Fact 2.2. Let $\delta \leq \lambda $ be ordinals such that $\delta $ is a cardinal. Then $|[\lambda ]^{<\delta }| = |\mathscr {P}_\delta (\lambda )| = |{}^{<\delta }\lambda |$ .

Proof. Let $\Phi : [\lambda ]^{<\delta } \rightarrow \mathscr {P}_\delta (\lambda )$ be defined by $\Phi (f) = \mathrm {rang}(f)$ . $\Phi $ is a bijection.

Let $\pi : \lambda \times \lambda \rightarrow \lambda $ be a bijection. For $f \in {}^{<\delta }\lambda $ , let $G_f = \{\pi (\alpha ,\beta ) : \alpha \in \mathrm {dom}(f) \wedge f(\alpha ) = \beta \}$ . Note that since $\mathrm {dom}(f) \in \delta $ and $\delta $ is a cardinal, $|G_f| < \delta $ . Thus $G_f \in \mathscr {P}_{\delta }(\lambda )$ . Define $\Psi : {}^{<\delta }\lambda \rightarrow \mathscr {P}_\delta (\lambda )$ by $\Psi (f) = G_f$ . $\Psi $ is an injection. The previous paragraph showed that there is a bijection of $\mathscr {P}_\delta (\lambda )$ into $[\lambda ]^{<\delta }$ and $[\lambda ]^{<\delta } \subseteq {}^{<\delta }\lambda $ . Thus there is an injection $\Psi : \mathscr {P}_\delta (\lambda ) \rightarrow {}^{<\delta }\lambda $ . By the Cantor–Schröder–Bernstein theorem, $|{}^{<\delta }\lambda | = |\mathscr {P}_\delta (\lambda )| = |[\lambda ]^{<\delta }|$ .

Fact 2.3. If $\delta \leq \lambda $ are ordinals and $\lambda $ is indecomposable, then $|IB(\delta ,\lambda )| = |[\lambda ]^\delta |$ .

Proof. For $f \in IB(\delta ,\lambda )$ , define $\Phi (f) \in [\lambda ]^\delta $ by recursion as follows. Suppose for all $\beta < \delta $ , $\Phi (f) \upharpoonright \beta $ has been defined and for all $\alpha < \beta $ , $\Phi (f)(\alpha ) \leq \sup (f \upharpoonright \alpha + 1) \cdot (\alpha + 1) < \lambda $ . Then $\sup (\Phi (f) \upharpoonright \beta ) \leq \sup (f \upharpoonright \beta ) \cdot \beta < \lambda $ since $\sup (f \upharpoonright \beta ) < \lambda $ and $\lambda $ is indecomposable. Let $\Phi (f)(\beta ) = \sup (\Phi (f) \upharpoonright \beta ) + f(\beta ),$ which is less than $\lambda $ since $\lambda $ is indecomposable. Then $\Phi (f)(\beta ) = \sup (\Phi (f) \upharpoonright \beta ) + f(\beta ) \leq \sup (f \upharpoonright \beta ) \cdot \beta + f(\beta ) \leq \sup (f \upharpoonright \beta + 1) \cdot (\beta + 1) < \lambda $ since $\lambda $ is indecomposable.

This defines $\Phi : IB(\delta ,\lambda ) \rightarrow [\lambda ]^\delta $ . Note that for all $\alpha < \delta $ , $f(\alpha )$ is the unique ordinal $\gamma $ so that $\Phi (f)(\alpha ) = \sup (\Phi (f) \upharpoonright \alpha ) + \gamma $ . Thus $\Phi $ is an injection. Thus $|IB(\delta ,\lambda )| \leq |[\lambda ]^\delta |$ . Since $[\lambda ]^\delta \subseteq IB(\delta ,\lambda )$ , $|[\lambda ]^\delta | \leq |IB(\delta ,\delta )|$ . By the Cantor–Schröder–Bernstein, $|[\lambda ]^\delta | = |IB(\delta ,\lambda )|$ .

Fact 2.4. Let $\delta \leq \lambda $ be ordinals such that $\lambda $ is indecomposable and $\delta \leq {\mathrm {cof}}(\lambda )$ . Then $|{}^\delta \lambda | = |[\lambda ]^\delta |$ .

Proof. Suppose $\delta \leq {\mathrm {cof}}(\lambda )$ . For all $f \in {}^\delta \lambda $ and $\alpha < \delta $ , $\sup (f \upharpoonright \alpha ) < \lambda $ . Thus ${}^\delta \lambda \subseteq B(\delta ,\lambda )$ . Thus $|{}^\delta \lambda | = |B(\delta ,\lambda )| = |[\lambda ]^\delta |$ by Fact 2.3.

Fact 2.5. Let $\delta \leq \lambda $ be ordinals such that $\lambda $ is indecomposable, ${\mathrm {cof}}(\delta ) = {\mathrm {cof}}(\lambda )$ , and $\delta < {\mathrm {cof}}(\lambda )^+$ . Then $|{}^\delta \lambda | = |[\lambda ]^\delta |$ .

Proof. Note that $|{}^\delta \lambda | = |{}^{{\mathrm {cof}}(\lambda )}\lambda |$ since $|\delta | = |{\mathrm {cof}}(\delta )|$ . By Fact 2.4, $|{}^{{\mathrm {cof}}(\lambda )}\lambda | = |[\lambda ]^{{\mathrm {cof}}(\lambda )}|$ . Thus $|{}^\delta \lambda | = |[\lambda ]^{{\mathrm {cof}}(\lambda )}|$ . Thus it suffices to produce an injection of $[\lambda ]^{{\mathrm {cof}}(\lambda )}$ into $[\lambda ]^\delta $ . Let $\rho : {\mathrm {cof}}(\lambda ) \rightarrow \delta $ . Since $\lambda $ is indecomposable, $\delta \cdot \lambda = \lambda $ . For each $\alpha < \lambda $ , let $\iota (\alpha )$ be the least $\beta < {\mathrm {cof}}(\lambda )$ so that $\alpha \leq \rho (\beta )$ . For $f \in [\lambda ]^{{\mathrm {cof}}(\lambda )}$ , let $\Phi (f) : \delta \rightarrow \lambda $ be defined by $\Phi (f)(\alpha ) = \delta \cdot f(\iota (\alpha )) + \alpha $ . One can check that for all $f \in [\lambda ]^{{\mathrm {cof}}(\lambda )}$ , $\Phi (f) \in [\lambda ]^\delta $ and $\Phi : [\lambda ]^{{\mathrm {cof}}(\lambda )} \rightarrow [\lambda ]^\delta $ is an injection.

Fact 2.6. If $\kappa $ is a measurable cardinal (has a $\kappa $ -complete nonprincipal ultrafilter on $\kappa $ ), then for all $\delta < \kappa $ , there is no injection of $\kappa $ into ${\mathscr {P}(\delta )}$ .

Proof. Suppose $\Phi : \kappa \rightarrow {\mathscr {P}(\delta )}$ is a function. Let $\mu $ be a $\kappa $ -complete nonprincipal ultrafilter on $\kappa $ . For each $\alpha < \delta $ and $i \in \{0,1\}$ , let $A^i_\alpha = \{\beta < \kappa : \Phi (\beta )(\alpha ) = i\}$ (where elements of ${\mathscr {P}(\delta )}$ are identified with elements of ${}^\delta 2$ ). For each $\alpha < \delta $ , let $i_\alpha $ be the unique $i \in \{0,1\}$ so that $A_\alpha ^{i_\alpha } \in \mu $ . Since $\mu $ is $\kappa $ -complete, $\bigcap _{\alpha < \delta } A_\alpha ^{i_\alpha } \in \mu $ . Let $f \in {}^\delta 2$ be defined by $f(\alpha ) = i_\alpha $ . Since $\mu $ is nonprincipal, let $\alpha _1 < \alpha _2 < \delta $ so that $\alpha _1,\alpha _2 \in \bigcap _{\alpha < \delta } A_\alpha ^{i_\alpha }$ . $\Phi (\alpha _1) = f = \Phi (\alpha _2)$ . Thus $\Phi $ is not an injection.

Fact 2.7. Assume $\mathsf {AD}$ . Boldface $\mathsf {GCH}$ holds at $\omega $ and $\omega _1$ .

Fact 2.8. Suppose $\lambda $ is cardinal and $\lambda $ does not inject into ${\mathscr {P}(\kappa )}$ for any $\kappa < \lambda $ . Then $\neg (|[\lambda ]^{{\mathrm {cof}}(\lambda )}| \leq |\bigcup _{\delta \leq \kappa < \lambda } [\kappa ]^\delta |)$ .

Proof. Suppose there is an injection $\Phi : [\lambda ]^{{\mathrm {cof}}(\lambda )} \rightarrow \bigcup _{\delta \leq \kappa < \lambda }[\kappa ]^\delta $ . Let $\tilde \Phi \subseteq [\lambda ]^{{\mathrm {cof}}(\lambda )} \times \lambda \times \lambda $ be defined by $(f,\alpha ,\beta ) \in \tilde \Phi $ if and only if $\alpha \in \mathrm {dom}(\Phi (f))$ and $\Phi (f)(\alpha ) = \beta $ . $L[\tilde \Phi ] \models \mathsf {ZFC}$ . In $L[\tilde \Phi ]$ , define $\Psi : [\lambda ]^{{\mathrm {cof}}(\lambda )} \rightarrow \bigcup _{\delta \leq \kappa < \lambda } [\kappa ]^\delta $ by $\Psi (f)(\alpha ) = \beta $ if and only if $\tilde \Phi (f,\alpha ,\beta )$ . Note $\Psi \in L[\tilde \Phi ]$ and $L[\tilde \Phi ] \models \Psi : [\lambda ]^{{\mathrm {cof}}(\lambda )} \rightarrow \bigcup _{\delta \leq \kappa < \lambda } [\kappa ]^\delta $ is an injection. If there are $\delta \leq \kappa < \lambda $ so that $L[\tilde \Phi ] \models \lambda \leq |[\kappa ]^\delta |$ , then there is an injection of $\lambda $ into $[\kappa ]^\delta \subseteq {\mathscr {P}(\kappa )}$ in the real world. This contradicts the assumption that $\lambda $ does not inject into ${\mathscr {P}(\kappa )}$ for any $\kappa < \lambda $ . Thus $L[\tilde \Phi ] \models |\bigcup _{\delta \leq \kappa < \lambda } [\kappa ]^\delta | = \lambda $ . By a theorem of $\mathsf {ZFC}$ , $L[\tilde \Phi ] \models |[\lambda ]^{{\mathrm {cof}}(\lambda )}| \geq \lambda ^+$ . It is impossible that $L[\tilde \Phi ] \models \Psi : [\lambda ]^{{\mathrm {cof}}(\lambda )} \rightarrow \bigcup _{\delta \leq \kappa < \lambda } [\kappa ]^\delta $ is an injection.

Fact 2.9. Suppose $\kappa $ is a regular cardinal and there is no injection of $\kappa $ into ${\mathscr {P}(\delta )}$ for any $\delta < \kappa $ . Then $|[\kappa ]^{<\kappa }| < |{\mathscr {P}(\kappa )}|$ .

Proof. It is clear that $|[\kappa ]^{<\kappa }| \leq |{\mathscr {P}(\kappa )}|$ . Since $\kappa $ is regular, $[\kappa ]^{<\kappa } = \bigcup _{\delta \leq \mu < \kappa } [\mu ]^\delta $ . By Fact 2.8, $\neg (|{\mathscr {P}(\kappa )}| = |[\kappa ]^\kappa | \leq |\bigcup _{\delta \leq \mu < \kappa } [\mu ]^\delta ]| = [\kappa ]^{<\kappa })$ .

Fact 2.10. Assume $\mathsf {AD}$ . $|[\omega _2]^{<\omega _2}| < |{\mathscr {P}(\omega _2)}|$ .

Proof. This follows from Facts 2.7 and 2.9.

Fact 2.11. Let $\delta \leq \lambda $ be ordinals such that ${\mathrm {cof}}(\lambda ) < {\mathrm {cof}}(\delta )$ and $\lambda $ does not inject into ${\mathscr {P}(\kappa )}$ for all $\kappa < \lambda $ . Then $|[\lambda ]^\delta | < |{}^\delta \lambda |$ .

Proof. It is clear that $[\lambda ]^\delta \subseteq {}^\delta \lambda $ . Since ${\mathrm {cof}}(\delta ) \neq {\mathrm {cof}}(\lambda )$ , $[\lambda ]^{\delta } = \bigcup _{\kappa < \lambda }[\kappa ]^\delta \subseteq \bigcup _{\mu \leq \kappa < \lambda }[\kappa ]^\mu $ . Define $\Psi : [\lambda ]^{{\mathrm {cof}}(\lambda )} \rightarrow {}^\delta \lambda $ by

$$ \begin{align*}\Psi(f)(\alpha) = \begin{cases} f(\alpha) & \quad \alpha < {\mathrm{cof}}(\lambda) \\ 0 & \quad {\mathrm{cof}}(\lambda) < \alpha \end{cases}.\end{align*} $$

$\Psi $ is an injection. Thus if there was an injection of ${}^{\delta }\lambda $ into $|[\lambda ]^\delta |$ , then there would be an injection of $[\lambda ]^{{\mathrm {cof}}(\lambda )}$ into $\bigcup _{\mu \leq \kappa < \lambda } [\kappa ]^\mu $ , which contradicts Fact 2.8.

Fact 2.13.

• • [Reference Chan, Jackson and Trang9] $(\mathsf {AD}) [{\omega _1}]^{<{\omega _1}}$ does not inject into ${}^\omega (\omega _\omega )$ .

• • [Reference Chan, Jackson and Trang9] $(\mathsf {AD} + \mathsf {DC}_{\mathbb {R}})$ $[{\omega _1}]^{<{\omega _1}}$ does not inject into ${}^\omega \mathrm {ON}$ , the class of $\omega $ -sequences of ordinals.

• • [Reference Chan, Jackson and Trang10] More generally, if $\kappa \rightarrow (\kappa )^{<\kappa }_2$ ( $\kappa $ is a weak partition cardinal), then $[\kappa ]^{<\kappa }$ does not inject into ${}^\lambda \mathrm {ON}$ , for all $\lambda < \kappa $ .

Fact 2.14. Assume $\mathsf {AD}$ . $|[\omega _2]^{\omega _1}| < |[\omega _2]^{<\omega _2}|$ .

Proof. Under $\mathsf {AD}$ , Martin showed that $\omega _2$ is a weak partition cardinal (that is, satisfies $\omega _2 \rightarrow _* (\omega _2)^{<\omega _2}_2$ ). The result follows from the third point in Fact 2.13.

Fact 3.4. $\mathfrak {G} : {}^\omega {\omega _1} \rightarrow {\mathrm {WO}}$ and for all $\alpha < {\omega _1}$ , if $f \in \mathsf {surj}_\alpha $ , then $\mathfrak {G}(f) \in {\mathrm {WO}}_\alpha $ .

Proof. Note that $(\mathrm {field}(\mathfrak {G}(f)), \mathcal {R}_{\mathfrak {G}(f)}) = (A_f, \mathcal {R}_{\mathfrak {G}(f)})$ is order isomorphic to $(f[A_f],<),$ where $<$ is the usual ordering on ${\omega _1}$ . Thus $\mathfrak {G}(f)$ does indeed belong to ${\mathrm {WO}}$ . Also if $f \in \mathsf {surj}_\alpha $ , then $f[A_f] = \alpha $ and thus $\mathfrak {G}(f) \in {\mathrm {WO}}_\alpha $ .

Fact 3.6. (Moschovakis coding lemma) Assume $\mathsf {AD}$ . Let $\Gamma $ be a pointclass closed under $\exists ^{\mathbb {R}}$ , $\wedge $ , and continuous preimages. Let $(P,\preceq )$ be a prewellordering in $\Gamma $ . Let $\kappa $ be the length of $(P,\preceq )$ and $\varphi : P \rightarrow \kappa $ be the associated surjective norm. If $R \subseteq P \times \mathbb {R}$ , then there is an $S \in \Gamma $ with the following property:

• • $S \subseteq R.$

• • For all $\alpha < \kappa $ , there exists a $p \in P$ and $x \in \mathbb {R}$ so that $\varphi (p) = \alpha $ and $R(p,x)$ if and only if there exists a $p \in P$ and $x \in \mathbb {R}$ so that $\varphi (p) = \alpha $ and $S(p,x)$ .

Fact 3.7. If $\kappa $ is a surjective image of $\mathbb {R}$ (i.e., $\kappa < \Theta $ ), then $\mathbb {R}$ surjects onto ${\mathscr {P}(\kappa )}$ .

Proof. Assume the setting of (1) that for all $n \in \omega $ , $|A_n| \leq |{}^{<\delta }\lambda |$ . Let $R \subseteq \omega \times \mathbb {R}$ be defined by $R(n,r)$ if and only if $T_{\Xi ^{-1}_r[Z]}$ is the graph of an injection of $A_n$ into ${}^{<\delta }\lambda $ . (Recall that $\Xi ^{-1}_r[Z]$ is the subset of $\mathbb {R}$ Lipchitz reducible to Z via the Lipschitz continuous function $\Xi _r$ and $T_{\Xi ^{-1}_r[Z]}$ was defined before the statement of Theorem 3.8.) By $\mathsf {AC}^{\mathbb {R}}_\omega $ , there is a sequence $\langle r_n : n \in \omega \rangle $ so that for all $n \in \omega $ , $R(n, r_n)$ . Thus for all $n \in \omega $ , $T_{\Xi _{r_n}^{-1}[Z]}$ is the graph of an injection $A_n$ into ${}^{<\delta }\lambda $ . Let $\Phi _n : A_n \rightarrow {}^{<\delta }\lambda $ be the injection whose graph is $T_{\Xi _{r_n}^{-1}[Z]}$ . For each $x \in \bigcup _{n \in \omega } A_n$ , let $\iota (x)$ be the least n so that $x \in A_n$ . Since $\omega \leq \lambda $ , let $\varsigma : \omega \times \lambda \rightarrow \lambda $ be a bijection. Define $\Phi : \bigcup _{n \in \omega } A_n \rightarrow {}^{<\delta }\lambda $ by letting $\Phi (x) \in [\lambda ]^{|\Phi _{\iota (x)}(x)|}$ be defined by $\Phi (x)(\gamma ) = \varsigma (\iota (x),\Phi _{\iota (x)}(x)(\gamma ))$ . Suppose $x \neq y$ . If $\iota (x) \neq \iota (y)$ , then $\Phi (x) \neq \Phi (y)$ since $\varsigma $ is a bijection. If $\iota (x) = \iota (y)$ with common value $n \in \omega $ , then $\Phi _n(x) \neq \Phi _n(y)$ since $\Phi _n$ is an injection. Then again $\Phi (x) \neq \Phi (y)$ since $\varsigma $ is an injection. This establishes that $\Phi $ is an injection.

In the setting of (2), in which for all $n \in \omega $ , $|A_n| \leq |{}^\delta \lambda |$ , the proof is essentially the same.

In the setting of (3), in which for all $n \in \omega $ , $|A_n| \leq |[\lambda ]^\delta |$ , observe that the bijection $\varsigma : \omega \times \lambda \rightarrow \lambda $ may be chosen with the property that for all $n \in \omega $ and $\alpha < \beta < \lambda $ , $\varsigma (n,\alpha ) < \varsigma (n,\beta )$ . (For instance, $\varsigma $ derived from the Gödel pairing function would have such property.) Then the resulting function $\Phi (x)$ defined as above would belong to $[\lambda ]^\delta $ .

Proof. Assume the setting of $(1)$ that for all $\alpha < {\omega _1}$ , $|A_\alpha | \leq |{}^{<\delta }\lambda |,$ where ${{\mathrm {cof}}(\delta ) \geq {\omega _1}}$ . Since $|{}^{<\delta } \lambda \setminus \{\emptyset \}| = |{}^{<\delta }\lambda |$ , injections from $A_\alpha $ into ${}^{<\delta }\lambda \setminus \{\emptyset \}$ will be considered to simplify notation.

Let ${\mathrm {WO}} \subseteq \mathbb {R}$ be the $\Pi _1^1$ set of reals coding wellorderings and ${\mathrm {ot}} : {\mathrm {WO}} \rightarrow {\omega _1}$ be the associated surjective norm given by the order type function. Define ${R \subseteq {\mathrm {WO}} \times \mathbb {R}}$ by $R(w,r)$ if and only if $T_{\Xi _r^{-1}[Z]}$ is the graph of an injection of $A_{{\mathrm {ot}}(w)}$ into ${{}^{<\delta }\lambda \setminus \{\emptyset \}}$ . $({\mathrm {WO}},{\mathrm {ot}})$ is a prewellordering which belongs to the pointclass $\boldsymbol {\Sigma }_2^1$ which is closed under continuous preimage, $\wedge $ , and $\exists ^{\mathbb {R}}$ . By the Moschovakis coding lemma (Fact 3.6), there is a $\boldsymbol {\Sigma }_2^1$ set $S \subseteq R$ so that for all $\alpha < {\omega _1}$ , there is a $w \in {\mathrm {WO}}_\alpha $ and $r \in \mathbb {R}$ so that $S(w,r)$ . Let $\leq _{\Pi _1^1} \in \Pi _1^1$ and $\leq _{\Sigma _1^1} \in \Sigma _1^1$ be the two norm relations which witness that $({\mathrm {WO}},{\mathrm {ot}})$ is a $\Pi _1^1$ -norm. Let $\tilde S(w,r)$ if and only if $w \in {\mathrm {WO}} \wedge (\exists v)(v \leq _{\Sigma _1^1} w \wedge w \leq _{\Sigma _1^1} v \wedge S(v,r))$ . $\tilde S \in \boldsymbol {\Sigma }_2^1$ and $\mathrm {dom}(\tilde S) = {\mathrm {WO}}$ . Since $\boldsymbol {\Sigma }_2^1$ has the scale property, let $\Lambda : {\mathrm {WO}} \rightarrow \mathbb {R}$ be a uniformization with the property that for all $w \in {\mathrm {WO}}$ , $\tilde S(w,\Lambda (w))$ . Thus for all $w \in {\mathrm {WO}}$ , $R(w,\Lambda (w))$ . For all $w \in {\mathrm {WO}}$ , $T_{\Xi ^{-1}_{\Lambda (w)}[Z]}$ is the graph of an injection of $A_{{\mathrm {ot}}(w)}$ into ${}^{<\delta }\lambda \setminus \{\emptyset \}$ . For each $w \in {\mathrm {WO}}$ , let $\Phi _w : A_{{\mathrm {ot}}(w)} \rightarrow {}^{<\delta }\lambda \setminus \{\emptyset \}$ be the injection whose graph is $T_{\Xi _{\Lambda (w)}^{-1}[Z]}$ .

For each $x \in \bigcup _{\alpha < {\omega _1}} A_\alpha $ , let $\iota (x)$ be the least $\alpha < {\omega _1}$ so that $x \in A_\alpha $ . Note that $|{}^{<\omega }{\omega _1}| = |{\omega _1}|$ . Let $\sigma : {\omega _1} \times {}^{<\omega }{\omega _1} \times \delta \times \lambda \rightarrow \lambda $ be a bijection. Define

$$ \begin{align*}\Upsilon(x) &= \{\sigma(\iota(x),p,\eta,\zeta) : (\exists \epsilon < \delta)(\forall^{*,\iota(x)}_p f)(\epsilon = \mathrm{dom}(\Phi_{\mathfrak{G}(f)}(x))\\& \qquad \wedge \eta < \epsilon \wedge \Phi_{\mathfrak{G}(f)}(x)(\eta) = \zeta)\}.\end{align*} $$

Observe that $\Upsilon (x) \in \mathscr {P}(\lambda )$ .

Fix $x \in \bigcup _{\alpha < {\omega _1}} A_\alpha $ . Let $K_x = \{p \in {}^{<\omega }\iota (x) : (\exists \eta ,\zeta )(\sigma (\iota (x),p,\eta ,\zeta ) \in \Upsilon (x)\}$ . If $p \in K_x$ , then there is a unique $\epsilon <\delta $ so that $(\forall ^{*,\iota (x)}_p f)(\mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) = \epsilon )$ . To see this, suppose $\epsilon , \hat \epsilon < \delta $ are such that $(\forall ^{*,\iota (x)}_p f)(\mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) = \epsilon )$ and $(\forall ^{*,\iota (x)}_p f)(\mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) = \hat \epsilon )$ . Let $A_0 = \{f \in N_p^{\iota (x)} : \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) = \epsilon \}$ and $A_1 = \{f \in N_p^{\iota (x)} : \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) = \hat \epsilon \}$ . $A_0$ and $A_1$ are comeager subsets of $N_p^{\iota (x)}$ . Thus $A_0 \cap A_1 \neq \emptyset $ . Let $h \in A_0 \cap A_1$ . Then $\epsilon = \mathrm {dom}(\Phi _{\mathfrak {G}(h)}(x)) = \hat {\epsilon }$ . Let $\epsilon ^x_p$ be this unique $\epsilon $ associated with x and p. Let $U_{x,p} = \{\eta < \epsilon ^x_p : (\exists \zeta )(\sigma (\iota (x),p,\eta ,\zeta ) \in \Upsilon (x)\}$ . Note that $|U_{x,p}| \leq |\epsilon _p^x|$ . If $\eta \in U_{x,p}$ , there is a unique $\zeta $ such that $\sigma (\iota (x),p,\eta ,\zeta ) \in \Upsilon (x)$ . To see this, suppose $\zeta _1,\zeta _2$ so that $\sigma (\iota (x),p,\eta ,\zeta _1),\sigma (\iota (x),p,\eta ,\zeta _2) \in \Upsilon (x)$ . Then $B_0 = \{f \in N^{\iota (x)}_p : \Phi _{\mathfrak {G}(f)}(x)(\eta ) = \zeta _1\}$ and $B_1 = \{f \in N^{\iota (x)}_p : \Phi _{\mathfrak {G}(f)}(x)(\eta ) = \zeta _2\}$ are comeager in $N^{\iota (x)}_p$ . $B_0 \cap B_1$ is comeager in $N^{\iota (x)}_p$ . Let $h \in B_0 \cap B_1$ . Then $\zeta _1 = \Phi _{\mathfrak {G}(h)}(x)(\eta ) = \zeta _2$ . Let $\zeta ^x_{p,\eta }$ be this unique $\zeta $ . Thus $\Upsilon (x) = \{\sigma (\iota (x),p,\eta , \zeta ^x_{p,\eta }) : p \in K_x \wedge \eta \in U_{x,p}\}$ . Thus $|\Upsilon (x)| \leq |\bigcup _{p \in K_x} U_{x,p}| \leq \sup \{|\epsilon ^x_{p}| : p \in K_x\} < \delta $ since $|K_x| \leq |{}^{<\omega }\iota (x)| = \omega $ because $\iota (x) < {\omega _1}$ and ${\mathrm {cof}}(\delta )> \omega $ . Thus $\Upsilon (x)$ has cardinality less than $\delta $ and hence $\Upsilon (x) \in \mathscr {P}_\delta (\lambda )$ . It has been shown that $\Upsilon : \bigcup _{\alpha < {\omega _1}} A_\alpha \rightarrow \mathscr {P}_\delta (\lambda )$ .

Next, one will show that for all $x \in \bigcup _{\alpha < {\omega _1}} A_\alpha $ , $\Upsilon (x) \neq \emptyset $ . Let $\alpha = \iota (x)$ . Let $E_1 : \mathsf {surj}_\alpha \rightarrow \delta $ be defined by $E_1(f) = \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x))$ . Since well-ordered unions of meager subsets of ${}^\omega \alpha $ is a meager subset of ${}^\omega \alpha $ and $\mathsf {surj}_\alpha $ is a comeager subset of ${}^\omega \alpha $ , there is some $\epsilon < \delta $ so that $E_1^{-1}[\{\epsilon \}]$ is nonmeager. Let $E_2 : E_1^{-1}[\{\epsilon \}] \rightarrow \lambda $ be defined by $E_2(f) = \Phi _{\mathfrak {G}(f)}(x)(0)$ . Again since $E_1^{-1}[\{\epsilon \}]$ is nonmeager and well-ordered unions of meager sets are meager, there is some $\zeta < \lambda $ so that $E_2^{-1}[\{\zeta \}]$ is nonmeager. By the Baire property, there is a $p \in {}^{<\omega }\alpha $ so that $E_2^{-1}[\{\zeta \}]$ is comeager in $N_p^\alpha $ . Then $\sigma (\alpha ,p,0,\zeta ) \in \Upsilon (x)$ . $\Upsilon (x) \neq \emptyset $ .

Next, it will be shown that $\Upsilon $ is an injection. Suppose $x \neq y$ . First, suppose $\iota (x) \neq \iota (y)$ . Above, it was shown that $\Upsilon (x) \neq \emptyset $ . Let $\sigma (\iota (x),p,\eta ,\zeta ) \in \Upsilon (x)$ . Since $\sigma $ is an injection and all elements of $\Upsilon (y)$ take the form $\sigma (\iota (y),\hat {p},\hat {\eta },\hat {\zeta })$ , $\Upsilon (x) \neq \Upsilon (y)$ . Next, suppose that $\iota (x) = \iota (y)$ and denote this common ordinal by $\alpha $ . Let $D = \{f \in \mathsf {surj}_\alpha : \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) \neq \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(y))\}$ . First suppose D is nonmeager. Consider $\varpi : D \rightarrow \delta \times \delta $ by $\varpi (f) = (\mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)), \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(y)))$ . Since a well-ordered union of meager sets is meager and D is not meager, there is some $\epsilon _1,\epsilon _2 < \delta $ so that $\varpi ^{-1}[\{(\epsilon _1,\epsilon _2)\}]$ is nonmeager. Without loss of generality, suppose $\epsilon _1 < \epsilon _2$ . Define $\varsigma : \varpi ^{-1}[\{(\epsilon _1,\epsilon _2)\}] \rightarrow \lambda $ by $\varsigma (f) = \Phi _{\mathfrak {G}(f)}(y)(\epsilon _1)$ . Since $\varpi ^{-1}[\{(\epsilon _1,\epsilon _2)\}]$ is nonmeager and well-ordered union of meager sets is meager, there is a $\zeta \in \lambda $ so that $\varsigma ^{-1}[\{\zeta \}]$ is nonmeager. By the Baire property, let $p \in {}^{<\omega }\alpha $ be such that $\varsigma ^{-1}[\{\zeta \}]$ is comeager in $N^{\alpha }_p$ . Then $\sigma (\alpha ,p,\epsilon _1,\zeta ) \in \Upsilon (y)$ . However, $\sigma (\alpha ,p,\epsilon _1,\zeta ) \notin \Upsilon (x)$ since $(\forall ^{*,\alpha }_p f)(\mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) = \epsilon _1)$ . In this case, $\Upsilon (x) \neq \Upsilon (y)$ . Finally, suppose ${}^\omega \alpha \setminus D$ is comeager. Let $\Sigma : {}^\omega \alpha \setminus D \rightarrow \delta $ be defined by $\Sigma (f) = \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(x)) = \mathrm {dom}(\Phi _{\mathfrak {G}(f)}(y))$ . Since ${}^\omega \alpha \setminus D$ is comeager, there is some $\epsilon < \delta $ so that $\Sigma ^{-1}[\{\epsilon \}]$ is nonmeager. Note that since $\Phi _{\mathfrak {G}(f)}$ is an injection for all $f \in \mathsf {surj}(\alpha )$ (because it is the injection whose graph is $T_{\Xi ^{-1}_{\Lambda (\mathfrak {G}(f))}}[Z]$ ), $\Phi _{\mathfrak {G}(f)}(x) \neq \Phi _{\mathfrak {G}(f)}(y)$ . Define $\Pi : \Sigma ^{-1}[\{\epsilon \}] \rightarrow \epsilon $ be defined by $\Pi (f)$ is the least $\eta < \epsilon $ so that $\Phi _{\mathfrak {G}(f)}(x)(\eta ) \neq \Phi _{\mathfrak {G}(f)}(y)(\eta )$ . Since $\Sigma ^{-1}[\{\epsilon \}]$ is nonmeager, there is an $\eta < \epsilon $ so that $\Pi ^{-1}[\{\eta \}]$ is nonmeager. Let $\Gamma : \Pi ^{-1}[\{\eta \}] \rightarrow \lambda \times \lambda $ be defined by $\Gamma (f) = (\Phi _{\mathfrak {G}(f)}(x)(\eta ), \Phi _{\mathfrak {G}(f)}(y)(\eta ))$ . Since $\Pi ^{-1}[\{\eta \}]$ is nonmeager, there are $\zeta _1, \zeta _2 \in \lambda $ with $\zeta _1 \neq \zeta _2$ so that $\Gamma ^{-1}[\{(\zeta _1,\zeta _2)\}]$ is nonmeager. Since all subsets of ${}^\omega \alpha $ have the Baire property, there is a $p \in {}^{<\omega }\alpha $ so that $\Gamma ^{-1}[\{(\zeta _1,\zeta _2)\}] $ is comeager in $N^{\alpha }_p$ . Then $\sigma (\alpha ,p,\eta ,\zeta _1) \in \Upsilon (x)$ and $\sigma (\alpha ,p,\eta ,\zeta _1) \notin \Upsilon (y)$ . Thus $\Upsilon (x) \neq \Upsilon (y)$ . It has been shown that $\Upsilon : \bigcup _{\alpha < {\omega _1}} A_\alpha \rightarrow \mathscr {P}_\delta (\lambda )$ is an injection. Fact 2.2 shows $|{}^{<\delta }\lambda | = |\mathscr {P}_\delta (\lambda )|$ .

Next assume the setting of (2). The following will sketch the necessary modifications. By the same argument as above, for each $w \in {\mathrm {WO}}$ , there is an injection $\Phi _w : A_{{\mathrm {ot}}(w)} \rightarrow {}^\delta \lambda $ . Let

$$ \begin{align*}K_x = \{(p,\eta) : p \in {}^{<\omega}\iota(x) \wedge \eta < \delta \wedge (\exists \zeta < \lambda)(\forall^{*,\iota(x)}_p f)(\Phi_{\mathfrak{G}(f)}(x)(\eta) = \zeta)\}.\end{align*} $$

For each $(p,\eta ) \in K_x$ , by the argument provided above, there is a unique $\zeta $ so that $(\forall ^{*,\iota (x)}_p f)(\Phi _{\mathfrak {G}(f)}(x)(\eta ) = \zeta )$ . Thus for each $(p,\eta ) \in K_x$ , let $\zeta ^x_{p,\eta }$ be this unique $\zeta $ . Note that $K_x \subseteq {}^{<\omega }\iota (x) \times \delta \subseteq {}^{<\omega } {\omega _1} \times \delta $ . Let $\tau : {}^{<\omega }{\omega _1} \times \delta \rightarrow \delta $ be a bijection. Let $\mu : {\omega _1} \times \lambda \rightarrow \lambda $ be a bijection. Define $\Upsilon : X \rightarrow {}^\delta \lambda $ by

$$ \begin{align*}\Upsilon(x)(\alpha) = \begin{cases} \mu(\iota(x),0) & \quad \tau^{-1}(\alpha) \notin K_x \\ \mu(\iota(x),\zeta^x_{p,\eta}) & \quad \tau^{-1}(\alpha) \in K_x \wedge \tau^{-1}(\alpha) = (p,\eta) \end{cases}.\end{align*} $$

Finally, one will to show $\Upsilon $ is an injection. Suppose $x,y \in \bigcup _{\alpha < {\omega _1}} A_\alpha $ and $x \neq y$ . If $\iota (x) \neq \iota (y)$ , then $\Upsilon (x) \neq \Upsilon (y)$ since $\mu $ is a bijection. Now suppose $\iota (x) = \iota (y)$ and let $\alpha $ denote this common ordinal. For all $f \in \mathsf {surj}_\alpha $ , $\Phi _{\mathfrak {G}(f)}(x) \neq \Phi _{\mathfrak {G}(f)}(y)$ . Let $\Sigma : \mathsf {surj}_\alpha \rightarrow \delta $ be defined by $\Sigma (f)$ is the least $\eta < \delta $ so that $\Phi _{\mathfrak {G}(f)}(x)(\eta ) \neq \Phi _{\mathfrak {G}(f)}(y)(\eta )$ . Since $\mathsf {surj}_\alpha $ is comeager in ${}^\omega \alpha $ and well-ordered unions of meager sets are meager, there is an $\eta < \delta $ so that $\Sigma ^{-1}[\{\eta \}]$ is nonmeager. Let $\Pi : \Sigma ^{-1}[\{\eta \}] \rightarrow \lambda \times \lambda $ be defined by $\Pi (f) = (\Phi _{\mathfrak {G}(f)}(x)(\eta ), \Phi _{\mathfrak {G}(f)}(y)(\eta ))$ . Since $\Sigma ^{-1}[\{\eta \}]$ is nonmeager, there is some $\zeta _1,\zeta _2 < \lambda $ so that $\zeta _1 \neq \zeta _2$ and $\Pi ^{-1}[\{(\zeta _1,\zeta _2)\}]$ is nonmeager. By the Baire property, let $p \in {}^{<\omega }\alpha $ so that $\Pi ^{-1}[\{(\zeta _1,\zeta _2)\}]$ is comeager in $N^\alpha _p$ . Let $\beta = \tau (p,\eta )$ . Then $\Upsilon (x)(\beta ) = \mu (\alpha , \zeta _1) \neq \mu (\alpha , \zeta _2) = \Upsilon (y)(\beta )$ . Thus $\Upsilon (x) \neq \Upsilon (y)$ . It has been shown that $\Upsilon $ is an injection.

Assume the setting of (3). Let $K_x$ , $\zeta ^x_{p,\eta }$ , and $\tau : {}^{<\omega }{\omega _1} \times \delta \rightarrow \delta $ be defined as in (2). The bijection $\mu : {\omega _1} \times \lambda \rightarrow \lambda $ can be chosen with the property that for all $\nu < {\omega _1}$ and $\gamma < \lambda $ , $\sup \{\mu (\nu ,\beta ) : \beta < \gamma \} < \lambda $ . Let $\Upsilon $ be defined as above in (2). For $x \in X$ , $\gamma < \delta $ , and $p \in {}^{<\omega }\iota (x)$ , let $P^x_{\gamma ,p} = \{\eta \in \delta : \tau (p,\eta ) < \gamma \wedge \tau (p,\eta ) \in K_x\}$ . For each $p \in {}^{<\omega }\iota (x)$ , let $F^x_{p,\gamma } = \{\zeta _{p,\eta }^x : \eta \in P^x_{\gamma ,p}\}$ . The claim is that $F^x_{p,\gamma }$ is bounded below $\lambda $ . To see this, suppose $F^x_{p,\gamma }$ is not bounded below $\lambda $ . For each $\eta \in P^x_{\gamma ,p}$ , let $Y^x_{p,\gamma ,\eta } = \{f \in N^{\iota (x)}_p : \Phi _{\mathfrak {G}(f)}(x)(\eta ) = \zeta ^x_{p,\eta }\}$ . Each $Y^x_{p,\gamma ,\eta }$ is comeager in $N^{\iota (x)}_p$ . Since well-ordered intersection of comeager subsets of $N^{\iota (x)}_p$ is comeager in $N^{\iota (x)}_p$ , $\bigcap _{\eta \in P^x_{\gamma , p}} Y^x_{p,\gamma ,\eta }$ is comeager in $N^{\iota (x)}_p$ and is in particular nonempty. Let $f \in \bigcap _{\eta \in P^x_{\gamma , p}} Y^x_{p,\gamma ,\eta }$ . Then $\sup (\Phi _{\mathfrak {G}(f)}(x) \upharpoonright \gamma ) \geq \sup \{\zeta ^x_{p,\eta } : \eta \in P^x_{\gamma ,p}\} = \sup (F^x_{p,\gamma }) = \lambda $ . Then since $\gamma < \delta $ , $\Phi _{\mathfrak {G}(f)}(x)(\gamma ) \geq \lambda $ and hence $\Phi _{\mathfrak {G}(f)}(x) \notin [\lambda ]^\delta $ . This is a contradiction. Thus for all $p \in {}^{<\omega }\iota (x)$ , $\sup (F^x_{p,\gamma }) < \lambda $ . Since ${\mathrm {cof}}(\lambda )> \omega $ and $|{}^{<\omega }\iota (x)| = \omega $ , $\sup \{\sup (F^x_{p,\gamma }) : p \in {}^{<\omega }\iota (x)\} < \lambda $ . Note that $\sup (\Upsilon (x) \upharpoonright \gamma ) \leq \sup \{\mu (\iota (x),\zeta ) : \zeta \in \bigcup _{p \in {}^{<\omega }\iota (x)} F^x_{p,\gamma }\} \leq \sup \{\mu (\iota (x),\zeta ) : \zeta < \sup \{\sup (F^x_{p,\gamma }) : p \in {}^{<\omega }\iota (x)\}\} < \lambda $ (by the property of chosen bijection $\mu $ ). This shows that $\Upsilon : \bigcup _{\alpha < {\omega _1}} A_\alpha \rightarrow IB(\delta ,\lambda )$ . $\Upsilon $ is an injection by the same argument as in (2). The result now follows from Fact 2.3.

Proof. Recall that Martin and Monk showed that $\mathsf {AD}$ and $\mathsf {DC}_{\mathbb {R}}$ prove that strict Lipschitz reduction is wellfounded. For each $\alpha < {\omega _1}$ , let $\beta _\alpha $ be the least $\beta $ so that there is some $B \in {\mathscr {P}(\mathbb {R})}$ with $\mathrm {rk}_L(B) = \beta $ and $T_{B}$ is the graph of an injection of $A_\alpha $ into ${}^{<\delta }\lambda $ . Since ${\mathrm {cof}}(\Theta )> {\omega _1}$ , $\sup \{\beta _\alpha : \alpha < {\omega _1}\} < \Theta $ . Let $Z \in {\mathscr {P}(\mathbb {R})}$ so that $\mathrm {rk}_L(Z) = \sup \{\beta _\alpha : \alpha < {\omega _1}\}$ . The result now follows from Theorem 3.9.

Proof. The argument is similar to the proof of Theorem 3.10 using Theorem 3.8.

Fact 3.12. (Woodin) Suppose $\mathsf {AD}^+$ and $V = L({\mathscr {P}(\mathbb {R})})$ . Then either $\mathsf {AD}_{\mathbb {R}}$ holds or there is a set of ordinals J so that $V = L(J,\mathbb {R})$ .

Fact 3.13. If $\mathsf {AD}^+$ , $\neg \mathsf {AD}_{\mathbb {R}}$ , and $V = L({\mathscr {P}(\mathbb {R})})$ , then $\Theta $ is regular.

Proof. By Fact 3.12, there is a set of ordinals J so that $V = L(J,\mathbb {R})$ . All sets in $L(J,\mathbb {R})$ are ordinal definable from J and an $r \in \mathbb {R}$ . For each $r \in \mathbb {R}$ and $\alpha < \Theta $ , if there is an $\mathrm {OD}_{\{J,r\}}$ surjection $\varpi : \mathbb {R} \rightarrow \alpha $ , then let $\varpi _{\alpha ,r} : \mathbb {R} \rightarrow \alpha $ be the least such surjection according to the canonical wellordering of $\mathrm {OD}_{\{J,r\}}$ . For each $\alpha < \Theta $ , let $\pi _\alpha : \mathbb {R} \rightarrow \alpha $ be defined by

$$ \begin{align*}\pi_\alpha(x) = \begin{cases} \varpi_{x^{[0]}}(x^{[1]}) & \quad \text{if there is an } \mathrm{OD}_{\{J,x^{[0]}\}} \text{ surjection of } \mathbb{R} \text{ onto } \alpha \\ 0 & \quad \text{otherwise}. \end{cases}\end{align*} $$

$\pi _\alpha $ is a surjection. Thus a sequence $\langle \pi _\alpha : \alpha < \Theta \rangle $ has been defined so that $\pi _\alpha : \mathbb {R} \rightarrow \alpha $ is a surjection for each $\alpha < \Theta $ . Now suppose ${\mathrm {cof}}(\Theta ) < \Theta $ . Let $\tau : \mathbb {R} \rightarrow {\mathrm {cof}}(\Theta )$ be a surjection. Define $\sigma : \mathbb {R} \rightarrow \Theta $ by $\sigma (x) = \pi _{\tau (x^{[0]})}(x^{[1]})$ . $\sigma $ is a surjection onto $\Theta ,$ which is impossible.

Fact 3.14. (Woodin) Assume $\mathsf {AD}^+$ and $\mathsf {AD}_{\mathbb {R}}$ . All sets of reals are Suslin.

Fact 3.15. (Woodin [Reference Steel and Trang27]) ( $\Sigma _1$ -reflection into Suslin and coSuslin) Assume $\mathsf {AD}^+$ and $V = L({\mathscr {P}(\mathbb {R})})$ . $\mathcal {S} \prec _{\Sigma _1} (V,\in )$ . (That is, $\mathcal {S}$ is a $\Sigma _1$ -elementary substructure of the universe V.)

Proof. Consider the setting of $(1)$ . Let $\varsigma : \mathbb {R} \rightarrow X$ be a surjection. Define an equivalence relation E on $\mathbb {R}$ by $x \ E \ y$ if and only if $\varsigma (x) = \varsigma (y)$ . Note that X is in bijection with . For each $\alpha < {\omega _1}$ , let $K_\alpha = \varsigma ^{-1}[A_\alpha ]$ and $E_\alpha = E \upharpoonright K_\alpha $ . Then and $A_\alpha $ is in bijection with . Injections of $A_\alpha $ into ${}^{<\delta }\lambda $ induce injections of into ${}^{<\delta }\lambda $ . Let be defined by $\pi (x) = [x]_E$ . Let $\varpi : \mathbb {R} \rightarrow {\mathscr {P}(\lambda )}$ be a surjection given by Fact 3.7. Then injections between and $[\lambda ]^{<\delta }$ can be coded by sets of reals through the coding $B \mapsto T_B$ described above. This shows that X and $\langle A_\alpha : \alpha < {\omega _1}\rangle $ with the property stated in setting (1) are in bijection with objects and with the properties in setting (1), which belong to $L({\mathscr {P}(\mathbb {R})})$ . It suffices to prove the theorem in $L({\mathscr {P}(\mathbb {R})})$ .

With this discussion in mind, one will now assume $\mathsf {AD}^+$ , $V = L({\mathscr {P}(\mathbb {R})})$ , and that X and $\langle A_\alpha : \alpha < {\omega _1}\rangle $ belong to $L({\mathscr {P}(\mathbb {R})})$ with the properties stated in (1). If ${\mathrm {cof}}(\Theta )> {\omega _1}$ , then the result follows from Theorem 3.10. Suppose ${\mathrm {cof}}(\Theta ) \leq {\omega _1}$ . Thus $\Theta $ is singular and hence $\mathsf {AD}_{\mathbb {R}}$ holds by Fact 3.13. Assume for the sake of contradiction that there is a set X and a sequence $\langle A_\alpha : \alpha < {\omega _1}\rangle $ satisfying (1) and $\neg (|\bigcup _{\alpha < {\omega _1}} A_\alpha | \leq |{}^{<\delta }\lambda |)$ . Let $Y = \bigcup _{\alpha < {\omega _1}} A_\alpha $ and thus $\neg (|Y| \leq |{}^{<\delta }\lambda |)$ . Since all sets of reals are Suslin and coSuslin by Fact 3.14 since $\mathsf {AD}^+$ and $\mathsf {AD}_{\mathbb {R}}$ hold, the sets Y, $\delta $ , and $\lambda $ are Suslin and coSuslin and hence belong to $\mathcal {S}$ .

Let $\psi $ be the following sentence with $\delta $ , $\lambda $ , and Y as a parameter: $\delta \leq \lambda < \dot \Theta $ and there exists a sequence $\langle \tilde A_\alpha : \alpha < {\omega _1}\rangle $ so that $Y = \bigcup _{\alpha < {\omega _1}} \tilde A_\alpha $ and for all $\alpha < {\omega _1}$ , $|A_\alpha | \leq |{}^{<\delta }\lambda |$ . ( $\dot \Theta $ is an abbreviation for the ordinal defined as the supremum of the ordinals which are surjective images of $\mathbb {R}$ .) Let $\mathfrak {T}$ be some sufficiently strong finite fragment of $\mathsf {ZF}$ . Let $\varphi $ be the following $\Sigma _1$ -sentence with Y, $\delta $ , $\lambda $ , and $\mathbb {R}$ as parameters: There exists a transitive set $M \models \mathfrak {T} + \mathsf {AD}$ so that $\mathbb {R} \subseteq M$ and $M \models \psi $ . Let $\preceq $ be a prewellordering of length $\lambda $ whose associated norm was used to define the surjection $\varpi : \mathbb {R} \rightarrow {\mathscr {P}(\lambda ),}$ which appears in the coding described before Theorem 3.8. Since $L({\mathscr {P}(\mathbb {R})}) \models $ “ $\mathfrak {T}$ , $\mathsf {AD}$ , and $\psi $ ” and using reflection on the hierarchy $\langle L_\alpha ({\mathscr {P}(\mathbb {R})}) : \alpha < \mathrm {ON}\rangle $ , there is an ordinal $\alpha \geq \Theta $ such that $L_\alpha ({\mathscr {P}(\mathbb {R})}) \models $ “ $\mathfrak {T}$ , $\mathsf {AD}$ , and $\psi $ ”. Thus $L({\mathscr {P}(\mathbb {R})}) \models \varphi $ as witnessed by $L_\alpha ({\mathscr {P}(\mathbb {R})})$ . By $\Sigma _1$ -reflection into Suslin and coSuslin (Fact 3.15), $\mathcal {S} \models \varphi $ . Let $M \in \mathcal {S}$ be a transitive set containing $\mathbb {R}$ so that $M \models \psi $ . Let $\langle \tilde A_\alpha : \alpha < {\omega _1}\rangle $ with $Y = \bigcup _{\alpha < {\omega _1}} \tilde A_\alpha $ witness the existential quantifier in $\psi $ . Since for each $\alpha < {\omega _1}$ , $M \models |\tilde A_\alpha | \leq |{}^{<\delta }\lambda |$ , $\mathbb {R} \subseteq M$ , satisfies $\mathsf {AD}$ , and has the prewellordering $\preceq $ used to code injections of subsets of Y into ${}^{<\delta }\lambda $ , there is some $B \in {\mathscr {P}(\mathbb {R})} \cap M$ so that $T_B$ codes the graph of an injection of $\tilde A_\alpha $ into ${}^{<\delta }\lambda $ . Since $M \in \mathcal {S}$ implies M is a surjective image of $\mathbb {R}$ , $\sup \{\mathrm {rk}_L(B) : B \in {\mathscr {P}(\mathbb {R})} \cap M\} < \Theta ^V$ . In the real world, let $Z \in {\mathscr {P}(\mathbb {R})}$ be such that $\mathrm {rk}_L(Z) \geq \sup \{\mathrm {rk}_L(B) : B \in {\mathscr {P}(\mathbb {R})} \cap M\}$ . Note that for all $\alpha < {\omega _1}$ , there is an $r \in \mathbb {R}$ so that $T_{\Xi _r^{-1}[Z]}$ codes the graph of an injection of $\tilde A_\alpha $ into $[\lambda ]^{<\delta }$ . Applying Theorem 3.9 in the real world to $\langle \tilde A_\alpha : \alpha < {\omega _1}\rangle $ , one has that $|Y| = |\bigcup _{\alpha < {\omega _1}} \tilde A_\alpha | \leq |{}^{<\delta }\lambda |$ . This contradicts the assumption that $\neg (|Y| \leq |{}^{<\delta }\lambda |)$ .

Proof. The proof follows the template of the proof of Theorem 3.16 using Theorem 3.8.

Proof. Suppose $\langle A_\alpha : \alpha < {\omega _1}\rangle $ is a sequence such that $[\omega _2]^{<\omega _2} = \bigcup _{\alpha < {\omega _1}} A_\alpha $ . Suppose for the sake of contradiction that for all $\alpha < {\omega _1}$ , $|A_\alpha | \leq |[\omega _2]^{{\omega _1}}|$ . By Theorem 3.16, $|[\omega _2]^{<\omega _2}| \leq |[\omega _2]^{{\omega _1}}|$ which violates Fact 2.14.

Proof. Suppose $\langle A_\alpha : \alpha < {\omega _1}\rangle $ is a sequence such that ${\mathscr {P}(\omega _2)} = \bigcup _{\alpha < {\omega _1}} A_\alpha $ . Suppose for the sake of contradiction that for all $\alpha < {\omega _1}$ , $|A_\alpha | \leq |[\omega _2]^{<\omega _2}|$ . By Theorem 3.16, $|{\mathscr {P}(\omega _2)}| \leq |[\omega _2]^{<\omega _2}|,$ which violates Fact 2.10.

Fact 3.20. [Reference Chan4] Assume $\mathsf {AD}^+$ .

1. (1) (ABCD Conjecture) Let $\alpha $ , $\beta $ , $\gamma $ , and $\delta $ be cardinals such that $\omega \leq \alpha \leq \beta < \Theta $ and $\omega \leq \gamma \leq \delta < \Theta $ . $|{}^\alpha \beta | \leq |{}^\gamma \delta |$ if and only if $\alpha \leq \gamma $ and $\beta \leq \delta $ .

2. (2) If $\kappa < \Theta $ is a cardinal and $\epsilon < \kappa $ , then $|{}^\epsilon \kappa | < |{}^{<\kappa }\kappa |$ .

Proof. (1) If $|A_\alpha | \leq |[\kappa ]^{<\kappa }| = |{}^{<\kappa }\kappa |$ , then $|{\mathscr {P}(\kappa )}| = |{}^{<\kappa }\kappa |$ by Theorem 3.16. Since $\mathsf {AD}^+$ implies boldface $\mathsf {GCH}$ below $\Theta $ , this would contradict Fact 2.9.

(2) If $|A_\alpha | \leq |{}^\epsilon \kappa |$ , then $|{}^{<\kappa }\kappa | = |{}^\epsilon \kappa |$ by Theorem 3.16. This would contradict Fact 3.20.

The proof of (3) is similar.