of measurable Woodin cardinals).

In summary, if ZFC plus the $\Omega$ conjecture is consistent with a proper class of $X$ cardinals, then the Axiom of Determinacy is consistent with a proper class of $X$ cardinals, for any $X\in \{\text{Woodin, Extendible, Superduperhuge, etc}\}$. Moreover the $\Omega$ conjecture is consistent with a proper class of Woodin cardinals assuming the existence of a proper class of Woodin cardinals is consistent.

An unpublished theorem of Woodin (which appears without proof as Theorem 7.35 in "In search of Ultimate L") states that if the $\Omega$ conjecture holds and there is a proper class of Woodin cardinals, then there is a partially ordered set $\mathbb P$ such that if $G\subseteq \mathbb P$ is a $V$-generic filter and $\mathbb R^* = (\mathbb R)^{V[G]}$, then $V(\mathbb R^*)$ is a model of AD. The hypotheses of this theorem are known to be consistent: in fact they hold in any model with a proper class of Woodins that satisfies Woodin's weakly homogeneous iteration hypothesis (WHIH), and this includes many of the current fine structure models by Steel's theorem that such models satisfy UBH; see Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Definition 10.4 and Theorem 10.151. The model $V(\mathbb R^*)$ will in this context satisfy that there is a proper class of Woodin cardinals. If there is a proper class of cardinals with some stronger large cardinal property in $V$, say a proper class of extendible cardinals for concreteness, then $V(\mathbb R^*)$ will have a proper class of extendible cardinals by standard lifting arguments; but the $\Omega$ conjecture is not known to be consistent with the existence of a proper class of extendible cardinals (or even a proper class of measurable Woodin cardinals).

An unpublished theorem of Woodin (which appears without proof as Theorem 7.35 in "In search of Ultimate L") states that if the $\Omega$ conjecture holds and there is a proper class of Woodin cardinals, then there is a partially ordered set $\mathbb P$ such that if $G\subseteq \mathbb P$ is a $V$-generic filter and $\mathbb R^* = (\mathbb R)^{V[G]}$, then $V(\mathbb R^*)$ is a model of AD. The hypotheses of this theorem are known to be consistent: in fact they hold in any model with a proper class of Woodins that satisfies Woodin's weakly homogeneous iteration hypothesis (WHIH), and this includes many of the current fine structure models by Steel's theorem that such models satisfy UBH; see Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Definition 10.4 and Theorem 10.151. The model $V(\mathbb R^*)$ will in this context satisfy AD plus a proper class of Woodin cardinals. If there is a proper class of cardinals with some stronger large cardinal property in $V$, say a proper class of extendible cardinals for concreteness, then $V(\mathbb R^*)$ will have a proper class of extendible cardinals by standard lifting arguments; but the $\Omega$ conjecture is not known to be consistent with the existence of a proper class of extendible cardinals (or even a proper class of measurable Woodin cardinals).

In summary, if ZFC plus the $\Omega$ conjecture is consistent with a proper class of $X$ cardinals, then the Axiom of Determinacy is consistent with a proper class of $X$ cardinals, for any $X\in \{\text{Woodin, Extendible, Superduperhuge, etc}\}$. Moreover the $\Omega$ conjecture is consistent with a proper class of Woodin cardinals assuming the existence of a proper class of Woodin cardinals is consistent.