Contemporary Mathematics A trichotomy theorem in natural models of AD+ Andr´es Eduardo Caicedo and Richard Ketchersid Abstract. Assume AD+ and that either V = L(P(R)), or V = L(T, R) for some set T ⊂ ORD. Let (X, ≤) be a pre-partially ordered set. Then exactly one of the following cases holds: (1) X can be written as a well-ordered union of pre-chains, or (2) X admits a perfect set of pairwise ≤-incomparable elements, α and the quotient partial order induced by (X, ≤) embeds into (2 , ≤lex) for ω some ordinal α, or (3) there is an embedding of 2 /E0 into (X, ≤) whose range consists of pairwise ≤-incomparable elements. By considering the case where ≤ is the diagonal on X, it follows that for any set X exactly one of the following cases holds: (1) X is well-orderable, or ω (2) X embeds the reals and is linearly orderable, or (3) 2 /E0 embeds into X. In particular, a set is linearly orderable if and only if it embeds into P(α) for some α. Also, ω is the smallest infinite cardinal, and {ω1, R} is a basis for the uncountable cardinals. Assuming the model has the form L(T, R) for some T ⊂ ORD, the result is a consequence of ZF + DCR together with the existence of a fine σ-complete measure on Pω1 (R) via an analysis of Vopˇenka-like forcing. It is known that in the models not covered by this case, ADR holds. The result then requires more of the theory of determinacy; in particular, that V = OD((< Θ)ω), and the existence and uniqueness of supercompactness measures on Pω1 (γ) for γ < Θ. As an application, we show that (under the same basic assumptions) Scheepers’s countable-finite game over a set S is undetermined whenever S is uncountable. Contents 1. Introduction 2 2. Preliminaries 5 3. AD+ 19 4. The dichotomy theorem 21 5. The countable-finite game in natural models of AD+ 27 6. Questions 29 References 30 2010 Mathematics Subject Classification. 03E60, 03E25, 03C20. + Key words and phrases. Determinacy, AD , ADR, ∞-Borel sets, ordinal determinacy, Vopˇenka forcing, Glimm-Effros dichotomy, countable-finite game. The first author wants to thank the National Science Foundation for partial support through grant DMS-0801189. c 2010 American Mathematical Society 1 2 ANDRES´ EDUARDO CAICEDO AND RICHARD KETCHERSID 1. Introduction This paper deals with consequences of the strengthening AD+ of the axiom of determinacy AD for the general theory of sets, not just for sets of reals or sets of sets of reals. Particular versions of our results were known either in L(R) or under the addi- tional assumption of ADR. They can be seen as generalizations of well-known facts in the theory of Borel equivalence relations. We consider “natural” models of AD+, namely, those that satisfy V = L(P(R)), although our results apply to a slightly larger class of models. The special form of V is used in the argument, not just consequences of determinacy. Although an acquaintance with determinacy is certainly desirable, we strive to be reasonably self-contained and expect the paper to be accessible to readers with a working understanding of forcing, and combinatorial and descriptive set theory. We state explicitly all additional results we require, and provide enough background to motivate our assumptions. Jech [15] and Moschovakis [25] are standard sources for notation and definitions. For basic consequences of determinacy, some of which we will use without comment, see Kanamori [16]. 1.1. Results. Our main result can be seen as a simultaneous generalization of the Harrington- Marker-Shelah [10] theorem on Borel orderings, the Dilworth decomposition the- orem of Foreman [7], the Glimm-Effros dichotomy of Harrington-Kechris-Louveau [9], and the dichotomy theorem of Hjorth [13]. Recall that a pre-partial ordering ≤ on a set X (also called a quasi-ordering on X) is a binary relation that is reflexive and transitive, though not necessarily ω anti-symmetric. Recall that E0 is the equivalence relation on 2 defined by xE0y ⇐⇒ ∃n ∀m ≥ n x(m)= y(m). Theorem 1.1. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Let (X, ≤) be a pre-partially ordered set. Then exactly one of the following holds: (1) X is a well-ordered union of ≤-pre-chains. (2) There are perfectly many ≤-incomparable elements of X, and there is an order preserving injection of the quotient partial order induced by X into α (2 , ≤lex) for some ordinal α. ω (3) There are 2 /E0 many ≤-incomparable elements of X. The argument can be seen in a natural way as proving two dichotomy theorems, Theorems 1.2 and 1.3. Theorem 1.2. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Let (X, ≤) be a pre-partially ordered set. Then either: (1) There are perfectly many ≤-incomparable elements of X, or else (2) X is a well-ordered union of ≤-pre-chains. Theorem 1.3. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Let (X, ≤) be a partially ordered set. Then either: ω (1) There are 2 /E0 many ≤-incomparable elements of X, or else α (2) There is an order preserving injection of (X, ≤) into (2 , ≤lex) for some ordinal α. A TRICHOTOMY THEOREM IN NATURAL MODELS OF AD+ 3 ω It is easy to see that R injects into 2 /E0, and it is well-known that, under ω determinacy, ω1 does not inject into R, and 2 /E0 is not linearly orderable and therefore cannot embed into any linearly orderable set. This shows that the cases displayed above are mutually exclusive. Theorem 1.2 generalizes a theorem of Foreman [7] where, among other results, it is shown (in ZF + AD + DCR) that if ≤ is a Suslin/co-Suslin pre-partial ordering of R without perfectly many incomparable elements, then R is a union of λ-many Suslin sets, each pre-linearly-ordered by ≤, where λ is least such that both ≤ and its complement are λ-Suslin. By considering the case ≤= {(x, x) : x ∈ X}, the following corollary, a gener- alization of the theorem of Silver [29] on co-analytic equivalence relations, follows immediately: Theorem 1.4. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Let X be a set. Then either: (1) R embeds into X, or else (2) X is well-orderable. The corollary gives us the following basis result for infinite cardinalities: Corollary 1.5. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Let S be an infinite set. Then: (1) ω embeds into S. (2) If κ is a well-ordered cardinal, and S is strictly larger than κ, then either + κ or κ ∪ R embeds into S. In particular, ω1 and R form a basis for the uncountable cardinals. Note that there are no assumptions in Theorems 1.2–1.4 on the set X. If, in Theorem 1.4, the set X is a quotient of R by, say, a projective equivalence relation, one can give additional information on the length of the well-ordering. This has been investigated by several authors including Harrington-Sami [11], Ditzen [5], Hjorth [12], and Schlicht [28]. Theorems 1.2 and 1.4 were our original results, and we consider Theorem 1.2 the main theorem of this paper. After writing a first version of the paper, we found Hjorth [13], where the version of Theorem 1.4 for L(R) is attributed to Woodin. Hjorth [13] investigates in L(R) what happens when alternative 1 in Theorem 1.4 holds but the quotient R/E0 does not embed into X; much remains to be explored in this area. We remark that the argument of Hjorth [13] easily combines with our techniques, so we in fact have Theorem 1.3, a simultaneous generalization of further results in Foreman [7], and the main result in Hjorth [13]. The following corollary is immediate: Corollary 1.6. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Let X be a set. Then either: ω (1) 2 /E0 embeds into X, or else (2) X embeds into P(α) for some ordinal α. In particular: Corollary 1.7. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Then a set is linearly orderable if and only if it embeds into P(α) for some ordinal α. 4 ANDRES´ EDUARDO CAICEDO AND RICHARD KETCHERSID Since it is slightly easier to follow, we arrange the exposition around the proof of Theorem 1.4, and then explain the easy adjustments to the argument that allow us to obtain Theorem 1.2, and the modifications required to the argument in Hjorth [13] to obtain Theorem 1.3. Weak versions of some of these results were known previously in the context of ADR. It is thanks to the use of ∞-Borel codes in our arguments that we can extend them in the way presented here. As an application of our results, we show: Theorem 1.8. Assume AD+ holds and either V = L(T, R) for some T ⊂ ORD, or else V = L(P(R)). Then the countable-finite game CF (S) is undetermined for all uncountable sets S.