The Axiom of Determinacy Implies Dependent Choices in Mice

We show that the Axiom of Dependent Choices, $\operatorname{DC}$, holds in countably iterable, passive premice $\mathcal{M}$ construced over their reals which satisfy the Axiom of Determinacy, $\operatorname{AD}$, in a $\operatorname{ZF}+\operatorname{DC}_{\mathbb{R}^{\mathcal{M}}}$ background universe. This generalizes an argument of Kechris for $L(\mathbb{R})$ using Steel's analysis of scales in mice. In particular, we show that for any $n \leq \omega$ and any countable set of reals $A$ so that $M_n(A) \cap \mathbb{R} = A$ and $M_n(A) \vDash \operatorname{AD}$, we have that $M_n(A) \vDash \operatorname{DC}$.


Introduction
We prove that in passive, countably iterable mice M constructed over their reals, AD, the Axiom of Determinacy, implies DC, the Axiom of Dependent Choices, working in a background universe which satisfies ZF + DC R M . Here we write R M = R ∩ M for the set of reals in M.
Recall that DC is the following statement: For every nonempty set X and every binary relation P on X, ∀a ∈ X∃b ∈ X P (a, b) ⇒ ∃f : ω → X ∀n P (f (n), f (n + 1)).
Moreover, DC R denotes DC restricted to the case where X = R and more generally, for some nonempty set Y , DC Y denotes DC restricted to the case where X = Y . Gödel's constructible universe over the reals L(R) is the closure of R under the definable power set operation. Kechris showed in [Ke84] that in L(R), the Axiom of Determinacy implies the Axiom of Dependent Choices. His proof is based on the analysis of scales in L(R) which was developed by Martin, Moschovakis, and Steel (see [MMS82], [Mo08], [MaSt08], and [St08a]). A generalization of [Ke84] and the analysis of scales to the Dodd-Jensen core model over R was shown by Cunningham in [Cu95]. We prove the following more general result for arbitrary mice building on the analysis of scales in mice from [St08b]. Note that, in contrast to Kechris's result for L(R), our result requires DC R M to hold in V in order to consider countable elementary substructures of M. We will make it clear in the proof where the countability of the model in question is used. For countable mice it is not necessary to assume DC R M , see Theorem 2.1. In particular, Theorem 1.1 holds for mice of the form M n (A) for some n ≤ ω and some countable set of reals A such that M n (A) ∩ R = A. This result is for example used in [AgMu], where the authors derive a model with ω + n Woodin cardinals from a model of the form M n (A) with M n (A) ∩ R = A which satisfies the Axiom of Determinacy.
Finally, we would like to thank the referee for the helpful comments and suggestions.

Countable mice in a ZF background universe
For simplicity, we first show the following version of Theorem 1.1 for countable mice and argue in the next section that this implies Theorem 1.1. As mentioned above, we do not require any form of choice in the background universe for this result.
For the definition of premice and (ω 1 + 1)-iterability we refer the reader to [St10], and to [MS94] and [SchStZe02] for more background. Moreover, we refer to [St08b] for the notion of X-premice for arbitrary sets X. First, we recall the notion of iterability we use in the statement of Theorem 1.1.
Definition 2.2. Let A be a set of reals and suppose M is an A-premouse. We say that M is countably iterable iff wheneverM is a countableĀ-premouse for a set of realsĀ and there is an elementary embedding π :M → M, thenM is (ω 1 + 1)-iterable.
To prove Theorem 2.1, we will show that the argument in the proof of [St08b,Theorem 4.1] which yields the existence of scales in M using M DC, can be used to show the existence of quasi-scales without using DC in M. Moreover, we sketch how we can adapt the argument from [Ke84] for V = L(R) to obtain M DC from these quasi-scales.
Following the notation in [St08b], we write K(R) for the model-theoretic union of all ω-sound, countably iterable premice over R which project to R. Using DC R , it is easy to show that any two such premice M and N line up, i.e. satisfy M N or N M. Therefore K(R) is well-defined.
Remark. If we consider premice M(R) constructed over all reals R = R V , e.g. M(R) = K(R) or M(R) = M 1 (R), it is easy to see that DC in V (and in fact, using the argument at the beginning of the proof of Theorem 2.1, even DC R in V ) already implies DC in M(R) as every function f : ω → R witnessing DC in V can be coded by a single real and is therefore already contained in M(R). But the same does not hold in general for models M as in Theorem 1.1 with R M R since if f : ω → R M is a function witnessing DC in V for reals in M for some relation P , it can be coded by a single real in V , but this real need not be in R M .
For the reader's convenience, we repeat parts of the arguments from [Ke84] and [St08b] to point out the modifications we need to make. We start by recalling the notions of quasi-norm and quasi-scale which go back to [Ke84].
(1) ≤ is a linear preordering on B, i.e. ≤ is reflexive, transitive, and for all x, y ∈ B, x ≤ y or y ≤ x, and (2) there is no infinite descending chain in <, where for x, y ∈ B, we write x < y iff x ≤ y and ¬(y ≤ x).
If we replace (2) in Definition 2.3 by "every nonempty subset of B has a ≤-least element", we obtain the usual definitions of norm and scale. Hence, under DC R every quasi-scale is a scale.
Lemma 2.5 (AC ω,R ). Suppose B is a nonempty set of reals and (≤ i ) i is a quasiscale on B. Let Γ be a pointclass containing B such that the relation is in Γ. Moreover, suppose that Γ is closed under recursive substitutions, ¬, ∧, ∨, and existential and universal quantification over R. Then B contains a real x such that {(n, m) ∈ ω × ω : x(n) = m} is in Γ. Now we turn to the proof of Theorem 2.1.
Proof of Theorem 2.1. Work in the countable mouse M and note that it suffices to prove DC R M since there is a definable surjection F : Let ξ < Ord M be a large enough limit ordinal such that P ∈ M|ξ and M|ξ is passive. We may assume that such a limit ordinal exists because the general case when Ord M need not be a limit of limit ordinals can be shown similarly using the S-hierarchy (see the end of the proof of Theorem 2.1 in [St08a]). Let α be the least ordinal below ξ such that M|α ≺ 1 M|ξ (in the sense of Definition 4.4 in [St08b]) and note that α is a limit ordinal. The statement is Σ 1 in the parameter R M as any f : ω → R M can be coded by a real. Therefore it follows that there is a counterexample to DC R M (in M) inside M|α. To finish the proof, we use the following lemma.
Applying Lemma 2.6 to the counterexample P above, we can define a function f : ω → R M by letting f (0) = a ∈ R M be arbitrary and f (n + 1) = F (f (n)). Then P (f (n), f (n + 1)) holds for all n, contradicting the choice of P . So it suffices to prove Lemma 2.6.
Proof of Lemma 2.6. The proof divides into three claims. The first claim uses fine structural arguments to obtain definability for the sets of reals in M|α. The key part of the argument is Claim 2, where we show the existence of quasi-scales. Finally, in Claim 3 we piece Claim 2 and Lemma 2.5 together to obtain a basis result which will imply the existence of a uniformizing function, as desired. Proof. Standard fine structural arguments show that M|α has a Σ 1 Skolem function which is Σ 1 definable in M|α (without parameters). As in the proof of Lemma 1.11 in [St08a] for L(R), this together with the fact the we chose α minimal with the property that M|α ≺ 1 M|ξ yields that there is a partial surjection h : R M ։ M|α such that the graph of h is Σ 1 definable in M|α from parameter R M . Hence, every set of reals in M|α is Σ 1 definable in M|α from parameters in R M ∪ {R M }, as desired.
Claim 2. Let B ⊆ R M be a set of reals which is Σ 1 -definable in M|α from some real parameter r and the parameter R M . Then there is a quasi-scale (≤ i ) i<ω on B which is also Σ 1 -definable in M|α from the parameters r and R M .
Proof. Here we use Steel's analysis of scales in mice (see [St08b]) under DC R and observe that it can be used to obtain a quasi-scale without any use of DC R . In order to show how to do this, we sketch parts of his argument below.
So let B ⊆ R M be a set of reals which is Σ 1 -definable over M|α with some real parameter r and parameter R M . Hence for some Σ 1 formula ϕ, for all x ∈ R M . Recall that α is a limit ordinal. If M|α satisfies "Θ exists", let α * = Θ M|α , otherwise let α * = α. Now write for each β < α * and x ∈ R M , By [St08b, Lemma 3.2], applied inside HOD x,Σ , where x is a real coding M|α and Σ is an iteration strategy for M, we obtain B = β<α * B β . Note that Σ is amenable to HOD x,Σ , so the (canonically well-ordered) fragment Σ ∩ HOD x,Σ is available within the model HOD x,Σ and witnesses iterability there. Moreover, HOD x,Σ is a model of the Axiom of Choice. Steel constructs in the proof of Theorem 4.1 in [St08b] a closed game representation x → G β x of B β for each β < α * . We briefly sketch the argument here to show that it can be done in our situation as well. First, recall the definition of a closed game representation, which was essentially introduced in [Mo08].
In particular, G x is continuously associated to x. We say x → G x is a closed game representation of B iff B is the set of all x such that Player I has a winning quasi-strategy in G x .
We now define a closed game representation x → G β x of B β for each β < α * . Fix β < α * and x. Let G β x be the following game: x 1 x 3 . . . The rules of the game ask Player I to play i 0 , i 1 , · · · ∈ {0, 1} in order to code a theory T in the language L pm ({ẋ i : i ∈ ω}) of premice with additional constant symbols {ẋ i : i ∈ ω} such that every model N * of T is well-founded. Furthermore, the players alternate playing reals x i , i ∈ ω, and Player I plays additional ordinals γ i , i ∈ ω. The theory ensures that for every model N * of T , for all i ∈ ω, (ẋ i ) N * = x i and the definable closure of {x i : i ∈ ω} in N * ↾ L pm is an elementary submodel N of N * ↾ L pm . By considering its transitive collapse we can assume that N is transitive. The winning conditions for Player I require that he plays the theory T such that N "V = K(R) + ϕ(x, r, R) + all of my proper initial segments do not satisfy ϕ(x, r, R)".
In addition, he is using the ordinals γ i to not only verify well-foundedness of N by embedding the ordinals into ωβ, but also to verify iterability of N by embedding the local HOD's of N into the local HOD's of M|β. This latter embedding corresponds to the embedding of the ordinals. This amount of details suffices for our sketch of the argument, the formal definition of G β x can be found in [St08b, Section 4]. Let B β k (x, u) ⇔ u is a position of length k from which Player I has a winning quasi-strategy in G β x . We aim to show that each B β k is in M|α and that the map (β, k) → B β k is Σ 1 definable over M|α with parameters r and R M . In order to do that, we consider honest positions in the game G β x , which are positions where Player I played the theory T up to this point according to the theory of an initial segment M|ξ of the true model M|β and the embeddings induced by the ordinals γ i according to an elementary embedding between the local HOD's of M|ξ and the local HOD's of the true model M|β.
Definition. We say a position u = ((i n , x 2n , γ n , x 2n+1 ) : n < k) in the game G β x is (β, x)-honest iff M|β ϕ(x, r, R) and if ξ ≤ β is least such that For the other implication, let σ be a winning quasi-strategy for Player I from a position u in G β x . Recall that R M is countable in V and consider a complete run ((i n , x 2n , γ n , x 2n+1 ) : n < ω) of G β x according to σ such that {x i : i ∈ ω} = R M . Moreover, consider the canonical model N associated to this run of G β x as above. We need to show that N is an initial segment of M|β.
This part of the proof uses a comparison argument. Recall that the standard proof of the comparison lemma (see for example Theorem 3.11 in [St10]) uses a reflection argument to a small elementary substructure and hence DC. But M|β and hence N is (ω 1 + 1)-iterable in V , so we can perform the comparison in HOD x,Σ,Σ ′ , where x is a real coding M|β and N , and Σ and Σ ′ are iteration strategies for M|β and N respectively. Similar as before, Σ and Σ ′ are amenable to HOD x,Σ,Σ ′ and their (canonically well-ordered) fragments Σ ∩ HOD x,Σ,Σ ′ and Σ ′ ∩ HOD x,Σ,Σ ′ witness iterability in HOD x,Σ,Σ ′ , which is a model of the Axiom of Choice. So there is no further assumption on M needed and we obtain that N is an initial segment of M|β, in fact that N = M|ξ, where ξ is least such that M|ξ ϕ(x, r, R), as in the proof of [St08b, Claim 4.3]. Now let (≤ β i ) i be the quasi-scale on B β constructed from the closed game representation as in 2.6 in [Ke84] using the fake sup, min, and fake inf method. Then (β, i) →≤ β i is Σ 1 definable over M|α with parameters r and R M as well, as desired.
Using Claim 2 together with Lemma 2.5 we can now show the following claim.
Claim 3. Every nonempty set of reals B in M|α which is Σ 1 -definable in M|α from a real parameter r and the parameter R M , contains an element x which is first-order definable in M|α from r and R M .
Proof. We will use Lemma 2.5 to pick an element out of a set of reals B in a definable way using a quasi-scale on B. Recall that AC ω,R M holds in M as a consequence of AD. To obtain Claim 3, apply Lemma 2.5 inside M|α to a nonempty set B ⊆ R M which is Σ 1 -definable in M|α from some real parameter r and the parameter R M , the quasi-scale on B obtained in Claim 2, and the pointclass Γ of all sets which are first-order definable in M|α from the parameters r and R M .
Claim 3 now implies Lemma 2.6. Suppose P is as in Lemma 2.6. By Claim 1 we can in addition assume that P is Σ 1 -definable in M|α from a parameter r ∈ R M and the parameter R M . We can define a uniformizing function F as follows. If for a real x, ¬∃yP (x, y), let F (x) = x. Otherwise, let F (x) be the least (with respect to a fixed enumeration of first-order formulae) real z which is first-order definable from x, r, and R M in M|α such that P (x, z). Then F ∈ M is the desired uniformization.
This finishes the proof of Theorem 2.1.

Uncountable mice with DC R M in the background
In this section we argue that instead of working with countable premice M we can work in a background universe which is a model of DC R M , i.e. we derive Theorem 1.1 as a corollary of Theorem 2.1.
Proof of Theorem 1.1. Let M be a passive, countably iterable R M -premouse such that M AD. Using DC R M in V , we can by the standard proof of the Löwenheim-Skolem Theorem consider a countable elementary substructure N of M. Then N is an (ω 1 + 1)-iterable R N -premouse and we can apply Theorem 2.1 to N . This yields N DC and hence M DC.
Finally, note that the statements in Theorem 1.1 and Theorem 2.1 are in fact equivalent by the following argument. Let M be a countable, passive, (ω 1 + 1)iterable R M -premouse such that M AD. Let Σ be an (ω 1 + 1)-iteration strategy for M and x M be a real coding M. Now apply Theorem 1.1 inside HOD xM,Σ , which is a model of the Axiom of Choice.
Using that for any countable set of reals A the Woodin cardinals in M n (A) are countable in V , we obtain the following corollary.
Corollary 3.1 (ZF). Let n ≤ ω and let A ∈ P ω1 (R). Suppose that M ♯ n (A) exists and is (ω 1 + 1)-iterable. Moreover, suppose that M n (A) ∩ R = A and M n (A) AD. Then M n (A) DC.