Applications of infinity-Borel codes to definability and definable cardinals
Abstract
Woodin introduced an extension of the axiom of determinacy, $\mathsf {AD}$, called $\mathsf {AD}^+$ which includes an assertion that all sets of reals have an $\infty $-Borel code. An $\infty $-Borel code is a pair $(\varphi ,S)$ where $\varphi $ is a formula and $S$ is a set of ordinals which provides a highly absolute definition for a set of reals. This paper will use $\mathsf {AD}^+$ and $\infty $-Borel codes to establish a property of ordinal definability analogous to a property for $\Sigma _1^1$ shown by Harrington–Shore–Slaman (2017). Under $\mathsf {AD}^+$, the paper will also use $\infty $-Borel codes to explore the cardinality of sets below $\mathscr {P}(\omega _1)$ which Woodin (2006) began investigating under $\mathsf {AD}_\mathbb {R}$ and $\mathsf {DC}$. The following summarizes the main results.
Assume $\mathsf {ZF} + \mathsf {AD}^+ + \mathsf {V {=} L}(\mathscr {P}(\mathbb {R}))$. If $H \subseteq \mathbb {R}$ has the property that there is a nonempty $\mathrm {OD}$ set of reals $K$ such that $H$ is $\mathrm {OD}_z$ for any $z \in K$, then $H$ is $\mathrm {OD}$.
Assume $\mathsf {ZF} + \mathsf {AD}^+ + \neg \mathsf {AD}_\mathbb {R} + \mathsf {V {=} L}(\mathscr {P}(\mathbb {R}))$. Then there is a cardinal strictly between $|[\omega _1]^{ \lt \omega _1}|$ and $|[\omega _1]^{\omega _1}| = |\mathscr {P}(\omega _1)|$.
Assume $\mathsf {ZF} + \mathsf {AD}^+$. Then $S_1 = \{f \in [\omega _1]^{ \lt \omega _1} : \sup (f) = \omega _1^{L[f]}\}$ does not inject into ${}^\omega \mathrm {ON}$, the class of $\omega $-sequences of ordinals. This implies $|\mathbb {R}| \lt |S_1|$ and $|[\omega _1]^\omega | \lt |[\omega _1]^{ \lt \omega _1}|$.
Assume $\mathsf {ZF} + \mathsf {AD}^+$. Let $X$ be a surjective image of $\mathbb {R}$ and let $\mathscr {P}_{\omega _1}(X) = \{A \subseteq X : |A| \lt \omega _1\}$. If $\omega _1 \leq |\mathscr {P}_{\omega _1}(X)|$, then $\omega _1 \leq |X|$. If $|\mathscr {P}(\omega _1)| = |[\omega _1]^{\omega _1}| \leq |\mathscr {P}_{\omega _1}(X)|$, then $|\mathbb {R} \sqcup \omega _1| \leq |X|$.
$\mathsf {ZF} + \mathsf {AD}_\mathbb {R}$ implies that the uncountable cardinals below $|\mathbb {R} \times \omega _1|$ are $\omega _1$, $|\mathbb {R}|$, $|\mathbb {R} \sqcup \omega _1|$, and $|\mathbb {R} \times \omega _1|$. An elaborate structure of cardinals below $|\mathbb {R} \times \omega _1|$ is described under the assumption of $\mathsf {ZF} + \mathsf {AD}^+ + \neg \mathsf {AD}_\mathbb {R} + \mathsf {V {=} L(\mathscr {P}(\mathbb {R}))}$.
