Preserving levels of projective determinacy by tree forcings Fabiana Castiblancoa,∗, Philipp Schlichtb aInstitut f¨ur Mathematische Logik und Grundlagenforschung, Universit¨at M¨unster, Einsteinstraße 62, 48149 M¨unster, Germany bSchool of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK Abstract We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings. Keywords: Forcing preservation theorems, tree forcing, Projective Determinacy, thin relations, sharps, mouse operator, uniform indiscernibles 1. Motivation A. Levy and R. Solovay [28] have shown that if κ is a measurable cardinal and P is a small P 3 forcing notion, i.e. |P| < κ, then κ remains measurable in the generic extension V . Since the existence of compact, supercompact and huge cardinals, among others, are characterized by the existence of certain elementary embeddings related to ultrapowers, variants of the Levy-Solovay 6 argument were performed in that cases showing that small forcing preserves these large cardinal properties as well. Other large cardinal notions are instead characterized by the existence of extender embeddings 9 rather than simple ultrapower embeddings. In this respect, Hamkins and Woodin [17] have shown that if κ is λ-strong then it would be also λ-strong in a generic extension obtained after forcing with a small poset.1 Hence, the strongness and Woodiness of a cardinal are also preserved by 12 small forcing. Many global consequences implied by the existence of large cardinals also are preserved after forcing with certain posets. For instance, the existence of x♯ for each set of ordinals x satisfying 15 sup x ⊂ κ is a known consequence of the existence of a Ramsey cardinal κ [25, Chapter 2, §9]. Moreover, from a Ramsey cardinal κ we obtain closure under sharps for reals in the universe and 1 also, we gain Σ3-absoluteness for small generic extensions. 18 This is closely related to the following well-known result: ♯ arXiv:1810.12079v5 [math.LO] 18 Mar 2021 Fact 1.1. Suppose that for every set of ordinals x, x exists. Let P be a forcing in V and let G be P-generic over V . Then also V [G] |= ∀x (x♯ exists). ♯ 21 If we consider the property that x exists for every real x, this preservation result is no longer true. In fact, R. David (cf. [8]) has shown that in the minimal model closed under sharps for reals ♯ 1 L there is a Σ3-forcing P adding a real with no sharp in the generic extension. 24 Nevertheless, if we restrict the complexity of the forcing, we obtain positive results. Further- # more, Schlicht [36, Lemma 3.11] has proved a more general statement: given n<ω, if Mn (x) ∗Corresponding author Email addresses: [email protected] (Fabiana Castiblanco), [email protected] (Philipp Schlicht) 1In fact, they prove as well that a λ-strong cardinal κ cannot be created via any forcing of cardinality < κ (except possibly in the case of ordinals λ with cof λ ≤|P| +). Preprint submitted to Annals of Pure and Applied Logic March 19, 2021 1 exists and is ω1-iterable for every x ∈ R, then every Σ2 provably c.c.c. forcing preserves the ♯ 27 existence and iterability of Mn(x). Thus, it is natural to ask whether we can extend this result 1 to the wider class of Σ2 proper forcing notions. This paper addresses the preservation problem above and its consequences when we consider 30 Sacks (S), Silver (V), Mathias(M), Miller (ML) and Laver (L) forcing. These forcing notions are 1 proper and their complexity is at most ∆2. We prove that for each natural number n, all the ♯ ω forcing notions in the set T = {S, V, M, L, ML} preserve Mn(x) for every x ∈ ω or equivalently, 1 33 every partial order in T preserves Πn+1-determinacy (cf. Theorem 4.5). As a consequence, from ♯ 1 the existence of Mn(x) for every real x we obtain that Σn+3-P-absoluteness holds for every P ∈ T (cf. Theorem 5.4). This gives a partial answer to [21, Question 7.3]. 36 With these results in hand, we can show that given n ∈ ω, every forcing notion in T does not 1 ♯ add any new orbits to ∆n+3-thin transitive relations if we assume the existence of Mn(x) for every real x (cf. Theorem 6.10). As a motivation to this main result, we show that all the forcing notions 39 in T do not change the value of the second uniform indiscernible u2, which partially answers [21, Question 7.4]. Acknowledgements. This work is part of the Ph.D. thesis of the first author. She would like to 42 thank Ralf Schindler for his permission to include the joint results in Section 6.1 and providing background on inner model theory. The authors would further like to thank the referee for the careful reading and numerous highly useful suggestions. 45 The first author gratefully acknowledges support from the SFB 878 program “Groups, Ge- ometry & Actions” financed by the Deutsche Forschungsgemeinschaft (DFG). This project has received funding from the European Union’s Horizon 2020 research and innovation programme 48 under the Marie Sk lodowska-Curie grant agreement No 794020 (IMIC) of the second author. The second author was partially supported by FWF grant number I4039. 2. Basic notions 51 Let R denote the set of real numbers. As usual, we identify R with the power set ℘(ω) of the set ω of natural numbers, with the Baire space ωω, with the Cantor space ω2 or with the set ↑ωω of strictly increasing functions from ω to ω, depending on the context. 54 We assume familiarity of the reader with the basic facts about forcing. For undefined notions, consult the texts [22] and [37]. For our purposes, a forcing notion P consists of an underlying set P together with a preorder ≤P on P and the induced incompatibility relation ⊥P. In this case, we 57 write P = hP, ≤P, ⊥Pi. We often identify the underlying set P with P itself. 2.1. Arboreal forcing Definition 2.1. Let n ≥ 1 be a natural number. Let M be a transitive model of ZFC and let 1 1 P ∈ M be a forcing notion. We say that M is (1-step) Σn-P-absolute iff for every Σn-formula ϕ and for every real a ∈ M, we have M |= ϕ(a) ⇐⇒ M P |= ϕ(a). P V V 1 1 Π This is equivalent to the expression M ≺Σn M . Similarly, we define (1-step) n-P-absoluteness. Definition 2.2. Let n ≥ 1 be a natural number. Let M be a transitive model of ZFC. We say 1 1 Σ 1 that M is n- correct (in V ) iff M ≺Σn V . In other words, for each Σn-formula ϕ and every real a ∈ M, M |= ϕ(a) ⇐⇒ V |= ϕ(a). 1 60 In a similar way, we define Πn-correctness. 1 ω 1 Definition 2.3. Let n<ω. We say that a forcing notion P = hP, ≤P, ⊥Pi is Σn if P ⊂ ω is Σn 1 ω ω and the order and incompatibility relations ≤P and ⊥P are Σn-subsets of ω × ω. In a similar 1 1 1 1 63 way, we define Πn forcing notions. In addition, we say that P is ∆n if it is both Σn and Πn. 1 Finally, we say that P is projective if and only if P is Σn for some n<ω. 2 1 Definition 2.4. We say that a forcing notion P is Suslin if and only if it is Σ1-definable. Also, 1 66 we say that P is co-Suslin if and only if it is Π1-definable. Definition 2.5. Let P = hP, ≤P, ⊥Pi be a poset definable by a projective formula with parameter ω M M M a ∈ ω. Let M be a transitive model of ZF containing the parameter a. Then, P , P , ≤P and M ⊥P denote the forcing notion P re-interpreted in M. Also, we say that P is absolute for M if M M ≤P = ≤P ∩ M and ⊥P = ⊥P ∩ M. 1 Definition 2.6. Let n ≥ 1 be a natural number. A forcing notion P is called provably ∆n if there 1 1 is a Σn-formula ϕ and a Πn-formula ψ such that ZFC ⊢ “ϕ and ψ define the same triple P = (P, ≤P, ⊥P)” . P Definition 2.7. Let P be a forcing notion. We say that a cardinal λ is sufficiently large if λ> 2| | and we write λ ≫ P. Definition 2.8. Let P be a forcing notion and let λ ≫ P. Let M ≺ H(λ) be an elementary substructure. A condition q ∈ P is (M, P)-generic iff for every dense set D ⊂ P such that D ∈ M, D ∩ M is predense below q. Equivalently, q ∈ P is (M, P)-generic if and only if P ˙ q V “G ∩ P ∩ M is P ∩ M-generic over M”. 69 Definition 2.9. A forcing notion P is proper iff for any λ ≫ P and for any countable elementary substructure M ≺ H(λ) with P ∈ M, every p ∈ P ∩ M has an extension q ≤P p which is (M, P)- generic.