Proof That $L(\mathbb {R})\models {\mathsf {KP}}$ is a theorem of ${\mathsf {KP}}$ . Let $\Phi $ be a $\Pi _1$ monotone operator in $L(\mathbb {R})$ , say, given by a $\Pi _1$ formula $\phi $ and a set a. Consider the operator $\Phi '$ given by

$$ \begin{align*}X\mapsto \{x\in\mathbb{R}:L(\mathbb{R})\models \phi(x,X,a)\},\end{align*} $$

i.e., by

$$ \begin{align*}X\mapsto \{x\in\mathbb{R}:\forall \beta\in {\mathsf{Ord}}\, L_\beta(\mathbb{R})\models \phi(x,X,a)\}.\end{align*} $$

This is an operator of the form

$$ \begin{align*}X\mapsto \{x\in\mathbb{R}: \phi'(x,X,a,\mathbb{R})\}\end{align*} $$

for some $\Pi _1$ formula $\phi '$ and is monotone, so the sequence $\{(\Phi ')^\alpha :\alpha \leq |\Phi '|\}$ exists. Moreover, the operator $\Phi '$ belongs to $L(\mathbb {R})$ and has the same least fixed point as $\Phi $ . Since all ordinals belong to $L(\mathbb {R})$ and $L(\mathbb {R})\models {\mathsf {KP}}$ , $\{(\Phi ')^\alpha :\alpha \leq |\Phi '|\}$ belongs to $L(\mathbb {R})$ , so $\Phi ^\infty \in L(\mathbb {R})$ , as desired.⊣

Proof To see that (2) implies (1), it suffices to show that $L_\beta (\mathbb {R})$ satisfies ${\mathsf {KP}}$ . First, by a reflection argument, we see that $L_\beta (\mathbb {R})$ satisfies “ $\mathcal {P}(\mathbb {R})$ does not exist,” so in $L_\beta (\mathbb {R})$ every set is a surjective image of $\mathbb {R}$ . Then, we see that $L_\beta (\mathbb {R})$ satisfies (boldface) $\Sigma _1$ -Separation for sets of reals and thus for arbitrary sets. $\Sigma _1$ -Separation is true because it is equivalent to $\Pi _1$ -Separation, and this follows from $\Pi _1{-}\mathsf {MI}_{\mathbb {R}}$ (instances of separation are inductions of length $1$ ). To prove $\Delta _0$ -Collection, suppose that

$$ \begin{align*}L_\beta(\mathbb{R})\models \forall x\in A\,\exists y\, \phi(x,y,p)\end{align*} $$

for some $\Delta _0$ formula $\phi $ . By $\Sigma _1$ -Separation, there is a transitive set $B \in L_\beta (\mathbb {R})$ such that $A,p \in B$ and

$$ \begin{align*}B\prec_1 L_\beta(\mathbb{R}).\end{align*} $$

(This follows e.g., from the argument of Steel [Reference Steel, Kechris, Löwe and Steel18, Lemma 1.11].) It follows that

$$ \begin{align*}L_\beta(\mathbb{R})\models \forall x\in A\,\exists y\in B\, \phi(x,y,p),\end{align*} $$

so $L_\beta (\mathbb {R})$ satisfies $\Delta _0$ -Collection.

Conversely, suppose there is a transitive model M of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ which contains all reals. By Lemma 2.1,

$$ \begin{align*}L(\mathbb{R})^M\models\Pi_1\text{-}\mathsf{MI}_{\mathbb{R}}.\end{align*} $$

Since M contains all reals (hence all strategies for Gale-Stewart games on $\mathbb {N}$ ), it follows that $L(\mathbb {R})^M = L_{{\mathsf {Ord}}\cap M}(\mathbb {R})$ and that

$$ \begin{align*}L_{{\mathsf{Ord}}\cap M}(\mathbb{R})\models{\mathsf{AD}}.\end{align*} $$

Letting $\beta $ be least such that $L_\beta (\mathbb {R}) \models \Pi _1$ - $\mathsf {MI}_{\mathbb {R}}$ , we have $\beta \leq {\mathsf {Ord}}\cap M$ , so

$$ \begin{align*}L_\beta(\mathbb{R})\models{\mathsf{AD}}.\end{align*} $$

Since $L_\beta (\mathbb {R})$ contains all reals, it satisfies ${\mathsf {DC}}_{\mathbb {R}}$ . By minimality, no $L_\gamma (\mathbb {R})$ is a model of separation, for $\gamma <\beta $ , and, in fact, for all such $\gamma $ , there is a subset of $\mathbb {R}$ not in $L_\gamma (\mathbb {R})$ which is definable over $L_\gamma (\mathbb {R})$ . Thus, for every $\gamma <\beta $ , there is a surjection

$$ \begin{align*}\rho:\mathbb{R}\twoheadrightarrow L_\gamma(\mathbb{R}),\end{align*} $$

which is definable over $L_\gamma (\mathbb {R})$ (see e.g., Steel [Reference Steel, Kechris, Löwe and Steel18]). Hence, every set in $L_\beta (\mathbb {R})$ is coded by a set of reals in $L_\beta (\mathbb {R})$ , so, by ${\mathsf {DC}}_{\mathbb {R}}$ ,

$$ \begin{align*}L_\beta(\mathbb{R})\models{\mathsf{DC}},\end{align*} $$

as desired.⊣

Proof Suppose otherwise for a contradiction and that $\zeta \ := \mathop {\mathrm {sup}}\limits _{n\in \mathbb {N}}\zeta _n < \beta $ . (We shall drop mention of the uniform parameter x.) Note first that for such $\zeta _n$ . $L_{ \zeta _n}\prec _{\Sigma _1} L_{\zeta }$ .

Then, in M, for arbitrarily large $\tilde \zeta < \zeta $ , also with $L_{\tilde \zeta }\prec _{\Sigma _1} L_{\zeta }$ , there are $\tilde s> \zeta $ with $L_{\tilde \zeta }\prec _{\Sigma _2} L_{\tilde s}$ . We note that any such $\tilde s$ must be in the illfounded part of M. Suppose not. By definition of $\zeta $ there will be some $\bar \zeta> \tilde \zeta $ with a $\Sigma _2$ -end-extension to some $\bar s$ where $L_{\bar \zeta }\prec _{\Sigma _2} L_{\bar s}$ and $\bar s$ certainly in the illfounded part of M (just take some sufficiently large $\zeta _m$ ). However now we have two overlapping $\Sigma _2$ -extendible pairs with $\tilde \zeta < \bar \zeta < \tilde s< \bar s$ .

Claim: $L_{\tilde \zeta }\prec _{\Sigma _2} L_{\bar \zeta }\prec _{\Sigma _2} L_{\bar s}$ .

Proof We only need to justify the first substructure relation. This is an exercise. (Any $\Sigma _2$ statement about $\vec x$ from $L_{\tilde \zeta }$ true in $L_{\bar \zeta }$ upwardly persists in being true in $L_\tau $ for any $\tau \in [\bar \zeta, \bar s]$ and so in particular is true in $L_{\tilde s}$ . By downward $\Sigma _2$ -reflection it is true in $L_{\tilde \zeta }$ .)⊣

Proof By a Barwise Compactness argument, let M be any illfounded model with wellfounded part equal to $L_{\bar \zeta }$ . By the double $\Sigma _2$ -end-extension configuration in the last claim we can repeatedly use $\Sigma _2$ -reflection and have that $L_{\tilde \zeta }$ has arbitrarily large $\Sigma _2$ -end-extensions $L_\tau $ with $\tau <\bar \zeta $ . By overspill it has such end-extensions $L_t$ for $t\in On^M$ with t in the illfounded part of M. However then it is easy to see that $\bar \zeta $ has an infinite nesting supported in M.⊣

Proof Assume towards a contradiction that ${\text {cof}}\,(\alpha ) = \omega $ . Since $\alpha <\Theta ^M$ , for every M-ordinal $\beta $ with $\alpha <\beta <\Theta ^M$ , there is a surjection

$$ \begin{align*}f:\mathbb{R}\twoheadrightarrow (L_\beta(\mathbb{R}))^M\end{align*} $$

in M. Choose such a $\beta $ and such an f. Letting

$$ \begin{align*}\{\alpha_n:n\in\mathbb{N}\}\end{align*} $$

be cofinal in $\alpha $ and $n\in \mathbb {N}$ , there is $x_n\in \mathbb {R}$ such that $f(x_n) = \alpha _n$ . Because $\mathbb {R}^M = \mathbb {R}$ , and the Axiom of Dependent Choices holds in V, there is some $x\in \mathbb {R}$ coding each $x_n$ . By $\Delta _0$ -collection, the image of $\{x_n:n\in \mathbb {N}\}$ under f, i.e., the sequence

$$ \begin{align*}\{\alpha_n:n \in\mathbb{N}\},\end{align*} $$

belongs to M, which is a contradiction.⊣

Proof Let $\gamma \leq \alpha _g$ be admissible, but $\Sigma _1$ -projectible to $\mathbb {R}$ Footnote ¹ and let $A\subset \mathbb {R}$ be $\Pi _1$ -definable over $L_\gamma (\mathbb {R})$ . Given $a \in \mathbb {R}$ , we define a $\boldsymbol \Sigma ^0_3$ game $G(a,A)$ (uniformly in a) on $\mathbb {R}$ with the property that Player II has a winning strategy in it if, and only if, $a \in A$ . Since the winning condition for Player II is $\boldsymbol \Pi ^0_3$ , this implies that all $\Pi _1^{L_{\alpha _g}(\mathbb {R})}$ subsets of $\mathbb {R}$ are $\Game ^{\mathbb {R}}\boldsymbol \Pi ^0_3$ . Since $\boldsymbol \Sigma ^0_3$ games on reals are determined, the dual class of $\Game ^{\mathbb {R}}\boldsymbol \Pi ^0_3$ is $\Game ^{\mathbb {R}}\boldsymbol \Sigma ^0_3$ , so this in turn implies that all $\Sigma _1^{L_{\alpha _g}(\mathbb {R})}$ subsets of $\mathbb {R}$ are $\Game ^{\mathbb {R}}\boldsymbol \Sigma ^0_3$ , which yields the result. This game we define combines the $\Sigma ^0_3$ games from [Reference Welch19] with the games on reals of Martin-Steel [Reference Martin, Steel, Kechris, Löwe and Steel15]. The new feature in this game is the appearance of the generic g––this seems unavoidable, by Lemma 3.5.

Let $\varphi $ be the formula defining A over $L_\gamma (\mathbb {R})$ from a parameter $x_A \in \mathbb {R}$ and let $\varphi _\gamma $ be a $\Sigma _1$ formula with one free variable such that for some real $x_\gamma $ , $\varphi _\gamma (x_\gamma )$ holds in $L_\gamma (\mathbb {R})$ but in no proper initial segment of $L_\gamma (\mathbb {R})$ .

Let $\mathcal {L}$ be the language of set theory with an additional collection of constants $\{\dot x_i:i\in \mathbb {N}\}$ and let $\{\psi _i:i\in \mathbb {N}\}$ enumerate all formulae in this language in such a way that $\psi _i$ contains only constants among $\dot x_0, \dot x_1, \ldots, \dot x_i$ . We also fix a ternary formula $\theta (\cdot,\cdot,\cdot )$ such that in any model of ${\mathsf {KP}} + V = L(\mathbb {R})$ , $\theta $ defines the graph of an $\mathbb {R}$ -parametrized family of wellorders the union of whose fields include all of V. More precisely, we assume that, provably in ${\mathsf {KP}} + V = L(\mathbb {R})$ ,

• • $\theta $ holds of a triple $(x,\alpha, a)$ only if $x\in \mathbb {R}$ and $\alpha $ is an ordinal and

• • for every set a there is a real x and an ordinal $\alpha $ such that $\theta (x,\alpha,a)$ .

Such a formula is found easily e.g., by appealing to Steel [Reference Steel, Kechris, Löwe and Steel18, Section 1]. Finally, we fix two recursive injections $n(\cdot )$ and $m(\cdot )$ from $\{\psi _i:i\in \mathbb {N}\}$ into the set of odd integers and whose ranges are disjoint.

In the game $G(A,x)$ , Players I and II together build a structure in the language of set theory containing certain real numbers and a partial function

$$ \begin{align*}f: \mathbb{N}\times\mathbb{N} \to \{\psi_i:i\in\mathbb{N}\}.\end{align*} $$

More specifically,

1. (1) During turn $2i+1$ , Player II plays a real number $x_{2i+1}$ and either “accepts” or “rejects” the formula $\psi _i$ .

2. (2) During turn $2i$ , Player I plays a real number $x_{2i}$ . In addition, Player I may play a pair $(j,k)$ and select a formula $\psi $ as the value of $f(j,k)$ , but only if the following conditions are met:

   1. (a) Player II has previously accepted the formula “the unique x satisfying $\psi $ is an ordinal” and

   2. (b) either $k = 0$ , or $f(j,k-1)$ has been defined previously, and, moreover, Player II has previously accepted the formula “the unique x satisfying $\psi $ is smaller than the unique x satisfying $f(j,k-1)$ .”

Let T be the collection of all formulae accepted by Player II. The rules of the game are:

1. (3) During the course of the game, Player II must play a, $x_\gamma $ , and $x_A$ . If so, let $i_a$ , $i_\gamma $ , and $i_A$ be such that $x_{i_a} = a$ , $x_{i_\gamma } = x_\gamma $ , and $x_{i_A} = x_A$ .

2. (4) T must be a complete, consistent theory containing the axioms:

   1. (a) ${\mathsf {KP}} + V = L(\mathbb {R})$ ;

   2. (b) $\dot x_i \in \mathbb {R}$ , for each $i\in \mathbb {R}$ ;

   3. (c) $\dot x_i(m) = k$ , but only if $x_i(m) = k$ ;

   4. (d) $\varphi _\gamma (\dot x_{i_\gamma })$ + “no proper initial segment of me satisfies $\varphi _\gamma (\dot x_{i_\gamma })$ ”; and

   5. (e) the Skolem axioms

      $$ \begin{align*}\exists x\in\mathbb{R}\, \chi(x) \to \chi(\dot x_{n(\chi)}),\end{align*} $$
      as well as
      $$ \begin{align*}\exists x\, \chi(x) \to \exists x\, \exists \alpha\in{\mathsf{Ord}}\, \Big(\theta(\dot x_{m(\chi)},\alpha, x) \wedge \chi(x)\Big), \end{align*} $$
      for every formula $\chi $ ;

   6. (f) $\varphi (\dot x_{i_a}, \dot x_{i_A})$ .

   Otherwise, Player II loses the game.

3. (5) If there is $j\in \mathbb {N}$ such that for every $k\in \mathbb {N}$ , the value of $f(j,k)$ is specified by Player I at some point in the game, then she wins; otherwise, Player II wins.

This is a game in which Player II attempts to construct a model that looks like $L_\gamma (\mathbb {R})$ and incorporates the reals specified by Player I. Because we demand that the theory contain the Skolem axioms, it will have a minimal model containing all reals played during the course of the game (and none other). We shall denote by $M_p$ the model associated to a run p of the game. Condition (3) states that, during this game, Player I simultaneously carries out infinitely many attempts to find an infinite descending sequence in Player II’s model, and wins if one of those attempts is successful.

Clearly this is a $\boldsymbol \Sigma ^0_3$ game on $\mathbb {R}$ for Player I (so a $\boldsymbol \Pi ^0_3$ game for Player II). We need to show that Player II has a winning strategy in the game if, and only if, $a \in A$ i.e., if, and only if,

(3.1) $$ \begin{align} L_\gamma(\mathbb{R})\models\varphi(a). \end{align} $$

If (3.1) in fact holds, then Player II has a simple winning strategy obtained by playing the theory of $L_\gamma (\mathbb {R})$ and making sure that if x is a real played during the course of the game, then every real definable from x in $L_\gamma (\mathbb {R})$ is also played at some point in the game. The point is that any winning strategy for Player II requires that she actually play the theory of $L_\gamma (\mathbb {R})$ ; we verify this now.

To see this, let $\sigma $ be a winning strategy for Player II. Suppose that there is a run of the game consistent with $\sigma $ in which Player II accepts a formula not satisfied by $L_\gamma (\mathbb {R})$ . Let X be a countable elementary substructure of some sufficiently large $V_\kappa $ such that $\sigma \in X$ and let

$$ \begin{align*}\pi: W \to X\end{align*} $$

be the collapse embedding. Write $\bar \sigma = \pi ^{-1}(\sigma )$ and $\bar \gamma = \pi ^{-1}(\gamma )$ . Let $p_0$ be a partial play in W which is consistent with $\bar \sigma $ and in which Player II has accepted a formula not satisfied by $L_{\bar \gamma }(\mathbb {R}^W)$ . Observe that $\bar \sigma $ is simply the restriction of $\sigma $ to partial plays belonging to W. Let $h\subset \text {Coll}(\omega,\mathbb {R}^W)$ be W-generic, with $h \in V$ . Thus, $\bar \gamma $ admits no h-nesting in $L[h]$ . (This is because of the elementarity between V and W: $\gamma $ admits no g-nesting in $V[g]$ and this is forced over V by a condition $g_0$ . Without loss of generality, $g_0$ belongs to X and thus to W, and h extends $g_0$ . The existence of a g-nesting over $\gamma $ is easily seen to be $\Sigma ^1_1(g, x_\gamma )$ where $x_\gamma $ is a real coding $\gamma $ , and thus first-order expressible over the next $(g,x_\gamma )$ -admissible.) Working in $L[h]$ , extend $p_0$ by having Player II play according to $\bar \sigma $ and having Player I play recursive reals, and selecting values for f as in the proof of [Reference Welch19, Theorem 4] (see the proof of (4) on pp. 426–428). The values of f are natural numbers, so every move by Player I results in an extension of $p_0$ which belongs to W and, hence, to the domain of $\bar \sigma $ , so that this procedure is well defined. Although it is not immediately clear whether the full play p belongs to W, it certainly belongs to $L[h]$ and thus to V. In addition, all initial segments of p are consistent with $\bar \sigma $ and hence with $\sigma $ , so that p is a play won by Player II. However, the proof of [Reference Welch20, Theorem 2] (appealing to that of [Reference Welch19, Theorem 4]) shows that p is a winning play for Player I unless $\bar \gamma $ admits an h-nesting in $L[h]$ , so we have reached a contradiction, thus proving the claim.

Suppose that $\sigma $ is a winning strategy for Player II in the game. To each $b \in \mathcal {P}_{\omega _1}(\mathbb {R})$ , we shall associate the play $p_b$ of the game that results from Player II playing according to $\sigma $ , and Player I enumerating the reals of b (in any fixed order) and not specifying any values of f. Denote by $M_b$ the minimal model of the theory whose reals are precisely the reals played in the course of $p_b$ (so in particular $b \subset \mathbb {R}^{M_b}$ ). Because $\sigma $ demands that Player II play the theory of $L_\gamma (\mathbb {R})$ , M is wellfounded and thus of the form $L_{\gamma _b}(\mathbb {R}^{M_b})$ . The proof of the subclaim on p. 116 of Martin–Steel [Reference Martin, Steel, Kechris, Löwe and Steel15] shows that if $c\in \mathcal {P}_{\omega _1}(\mathbb {R})$ and $b\subset c$ , then $\mathbb {R}^{M_b} \subset \mathbb {R}^{M_c}$ and there is a (unique) elementary embedding from $L_{\gamma _b}(\mathbb {R}^{M_b})$ into $L_{\gamma _c}(\mathbb {R}^{M_c})$ obtained by mapping each real to itself and each element of $L_{\gamma _b}(\mathbb {R}^{M_b})$ (which is definable from some real $x \in \mathbb {R}^{M_b}$ ) to the element of $L_{\gamma _c}(\mathbb {R}^{M_c})$ that has the same definition. The directed system

$$ \begin{align*}\mathcal{D} = \{L_{\gamma_b}(\mathbb{R}^{M_b}):b\in\mathcal{P}_{\omega_1}(\mathbb{R})\}\end{align*} $$

is countably closed, so its direct limit is wellfounded and hence of the form $L_{\gamma ^*}(\mathbb {R})$ . Because of rule (4d) of the game,

so $\gamma ^* = \gamma $ . By rule (4f), $L_\gamma (\mathbb {R})\models \varphi (a)$ . This completes the proof of the lemma.⊣

Proof Before proving the theorem, let us mention the following lemma which will be used repeatedly without mention:

Proof We prove the part of the lemma which concerns ${\mathsf {KP}}$ . Suppose $g\subset \text {Coll}(\omega,\mathbb {R})$ is $L(\mathbb {R})$ -generic and $L_\alpha [g]\models {\mathsf {KP}}$ . Since $\mathbb {R} \in L_\alpha [g]$ , $L_\alpha (\mathbb {R})\models {\mathsf {KP}}$ . It is verified in the course of the proof of Mathias [Reference Mathias16, Theorem 10.17] that this implies

Now, suppose that g is $L(\mathbb {R})$ -generic and

Since g is $L(\mathbb {R})$ -generic, it decides every formula in the forcing language, in the sense that for every $\varphi $ , there is a condition in g forcing $\varphi $ or $\lnot \varphi $ . By combining Mathias [Reference Mathias16, Theorem 10.11] and Mathias [Reference Mathias16, Proposition 10.12], we see that $L_\alpha [g]\models {\mathsf {KP}}$ .

To obtain the part of the lemma which concerns ${\mathsf {KPi}}$ , we simply apply the first part to every $\mathbb {R}$ -admissible ordinal ${\leq }\ \alpha $ .⊣

We begin with the proof of (1). It is based on the argument from [Reference Aguilera and Müller4] for showing that the $\Game ^{\mathbb {R}}\Pi ^1_{n+1}$ subsets of $\mathbb {R}$ are all $\Sigma _1$ -definable over the least inner model of ${\mathsf {ZF}}$ with n Woodin cardinals containing the reals. The details are made simpler by the fact that $\Gamma \subset \Delta ^1_1$ , but harder by the fact that we work with fragments of ${\mathsf {ZFC}}$ . We shall use the notation from [Reference Aguilera and Müller4].

Let G be a game of length $\omega $ on $\mathbb {R}$ with payoff in $\boldsymbol \Gamma $ . For simplicity, let us assume the payoff is in $\Gamma $ . Given $A\subset \mathbb {R}$ , denote by $G_A$ the variant of the game G in which each player is required to choose each individual move from the elements of A. If A is countable and $x_A$ is a real coding A, then $G_A$ can be simulated by a game on natural numbers, which we denote by $G(x_A)$ . We first show:

Thus if Player I has a winning strategy in G, then she has one in $G_A$ for almost every A. It follows that if $x_A$ codes A, then Player I has a winning strategy in $G(x_A)$ . By hypothesis, there is one such strategy in $L_{\gamma _{x_A}}[x_A]$ . Let $\psi (x_A)$ be the $\Sigma _1$ formula expressing in ${\mathsf {KPi}}$ that there is a winning strategy for Player I for the game $G(x_A)$ .

Proof Suppose Player I has a winning strategy in G and let W be the transitive collapse of a countable elementary substructure of some large $V_\kappa $ . Let $A = \mathbb {R}^W$ . By the previous claim, Player I has a winning strategy in $G_A$ . Let $h\subset \text {Coll}(\omega,A)$ be W-generic. Thus, h codes A, so

$$ \begin{align*}L_{\gamma_h}[h]\models \psi(h).\end{align*} $$

Now, $\psi $ is a $\Sigma _1$ formula. By a theorem of Mathias [Reference Mathias16], every formula true in $L_{\gamma _h}[h]$ is forced; it follows that

The elementarity between W and V yields that

as desired. Suppose now that Player I does not have a winning strategy in G. Then, by Borel determinacy, Player II has a winning strategy in G. Repeating the above argument, we see that

It follows that we cannot possibly have

$$ \begin{align*}\Vdash^{L_{\gamma_{\mathbb{R}}}(\mathbb{R})}_{\text{Coll}(\omega,\mathbb{R})}\psi(g),\end{align*} $$

for $\psi $ is a $\Sigma _1$ formula, so otherwise by forcing over $L(\mathbb {R})$ we should be able to construct winning strategies for both players in $G(g)$ , which is absurd.

Since $\psi $ is $\Sigma _1$ , this last claim completes the proof of (1).

Let us proceed now to the proof of (2); we sketch it. Let $\gamma = \gamma _{\mathbb {R}}$ . The argument is similar to the one used in [Reference Aguilera and Müller4] to prove Projective Determinacy for games of length $\omega ^2$ . The difference is that we have not shown in general that

$$ \begin{align*}\Sigma_1^{L_{\gamma}(\mathbb{R})}\subset\Game^{\mathbb{R}}\boldsymbol\Gamma,\end{align*} $$

although we conjecture that it is true, under the hypotheses of the theorem.

Suppose that $L_{\gamma }(\mathbb {R})\models {\mathsf {AD}}$ . By the Kechris–Woodin determinacy transfer theorem [Reference Kechris and Woodin13], all subsets of $\mathbb {R}$ which are $\Sigma _1$ -definable over $L_{\gamma }(\mathbb {R})$ are determined.Footnote ² By (1), all sets in $\Game ^{\mathbb {R}}\boldsymbol \Gamma $ are determined. Applying Steel [Reference Steel, Kechris, Löwe and Steel18, Theorem 2.1] to $L_{\gamma }(\mathbb {R})$ , we see that $\Sigma _1^{L_{\gamma }(\mathbb {R})}$ has the scale property. Because $L_{\gamma }(\mathbb {R})$ is admissible by hypothesis, $\Sigma _1^{L_\gamma (\mathbb {R})}$ is closed under real universal quantification, so, by a well-known theorem of Moschovakis (see [Reference Moschovakis17, 4E.7]), every relation in $\Sigma _1^{L_\gamma (\mathbb {R})}$ (hence every relation in $\Game ^{\mathbb {R}}\boldsymbol \Gamma $ ) can be uniformized by a function in $\Sigma _1^{L_\gamma (\mathbb {R})} $ .

Now, the determinacy of $\boldsymbol \Gamma $ games of length $\omega ^2$ follows by the following general fact:

Let $\Lambda = \boldsymbol \Gamma $ and $\Pi = \Sigma _1^{L_\gamma (\mathbb {R})}$ . Since $\gamma $ is $\mathbb {R}$ -admissible, $\Sigma _1^{L_\gamma (\mathbb {R})}$ is closed under real quantifiers and countable unions. Thus, the hypotheses of the lemma are satisfied, so every game of length $\omega ^2$ on $\mathbb {N}$ with payoff in $\boldsymbol \Gamma $ is determined. This completes the proof of the theorem.⊣

Proof Putting together Theorems 2.1, 3.11, and 4.2 of Hachtman [Reference Hachtman9], we see that $\beta $ is least such that

$$ \begin{align*}L_\beta[g]\models\boldsymbol\Pi^1_2\text{-}\mathsf{MI}_{\mathbb{N}}.\end{align*} $$

Since $\boldsymbol \Pi ^1_2$ is the same as $\Pi _1$ over $H(\omega _1)$ (with parameters), provably in ${\mathsf {KPi}}$ , $\beta $ is least such that

$$ \begin{align*}L_\beta[g]\models\Pi_1\text{-}\mathsf{MI}_{\mathbb{N}}.\end{align*} $$

Using the fact that in $L_\beta [g]$ there is a bijection between $\mathbb {N}$ and $\mathbb {R}^V$ , it is easy to show, by an argument like the one of Lemma 2.1, that

$$ \begin{align*}L_{\beta}(\mathbb{R})\models\Pi_1\text{-}\mathsf{MI}_{\mathbb{R}}.\end{align*} $$

To prove the converse, let $\eta $ be least such that

$$ \begin{align*}L_{\eta}(\mathbb{R})\models\Pi_1\text{-}\mathsf{MI}_{\mathbb{R}}.\end{align*} $$

We show that

$$ \begin{align*}L_\eta[g]\models \Pi_1\text{-}\mathsf{MI}_{\mathbb{N}}.\end{align*} $$

Thus, let $\Phi $ be a $\Pi _1$ operator in $L_\eta [g]$ , say, given by

$$ \begin{align*}x\mapsto \{n\in\mathbb{N}: \phi(n,x,a)\}\end{align*} $$

for some set $a\in L_\eta [g]$ . Working in $L_\eta (\mathbb {R})$ , let $\tau $ be a $\text {Coll}(\omega,\mathbb {R})$ -name for a and let $p_0$ be a condition forcing that $\Phi $ is monotone. We consider the following operator $\Psi $ that inductively constructs a name for a set of natural numbers:

$$ \begin{align*}X\mapsto \{(p,n): p \leq p_0\, \&\, p\Vdash \phi(n,X[g],\tau[g])\}.\end{align*} $$

Since $\phi $ is $\Pi _1$ , $\Psi $ is a $\Pi _1$ operator using $\tau $ , $p_0$ and $\text {Coll}(\omega,\mathbb {R})$ as parameters. It is monotone, for if $X\subset Y$ , and $(p,n)\in \Psi (X)$ , then $p \leq p_0$ and $p\Vdash \phi (n,X[g],\tau [g])$ . But then p forces that $\phi $ is monotone, so that $p\Vdash \phi (n,Y[g],\tau [g])$ ; thus, $(p,n) \in Y$ . It follows that $\Psi $ has a least fixed point, $\Psi ^\infty $ . Since $\Psi ^\infty \in L_\eta [g]$ , $L_\eta [g]$ can compute $\Psi ^\infty [g]$ and it is easy to see that

$$ \begin{align*}\Psi^\infty[g] = \Phi^\infty,\end{align*} $$

so $L_\eta [g]$ satisfies $\Pi _1$ - $\mathsf {MI}_{\mathbb {N}}$ , as claimed.⊣

Proof Let $g\subset \text {Coll}(\omega,\mathbb {R})$ be $L(\mathbb {R})$ -generic. By Corollary 2.2 and Lemma 5.1, it suffices to prove that the following are equivalent:

1. (1) $\boldsymbol \Sigma ^0_3$ games of length $\omega ^2$ with moves in $\mathbb {N}$ are determined and

2. (2) $L_{\beta _g}(\mathbb {R})\models {\mathsf {AD}}$ , where $\beta _g$ is computed in $L[g]$ .

Since

$$ \begin{align*}L_{\alpha_g}[g]\prec_{\Sigma_1}L_{\beta_g}[g],\end{align*} $$

it follows that

$$ \begin{align*}L_{\alpha_g}(\mathbb{R})\prec_{\Sigma_1}L_{\beta_g}(\mathbb{R}),\end{align*} $$

and thus $L_{\beta _g}(\mathbb {R}) \models {\mathsf {AD}}$ if, and only if, $L_{\alpha _g}(\mathbb {R}) \models {\mathsf {AD}}$ . If $\boldsymbol \Sigma ^0_3$ games of length $\omega ^2$ are determined, then $\Game ^{\mathbb {R}}\boldsymbol \Sigma ^0_3$ games of length $\omega $ are determined, so $L_{\beta _g}(\mathbb {R})\models {\mathsf {AD}}$ by Lemma 3.6.

Conversely, suppose $L_{\beta _g}(\mathbb {R})\models {\mathsf {AD}}$ . We apply Theorem 4.1 with $\Gamma = \Sigma ^0_3$ . $\boldsymbol \Sigma ^0_3$ has the scale property by a theorem of Kechris [Reference Kechris12]. Using a universal $\Pi ^1_2$ formula, there is a first-order formula expressing $\boldsymbol \Pi ^1_2$ -MI over every model of ${\mathsf {KP}}$ , and we choose this as the formula $\phi $ in the statement of Theorem 4.1. All $ \Sigma ^0_3(x)$ games for which Player I has a winning strategy have winning strategies in $L_{\alpha _x+1}[x]$ by [Reference Welch19], and in particular in $L_{\gamma _x}[x]$ , since $\gamma _x = \beta _x$ by Hachtman [Reference Hachtman9].

By Lemma 5.1, $\beta _g$ does not depend on g, so it is least such that

$$ \begin{align*} \Vdash^{L(\mathbb{R} )}_{\text{Coll}(\omega,\mathbb{R})} L_{\beta_g}[g]\models \boldsymbol\Pi^1_2\text{-MI}.\end{align*} $$

We have verified all hypotheses, so all $\boldsymbol \Sigma ^0_3$ games of length $\omega ^2$ on $\mathbb {N}$ are determined by Theorem 4.1.⊣