SIGMA-PRIKRY FORCING III: DOWN to ℵ 1. Introduction Many Natural Questions Cannot Be Resolved by the Standard Mathematic

SIGMA-PRIKRY FORCING III: DOWN TO ℵ휔

ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Abstract. We prove the consistency of the failure of the singular cardinals hypothesis at ℵ휔 together with the reflection of all stationary subsets of ℵ휔+1. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.

1. Introduction Many natural questions cannot be resolved by the standard mathematical axioms (ZFC); the most famous example being Hilbert’s first problem, the continuum hypothesis (CH). At the late 1930’s, Gödelconstructed an inner model of set theory [Göd40]in which the generalized continuum hypothesis (GCH) holds, demonstrating, in particular, that CH is consistent with ZFC. Then, in 1963, Cohen invented the method of forcing [Coh63] and used it to prove that ¬CH is, as well, consistent with ZFC. In an advance made by Easton [Eas70], it was shown that any reasonable behavior of the continuum function 휅 ↦→ 2휅 for regular cardinals 휅 may be materialized. In a review on Easton’s paper for AMS Mathematical Reviews, Azriel Lévywrites:

The corresponding question concerning the singular ℵ훼’s is still open, and seems to be one of the most difficult open problems of set theory in the post-Cohen era. It is, e.g., ℵ푛 . unknown whether for all 푛(푛 < 휔 → 2 = ℵ푛+1) implies ℵ휔 2 = ℵ휔+1 or not. A preliminary finding of Bukovský[Buk65] (and independently Hechler) suggested that singular cardinals may indeed behave differently, but it was only around 1975, with Silver’s theorem [Sil75] and the pioneering work of Galvin and Hajnal [GH75], that it became clear that singular cardinals obey much deeper constraints. This lead to the formulation of the singular cardinals hypothesis (SCH) as a (correct) relativization of GCH to singular cardinals, and ultimately to Shelah’s pcf theory [She92, She00]. Shortly after Silver’s discovery, advances in inner model theory due to Jensen (see [DJ75]) provided a covering lemma between Gödel’soriginal model of GCH and many other models of set theory, thus establishing that any consistent

Date: September 1, 2021. 1 2 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA failure of SCH must rely on an extension of ZFC involving large cardinals axioms. Compactness is the phenomenon where if a certain property holds for every strictly smaller substructure of a given object, then it holds for the object itself. Countless results in topology, graph theory, algebra and logic demon- strate that the first infinite cardinal is compact. Large cardinals axioms are compactness postulates for the higher infinite. A crucial tool for connecting large cardinals axioms with singular cardinals was introduced by Prikry in [Pri70]. Then Silver (see [Men76]) constructed a model of ZFC whose extension by Prikry’s forcing gave the first universe of set theory with a singular strong limit cardinal 휅 such that 2휅 > 휅+. Shortly after, Magidor [Mag77a] proved that the same may be achieved at level of the very first singular cardinal, that is, 휅 = ℵ휔. Finally, in 1977, Magidor answered the question from Lévy’sreview in the affirmative: Theorem 1 (Magidor, [Mag77b]). Assuming the consistency of a supercom- ℵ푛 pact cardinal and a huge cardinal above it, it is consistent that 2 = ℵ푛+1 ℵ휔 for all 푛 < 휔, and 2 = ℵ휔+2. Later works of Gitik, Mitchell, and Woodin pinpointed the optimal large cardinal hypothesis required for Magidor’s theorem (see [Git02, Mit10]). Note that Theorem 1 is an incompactness result; the values of the power- set function are small below ℵ휔, and blow up at ℵ휔. In a paper from 1982, Magidor obtained a result of an opposite nature, asserting that stationary reflection — one of the most canonical forms of compactness — may hold at the level of the successor of the first singular cardinal: Theorem 2 (Magidor, [Mag82]). Assuming the consistency of infinitely many supercompact cardinals, it is consistent that every stationary subset of 1 ℵ휔+1 reflects. Ever since, it remained open whether Magidor’s compactness and incompactness results may co-exist. The main tool for obtaining Theorem 1 (and the failures of SCH, in general) is Prikry-type forcing (see Gitik’s survey [Git10]), however, adding Prikry sequences at a cardinal 휅 typically implies the failure of reflection at 휅+. On the other hand, Magidor’s proof of Theorem 2 goes through Lévy- collapsing 휔-many supercompact cardinals to become the ℵ푛’s, and in any such model SCH would naturally hold at the supremum, ℵ휔. Various partial progress to combine the two results was made along the way. Cummings, Foreman and Magidor [CFM01] investigated which sets can reflect in the classical Prikry generic extension. In his 2005 disserta- tion [Sha05], Sharon analyzed reflection properties of extender-based Prikry forcing (EBPF, due to Gitik and Magidor [GM94]) and devised a way to

1 That is, for every subset 푆 ⊆ ℵ휔+1, if for every ordinal 훼 < ℵ휔+1 (of uncountable cofinality), there exists a closed and unbounded subset of 훼 disjoint from 푆, then there exists a closed and unbounded subset of ℵ휔+1 disjoint from 푆. SIGMA-PRIKRY FORCING III 3 kill one non-reflecting stationary set, again in a Prikry-type fashion. He then described an iteration to kill all non-reflecting stationary sets, but the exposition was incomplete. In the other direction, works of Solovay [Sol74], Foreman, Magidor and Shelah [FMS88], Veliˇcković[Vel92], Todorˇcević[Tod93], Foreman and Todorˇcević [FT05], Moore [Moo06], Viale [Via06], Rinot [Rin08], Shelah [She08], Fuchino and Rinot [FR11], and Sakai [Sak15] add up to a long list of compactness principles that are sufficient to imply the SCH. In [PRS19], we introduced a new class of Prikry-type forcing called Σ- Prikry and showed that many of the standard Prikry-type forcing for vi- olating SCH at the level of a singular cardinal of countable cofinality fits into this class. In addition, we verified that Sharon’s forcing for killing a single non-reflecting stationary set fits into this class. Then, in [PRS20], we devised a general iteration scheme for Σ-Prikry forcing. From this, we constructed a model of the failure of SCH at 휅 with stationary reflection at 휅+; we first violate the SCH using EBPF and then carry out an iteration of length 휅++ of the Σ-Prikry posets to kill all non-reflecting stationary subsets of 휅+. Independently, and around the same time, Ben-Neria, Hayut and Unger [OHU19] also obtained the consistency of the failure of SCH at 휅 with stationary reflection at 휅+. Their proof differs from ours in quite a few aspects; we mention just two of them. First, instead of EBPF, they violate SCH by using Gitik’s very recent forcing [Git19a] which is also applicable to cardinals of uncountable cofinality. Second, they cleverly avoid the need to carry out iterated forcing, by invoking iterated ultrapowers, instead. An even simpler proof was then given by Gitik in [Git19c]. Still, in all of the above, the constructions are for a singular cardinal 휅 that is very high up; more precisely, 휅 is a limit of inaccessible cardinals. Obtaining a similar construction for 휅 = ℵ휔 is quite more difficult, as it involves interleaving collapses. This makes key parts of the forcing no longer closed, and closure is an essential tool to make use of the indestructibility of the supercompact cardinals when proving reflection. In this paper, we extend the machinery developed in [PRS19, PRS20] to support interleaved collapses, and show that this new framework captures Gitik’s EBPF with interleaved collapses [Git19b]. The new class is called (Σ,⃗S)-Prikry. Finally, by running our iteration of (Σ,⃗S)-Prikry forcings over a suitable ground model, we establish that Magidor’s compactness and incompactness results can indeed co-exist:

Main Theorem. Assuming the consistency of infinitely many supercompact cardinals, it is consistent that all of the following hold:

ℵ푛 (1) 2 = ℵ푛+1 for all 푛 < 휔; ℵ휔 (2) 2 = ℵ휔+2; (3) every stationary subset of ℵ휔+1 reflects. 4 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

1.1. Organization of this paper. In Section 2, we introduce the concepts of nice projection and suitability for reflection. In Section 3, we introduce the class of (Σ,⃗S)-Prikry forcing and prove some of their main properties. In Section 4, we prove that Gitik’s Extender Based Prikry Forcing with Collapses (EBPFC) fits into the (Σ,⃗S)-Prikry framework. Here we also analyze the preservation of cardinals in the corresponding generic extension and show that EBPFC is suitable for reflection. In Section 5, we introduce the notion of nice forking projection, a strengthening of the concept of forking projection from Part I of this series. We show that a graded poset admitting an exact forking projection to a (Σ,⃗S)-Prikry poset is not far from being (Σ,⃗S)-Prikry on its own. The section concludes with a sufficient condition for exact forking projections to preserve suitability for reflection. In Section 6, we revisit the functor A(·, ·) from Part II of this series, improving the main result of [PRS20, S4]. Specifically, we prove that, for every (Σ,⃗S)-Prikry forcing P and every P-name 푇˙ for a fragile stationary set, the said functor produces a (Σ,⃗S)-Prikry forcing A(P, 푇˙ ) admitting a nice forking projection to P and killing the stationarity of 푇˙ . In Section 7, we improve one of the main result from Part II of this series, showing that, modulo necessary variations, the very same iteration scheme from [PRS20, S3] is also adequate for (Σ,⃗S)-Prikry forcings. In Section 8, we present the primary application of our framework. The proof of the Main Theorem may be found there. 1.2. Notation and conventions. Our forcing convention is that 푝 ≤ 푞 means that 푝 extends 푞. We write P ↓ 푞 for {푝 ∈ P | 푝 ≤ 푞}. Denote 휇 휇 휇 퐸휃 := {훼 < 휇 | cf(훼) = 휃}. The sets 퐸<휃 and 퐸>휃 are defined in a similar fashion. For a stationary subset 푆 of a regular uncountable cardinal 휇, 휇 we write Tr(푆) := {훿 ∈ 퐸>휔 | 푆 ∩ 훿 is stationary in 훿}. 퐻휈 denotes the collection of all sets of hereditary cardinality less than 휈. For every set of ordinals 푥, we denote cl(푥) := {sup(푥 ∩ 훾) | 훾 ∈ Ord, 푥 ∩ 훾 ̸= ∅}, and 휇 + acc(푥) := {훾 ∈ 푥 | sup(푥 ∩ 훾) = 훾 > 0}. We write CH휇 to denote 2 = 휇 and GCH<휈 as a shorthand for CH휇 holds for every infinite cardinal 휇 < 휈. For a sequence of maps ⃗휛 = ⟨휛푛 | 푛 < 휔⟩ and yet a another map 휋 such ⋂︀ that Im(휋) ⊆ 푛<휔 dom(휛푛), we let ⃗휛 ∙ 휋 denote ⟨휛푛 ∘ 휋 | 푛 < 휔⟩. 2. Nice projections and reflection Definition 2.1. Given a poset P = (푃, ≤) with greatest element 1l and a 휛 휛 map 휛 with dom(휛) ⊇ 푃 , we derive a poset P := (푃, ≤ ) by letting 푝 ≤휛 푞 iff (푝 = 1l or (푝 ≤ 푞 and 휛(푝) = 휛(푞))). Definition 2.2. For two notions of forcing P = (푃, ≤) and S = (푆, ⪯) with maximal elements 1lP and 1lS, respectively, we say that a map 휛 : 푃 → 푆 is a nice projection from P to S iff all of the following hold: SIGMA-PRIKRY FORCING III 5

(1) 휛(1lP) = 1lS; (2) for any pair 푞 ≤ 푝 of elements of 푃 , 휛(푞) ⪯ 휛(푝); (3) for all 푝 ∈ 푃 and 푠 ⪯ 휛(푝), the set {푞 ∈ 푃 | 푞 ≤ 푝 ∧ 휛(푞) ⪯ 푠} admits a ≤-greatest element, which we denote by 푝 + 푠. Moreover, 푝 + 푠 has the additionally property that 휛(푝 + 푠) = 푠;2 (4) for every 푞 ≤ 푝 + 푠, there is 푝′ ≤휛 푝 such that 푞 = 푝′ + 휛(푞); In particular, the map (푝′, 푠′) ↦→ 푝′ + 푠′ forms a projection from 휛 (P ↓ 푝) × (S ↓ 푠) onto P ↓ 푝. Example 2.3. If P is a product of the form S × T, then the map (푠, 푡) ↦→ 푠 forms an nice projection from P to S. Note that the composition of nice projections is again a nice projection.

Definition 2.4. Let P = (푃, ≤) and S = (푆, ⪯) be two notions of forcing and 휛 : 푃 → 푆 be a nice projection. For an S-generic ﬁlter 퐻, we deﬁne the quotient forcing P/퐻 := (푃/퐻, ≤P/퐻 ) as follows: ∙ 푃/퐻 := {푝 ∈ 푃 | 휛(푝) ∈ 퐻};

∙ for all 푝, 푞 ∈ 푃/퐻, 푞 ≤P/퐻 푝 iﬀ there is 푠 ∈ 퐻 with 푠 ⪯ 휛(푞) such that 푞 + 푠 ≤ 푝.

Remark 2.5. In a slight abuse of notation, we tend to write P/S when refer- ring to a quotient as above, without specifying the generic for S or the map 휛. By standard arguments, P is isomorphic to a dense subposet of S * P/S (see [Abr10, p. 337]).

Lemma 2.6. Suppose that 휛 : P → S is a nice projection. Let 푝 ∈ 푃 and set 푠 := 휛(푝). For any condition 푎 ∈ S ↓ 푠, define an ordering ≤푎 over 휛 3 P ↓ 푝 by letting 푝0 ≤푎 푝1 iff 푝0 + 푎 ≤ 푝1 + 푎. Then: 휛 (1) (S ↓ 푎) × ((P ↓ 푝), ≤푎) projects to P ↓ (푝 + 푎), and 휛 휛 ′ 4 (2) ((P ↓ 푝), ≤푎) projects to ((P ↓ 푝), ≤푎′ ) for all 푎 ⪯ 푎. 휛 휛 (3) If P contains a 훿-closed dense set, then so does ((P ↓ 푝), ≤푎). 휛 Proof. Note that for 푝0, 푝1 in P ↓ 푝: 휛 휛 휛 ∙ 푝0 ≤푠 푝1 iff 푝0 ≤ 푝1, and so (P ↓ 푝, ≤푠) is simply P ↓ 푝; ′ ∙ if 푝0 ≤푎 푝1, then 푝0 ≤푎′ 푝1 for any 푎 ⪯ 푎; 휛 ∙ in particular, if 푝0 ≤ 푝1, then 푝0 ≤푎 푝1 for any 푎 in S ↓ 푠. The first projection is given by (푎′, 푟) ↦→ 푟 + 푎′, and the second projection is given by the identity. 휛 For the last statement, denote P푎 := ((P ↓ 푝), ≤푎) and let 퐷 be a 훿- 휛 휛 closed dense subset of P . We claim that 퐷푎 := {푟 ∈ P ↓ 푝 | 푟 + 푎 ∈ 퐷} is 휛 휛 a 훿-closed dense subset of P푎. For the density, if 푟 ∈ P ↓ 푝, let 푞 ≤ 푟+푎 be in 퐷. Then, by Clause (4) of Definition 2.2, 푞 = 푟′ +푎 for some 푟′ ≤휛 푟, and

2By convention, a greatest element, if exists, is unique. 3 Strictly speaking, ≤푎 is reflexive and transitive, but not asymmetric. But this isalso always the case, for instance, in iterated forcing. 4 휛 휛 Taking 푎 = 푠 we have in particular that P ↓ 푝 projects to ((P ↓ 푝), ≤푎′ ). 6 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

′ ′ so 푟 ≤푎 푟 and 푟 ∈ 퐷푎. For the closure, suppose that ⟨푝푖 | 푖 < 휏⟩ is a ≤P푎 - decreasing sequence in 퐷푎 for some 휏 < 훿. Setting 푞푖 := 푝푖 + 푎, that means 휛 that ⟨푞푖 | 푖 < 휏⟩ is a ≤ -decreasing sequence in 퐷 and so has a lower bound 푞. More precisely, 푞 ∈ 퐷 and for each 푖, 푞 ≤ 푞푖 and 휛(푞) = 휛(푞푖) = 푎. Let 푝* ≤휛 푝, be such that 푝* + 푎 = 푞. Here again we use Clause (4) of * Definition 2.2. Then 푝 ∈ 퐷푎, which is the desired ≤P푎 -lower bound. The next lemma clarifies the relationship between the different generic extensions that we will be considering:

Lemma 2.7. Suppose that 휛 : P → S, 푝 ∈ 푃 , 푠 := 휛(푝) and ≤푎 for 푎 in S ↓ 푠 are as in the above lemma. Let 퐺 be P-generic with 푝 ∈ 퐺. * 휛 Next, let 퐻 × 퐺 be ((S ↓ 푠) × (P ↓ 푝))/퐺-generic over 푉 [퐺]. For each 휛 * 푎 ∈ 퐻, let 퐺푎 be the ((P ↓ 푝), ≤푎)-generic filter obtained from 퐺 . Then: * * (1) For any 푎 ∈ 퐻, 푉 [퐺] ⊆ 푉 [퐻 ×퐺푎] ⊆ 푉 [퐻 ×퐺 ], and 퐺 ⊇ 퐺푎 ⊇ 퐺 ; ′ (2) For any pair 푎 ⪯ 푎 of elements of 퐻, 푉 [퐻 × 퐺푎′ ] ⊆ 푉 [퐻 × 퐺푎], and 퐺푎′ ⊇ 퐺푎; 휛 ⋃︀ (3) 퐺 ∩ (P ↓ 푝) = 푎∈퐻 퐺푎. * 휛 * Proof. For notational convenience, denote P := P ↓ 푝 and S := S ↓ 푠. The first two items follow from the corresponding choices of the projec- ⋃︀ * tions in Lemma 2.6. For the third item, first note that 푎∈퐻 퐺푎 ⊆ P ∩ 퐺. * * Suppose that 푟 ∈ 퐺 ∩ P . In 푉 [퐻], define * * * 5 퐷 := {푟 ∈ 푃 | (∃푎 ∈ 퐻)(푟 ≤푎 푟 ) ∨ 푟 ⊥P/퐻 푟 }. * Claim 2.7.1. 퐷 is a dense set in P . * * Proof. Let 푟 ∈ 푃 . If 푟 ⊥P/퐻 푟 , then 푟 ∈ 퐷, and so we are done. So * suppose that 푟 and 푟 are compatible in P/퐻. Let 푞 ∈ 푃/퐻 be such that, * ′ 푞 ≤P/퐻 푟 and 푞 ≤P/퐻 푟 . Let 푎 ⪯ 휛(푞) in 퐻 be such that 푞 + 푎 ≤ 푟, 푟 . By Definition 2.2(4) and exactness of 휛 we may let 푟′ ≤휛 푟, be such that ′ ′ * ′ 푟 + 푎 = 푞 + 푎. In particular, 푟 ≤푎 푟 , and so 푟 ∈ 퐷. * * * Now let 푟 ∈ 퐷 ∩ 퐺 . Since both 푟, 푟 ∈ 퐺, it must be that 푟 ≤푎 푟 for * * some 푎 ∈ 퐻. And since 푟 ∈ 퐺 ⊆ 퐺푎, we get that 푟 ∈ 퐺푎. Lemma 2.8. Suppose that 휛 : P → S is an exact nice projection and that 훿 < 휅 are infinite regular cardinals for which the following hold: 휛 (1) |S| < 훿 and P contains a 훿-directed-closed dense subset; (2) After forcing with × 휛, 훿 and 휅 remain regular; S P 휛 휅 P P 휅 6 (3) 퐸<훿 is the same as computed in 푉 and in 푉 and 푉 |= “퐸<훿 ∈ 퐼[휅]”. Let 푝 ∈ P and set 푠 := 휛(푝). Then, for any P-generic 퐺 with 푝 ∈ 퐺, the 휛 휅 푉 [퐺] quotient ((S ↓ 푠) × (P ↓ 푝))/퐺 preserves stationary subsets of (퐸<훿) .

5 * Since 푟 ∈ 푃 then 휛(푟) = 휛(푝) ∈ 퐻 and thus 푟 is a condition in P/퐻. 6For the definition of the ideal 퐼[휅] see [She94, Definition 2.3]. SIGMA-PRIKRY FORCING III 7

* 휛 * Proof. For the scope of the proof denote P := P ↓ 푝 and S := S ↓ 푠. * * * Let 퐻 × 퐺 be (S × P )/퐺-generic over 푉 [퐺]. For each 푎 ∈ 퐻, let 퐺푎 be 휛 * the ((P ↓ 푝), ≤푎)-generic, obtained from 퐺 . For notational convenience 휛 we will also denote P푎 := ((P ↓ 푝), ≤푎). Combining Clause (1) of our assumptions with Lemma 2.6 we have that P푎 contains a 훿-closed dense subset, hence it is 훿-strategically-closed. Standard arguments imply that * 7 P /퐺푎 is 훿-strategically-closed over 푉 [퐺푎]. 휅 Suppose for contradiction that 푉 [퐺] |= “푇 ⊆ 퐸<훿 is a stationary set”, * but that 푇 is nonstationary in 푉 [퐻 × 퐺 ]. Since |S| < 훿, Clause (3) above * 휅 푉 휅 푉 [퐺 ] 휅 푉 [퐺] 휅 푉 [퐺푎] and Lemma 2.7(1) yield (퐸<훿) = (퐸<훿) = (퐸<훿) = (퐸<훿) for 휅 all 푎 ∈ 퐻. Thus, we can unambiguously denote this set by 퐸<훿.

Claim 2.8.1. Let 푎 ∈ 퐻. Then, 푇 is non-stationary in 푉 [퐻 × 퐺푎].

Proof. Otherwise, if 푇 was stationary in 푉 [퐻][퐺푎], then since |S| < 휅, ′ 휅 * ˙ 푇 := {훼 ∈ 퐸<훿 | (∃푟 ∈ 퐺푎)(푏, 푟) S ×P푎 훼 ∈ 푇 } is a stationary set lying in 푉 [퐺푎], where 푏 ∈ 퐻. Combining Lemma 2.7(1) 푉 [퐺] 푉 [퐻×퐺 ] 푉 [퐺 ] with the fact that S is small we have 퐼[휅] ⊆ 퐼[휅] 푎 ⊆ 퐼[휅] 푎 . 휅 푉 [퐺푎] Thus, Clause (3) of our assumption yields 퐸<훿 ∈ 퐼[휅] . Now, since * * P /퐺푎 is 훿-strategically closed in 푉 [퐺푎], by Shelah’s theorem [She79], P /퐺푎 휅 ′ * preserves stationary subsets of 퐸<훿 hence 푇 remains stationary in 푉 [퐺 ]. ′ Once again, since S is a small forcing 푇 remains stationary in the further generic extension 푉 [퐻 × 퐺*]. This is a contradiction with 푇 ′ ⊆ 푇 and our * assumption that 푇 was non-stationary in 푉 [퐻 × 퐺 ].

Then for every 푎 ∈ 퐻, let 퐶푎 be a club in 푉 [퐻 × 퐺푎], disjoint from 푇 . ˙ Since S is a small forcing, we may assume that 퐶푎 ∈ 푉 [퐺푎]. Let 퐶푎 be a P푎-name for this club such that ˙ ∙ 푝 P푎 “퐶푎 is a club”, and 8 * ˙ ˙ ∙ (푎, 푝) (S ×P푎) “퐶푎 ∩ 푇 = ∅”. Since S is a small forcing, we may ﬁx some 푎 ∈ 퐻, such that 휅 ˙ 푇푎 := {훼 ∈ 퐸<훿 | ∃푟 ∈ 퐺[푟 ≤ 푝, 휛(푟) = 푎 & 푟 P 훼ˇ ∈ 푇 ]} is stationary in 푉 [퐺]. * Claim 2.8.2. There is a condition 푝 ≤푎 푝 and an ordinal 훾 < 휅 such that * * ˙ ˙ (푎, 푝 ) (S ×P푎) 훾 ∈ 퐶푎 ∩ 푇 .

Proof. Work ﬁrst in 푉 [퐺]. Let 푀 be an elementary submodel of 퐻휃 (for a large enough regular cardinal 휃), such that:

∙ 푀 contains all the relevant objects, including 퐶˙ 푎 and (푎, 푝); ∙ 훾 := 푀 ∩ 휅 ∈ 푇푎

7 Note that 훿 is still regular in 푉 [퐺푎], as P푎 being 훿-strategically-closed over 푉 . 8 ˙ ˙ * Here we identify 퐶푎 and 푇 with a (S × P푎)-name in the natural way (cf. Lemma 2.7(1)). 8 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

푉 Let 휒 = cf (훾) and ⟨훾푖 | 푖 < 휒⟩ ∈ 푉 be an increasing sequence with limit 휅 훾. As 훾 ∈ 푇푎 ⊆ 퐸<훿, 휒 < 훿. Also, since 훾 ∈ 푇푎, we may fix some 푟 ∈ 퐺 ˙ with 휛(푟) = 푎 such that 푟 P 훾ˇ ∈ 푇 . Using Clause (4) of Definition 2.2, let * * * * 푟 be a condition in P such that 푟 + 푎 = 푟. Note that also 푟 ∈ 퐺. ′ ′ ′ Below, for a condition 푝 ∈ P푎, we say that 푝 ∈ P푎/퐺 if 푝 + 푎 ∈ 퐺. Then ′ ˙ ′ for 푞 ∈ P, 푞 P 푝 ∈ P푎/퐺 iff 푞 ≤P 푝 + 푎. ′ Since 푝 forces 퐶˙ 푎 is a club, for all 훽 < 휅, there is 훽 ≤ 훼 < 휅, and 푝 ≤푎 푝 ˙ ′ forcing 훼 ∈ 퐶푎. And by density, we can find such 푝 ∈ P푎/퐺. Then by ′ elementarity and since 푝 ∈ 푀, for all 푖 < 휒, there is 훼 ∈ 푀 ∖ 훾푖, and 푝 ≤푎 푝 ′ ′ ˙ in 푀, such that 푝 ∈ P푎/퐺 and 푝 P푎 훼 ∈ 퐶푎. Fix a name 푀˙ and, by strengthening 푟* if necessary, suppose that for some 푎′ ≤ 푎 in 퐻, 푟* + 푎′ forces that the above properties hold. In particular, for * ′ ′ ′ all 푖 < 휒 and 푞 ≤P 푟 + 푎 , then there are 푞 ≤P 푞, 푝 ≤푎 푝 and 훼 ≥ 훾푖, such ′ ′ ′ ′ ˙ ˙ ′ ˙ that 푞 ≤P 푝 + 푎, 푞 P “푝 ∈ 푀, 훼 ∈ 푀” and 푝 P푎 훼 ∈ 퐶푎. ′ * ′ Breaking this down, we get that for all 푖 < 휒, if 푟 ≤푎 푟 and 푏 ≤S 푎 , ′ ′ ′ ′ ′ ′ then there are 푏 ≤S 푏, 푞 ≤푎 푟 , 푝 ≤푎 푝 and 훾푖 ≤ 훼 < 휅, such that 푞 ≤푎 푝 , ′ ′ ′ ˙ ˙ ′ ˙ 푞 + 푏 P “푝 ∈ 푀, 훼 ∈ 푀” and 푝 P푎 훼 ∈ 퐶푎. The later also gives that ′ ˙ 푞 P푎 훼 ∈ 퐶푎. In other words we have the following for each 푖: ′ * ′ ′ ′ ′ (†) for all 푟 ≤푎 푟 and 푏 ≤S 푎 , there are 푞 ≤푎 푟 , 푏 ≤S 푏, and 훼 < 휅, ′ ′ ˙ ′ ˙ such that 푞 + 푏 P “훼 ∈ 푀 ∖ 훾푖” and 푞 P푎 훼 ∈ 퐶푎. Then, since the closure of ≤푎 is more than |S|, we get that for each 푖: ′ * ′ ′ (††) for all 푟 ≤푎 푟 , there are 푞 ≤푎 푟 , and a dense 퐷 ⊂ S, such that for ′ ˙ ′ ˙ all 푏 ∈ 퐷, there is 훼 < 휅, with 푞 + 푏 P “훼 ∈ 푀 ∖ 훾푖” and 푞 P푎 훼 ∈ 퐶푎. Working in 푉 , construct a ≤푎-decreasing sequences ⟨푞푖 | 푖 ≤ 휒⟩ of condi- * tions in P푎 below 푟 and a family ⟨퐷푖 | 푖 < 휒⟩ of dense sets of S with the following properties: For each 푖 < 휒 and 푏 ∈ 퐷푖, there is 훼 < 휅, such that: ˙ (1) 푞푖+1 + 푏 P 훼 ∈ 푀 ∖ 훾푖, and ˙ (2) 푞푖+1 P푎 훼 ∈ 퐶푎. At successor stages we use (††), and at limit stages we take lower bounds. * * * Let 푝 := 푞휒. Since we can find 푝 as above ≤푎-densely often below 푟 , we may assume that 푝* + 푎 ∈ 퐺. Now go back to 푉 [퐺]. For each 푖 < 휒, let 푏푖 ∈ 퐷푖 ∩ 퐻, where recall that 퐻 is the induced S-generic from 퐺. Also, let 훼푖 witness that 푏푖 ∈ 퐷푖. * ˙ Then 푝 P푎 훼푖 ∈ 퐶푎, and in 푉 [퐺], 훼푖 ∈ 푀 ∖ 훾푖 (since 푞푖+1 + 푏푖 ∈ 퐺), so * ˙ 훾 = sup푖 훼푖. It follows that 푝 P푎 훾 ∈ 퐶푎. * * * ˙ Finally, as 푝 + 푎 ≤ 푟 + 푎 = 푟, we have that 푝 + 푎 P 훾 ∈ 푇 . Recall that * ′ ′ ′ ′ the projection from S ×P푎 to P is witnessed by (푎 , 푝 ) ↦→ 푝 +푎 (Lemma 2.6), * * * ˙ * ˙ ˙ hence it follows that (푎, 푝 ) S ×P푎 훾 ∈ 푇 . So, (푎, 푝 ) S ×P푎 훾ˇ ∈ 푇 ∩퐶푎. Choose 푝* as in the above lemma. That gives a contradiction with

* ˙ ˙ (푎, 푝) (S ×P푎) 퐶푎 ∩ 푇 = ∅. Definition 2.9. For stationary subsets ∆, Γ of a regular uncountable cardinal 휇, Reﬂ(∆, Γ) asserts that for every stationary subset 푇 ⊆ ∆, there 휇 exists 훾 ∈ Γ ∩ 퐸>휔 such that 푇 ∩ 훾 is stationary in 훾. SIGMA-PRIKRY FORCING III 9

We end this section by establishing a suﬃcient condition for Reﬂ(...) to hold in generic extensions; this will play a crucial role at the end of Section 5.

Definition 2.10. For inﬁnite cardinals 휏 < 휎 < 휅 < 휇, we say that (P, S, 휛) is suitable for reflection with respect to ⟨휏, 휎, 휅, 휇⟩ iﬀ all the following hold: (1) P and S are nontrivial notions of forcing; 휛 (2) 휛 : P → S is an exact nice projection and P contains a 휎-directed- closed dense subset;9 휛 ++ (3) In any forcing extension by P or S × P , |휇| = cf(휇) = 휅 = 휎 ; + (4) For any 푠 ∈ 푆 ∖ {1lS}, there is a cardinal 훿 with 휏 < 훿 < 휎, such ∼ that S ↓ 푠 = Q × Col(훿, <휎) for some notion of forcing Q of size < 훿.

Lemma 2.11. Let (P, S, 휛) be suitable for reflection with respect to ⟨휏, 휎, 휅, 휇⟩. 휛 Suppose 휎 is a supercompact cardinal indestructible under forcing with P . P 휇 휇 Then 푉 |= Reﬂ(퐸≤휏 , 퐸<휎+ ).

P 휅 휅 Proof. By Definition 2.10(3), it suffices to prove that 푉 |= Refl(퐸≤휏 , 퐸<휎+ ). 휅 Let 퐺 be P-generic. In 푉 [퐺], let 푇 be a stationary subset of 퐸≤휏 . Suppose for simplicity that this is forced by the empty condition.

Claim 2.11.1. Let 푝 ∈ 퐺 be such that 푠 := 휛(푝) strictly extends 1lS. 휛 Then, the quotient ((S ↓ 푠) × (P ↓ 푝))/퐺 preserves the stationarity of 푇 . Proof. Using Clauses (1) and (2) of Definition 2.10, let us pick any 푝 ∈ 퐺 for which 푠 := 휛(푝) strictly extends 1lS. Back in 푉 , using Clause (4) of + Definition 2.10, fix a cardinal 훿 with 휏 < 훿 < 휎, a notion of forcing Q of size < 훿, and an isomorphism 휄 from S ↓ 푠 to Q×Col(훿, <휎). Let 휄0, 휄1 denote the ′ ′ ′ unique maps to satisfy 휄(푠 ) = (휄0(푠 ), 휄1(푠 )). By Example 2.3, 휄0 and 휄1 are exact nice projections. Set 휋 := (휄0 ∘ 휛) (P ↓ 푝) and 휚 := (휄1 ∘ 휛) (P ↓ 푝), so that 휋 and 휚 are nice projection from P ↓ 푝 to Q and from P ↓ 푝 to Col(훿, <휎), respectively. Note that the definition of 휋 depends on our choice of Q, which depends on our choice of 푝, and formally we defined 휋 as a 휋 projection from P ↓ 푝 to Q. In an slight abuse of notation we will write P 휋 휋 휋 10 rather than (P ↓ 푝) . More precisely, P := ({푞 ∈ 푃 | 푞 ≤ 푝}, ≤ ). Subclaim 2.11.1.1. 휛 휋 (i) (S ↓ 푠) × (P ↓ 푝) projects onto Q × (P ↓ 푝), and that projects onto P ↓ 푝; 휛 휋 (ii) (S ↓ 푠) × (P ↓ 푝) projects onto Q × (P ↓ 푝), and that projects onto 휋 P ↓ 푝; 휛 휛 (iii) (S ↓ 푠) × (P ↓ 푝) projects onto Col(훿, <휎) × (P ↓ 푝), and that 휋 projects onto P ↓ 푝.

9In particular, we assume that 휎 is a regular cardinal. 10Recall Definition 2.1. 10 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

′ ′ ′ ′ ′ Proof. (i) For the first part, the map (푠 , 푝 ) ↦→ (휄0(푠 ), 푝 + 휄1(푠 )) is such a projection, where + operation is computed with respect to the nice projection 휚. For the second part, the map (푞′, 푝′) ↦→ 푝′ +푞′ gives such a projection, where the + operation is computed with respect to 휋.11 (ii) For the second part, the map (푞′, 푝′) ↦→ 푝′ is such a projection. ′ ′ ′ ′ (iii) For the first part, the map (푠 , 푝 ) ↦→ (휄1(푠 ), 푝 ) is such a projection. For the second part, the map (푐′, 푝′) ↦→ 푝′ + 푐′ is such a projection, where the + operation is with respect to the nice projection 휚. By Definition 2.10(3), in all forcing extensions with posets from Clause (i), 휅 is a cardinal which is the double successor of 휎. But then, since |Q| < 휅, it follows from the second part of Clause (ii) that 휅 is the double successor 휋 of 휎 in forcing extensions by P ↓ 푝. Actually, in any forcing extension by 휋 12 P . Altogether, in all forcing extensions with posets from the preceding subclaim, 휅 is the double successor of 휎. * 휋 Let 퐺푞 ×퐺 be (Q×(P ↓ 푝))/퐺-generic over 푉 [퐺]. Next, we want to use * Lemma 2.8 to show that 푇 remains stationary in 푉 [퐺푞 × 퐺 ], so we have to verify its assumptions hold. The next claim along with Definition 2.10(4) yields Clause (1) of Lemma 2.8: 휋 Subclaim 2.11.1.2. P contains a 훿-directed-closed dense set. 휛 Proof. Let 퐷 ⊆ P be the 휎-directed-closed dense subset given by Clause (2) of our assumptions.13 We claim that the set ′ 휚 퐷 := {푞 + 푐 | 푞 ∈ 퐷, 푞 ≤ 푝, 푐 ≤Col(훿,<휎) 휚(푝)} 휋 is a 훿-directed-closed dense subset of P . Here 푞+푐 is computed with respect to the projection map 휚 : P ↓ 푝 → Col(훿, < 휎). 휋 ′ 휚 For density, if 푞 ∈ P , since 휚 is a nice projection, let 푞 ≤ 푝 be with 푞′ +휚(푞) = 푞. Now, let 푞′′ ≤휛 푞′ be in 퐷. In particular, 휚(푞′′) = 휚(푝), so that 푞′′ + 휚(푞) is well-defined. Then 푞′′ + 휚(푞) ≤휋 푞, 푞′′ ≤휚 푝 and 푞′′ + 휚(푞) ∈ 퐷′. 휋 For directed closure, suppose that 휈 < 훿 and {푝훼 | 훼 < 휈} is a ≤ - ′ directed set in 퐷 . For each 훼 < 휈, write 푝훼 = 푞훼 + 푐훼, where 푞훼 ∈ 퐷, 휚(푞훼) = 휚(푝), and 푐훼 = 휚(푝훼) ∈ Col(훿, <휎). Note that for each 훼 < 휈, 휋 휋(푞훼) = 휋(푝훼). Note that 푝훼 and 푝훽 being ≤ -compatible implies that 푞훼 and 푞훽 are ≤-compatible and also that 휋(푞훼) = 휋(푞훽), hence 푞훼 and 푞훽 are 휛 actually ≤ -compatible. Clearly, it also yields 푐훼 ∪ 푐훽 ∈ Col(훿, <휎). 휛 Then {푞훼 | 훼 < 휈} is a ≤ -directed set in 퐷 of size < 휅, so we may let 휛 푞 ∈ 퐷 be a ≤ -lower bound for {푞훼 | 훼 < 휈}. In particular, 휚(푞) = 휚(푝) ⋃︀ and 휋(푞) = 휋(푝훼) for all 훼 < 휈. Additionally, let 푐 := 훼<휈 푐훼 ∈ Col(훿, <휎). ′ 휋 Then 푞 + 푐 ∈ 퐷 is the desired ≤ -lower bound for {푝훼 | 훼 < 휈}.

11See Clause (4) of Definition 2.2(4) regarded with respect to 휋. 12Note that Subclaim 2.11.1.1 remains valid even if we replace 푝 by any 푝′ ≤ 푝. For ′ 휛 ′ instance, regarding Clause (i), we will then have that (S ↓ 휛(푝 ))×(P ↓ 푝 ) projects onto ′ 휋 ′ ′ (Q ↓ 휋(푝 )) × (P ↓ 푝 ) and that this latter projects onto P ↓ 푝 . 13I.e., 퐷 is dense and 휎-directed-closed with respect to ≤휛. SIGMA-PRIKRY FORCING III 11

Clause (2) of Lemma 2.8 easily follows combining the above subclaim, the 휋 ++ fact that P forces “휅 = 휎 ” and Clause (4) of Definition 2.10. Finally, for 휅 Clause (3) we argue as follows: first, the above subclaim implies that 퐸<훿 휋 is computed in the same way in 푉 and 푉 P . Second, Clauses (3) and (4) of ++ P 휅 휎 ++ Definition 2.10 and [She91, Lemma 4.4] yield 푉 |= 퐸<훿 ⊆ 퐸<휎+ ∈ 퐼[휎 ]. * Thereby, 푇 remains stationary in 푉 [퐺푞 × 퐺 ]. As Q is small, we may fix 푇 ′ ⊆ 푇 such that 푇 ′ is in 푉 [퐺*] and moreover stationary in 푉 [퐺*]. As * ′ 휎++ ++ 휋 휛 established earlier, 푉 [퐺 ] |= “푇 ⊆ 퐸<휎+ ∈ 퐼[휎 ]”. Since both P and P 휛 * are 훿-strategically closed (actually P is more), we have that, in 푉 [퐺 ], the 휛 * quotient (Col(훿, <휎) × (P ↓ 푝))/퐺 is also 훿-strategically closed. So, again 휛 * ′ it follows that (Col(훿, <휎) × (P ↓ 푝))/퐺 preserves the stationarity of 푇 . ∼ Finally, since S ↓ 푠 = Q × Col(훿, <휎), the quotient 휛 휛 ((S ↓ 푠) × (P ↓ 푝))/(Col(훿, <휎) × (P ↓ 푝)) ′ is isomorphic to Q, which is a small of forcing. Altogether, 푇 (and hence also 푇 ) remains stationary in the generic extension 푉 [퐺].

Let 푝 ∈ 퐺 be such that 푠 := 휛(푝) strictly extends 1lS. Let 퐻 be the * * generic filter for S induced by 휛 and 퐺. Let 퐺 be such that 퐻 × 퐺 is 휛 generic for ((S ↓ 푠) × (P ↓ 푝))/퐺. By the above claim, 푇 is still stationary * 휎++ 푉 [퐺*][퐻] in 푉 [퐺 ][퐻]. Also, by Definition 2.10(3), 푇 ⊆ (퐸≤휏 ) . 휛 * Using that 휎 is a supercompact indestructible under P , let (in 푉 [퐺 ]) 푗 : 푉 [퐺*] → 푀 be a 휅-supercompact embedding with crit(푗) = 휎. We shall want to lift this embedding to 푉 [퐺*][퐻]. ∼ Work below the condition 푠 that we fixed earlier. Recall that S ↓ 푠 = + Q × Col(훿, <휎) for some poset Q of size < 훿 with 휏 < 훿 < 휎. So, 퐻 may be seen as a product of two corresponding generics, 퐻 = 퐻0 × 퐻1. For the ease of notation, put C := Col(훿, <휎). Since Q has size < 훿 < crit(푗), we can lift 푗 to an embedding * ′ 푗 : 푉 [퐺 ][퐻0] → 푀 . Then we lift 푗 again to get 푗 : 푉 [퐺*][퐻] → 푁 * in an outer generic extension of 푉 [퐺 ][퐻] by 푗(C)/퐻1. Since 푗(C)/퐻1 is ′ 훿-closed in 푀 [퐻1] and this latter model is closed under 휅-sequences in * * 푉 [퐺 ][퐻], then 푗(C)/퐻1 is also 훿-closed in 푉 [퐺 ][퐻]. Set 훾 := sup(푗“휅). Clearly, 푗(푇 ) ∩ 훾 = 푗“푇 . Note that, by virtue of the collapse 푗(C), 푁 |= “|휅| = 훿 & cf(훾) ≤ cf(|휅|) = 훿 < 푗(휎)”. Once again, [She91, Lemma 4.4] Definition 2.10(3) together yield 휎++ 푉 [퐺*][퐻] 휎++ 푉 [퐺*][퐻] ++ 푉 [퐺*][퐻] 푇 ⊆ (퐸≤휏 ) ⊆ (퐸<휎+ ) ∈ 퐼[휎 ] . As customary, Shelah’s theorem (c.f. [She79]) along with the 훿-closedness of * 푗(C)/퐻1 in 푉 [퐺 ][퐻] imply that this latter forcing preserves the stationarity 12 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA of 푇 . Now, a standard argument shows that that 푗(푇 ) ∩ 훾 is stationary in 푗(휅) 푁. Thus, “∃훼 ∈ 퐸<푗(휎)(푗(푇 ) ∩ 훼 is stationary in 훼)” holds in 푁. So, by elementarity, in 푉 [퐺*][퐻], 푇 reflects at a point of cofinality <휎+.14 Since reflection is downwards absolute, it follows that 푇 reflects at a point of + cofinality < 휎 in 푉 [퐺], as wanted.

3. (Σ,⃗S)-Prikry forcings We commence by recalling a few concepts from [PRS20, S2]. Definition 3.1. A graded poset is a pair (P, ℓ) such that P = (푃, ≤) is a poset, ℓ : 푃 → 휔 is a surjection, and, for all 푝 ∈ 푃 : ∙ For every 푞 ≤ 푝, ℓ(푞) ≥ ℓ(푝); ∙ There exists 푞 ≤ 푝 with ℓ(푞) = ℓ(푝) + 1.

Convention 3.2. For a graded poset as above, we denote 푃푛 := {푝 ∈ 푃 | ℓ(푝) = 푛} and P푛 := (푃푛 ∪ {1l}, ≤). In turn, P≥푛 and P>푛 are defined 푝 analogously. We also write 푃푛 := {푞 ∈ 푃 | 푞 ≤ 푝, ℓ(푞) = ℓ(푝) + 푛}, and sometimes write 푞 ≤푛 푝 (and say that 푞 is an 푛-step extension of 푝) rather 푝 than writing 푞 ∈ 푃푛 . 푟 A subset 푈 ⊆ 푃 is said to be 0-open set iff, for all 푟 ∈ 푈, 푃0 ⊆ 푈. Now, we define the (Σ,⃗S)-Prikry class, a class broader than Σ-Prikry from [PRS20, Definition 2.3]. Definition 3.3. Suppose:

(훼)Σ= ⟨휎푛 | 푛 < 휔⟩ is a non-decreasing sequence of regular uncountable cardinals, converging to some cardinal 휅; ⃗ (훽) S = ⟨S푛 | 푛 < 휔⟩ is a sequence of notions of forcing, S푛 = (푆푛, ⪯푛), with |푆푛| < 휎푛; (훾) P = (푃, ≤) is a notion of forcing with a greatest element 1l; + (훿) 휇 is a cardinal such that 1l P 휇ˇ =휅 ˇ ; (휀) ℓ : 푃 → 휔 and 푐 : 푃 → 휇 are functions;15 (휁) ⃗휛 = ⟨휛푛 | 푛 < 휔⟩ is a sequence of functions. We say that (P, ℓ, 푐, ⃗휛) is (Σ,⃗S)-Prikry iﬀ all of the following hold: (1)( P, ℓ) is a graded poset; ˚ (2) For all 푛 < 휔, P푛 := (푃푛 ∪ {1l}, ≤) contains a dense subposet P푛 which is countably-closed; 푝 푞 (3) For all 푝, 푞 ∈ 푃 , if 푐(푝) = 푐(푞), then 푃0 ∩ 푃0 is non-empty; (4) For all 푝 ∈ 푃 , 푛, 푚 < 휔 and 푞 ≤푛+푚 푝, the set {푟 ≤푛 푝 | 푞 ≤푚 푟} contains a greatest element which we denote by 푚(푝, 푞).16 In the special case 푚 = 0, we shall write 푤(푝, 푞) rather than 0(푝, 푞);17

14Actually, at a point of cofinality < 휎. 15In some applications 푐 will be a function from 푃 to some canonical structure of size <휇 휇, such as 퐻휇 (assuming 휇 = 휇). 16By convention, a greatest element, if exists, is unique. 17Note that 푤(푝, 푞) is the weakest extension of 푝 above 푞. SIGMA-PRIKRY FORCING III 13

(5) For all 푝 ∈ 푃 , the set 푊 (푝) := {푤(푝, 푞) | 푞 ≤ 푝} has size < 휇; (6) For all 푝′ ≤ 푝 in 푃 , 푞 ↦→ 푤(푝, 푞) forms an order-preserving map from 푊 (푝′) to 푊 (푝); (7) Suppose that 푈 ⊆ 푃 is a 0-open set. Then, for all 푝 ∈ 푃 and 푛 < 휔, 0 푞 푞 there is 푞 ≤ 푝, such that, either 푃푛 ∩ 푈 = ∅ or 푃푛 ⊆ 푈; (8) For all 푛 < 휔, 휛푛 is a nice projection from P≥푛 to S푛, such that, for any integer 푘 ≥ 푛, 휛푛 P푘 is again a nice projection; ˚ ˚휛푛 휛푛 (9) For all 푛 < 휔, if P푛 is a witness for Clause (2) then P푛 is a ≤ - 휛푛 휛푛 18 dense and 휎푛-directed-closed subposet of P푛 := (푃푛 ∪ {1l}, ≤ ). Convention 3.4. We derive yet another ordering ≤⃗휛 of the set 푃 , letting ⃗휛 ⋃︀ 휛푛 ⃗휛 ≤ := 푛<휔 ≤ . Simply put, this means that 푞 ≤ 푝 iﬀ (푝 = 1l), or, 0 (푞 ≤ 푝, ℓ(푝) = ℓ(푞) and 휛ℓ(푝)(푝) = 휛ℓ(푞)(푞)).

Convention 3.5. We say that (P, ℓ, 푐) has the Linked0-property if it witnesses Clause (3) above. Similarly, we will say that (P, ℓ) has the Complete Prikry Property (CPP) if it witnesses Clause (7).

Any Σ-Prikry triple (P, ℓ, 푐) can be regarded as a (Σ,⃗S)-Prikry forcing ⃗ (P, ℓ, 푐, ⃗휛) by letting S := ⟨(푛, {1lP}) | 푛 < 휔⟩ and ⃗휛 be the sequence of trivial ⃗ projections 푝 ↦→ 1lP. Conversely, any (Σ, S)-Prikry quadruple (P, ℓ, 푐, ⃗휛) with ⃗S and ⃗휛 as above witnesses that (P, ℓ, 푐) is Σ-Prikry. In particular, all the forcings from [PRS19, S3] are examples of (Σ,⃗S)-Prikry forcings. In Section 4, we will add a new example to this list by showing that Gitik’s EPBFC (The long Extender-Based Prikry forcing with Collapses [Git19b]) falls into the (Σ,⃗S)-Prikry framework. Throughout the rest of the section, assume that (P, ℓ, 푐, ⃗휛) is a (Σ,⃗S)- Prikry quadruple. We shall spell out some basic features of the components of the quadruple, and work towards proving Lemma 3.14 that explains how bounded sets of 휅 are added to generic extensions by P. Lemma 3.6 (The 푝-tree). Let 푝 ∈ 푃 . (1) For every 푛 < 휔, 푊푛(푝) is a maximal antichain in P ↓ 푝; (2) Every two compatible elements of 푊 (푝) are comparable; (3) For any pair 푞′ ≤ 푞 in 푊 (푝), 푞′ ∈ 푊 (푞); (4) 푐 푊 (푝) is injective. Proof. The proof of [PRS19, Lemma 2.8] goes through. We commence by introducing the notion of coherent sequence of nice projections, which will be important in Section 6. Definition 3.7. The sequence of nice projections ⃗휛 is called coherent if:

(1) for all 푛 < 휔, if 푝 ∈ 푃≥푛 then 휛푛“푊 (푝) = {휛푛(푝)}; (2) for all 푛 ≤ 푚 < 휔, 휛푚 factors through 휛푛; i.e., there is a map 휋푚,푛 : S푚 → S푛 such that 휛푛(푝) = 휋푚,푛(휛푚(푝)) for all 푝 ∈ 푃≥푚.

18 휛푛 More verbosely, for every 푝 ∈ 푃푛 there is 푞 ∈ 푃˚푛 such that 푞 ≤ 푝 (see Notation 2.1). 14 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Lemma 3.8. Assume that ⃗휛 is coherent. For every 푛 < 휔, if 푝 ∈ 푃≥푛 and 푡 ⪯푛 휛푛(푝) the following hold: (1) for each 푞 ∈ 푊 (푝 + 푡), 푞 = 푤(푝, 푞) + 푡; (2) for each 푞 ∈ 푊 (푝), 푤(푝 + 푡, 푞 + 푡) = 푞 + 푡; (3) for each 푚 < 휔, 푊푚(푝 + 푡) = {푞 + 푡 | 푞 ∈ 푊푚(푝)}. (4) 푝 + 푡 = 푝 + 휛ℓ(푝)(푝 + 푡);

Proof. (1) Let 푞 ∈ 푊 (푝+푡). By virtue of Definition 3.7(1), we have 휛푛(푞) = 0 휛푛(푝 + 푡) = 푡. This, together with 푞 ≤ 푤(푝, 푞), implies that 푤(푝, 푞) + 푡 is well-defined and also that 푞 ≤0 푤(푝, 푞) + 푡. On the other hand, 푞 ≤0 푤(푝, 푞) + 푡 ≤ 푝 + 푡, hence 푤(푝 + 푡, 푤(푝, 푞) + 푡) and 푞 are two compatible conditions in 푊 (푝 + 푡) that have the same length. By Lemma 3.6(1) it follows that 푞 = 푤(푝 + 푡, 푤(푝, 푞) + 푡), hence 푤(푝, 푞) + 푡 ≤0 푞, as desired. (2) By Definition 3.7(1), 푞 ≤휛푛 푝, hence 푞 + 푡 is well-defined and so 푤(푝 + 푡, 푞 + 푡) belongs to 푊 (푝 + 푡). Combining Clause (1) above with [PRS19, Lemma 2.9] we obtain the following chain of equalities: 푤(푝 + 푡, 푞 + 푡) = 푤(푝, 푤(푝 + 푡, 푞 + 푡)) + 푡 = 푤(푝, 푞 + 푡) + 푡. Now, combine Lemma 3.9 with 푞 ∈ 푊 (푝) to infer that 푤(푝, 푞 + 푡) = 푞. Altogether, this shows that 푤(푝 + 푡, 푞 + 푡) = 푞 + 푡. (3) The left-to-right inclusion is given by (1) and the converse by (2). (4) Note that 푝 + 푡 ≤ 푝 + 휛ℓ(푝)(푝 + 푡). Conversely, by using Clause (2) of Definition 3.7 we have that 휛푛(푝 + 휛ℓ(푝)(푝 + 푡)) = 휛푛(푝 + 푡) = 푡.

Lemma 3.9. Let 푝 ∈ 푃 . Then for each 푞 ∈ 푊 (푝), 푛 ≤ ℓ(푞) and 푡 ⪯푛 휛푛(푞), 푤(푝, 푞 + 푡) = 푤(푝, 푞). Proof. Note that 푤(푝, 푞 + 푡) and 푤(푝, 푞) are two compatible conditions in 푊 (푝) with the same length. In eﬀect, Lemma 3.6(1) yields the desired. Proposition 3.10. For every condition 푝 in P and an ordinal 훼 < 휅, there ′ exists an extension 푝 ≤ 푝 such that 휎ℓ(푝′) > 훼.

Proof. Let 푝 and 훼 be as above. Since 훼 < 휅 = sup푛<휔 휎푛, we may find some 푛 < 휔 such that 훼 < 휎푛. By Definition 3.3(1), (P, ℓ) is a graded poset, so by possibly iterating the second bullet of Definition 3.1 finitely many times, we may find an extension 푝′ ≤ 푝 such that ℓ(푝′) ≥ 푛. As Σ is non-decreasing, ′ 푝 is as desired. As in the context of Σ-Prikry forcings, also here, the CPP implies the Prikry Property (PP) and the Strong Prikry Property (SPP). Lemma 3.11. Let 푝 ∈ 푃 . (1) Suppose 휙 is a sentence in the language of forcing. Then there is 푝′ ≤0 푝, such that 푝′ decides 휙; (2) Suppose 퐷 ⊆ 푃 is a 0-open set which is dense below 푝. Then there ′ 0 푝′ 19 is 푝 ≤ 푝, and 푛 < 휔, such that 푃푛 ⊆ 퐷.

19 푞 Note that if 퐷 is moreover open, then 푃푚 ⊆ 퐷 for all 푚 ≥ 푛. SIGMA-PRIKRY FORCING III 15

′ ˚휛ℓ(푝) Moreover, we can let 푝 above to be a condition from Pℓ(푝) ↓ 푝. Proof. We only give the proof of (1), the proof of (2) is similar. Fix 휙 and 푝. + − Put 푈휙 := {푞 ∈ 푃 | 푞 P 휙} and 푈휙 := {푞 ∈ 푃 | 푞 P ¬휙}. Both of these are 0-open, so applying Clause (7) of Deﬁnition 3.3 twice, we get following: Claim 3.11.1. For all 푞 ∈ 푃 and 푛 < 휔, there is 푞′ ≤0 푞, such that either 푞′ 푞′ all 푟 ∈ 푃푛 decide 휙 the same way, or no 푟 ∈ 푃푛 decides 휙. 0 Now using the claim construct a ≤ decreasing sequence ⟨푝푛 | 푛 < 휔⟩ below 푝. By using Clause (2) of Deﬁnition 3.3 we may additionally assume ˚ ′ 0 that these are conditions in Pℓ(푝). Letting 푝 a ≤ -lower bound for this 0 sequence we obtain ≤ -extension of 푝 deciding 휙.

Corollary 3.12. Let 푝 ∈ 푃 and 푠 ⪯ℓ(푝) 휛ℓ(푝)(푝). (1) Suppose 휙 is a sentence in the language of forcing. Then there is ′ ⃗휛 ′ ′ ′ 푝 ≤ 푝 and 푠 ⪯ℓ(푝) 푠 such that 푝 + 푠 decides 휙; (2) Suppose 퐷 ⊆ 푃 is a 0-open set which is dense below 푝. Then there ′ ⃗휛 ′ 푝′+푠′ are 푝 ≤ 푝, 푠 ⪯ℓ(푝) 푠 and 푛 < 휔 such that 푃푛 ⊆ 퐷. ′ ˚휛ℓ(푝) Moreover, we can let 푝 above to be a condition from Pℓ(푝) ↓ 푝. Proof. We only show (1) as (2) is similar. By Lemma 3.11, let 푞 ≤0 푝 + 푠 deciding 휙. By Definition 3.3(8) the map 휛푛 is a nice projection, hence ′ ⃗휛 ′ ′ ′ there is 푝 ≤ 푝 and 푠 ⪯ℓ(푝) 푠 such that 푝 + 푠 = 푞 (Definition 2.2(4)). The ˚휛ℓ(푝) 휛ℓ(푝) moreover part follows from density of Pℓ(푝) in Pℓ(푝) (Definition 3.3(9)). Working a bit more we can obtain the following:

Lemma 3.13. Let 푝 ∈ 푃 . Set ℓ := ℓ(푝) and 푠 := 휛푛(푝). (1) Suppose 휙 is a sentence in the language of forcing. Then there is ⃗휛 푞 ≤ 푝 such that 퐷휙,푞 := {푡 ⪯ℓ 푠 | (푞 + 푡 P 휙) or (푞 + 푡 P ¬휙)} is dense in Sℓ ↓ 푠; (2) Suppose 퐷 ⊆ 푃 is a 0-open set. Then there is 푞 ≤⃗휛 푝 such that 푞+푡 푞+푡 푈퐷,푞 := {푡 ⪯ℓ 푠 | ∀푚 < 휔 (푃푚 ⊆ 퐷 or 푃푚 ∩ 퐷 = ∅)} is dense in Sℓ ↓ 푠. (3) Suppose 퐷 ⊆ 푃 is a 0-open set which is dense below 푝. Then there ⃗휛 푞+푡 is 푞 ≤ 푝 such that 푈퐷,푞 := {푡 ⪯ℓ 푠 | ∃푚 < 휔 푃푚 ⊆ 퐷} is dense in Sℓ ↓ 푠. ˚휛ℓ Moreover, 푞 above belongs to Pℓ ↓ 푝.

Proof. (1) By Deﬁnition 3.3(훽), let us ﬁx some cardinal 휃 < 휎ℓ along with an injective enumeration ⟨푠훼 | 훼 < 휃⟩ of the conditions in Sℓ ↓ 푠, such that 훼 푠0 = 푠. We will construct by recursion two sequences of conditions ⃗푝 = ⟨푝 | 훼 < 휃⟩ and ⃗푠 = ⟨푠훼 | 훼 < 휃⟩ for which all of the following hold: ⃗휛 ˚휛ℓ (a) ⃗푝 is a ≤ -decreasing sequence of conditions in Pℓ below 푝; (b) ⃗푠 is a sequence of conditions below 푠; 16 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

훼 훼 훼 (c) for each 훼 < 휃, 푠 ⪯푛 푠훼 and 푝 + 푠 ‖P 휙. To see that this will do, assume for a moment that there are sequences ⃗푝 ⃗휛 ˚휛ℓ and ⃗푠 as above. Since 휃 < 휎ℓ, we may find a ≤ -lower bound 푞 for ⃗푝 in Pℓ . ⃗휛 In particular, 푞 ≤ 푝. We claim that 퐷휙,푞 is dense in Sℓ ↓ 푠. To this end, ′ ′ let 푠 ⪯ℓ 푠 be arbitrary. Find 훼 < 휃 such that 푠 = 푠훼. By the hypothesis, 훼 훼 훼 훼 푠 ⪯ℓ 푠훼 and 푝 + 푠 decides 휙, hence 푞 + 푠 also decides it. In particular, 훼 ′ 푠 is an extension of 푠 belonging to 퐷휙,푞. Claim 3.13.1. There are sequences ⃗푝 and ⃗푠 as above. Proof. We construct the two sequences by recursion on 훼 < 휃. For the base case, appeal to Corollary 3.12(1) with 푝 and 푠, and retrieve 푝0 ≤⃗휛 푝 and 0 ˚휛ℓ 0 0 푠 ⪯푛 푠 such that 푝0 ∈ 푃ℓ and 푝 + 푠 indeed decides 휙. 훾 훾 I Assume 훼 = 훽 + 1 and that ⟨푝 | 훾 ≤ 훽⟩ and ⟨푠 | 훾 ≤ 훽⟩ have 훽 훽 been already defined. Since 푠훼 ⪯ℓ 푠 = 휛ℓ(푝 ), it follows that 푝 + 푠훼 is a 훽 legitimate condition in 푃ℓ. Appealing to Corollary 3.12(1) with 푝 and 푠훼, 훼 ⃗휛 훽 훼 훼 ˚휛ℓ 훼 훼 let 푝 ≤ 푝 and 푠 ⪯ℓ 푠훼 be such that 푝 ∈ 푃ℓ and 푝 + 푠 decides 휙. 훽 훽 I Assume 훼 ∈ acc(휃) and that the sequences ⟨푝 | 훽 < 훼⟩ and ⟨푠 | 훽 < 훼⟩ have already been defined. Appealing to Definition 3.3(9), let 푝* be a ≤⃗휛-lower bound for ⟨푝훽 | 훽 < 훼⟩. Finally, obtain 푝훼 ∈ 퐷 and 푠훼 by * appealing to Corollary 3.12(1) with respect to 푝 and 푠훼. This completes the proof of Clause (1). The proof of Clauses (2) and (3) is similar by amending suitably Clause (c) above. For instance, for Clause (2) we require the following in Clause (c): for each 훼 < 휃 and 푛 < 휔, 훼 푝훼+푠훼 푝훼+푠훼 푠 ⪯푛 푠훼 and either 푃푛 ⊆ 퐷 or 푃푛 ∩ 퐷 = ∅. For the verification of this new requirement we combine Clauses (2), (7) and (8) of Definition 3.3 with Definition 2.2(4). Similarly, to prove Clause (3) of the lemma one uses Clause (2) of Corollary 3.12. We now arrive at the main result of the section: Lemma 3.14 (Analysis of bounded sets).

(1) If 푝 ∈ 푃 forces that 푎˙ is a P-name for a bounded subset 푎 of 휎ℓ(푝), then 푎 is added by Sℓ(푝). In particular, if 푎˙ is a P-name for a bounded subset 푎 of 휅, then, for any large enough 푛 < 휔, 푎 is added by S푛; (2) P preserves 휅. Moreover, if 휅 is a strong limit, it remains so; (3) For every regular cardinal 휈 ≥ 휅, if there exists 푝 ∈ 푃 for which ⃗휛 20 푝 P cf(휈) < 휅, then there exists 푞 ≤ 푝 with |푊 (푞)| ≥ 휈; + (4) Suppose 1l P “휅 is singular”. Then 휇 = 휅 if and only if, for all 푝 ∈ 푃 , |푊 (푝)| ≤ 휅. Proof. (1) The “in particular” part follows from the ﬁrst part together with Proposition 3.10. Thus, let us suppose that 푝 is a given condition forcing that푎 ˙ is a name for a subset 푎 of some cardinal 휃 < 휎ℓ(푝).

20For future reference, we point out that this fact relies only on clauses (1), (5), (7), (8) and (9) of Definition 3.3. SIGMA-PRIKRY FORCING III 17

For each 훼 < 휃, denote the sentence “ˇ훼 ∈ 푎˙” by 휙훼. Set 푛 := ℓ(푝) and 푠 := 휛푛(푝). Combining Deﬁnition 3.3(9) with Lemma 3.13(1), we may recursively obtain a ≤휛푛 -decreasing sequence of conditions ⃗푝 = ⟨푝훼 | 훼 < 휃⟩ 훼 휛푛 훼 with a lower bound, such that, for each 훼 < 휃, 푝 ≤ 푝 and 퐷휙훼,푝 is dense 휛푛 in S푛 ↓ 푠. Then let 푞 ∈ 푃푛 be ≤ -below all elements of ⃗푝. It follows that for every 훼 < 휃,

퐷휙훼,푞 = {푡 ⪯푛 푠 | (푞 + 푡 P 휙훼) or (푞 + 푡 P ¬휙훼)} is dense in S푛 ↓ 푠. Now, let 퐺 be a P-generic ﬁlter with 푝 ∈ 퐺. Let 퐻푛 be the S푛-generic ﬁlter induced by 휛푛 from 퐺, and work in 푉 [퐻푛]. It follows that, for every 훼 < 휃, for some 푡 ∈ 퐻푛, either (푞 + 푡 P 훼ˇ ∈ 푎˙) or (푞 + 푡 P 훼ˇ ∈ / 푎˙). Set

푏 := {훼 < 휃 | ∃푡 ∈ 퐻푛[푞 + 푡 P 훼ˇ ∈ 푎˙]}. ⃗휛 As 푞 ≤ 푝, we infer that 휛푛(푞) = 휛푛(푝) = 푠 ∈ 퐻푛, so that 푞 ∈ 푃/퐻푛.

Claim 3.14.1. 푞 P/퐻푛 푏 =푎 ˙ 퐻푛 .

Proof. Clearly, 푞 P/퐻푛 푏 ⊆ 푎˙ 퐻푛 . For the converse, let 훼 < 휃 and 푟 ≤P/퐻푛 푞 be such that 푟 P/퐻푛 훼ˇ ∈ 푎˙ 퐻푛 . By the very Definition 2.4, there is 푡0 ∈ 퐻푛 with 푡0 ⪯푛 휛푛(푟) such that 푟 + 푡0 ≤ 푞. By extending 푡 if necessary, we may moreover assume that 푟 + 푡0 P 훼ˇ ∈ 푎˙. Set 푞0 := 푟 + 푡0. By the choice of 푞, there is 푡1 ∈ 퐻푛 such that 푞 + 푡1 ‖P 훼ˇ ∈ 푎˙. Set 푞1 := 푞 + 푡1. Let 푡 ∈ 퐻푛 be such that 푡 ⪯푛 푡0, 푡1. Recalling Definition 3.3(9), 휛푛 is nice, so 푡 ⪯푛 휛푛(푞0), 휛푛(푞1). By Definition 2.2(4), 푞0 + 푡 witnesses the compatibility of 푞0 and 푞1, hence 푞 + 푡1 P 훼ˇ ∈ 푎˙, and thus 훼 ∈ 푏.

Altogether,푎 ˙ 퐺 ∈ 푉 [퐻푛]. (2) If 휅 were to be collapsed, then, by Clause (1), it would have been collapsed by S푛 for some 푛 < 휔. However, S푛 is a notion of forcing of size < 휎푛 ≤ 휅. Next, suppose towards a contradiction that 휅 is strong limit cardinal, 휃 and yet, for some P-generic filter 퐺, for some 휃 < 휅, 푉 [퐺] |= 2 ≥ 휅. For each 푛 < 휔, let 퐻푛 be the S푛-generic filter induced by 휛푛 from 퐺. Using 푉 [퐺] Clause (1), for every 푎 ∈ 풫 (휃), we fix 푛푎 < 휔 such that 푎 ∈ 푉 [퐻푛푎 ]. I If 휅 is regular, then there must exist some 푛 < 휔 for which |{푎 ∈ 푉 [퐺] 풫 (휃) | 푛푎 = 푛}| ≥ 휅. However S푛 is a notion of forcing of some size 휆 < 휅, and so by counting nice names, we see it cannot add more than 휃휆 many subsets to 휃, contradicting the fact that 휅 is strong limit. I If 휅 is not regular, then Σ is not eventually constant, and cf(휅) = 휔, so that, by König’slemma, 푉 [퐺] |= 2휃 ≥ 휅+. It follows that exists some 푛 < 휔 푉 [퐺] for which |{푎 ∈ 풫 (휃) | 푛푎 = 푛}| > 휅, leading to the same contradiction. (3) Suppose 휃, 휈 are regular cardinals with 휃 < 휅 ≤ 휈, 푓˙ is a P-name for a function from 휃 to 휈, and 푝 ∈ 푃 is a condition forcing that the image of 푓˙ is cofinal in 휈. Denote 푛 := ℓ(푝) and 푠 := 휛푛(푝). By Proposition 3.10, we may extend 푝 and assume that 휎푛 > 휃. 18 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA ˙ ˇ For all 훼 < 휃, set 퐷훼 := {푟 ≤ 푝 | ∃훽 < 휈, 푟 P 푓(ˇ훼) = 훽}. As 퐷훼 is 0-open and dense below 푝, by combining Lemma 3.13(3) with the 휎푛- ˚휛푛 directed closure of P푛 (see Definition 3.3(9)), we may recursively define a ≤⃗휛-decreasing sequence of conditions ⟨푞훼 | 훼 ≤ 휃⟩ below 푝 such that, for 휃 훼 every 훼 < 휃, 푈퐷훼,푞 is dense in S푛 ↓ 푠. Set 푞 := 푞 , and note that 푞+푡 푈퐷훼,푞 := {푡 ⪯푛 푠 | ∃푚 < 휔[푃푛 ⊆ 퐷훼]} is dense in S푛 ↓ 푠 for all 훼 < 휃. In particular, the above sets are non-empty.

For each 훼 < 휃, let us ﬁx 푡훼 ∈ 푈퐷훼,푞 and 푚훼 < 휔 witnessing this. We now 푞+푡훼 ˙ ˇ show that |푊 (푞)| ≥ 휈. Let 퐴훼 := {훽 < 휈 | ∃푟 ∈ 푃푚훼 [푟 P 푓(ˇ훼) = 훽]}. By Lemma 3.6(1), we have ˙ ˇ 퐴훼 = {훽 < 휈 | ∃푟 ∈ 푊푚훼 (푞 + 푡훼)[푟 P 푓(ˇ훼) = 훽]}. ⋃︀ Let 퐴 := 훼<휃 퐴훼. Then, ∑︁ 21 |퐴| ≤ |푊푚(푞 + 푡)| ≤ max{ℵ0, |푆푛|} · |푊 (푞)|.

푚<휔,푡⪯푛푠 Also, by clauses (훼) and (훽) of Definition 3.3 and our assumption on 휈, max{ℵ0, |푆푛|} < 휎푛 < 휈. It follows that if |푊 (푞)| < 휈, then |퐴| < 휈, and so sup(퐴) < 휈. Thus, 푞 forces that the range of 푓˙ is bounded below 휈, which leads us to a contradiction. Therefore, |푊 (푞)| ≥ 휈, as desired. (4) The left-to-right implication is obvious using Definition 3.3(5). Next, suppose that, for all 푝 ∈ 푃 , |푊 (푝)| ≤ 휅. Towards a contradiction, suppose that there exist 푝 ∈ 푃 forcing that 휅+ is collapsed. Denote 휈 := 휅+. As by assumption 1l P “휅 is singular”, this means that 푝 P cf(휈) < 휅, contradicting Clause (3) of this lemma. We end this section recalling the concept of property 풟. This notion was introduced in [PRS20, S2] and usually captures how various forcings satisfy the Complete Prirky Property (i.e., Clause (7) of Definition 3.3): Definition 3.15 (Diagonalizability game). Given 푝 ∈ 푃 , 푛 < 휔, and a good enumeration ⃗푟 = ⟨푟휉 | 휉 < 휒⟩ of 푊푛(푝), we say that ⃗푞 = ⟨푞휉 | 휉 < 휒⟩ is diagonalizable (with respect to ⃗푟) iff the two hold: 0 (a) 푞휉 ≤ 푟휉 for every 휉 < 휒; ′ 0 ′ ′ ′ 0 (b) there is 푝 ≤ 푝 such that for every 푞 ∈ 푊푛(푝 ), 푞 ≤ 푞휉, where 휉 is ′ the unique index to satisfy 푟휉 = 푤(푝, 푞 ). Besides, if 퐷 is a dense subset of , (푝, ⃗푟, 퐷) is a game of length 휒 PℓP(푝)+푛 aP between two players I and II, defined as follows: 0 ∙ At stage 휉 < 휒, I plays a condition 푝휉 ≤ 푝 compatible with 푟휉, and 0 then II plays 푞휉 ∈ 퐷 such that 푞휉 ≤ 푝휉 and 푞휉 ≤ 푟휉; ∙ I wins the game iff the resulting sequence ⃗푞 = ⟨푞휉 | 휉 < 휒⟩ is diagonalizable. In the special case that 퐷 is all of , we omit it, writing (푝, ⃗푟). PℓP(푝)+푛 aP 21 Observe that, for each 푡 ⪯푛 푠, |푊 (푞 + 푡)| ≤ |푊 (푞)|. SIGMA-PRIKRY FORCING III 19

Definition 3.16 (Property 풟). We say that (P, ℓP) has property 풟 iﬀ for any 푝 ∈ 푃 , 푛 < 휔 and any good enumeration ⃗푟 = ⟨푟휉 | 휉 < 휒⟩ of 푊푛(푝), I 22 has a winning strategy for the game aP(푝, ⃗푟).

4. Extender Based Prikry Forcing with collapses In this section we present Gitik’s notion of forcing from [Git19b], and analyze its properties. Gitik came up with this notion of forcing in September 2019, during the week of the 15th International Workshop on Set Theory in Luminy, after being asked by the second author whether it is possible to interleave collapses in the Extender Based Prikry Forcing (EBPF) with long extenders [GM94, S3]. The following theorem summarizes the main properties of the generic extensions by Gitik’s forcing: Theorem 4.1 (Gitik). All of the following hold in 푉 P: (1) All cardinals ≥ 휅 are preserved; (2) 휅 = ℵ휔, 휇 = ℵ휔+1 and 휆 = ℵ휔+2; (3) ℵ휔 is a strong limit cardinal;

(4) GCH<ℵ휔 , provided that 푉 |= GCH<휅; ℵ휔 (5) 2 = ℵ휔+2, hence the SCHℵ휔 fails. For people familiar with the forcing many of the proofs in this section can be skipped. But since the forcing notion is fairly new, we include the details of some of its properties for posterity. Also, for us it is important to verify the existence of various nice projections and reﬂections properties in Corollary 4.30 and Lemma 4.32. Unlike the exposition of this forcing from [Git19b], the exposition here shall not assume the GCH. Setup 4. Throughout this section our setup will be as follows:

∙ ⃗휅 = ⟨휅푛 | 푛 < 휔⟩ is a strictly increasing sequence of cardinals; + + ∙ 휅−1 := ℵ0, 휅 := sup푛<휔 휅푛, 휇 := 휅 and 휆 := 휇 ; ∙ 휇<휇 = 휇 and 휆<휆 = 휆; ∙ for each 푛 < 휔, 휅푛 is (휆 + 1)-strong; + 23 ∙ Σ := ⟨휎푛 | 푛 < 휔⟩, where, for each 푛 < 휔, 휎푛 := (휅푛−1) ;

In particular, we are assuming that, for each 푛 < 휔, there is a (휅푛, 휆 + 1)- extender 퐸푛 whose associated embedding 푗푛 : 푉 → 푀푛 is such that 푀푛 is a 휅푛 transitive class, 푀푛 ⊆ 푀푛, 푉휆+1 ⊆ 푀푛 and 푗푛(휅푛) > 휆. For each 푛 < 휔, and each 훼 < 휆, set

퐸푛,훼 := {푋 ⊆ 휅푛 | 훼 ∈ 푗푛(푋)}.

Note that 퐸푛,훼 is a non-principal 휅푛-complete ultraﬁlter over 휅푛, provided that 훼 ≥ 휅푛. Moreover, in the particular case of 훼 = 휅푛, 퐸푛,휅푛 is also

22 In a mild abuse of terminology, we often say that (P, ℓ, 푐, ⃗휛) has property 풟 whenever the pair (P, ℓ) has property 풟. 23 In particular, 휎0 = ℵ1. 20 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA normal. For ordinals 훼 < 휅푛 the measures 퐸푛,훼 are principal so the only reason to consider them is for a more neat presentation.

For each 푛 < 휔, we shall consider an ordering ≤퐸푛 over 휆, as follows: Definition 4.2. For each 푛 < 휔, set

휅푛 ≤퐸푛 := {(훽, 훼) ∈ 휆 × 휆 | 훽 ≤ 훼, ∧ ∃푓 ∈ 휅푛 푗푛(푓)(훼) = 훽}.

It is routine to check that ≤퐸푛 is reflexive, transitive and antisymmetric, hence (휆, ≤퐸푛 ) is a partial order. In case 훽 ≤퐸푛 훼, we shall fix in advance a witnessing map 휋훼,훽 : 휅푛 → 휅푛. Also, in the special case where 훼 = 훽, by convention, 휋훼,훼 =: id. Observe that ≤퐸푛 (휅푛 × 휅푛) is exactly the ∈-order over 휅푛 so that when we refer to ≤퐸푛 we will really be speaking about the restriction of this order to 휆 ∖ 휅푛. The most notable property of the poset <휅푛 (휆, ≤퐸푛 ) is that it is 휅푛-directed: that is, for every 푎 ∈ [휆] there is 훼 < 휆 such that 훽 ≤퐸푛 훼 for all 훽 ∈ 푎. This and other nice features of (휆, ≤퐸푛 ) are proved at the beginning of [Git10, S2] under the unnecessary assumption of the GCH. A proof without GCH may be found in [?, S10.2]. Remark 4.3. For future reference, it is worth mentioning that all the relevant properties of (휆, ≤퐸푛 ) reflect down to (휇, ≤퐸푛 휇 × 휇). In particular, it is <휅푛 * * true that every 푎 ∈ [휆] may be enlarged to an 푎 such that 휅푛, 휇 ∈ 푎 * and 푎 ∩ 휇 contains a ≤퐸푛 -greatest. For details, see [Git10, S2]. 4.1. The forcing. Before giving the definition of Gitik’s forcing we shall first introduce the basic building block modules Q푛0 and Q푛1. To that effect, for each 푛 < 휔, let us fix 푠푛 : 휅푛 → 휅푛 be a map representing 휇 in the normal ultrapower 퐸푛,휅푛 . Specifically, for each 푛 < 휔, 푗푛(푠푛)(휅푛) = 휇.

Definition 4.4. For each 푛 < 휔, define Q푛1, Q푛0 and Q푛 as follows: 푝 푝 푝 0푝 1푝 2푝 (0)푛 Q푛0 := (푄푛0, ≤푛0) is the set of 푝 := (푎 , 퐴 , 푓 , 퐹 , 퐹 , 퐹 ), where: 푝 푝 푝 * (1)( 푎 , 퐴 , 푓 ) is in the 푛0-module 푄푛0 from the Extender Based Prikry Forcing (EBPF) as defined in [Git10, Definition 2.6]. 푝 푝 Moreover, we require that 휅푛, 휇 ∈ 푎 and that 푎 ∩ 휇 contains 푝 24 a ≤퐸푛 -greatest element denoted by mc(푎 ∩ 휇); 푖푝 푝 푖푝 (2) for 푖 < 3, dom(퐹 ) = 휋mc(푎푝),mc(푎푝∩휇)[퐴 ], and for 휈 ∈ dom(퐹 ), 푝 setting 휈0 := 휋mc(푎 ∩휇),휅푛 (휈), we have: 0푝 (a) 퐹 (휈) ∈ Col(휎푛, <휈0); 1푝 (b) 퐹 (휈) ∈ Col(휈0, 푠푛(휈0)); 2푝 ++ (c) 퐹 (휈) ∈ Col(푠푛(휈0) , <휅푛). 푞 푞 푞 The ordering ≤ is defined as follows: 푞 ≤ 푝 iff (푎 , 퐴 , 푓 ) ≤ * 푛0 푛0 Q푛0 (푎푝, 퐴푝, 푓 푝) as in [Git10, Definition 2.7], and for each 휈 ∈ dom(퐹 푖푞), 푖푞 푖푝 ′ ′ 퐹 (휈) ⊇ 퐹 (휈 ), where 휈 := 휋mc(푎푞∩휇),mc(푎푝∩휇)(휈). 푝 푝 0푝 1푝 2푝 (1)푛 Q푛1 := (푄푛1, ≤푛1) is the set of 푝 := (푓 , 휌 , ℎ , ℎ , ℎ ), where:

24 푝 푝 푝 * 푝 Recall that (푎 , 퐴 , 푓 ) ∈ 푄푛0 in particular implies that 푎 contains a ≤퐸푛 -greatest element, which is typically denoted by mc(푎푝). Note that since 휇 ∈ 푎푝 then mc(푎푝) is 푝 always strictly ≤퐸푛 -larger than mc(푎 ∩ 휇). SIGMA-PRIKRY FORCING III 21

푝 ≤휅 (1) 푓 is a function from some 푥 ∈ [휆] to 휅푛; 푝 (2) 휌 < 휅푛 inaccessible; 0푝 푝 (3) ℎ ∈ Col(휎푛, <휌 ); 1푝 푝 푝 (4) ℎ ∈ Col(휌 , 푠푛(휌 )); 2푝 푝 ++ (5) ℎ ∈ Col(푠푛(휌 ) , <휅푛). 푞 푝 푝 푞 The ordering ≤푛1 is defined as follows: 푞 ≤푛1 푝 iff 푓 ⊇ 푓 , 휌 = 휌 , and for 푖 < 3, ℎ푖푞 ⊇ ℎ푖푝. (2)푛 Set Q푛 := (푄푛0 ∪ 푄푛1, ≤푛) where the ordering ≤푛 is defined as follows: for each 푝, 푞 ∈ 푄푛, 푞 ≤푛 푝 iff (1) either 푝, 푞 ∈ 푄푛푖, some 푖 ∈ {0, 1}, and 푞 ≤푛푖 푝, or 푝 (2) 푞 ∈ 푄푛1, 푝 ∈ 푄푛0 and, for some 휈 ∈ 퐴 , 푞 ≤푛1 푝y⟨휈⟩, where y 푝 푝 0푝 1푝 2푝 푝 ⟨휈⟩ := (푓 ∪ {⟨훽, 휋mc(푎푝),훽(휈)⟩ | 훽 ∈ 푎 ⟩}, 휈¯0, 퐹 (¯휈), 퐹 (¯휈), 퐹 (¯휈)),

and휈 ¯ = 휋mc(푎푝),mc(푎푝∩휇)(휈).

Remark 4.5. For each 푛 < 휔 and all 훼 ≥ 휅푛 we have + {휌 < 휅푛 | (휅푛−1) < 휌 < 푠푛(휌) < 휅푛 & 휌 inaccessible} ∈ 퐸푛,훼.

<휅푛 Similarly, for 푎 ∈ [휆] as in (1)푛 above and 퐴 ∈ 퐸푛,mc(푎), 푝 + (⋆) {휌 ∈ 휋mc(푎),mc(푎∩휇)“퐴 | |{휈 ∈ 퐴푛 | 휈¯0 = 휌0}| ≤ 푠푛(휌0) } ∈ 퐸푛,mc(푎∩휇). We will only use these measures of the extenders, and by restricting to a measure one set, we assume that the above is always the case for all 휌 < 휅푛 that we ever consider. Similarly, we may also assume that 푠푛(휌) is regular 푝 <휌푝 푝 (actually the successor of a singular) and that 푠푛(휌 ) = 푠푛(휌 ). The reason we consider conditions witnessing Clause (⋆) above is related with the verification of property 풟 and CPP (cf. lemmas 4.20 and 4.21). Essentially, when we describe the moves of I and II we would like to be able to take lower bounds of the top-most collapsing maps appearing in conditions played by II. Namely, we would like to take lower bounds of the ℎ2푞휉 ’s. Assuming (⋆) we will have that the number of maps that need to be + amalgamated is at most 푠푛(휈0) , hence less than the completeness of the ++ top-most Lèvycollapse Col(푠푛(휈0) , < 휅푛+1). 푖푝 Remark 4.6. The reason Gitik makes 퐹푛 dependent on the partial extender 퐸푛 휇 rather than on the full extender 퐸푛 is related with the verification 0푝 1푝 2푝 of the chain condition. Indeed, in that way the triple ⟨퐹푛 , 퐹푛 , 퐹푛 ⟩ will represent three (partial) collapsing functions as computed in the ultrapower by 퐸푛 휇. Observe that, from the perspective of 푉 , these collapses have size <휇, hence there cannot be 휇+-many incompatible conditions. In particular, this will guarantee that the map given in Definition 4.10 has range 퐻휇. Having all necessary building blocks, we can now define the poset P. Definition 4.7. The Extender Based Prikry Forcing with collapses (EBPFC) is the poset P := (푃, ≤) defined by the following clauses: ∏︀ ∙ Conditions in 푃 are sequences 푝 = ⟨푝푛 | 푛 < 휔⟩ ∈ 푛<휔 푄푛. 22 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

∙ For all 푝 ∈ 푃 , – There is 푛 < 휔 such that 푝푛 ∈ 푄푛0; 푝푛 푝푚 – For every 푛 < 휔, if 푝푛 ∈ 푄푛0 then 푝푚 ∈ 푄푚0 and 푎 ⊆ 푎 , for every 푚 ≥ 푛. ∙ For all 푝, 푞 ∈ 푃 , 푝 ≤ 푞 iﬀ 푝푛 ≤푛 푞푛, for every 푛 < 휔.

Definition 4.8. ℓ : 푃 → 휔 is deﬁned by letting for all 푝 = ⟨푝푛 | 푛 < 휔⟩,

ℓ(푝) := min{푛 < 휔 | 푝푛 ∈ 푄푛0}.

Notation 4.9. Given 푝 ∈ 푃 , 푝 = ⟨푝푛 | 푛 < 휔⟩, we will typically write 푝 푝 0푝 1푝 2푝 푝 푝 푝 0푝 1푝 2푝 푝푛 = (푓푛, 휌푛, ℎ푛 , ℎ푛 , ℎ푛 ) for 푛 < ℓ(푝), and 푝푛 = (푎푛, 퐴푛, 푓푛, 퐹푛 , 퐹푛 , 퐹푛 ) 푝 for 푛 ≥ ℓ(푝). Also, for each 푛 ≥ ℓ(푝), we shall denote 훼푝푛 := mc(푎푛 ∩ 휇). + We already have (P, ℓ) and we will eventually check that 1l P 휇ˇ =휅 ˇ ⃗ (Corollary 4.24). Next, we introduce sequences S = ⟨S푛 | 푛 < 휔⟩ and ⃗휛 = ⟨휛푛 | 푛 < 휔⟩, and a map 푐 : 푃 → 퐻휇 such that (P, ℓ, 푐, ⃗휛) will be a (Σ,⃗S)-Prikry forcing having property 풟. As in [?, S10.2], using 휇휅 = 휇 and 2휇 = 휆, we ﬁx a sequence of functions ⟨푒푖 | 푖 < 휇⟩ from 휆 to 휇 such that, for all 푥 ∈ [휆]휅 and every function 푒 : 푥 → 휇, there exists 푖 < 휇 with 푒 ⊆ 푒푖.

Definition 4.10. For every condition 푝 = ⟨푝푛 | 푛 < 휔⟩ in P, deﬁne a 25 sequence of indices ⟨푖(푝푛) | 푛 < 휔⟩ as follows: {︃ min{푖 < 휇 | 푓 ⊆ 푒푖}, if 푛 < ℓ(푝); 푖(푝푛) := 푖 푝 푖 푝 푝 min{푖 < 휇 | 푒 푎푛 = 0 & 푒 dom(푓푛) = 푓푛 + 1}, if 푛 ≥ ℓ(푝).

Deﬁne a map 푐 : 푃 → 퐻휇, by letting for any condition 푝 = ⟨푝푛 | 푛 < 휔⟩, 푝 ⃗ 푝 ⃗ 푝 푐(푝) := (ℓ(푝), ⟨휌푛 | 푛 < ℓ(푝)⟩, ⟨푖(푝푛) | 푛 < 휔⟩, ⟨ℎ푛 | 푛 < ℓ(푝)⟩, ⟨퐺푛 | 푛 ≥ ℓ(푝)⟩), ⃗ 푝 푖푝 ⃗ 푝 푖푝 where ℎ푛 := ⟨ℎ푛 | 푖 < 3⟩ and 퐺푛 := ⟨푗푛(퐹푛 )(훼푝푛 ) | 푖 < 3⟩. Definition 4.11. For each 푛 < 휔, set {︃ {1l}, if 푛 = 0; 푆푛 := 푝 0푝 1푝 2푝 {⟨(휌푘, ℎ푘 , ℎ푘 , ℎ푘 ) | 푘 < 푛⟩ | 푝 ∈ 푃푛}, if 푛 ≥ 1.

For 푛 ≥ 1 and 푠, 푡 ∈ 푆푛, write 푠 ⪯푛 푡 iﬀ there are 푝, 푞 ∈ 푃푛 with 푝 ≤ 푞 witnessing, respectively, that 푠 and 푡 are in 푆푛. ⃗ Denote S푛 := (푆푛, ⪯푛) and set S := ⟨S푛 | 푛 < 휔⟩.

Remark 4.12. Observe that |푆푛| < 휎푛. Moreover, for each 푠 ∈ 푆푛 ∖ {1lS푛 }, ∼ S푛 ↓ 푠 = Col(훿, <휅푛−1)×Q, where Q is a notion of a forcing of size < 훿 such that 휎푛−1 < 훿 < 휅푛−1. Speciﬁcally, if 푝 ∈ 푃푛 is the condition from which 푠 푝 ++ arises, then 훿 = 푠푛−1(휌푛−1) and Q is a product 푝 푝 푝 R × Col(휎푛−1, <휌푛−1) × Col(휌푛−1, 푠푛−1(휌푛−1)),

25Here 0 stands for the constant map with value 0. SIGMA-PRIKRY FORCING III 23

26 where R is a notion of forcing of size ≤ 휅푛−2. Also, by combining Easton’s lemma with a counting of nice names, if the GCH holds below 휅, then S푛 ↓ 푠 preserves this behavior of the power set function for each 푠 ∈ 푆푛 ∖ {1lS푛 }. On another note, observe that the the map (푞, 푠) ↦→ 푞 + 푠 yields an 휛푛 27 isomorphism between (S푛 ↓ 휛푛(푝)) × (P푛 ↓ 푝) and P푛 ↓ 푝.

Definition 4.13. For each 푛 < 휔, deﬁne 휛푛 : 푃≥푛 → 푆푛 as follows: {︃ {1l}, if 푛 = 0; 휛푛(푝) := 푝 0푝 1푝 2푝 ⟨(휌푘, ℎ푘 , ℎ푘 , ℎ푘 ) | 푘 < 푛⟩, if 푛 ≥ 1.

Set ⃗휛 := ⟨휛푛 | 푛 < 휔⟩. The next lemma collects some useful properties about the 푛0-modules of EBPFC (i.e, the Q푛0’s) and reveals some of their connections with the * corresponding modules of EBPF (i.e, then Q푛0’s). Lemma 4.14. Let 푛 < 휔. All of the following hold: * (1) P푛 projects to Q푛0, and this latter projects to Q푛0. * (2) Q푛0 is 휅푛-directed-closed, while Q푛0 is 휎푛-directed-closed. (3) S푛 satisfies the (휅푛−1)-cc. 푝 푝 푝 0푝 1푝 2푝 Proof. (1) The map 푝 ↦→ (푎푛, 퐴푛, 푓푛, 퐹푛 , 퐹푛 , 퐹푛 ) is a projection between 0 1 2 P푛 and Q푛0. Similarly, (푎, 퐴, 푓, 퐹 , 퐹 , 퐹 ) ↦→ (푎, 퐴, 푓) defines a projection * between Q푛0 and Q푛0. * (2) The argument for the 휅푛-directed-closedness of Q푛0 is given in [?, Lemma 10.2.40]. Let 퐷 ⊆ Q푛0 be a directed set of size <휎푛 and denote by 휚푛 * the projection between Q푛0 and Q푛0 given in the proof of item (1). Clearly, * 휚푛[퐷] is a directed subset of Q푛0 of size <휎푛, so that we may let (푎, 퐴, 푓) be a ≤ * -lower bound for it. By ≤ * -extending (푎, 퐴, 푓) we may assume that Q푛0 Q푛0 휅푛, 휇 ∈ 푎 and that 푎∩휇 contains a ≤퐸푛 -greatest element. Set 훼 := mc(푎∩휇). 푖 ⋃︀ 푖푝 For each 푖 < 3 and each 휈 ∈ 휋mc(푎)훼[퐴], define 퐹 (휈) := 푝∈퐷 퐹 (휋훼,훼푝 (휈)). 0 1 2 Finally, (푎, 퐴, 푓, 퐹 , 퐹 , 퐹 ) is a condition in Q푛0 extending every 푝 ∈ 퐷. (3) This is immediate from the definition of S푛 (Definition 4.11).

4.2. EBPFC is (Σ,⃗S)-Prikry. We verify that (P, ℓ, 푐, ⃗휛) is a (Σ,⃗S)-Prikry having property 풟. To that eﬀect, we go over the clauses of Deﬁnition 3.3.

Convention 4.15. For every sequence {퐴푘}푖≤푘≤푗 such that each 퐴푘 is a ∏︀푗 subset of 휅푘, we shall identify 푘=푖 퐴푘 with its subset consisting only of the sequences that are moreover increasing.

Definition 4.16. Let 푝 = ⟨푝푛 | 푛 < 휔⟩ ∈ 푃 . Deﬁne: ∙ 푝y∅ := 푝;

26 In the particular case where 푛 = 1 the poset R is trivial. 27In general terms the above map simply defines a projection (see Definition 2.2(4)) but in the particular case of the EBPFC it moreover gives an isomorphism. 24 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

푝 y ∙ For every 휈 ∈ 퐴ℓ(푝), 푝 ⟨휈⟩ is the unique condition 푞 = ⟨푞푛 | 푛 < 휔⟩, such that for each 푛 < 휔: {︃ 푝푛, if 푛 ̸= ℓ(푝) 푞푛 = y 푝ℓ(푝) ⟨휈⟩, otherwise. ∏︀푚+1 푝 ∙ Inductively, for all 푚 ≥ ℓ(푝) and ⃗휈 = ⟨휈ℓ(푝), . . . , 휈푚, 휈푚+1⟩ ∈ 푛=ℓ(푝) 퐴푛, set 푝y⃗휈 := (푝y⃗휈 (푚 + 1))y⟨휈푚+1⟩. Fact 4.17. Let 푝, 푞 ∈ 푃 . 0 ∙ 푞 ≤ 푝 iff ℓ(푝) = ℓ(푞) and 푞 ≤푛 푝, for each 푛 < 휔; ∏︀ℓ(푞)−1 푝 0 y ∙ 푞 ≤ 푝 iff there is ⃗휈 ∈ 푛=ℓ(푝) 퐴푛 such that 푞 ≤ 푝 ⃗휈; ∙ The sequence ⃗휈 above is uniquely determined by 푞. Specifically, for 푞 푝 each 푛 ∈ [ℓ(푝), ℓ(푞)), 휈푛 = 푓푛(mc(푎푛)). By the very definition of the EBPFC (Definition 4.7) and the function ℓ (Definition 4.8), (P, ℓ) is a graded poset, hence (P, ℓ, 푐, ⃗휛) witnesses Clause (1). Also, combining Lemma 4.14(2) with the fact that all of the L‘evy collapses considered are at least ℵ1-closed, Clause (2) follows:

Lemma 4.18. For all 푛 < 휔, P푛 is ℵ1-closed. We now verify that the map of Definition 4.10 witnesses Clause (3): 푝 푞 Lemma 4.19. For all 푝, 푞 ∈ 푃 , if 푐(푝) = 푐(푞), then 푃0 ∩ 푃0 is non-empty. Proof. Let 푝, 푞 ∈ 푃 and assume that 푐(푝) = 푐(푞). By Definition 4.10, we 푝 푞 푝 have ℓ(푝) = ℓ(푞) and 휌푛 = 휌푛 for all 푛 < ℓ(푝). Set ℓ := ℓ(푝) and 휌푛 := 휌푛 for 푝 푞 푝 푞 each 푛 < ℓ. Also, 푐(푝) = 푐(푞) yields ⃗ℎ푛 = ⃗ℎ푛 for each 푛 < ℓ and 퐺⃗ 푛 = 퐺⃗ 푛, ⃗ ⃗ 푝 ⃗ 0 1 2 for each 푛 ≥ ℓ. For 푛 < ℓ, put ℎ푛 := ℎ푛 and denote ℎ푛 = (ℎ푛, ℎ푛, ℎ푛). 푝 푞 We now define a condition 푟 witnessing that 푃0 ∩ 푃0 is non-empty. 푝 푞 푖 I If 푛 < ℓ then 푐(푝) = 푐(푞) implies 푖 = 푖(푝푛) = 푖(푞푛), and so 푓푛 ∪ 푓푛 ⊆ 푒 . 푝 푞 0 1 2 Set 푟푛 := (휌푛, 푓푛 ∪ 푓푛, ℎ푛, ℎ푛, ℎ푛). Clearly, 푟푛 ∈ 푄푛1. 푟 I If 푛 ≥ ℓ, put 푎ℓ−1 := ∅ and argue by recursion as follows: Since 푐(푝) = 푐(푞) implies 푖 = 푖(푝푛) = 푖(푞푛), arguing as in [?, Lemma 10.2.41] it 푝 푞 푞 푝 푝 푝 푝 follows that 푎푛 ∩dom(푓푛) = 푎푛 ∩dom(푓푛) = ∅. This implies that (푎푛, 퐴푛, 푓푛) 푞 푞 푞 * 푟 푟 푟 and (푎푛, 퐴푛, 푓푛) are two compatible conditions in Q푛0. Let (푎푛, 퐴푛, 푓푛) be in * 푟 푟 Q푛0 witnessing this and such that 푎푛−1 ⊆ 푎푛. As usual, we may assume that 푟 푟 휅푛, 휇 ∈ 푎푛 and that 푎푛 ∩휇 has a ≤퐸푛 -maximal element 훼푟푛 (see Remark 4.3). 푝 푞 Let us now define the 퐹 -part of 푟푛. Since 퐺⃗ 푛 = 퐺⃗ 푛, for each 푖 < 3, 푖푝 푖푞 푖푝 푖푝 푗푛(퐹푛 )(훼푝푛 ) = 푗푛(퐹푛 )(훼푞푛 ). Also 푗푛(퐹푛 )(훼푝푛 ) = 푗푛(퐹푛 ∘ 휋훼푟푛 ,훼푝푛 )(훼푟푛 ). 푞푖 Similarly, the same applies for 푗푛(퐹푛 )(훼푞푛 ). Thus,

퐸푛,훼푟푛 푖푝 푖푞 ∀ 휈 (퐹푛 (휋훼푟푛 ,훼푝푛 (휈)) = 퐹푛 (휋훼푟푛 ,훼푞푛 (휈))) 푟 푖푟 푟 holds. Shrinking 퐴 appropriately, we deﬁne 퐹 with domain 휋 푟 [퐴 ] 푛 푛 mc(푎푛),훼푟푛 푛 푖푝 푟 푟 푟 0푟 1푟 2푟 as 휈 ↦→ 퐹푛 (휋훼푟푛 ,훼푝푛 (휈)). Clearly, 푟푛 := (푎푛, 퐴푛, 푓푛, 퐹푛 , 퐹푛 , 퐹푛 ) ∈ 푄푛0 and it is routine to check that 푟푛 ≤푛0 푝푛, 푞푛. SIGMA-PRIKRY FORCING III 25

Finally, putting 푟 := ⟨푟푛 | 푛 < 휔⟩ we get a condition in P witnessing that 푝 푞 the set 푃0 ∩ 푃0 is non-empty. The verification of Clauses (4), (5) and (6) is the same as in [?, Lemma 10.2.45, 10.2.46 and 10.2.47], respectively. It is worth saying that regarding Clause (5) we actually have that |푊 (푝)| ≤ 휅 for each 푝 ∈ 푃 . We now show that (P, ℓ) has property 풟 and that it satisfies Clause (7). Lemma 4.20. (P, ℓ) has property 풟. Proof. Let 푝 ∈ 푃 , 푛 < 휔 and ⃗푟 be a good enumeration of 푊푛(푝). Our aim is to show that I has a winning strategy in the game aP(푝, ⃗푟). To enlighten the exposition we just give details for the case when 푛 = 1. The general argument can be composed using the very same ideas. 0 1 2 a 0 1 2 Write 푝 = ⟨(푓푛, 휌푛, ℎ푛, ℎ푛, ℎ푛) | 푛 < ℓ⟩ ⟨(푎푛, 퐴푛, 푓푛, 퐹푛 , 퐹푛 , 퐹푛 ) | 푛 ≥ ℓ⟩. By Fact 4.17, we can identify ⃗푟 with ⟨휈휉 | 휉 < 휅ℓ⟩, a good enumeration of 퐴ℓ. Specifically, for each 휉 < 휅ℓ we have that 푟휉 = 푝y⟨휈휉⟩. Using this enumeration we define a sequence ⟨(푝휉, 푞휉) | 휉 < 휅ℓ⟩ of moves in aP(푝, ⃗푟). 0 To begin with, I plays 푝0 := 푝 and in response II plays some 푞0 ≤ 푟0 with 푞0 ≤ 푝0. Note that this move is possible, as 푝0 and 푟0 are compatible. Suppose by induction that we have defined a sequence ⟨(푝휂, 푞휂) | 휂 < 휉⟩ of moves in aP(푝, ⃗푟) which moreover satisfies the following: (1) For each 푛 < ℓ the following hold: 푝휉 0푝휉 0 1푝휉 1 2푝휉 2 (a) for all 휂 < 휉, 휌푛 = 휌푛, ℎ푛 = ℎ푛, ℎ푛 = ℎ푛, ℎ푛 = ℎ푛; 푞휁 푝휂 (b) for all 휁 < 휂 < 휉, 푓푛 ⊆ 푓푛 ; (2) For all 휁 < 휂 < 휉 and 푛 > ℓ,(푞휂)푛 ≤푛0 (푝휂)푛 ≤푛0 (푞휁 )푛; (3) For all 휂 < 휉: 푝휉 푝휉 0푝휉 0 1푝휉 1 (a) 푎ℓ = 푎ℓ, 퐴ℓ = 퐴ℓ, 퐹ℓ = 퐹ℓ and 퐹ℓ = 퐹ℓ ; 2푞휁 2푞휂 (b) for each 휁 < 휂, if (¯휈휁 )0 = (¯휈휂)0 then ℎℓ ⊆ ℎℓ . Let us show how to define the 휉th move of I:

Successor case: Suppose 휉 = 휂 + 1. Then put 푝휉 := ⟨(푝휉)푛 | 푛 < 휔⟩, where ⎧ 푞 (푓 휂 , 휌 , ℎ0 , ℎ1 , ℎ2 ), if 푛 < ℓ; ⎨⎪ 푛 푛 푛 푛 푛 푞휂 0 1 2휉 (푝휉)푛 := (푎푛, 퐴푛, 푓푛 ∖ 푎푛, 퐹푛 , 퐹푛 , 퐹 ), if 푛 = ℓ; ⎪ ⎩(푞휂)푛, if 푛 > ℓ. 2휉 Here 퐹 denotes the map with domain 휋 “퐴ℓ deﬁned as mc(푎ℓ),훼푝ℓ {︃ 2 ⋃︀ 푞휁 ,2 휉,2 퐹ℓ (¯휈) ∪ {ℎℓ | 휁 < 휉, (¯휈휁 )0 = (¯휈휉)0}, if 휈 = 휈휉; 퐹 (¯휈) := 2 퐹ℓ (¯휈), otherwise. By Clauses (3) of the induction hypothesis and our comments in Remark 4.5, 2휉 퐹 is a function. A moment’s reﬂection makes clear that 푝휉 is a condition 0 in P witnessing (1) and (3)(a) above. Also, 푝휉 ≤ 푝 and 푝휉 is compatible 28 0 with 푟휉, hence it is a legitimate move for I. In response, II plays 푞휉 ≤ 푟휉

28 y Note that 푝휉 ⟨휈휉⟩ ≤ 푝휉, 푟휉. 26 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA such that 푞휉 ≤ 푝휉. In particular, for each 푛 > ℓ,(푞휉)푛 ≤푛0 (푝휉)푛 ≤푛0 (푞휂)푛, 2휉 2푞휉 and also 퐹 (¯휈휉) ⊆ ℎℓ . This combined with the induction hypothesis yield Clause (2) and (3)(b), which completes the successor case.

Limit case: In the limit case we put 푝휉 := ⟨(푝휉)푛 | 푛 < 휔⟩, where

⎧ 푞휂 (⋃︀ 푓 , 휌 , ℎ0 , ℎ1 , ℎ2 ), if 푛 < ℓ; ⎨⎪ 휂<휉 푛 푛 푛 푛 푛 ⋃︀ 푞휂 0 1 2휉 (푝휉)푛 := (푎푛, 퐴푛, 휂<휉(푓푛 ∖ 푎푛), 퐹푛 , 퐹푛 , 퐹 ), if 푛 = ℓ; ⎪ * ⎩(푞휉 )푛, if 푛 > ℓ.

2휉 * Here, 퐹 is deﬁned as before and (푞휉 )푛 is a lower bound for the sequence ⟨(푝휂)푛 | 휂 < 휉⟩. Note that this choice is possible because the orderings ≤푛0 are 휎ℓ+1-directed-closed. Once again, 푝휉 is a legitimate move for I and, in response, II plays 푞휉. It is routine to check that (1)–(3) above hold.

After this process we get a sequence ⟨(푝휉, 푞휉) | 휉 < 휅ℓ⟩. We next show * 0 how to form a condition 푝 ≤ 푝 diagonalizing ⟨푞휉 | 휉 < 휅ℓ⟩. ′ Note that by shrinking 퐴ℓ to some 퐴ℓ we may assume that there are *0 *1 *2 푖푞휉 *푖 ′ maps ⟨(ℎ푛 , ℎ푛 , ℎ푛 ) | 푛 < ℓ⟩ such that ℎ푛 = ℎ푛 for all 휈휉 ∈ 퐴ℓ and 푖 < 3. ′ 0푞휈 1푞휈 29 Next, deﬁne a map 푡 with domain 퐴ℓ such that 푡(휈) := ⟨ℎℓ , ℎℓ ⟩. Since 푀 퐸ℓ 푗ℓ(푡)(mc(푎ℓ)) ∈ 푉휅+1 we can argue as in [Git19b, Claim 1] that there is ′ ′ * 훼 < 휇 and a map 푡 such that 푗ℓ(푡)(mc(푎ℓ)) = 푗ℓ(푡 )(훼). Now let 푎ℓ be such * that 푎ℓ ∪ {훼} ⊆ 푎ℓ witnessing Clause (1) of Deﬁnition 4.4(0)푛. Then, ′ 퐴 := {휈 < 휅 | 푡 ∘ 휋 * (휈) = 푡 ∘ 휋 * ∘ 휋 * * (휈)} ℓ mc(푎ℓ ),mc(푎ℓ) mc(푎ℓ ∩휇),훼 mc(푎ℓ ),mc(푎ℓ ∩휇)

* −1 ′ is 퐸ℓ,mc(푎*)-large. Set 퐴ℓ := 퐴 ∩ 휋 * 퐴ℓ and ℓ mc(푎ℓ ),mc(푎ℓ) ′ * 푡ˆ:= (푡 ∘ 휋 * ) 휋 * * “퐴 . mc(푎ℓ ∩휇),훼 mc(푎ℓ ),mc(푎ℓ ∩휇) ℓ * ′ * Note that 휋 * “퐴 ⊆ 퐴 ⊆ 퐴 . Also, for each 휈 ∈ 퐴 , mc(푎ℓ ),mc(푎ℓ) ℓ ℓ ℓ ℓ

0푞휈˜ 1푞휈˜ 푡ˆ(휋 * * (휈)) = 푡(˜휈) = ⟨ℎ , ℎ ⟩, mc(푎ℓ ),mc(푎ℓ ∩휇) ℓ ℓ *,푖 where휈 ˜ := 휋 * (휈). For each 푖 < 2, deﬁne a map 퐹 with domain mc(푎ℓ ),mc(푎ℓ) ℓ * * 휋 * “퐴 , such that for each 휈 ∈ 퐴 , mc(푎ℓ ),mc(푎ℓ) ℓ ℓ

*,푖 푖푞휈˜ 퐹 (휋 * * (휈)) := ℎ . ℓ mc(푎ℓ ),mc(푎ℓ ∩휇) ℓ *,2 Similarly, deﬁne 퐹ℓ by taking lower bounds over the stages of the inductive * construction mentioning ordinals 휈휂 ∈ 휋 * “퐴 ; i.e., mc(푎ℓ ),mc(푎ℓ) ℓ

*,2 ⋃︁ 2푞휈 * 퐹 (휋 * * (휈)) := {ℎ 휂 | 휈휂 ∈ 휋 * “퐴 , (휈˜¯)0 = (¯휈휂)0}. ℓ mc(푎ℓ ),mc(푎ℓ ∩휇) mc(푎ℓ ),mc(푎ℓ) ℓ

29 In a slight abuse of notation, here we are identifying 푞휈 with 푞휉, where 휈 = 휈휉. SIGMA-PRIKRY FORCING III 27

* * Next, deﬁne 푝 := ⟨푝푛 | 푛 < 휔⟩, where ⎧ ⋃︀ 푞휉 *0 *1 *2 ( 푓푛 , 휌푛, ℎ , ℎ , ℎ ), if 푛 < ℓ; ⎨⎪ 휉<휅ℓ 푛 푛 푛 푝* := (푎* , 퐴* , 푓 푝 ∪ ⋃︀ (푓 푞휂 ∖ 푎* ), 퐹 *0, 퐹 *1, 퐹 *2), if 푛 = ℓ; 푛 푛 푛 푛 휉<휅ℓ 푛 푛 푛 푛 푛 ⎪ * ⎩푞푛, if 푛 > ℓ. * and 푞푛 is a ≤푛0-lower bound for ⟨(푞휉)푛 | 휉 < 휅ℓ⟩. * Claim 4.20.1. 푝 is a condition in P diagonalizing ⟨푞휉 | 휉 < 휅ℓ⟩. Proof. Clearly, 푝* ∈ 푃 and it is routine to check that 푝* ≤0 푝. * * * * Let 푠 ∈ 푊1(푝 ) and 휈 ∈ 퐴 be with 푠 = 푝 y⟨휈⟩. Since 휋 * “퐴 ℓ mc(푎ℓ ),mc(푎ℓ) ℓ is contained in 퐴ℓ there is some 휉 < 휅ℓ such that휈 ˜ = 휈휉. Note that * 0 푤(푝, 푠) = 푝y⟨휈휉⟩, hence we need to prove that 푝 y⟨휈⟩ ≤ 푞휉. Note that for 푖푞휉 *푖 this it is enough to show that ℎ ⊆ 퐹 (휋 * * (휈)) for 푖 < 3. And, ℓ ℓ mc(푎ℓ ),mc(푎ℓ ∩휇) *푖 of course, this follows from our deﬁnition of 퐹ℓ and the fact that휈 ˜ = 휈휉.

The above shows that I has a winning strategy for the game aP(푝, ⃗푟). Lemma 4.21. (P, ℓ) has the CPP. Proof. Fix 푝 ∈ 푃 , 푛 < 휔 and 푈 a 0-open set. Set ℓ := ℓ(푝). Claim 4.21.1. There is 푞 ≤0 푝 such that if 푟 ∈ 푃 푞 ∩ 푈 then 푤(푞, 푟) ∈ 푈. 푛 Proof. For each 푛 < 휔 and a good enumeration ⃗푟 := ⟨푟휉 | 휉 < 휒⟩ of 푊푛(푝) 0 appeal to Lemma 4.20 and find 푝푛 ≤ 푝 such that 푝푛 diagonalizes a sequence 푛 ⟨푞휉 | 휉 < 휒⟩ of moves of II which moreover satisfies 푛 푟휉 푛 푃0 ∩ 푈 ̸= ∅ =⇒ 푞휉 ∈ 푈. 0 Appealing iteratively to Lemma 4.20 we arrange ⟨푝푛 | 푛 < 휔⟩ to be ≤ - decreasing, and by Definition 3.32 we find 푞 ≤0 푝 a lower bound for it. 푛 Let 푟 ≤ 푞 be in 퐷 and set 푛 := ℓ(푟) − ℓ(푞). Then, 푟 ≤ 푝푛 and so 0 0 푛 푛 푟 ≤ 푤(푝푛, 푟) ≤ 푞휉 for some 휉. This implies that 푞휉 ∈ 푈. Finally, since 0 0 푛 푤(푞, 푟) ≤ 푤(푝푛, 푟) ≤ 푞휉 we infer from 0-openess of 푈 that 푤(푞, 푟) ∈ 푈. Let 푞 ≤0 푝 be as in the conclusion of Claim 4.21.1. We define by induction 0 a ≤ -decreasing sequence of conditions ⟨푞푛 | 푛 < 휔⟩ such that for each 푛 < 휔

(⋆)푛 푊푛(푞푛) ⊆ 푈 or 푊푛(푞푛) ∩ 푈 = ∅. The cases 푛 ≤ 1 are easily handled and the cases 푛 ≥ 3 are similar to the case 푛 = 2. So, let us simply describe how do we proceed in this latter case. 푞1 Suppose that 푞1 has been defined. For each 휈 ∈ 퐴ℓ , define + 푞1 y − 푞1 + 퐴휈 := {훿 ∈ 퐴ℓ+1 | 푞1 ⟨휈, 훿⟩ ∈ 푈} and 퐴휈 := 퐴ℓ+1 ∖ 퐴휈 . + + 30 − If 퐴휈 is large then set 퐴휈 := 퐴휈 . Otherwise, define 퐴휈 := 퐴휈 . Put + 푞1 + − 푞1 − + 퐴 := {휈 ∈ 퐴ℓ | 퐴휈 = 퐴휈 }, and 퐴 := {휈 ∈ 퐴ℓ | 퐴휈 = 퐴휈 }. If 퐴 is 30 More explicitly, 퐸 푞1 -large. ℓ+1,mc(푎ℓ+1) 28 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

+ − 0 large we let 퐴ℓ := 퐴 and otherwise 퐴ℓ := 퐴 . Finally, let 푞2 ≤ 푞1 be such 푞2 푞2 ⋂︀ that 퐴 := 퐴ℓ and 퐴 := 푞1 퐴휈 Then, 푞2 witnesses (⋆)2. ℓ ℓ+1 휈∈퐴ℓ 0 Once we have defined ⟨푞푛 | 푛 < 휔⟩, let 푞휔 be a ≤ -lower bound for it (cf. Definition 3.3(2)). It is routine to check that, for each 푛 < 휔, the condition 푞휔 witnesses the alternative (⋆)푛, hence it is as desired. Let us dispose with the verification of Clauses (8) and (9):

Lemma 4.22. For all 푛 < 휔, the map 휛푛 is a nice projection from P≥푛 to S푛 such that, for all 푘 ≥ 푛, 휛푛 P푘 is again a nice projection to S푛. Moreover, the sequence of nice projections ⃗휛 is coherent.31

Proof. Fix some 푛 < 휔. By definition, 휛푛(1lP) = 1lS푛 and it is not hard to check that it is order-preserving. Let 푝 ∈ 푃≥푛 and 푠 ⪯푛 휛푛(푝). 푝 0 1 2 Then 푠 = ⟨(휌푘, ℎ푘, ℎ푘, ℎ푘) | 푘 < 푛⟩, and we define 푟 := ⟨푟푘 | 푘 < 휔⟩ as {︃ 푝 푝 0 1 2 (휌푘, 푓푘 , ℎ푘, ℎ푘, ℎ푘), if 푘 < 푛; 푟푘 := 푝푘, otherwise. 0 It is not hard to check that 푟 ≤ 푝 and 휛푛(푟) = 푠. Actually, 푟 is the greatest such condition, hence 푟 = 푝 + 푠. This yields Clause (3) of Definition 2.2. For the verification of Clause (4) of Definition 2.2, let 푞 ≤0 푝 + 푠 and ′ ′ define a sequence 푝 := ⟨푝푛 | 푛 < 휔⟩ as follows:

{︃ 푝 푝 0푝 1푝 2푝 ′ (휌푘, 푓푘 , ℎ푘 , ℎ푘 , ℎ푘 ), if 푘 < 푛; 푝푘 := 푞푘, otherwise. ′ ′ Note that 푝 ∈ 푃 and 푝 + 휛푛(푞) = 푞. Thus, Clause (4) follows. Altogether, the above shows that 휛푛 is an exact nice projection. Similarly, one shows that 휛푛 P푘 is a nice projection for each 푘 ≥ 푛. Finally, the moreover part of the lemma follows from the deﬁnition of 휛푛 and the fact y ∏︀ℓ(푝)+|⃗휈|−1 푝 that 푊 (푝) = {푝 ⃗휈 | ⃗휈 ∈ 푘=ℓ(푝) 퐴푘}.

휛푛 32 Lemma 4.23. For each 푛 < 휔, P푛 is 휎푛-directed-closed.

휛0 Proof. Since P0 = {1l} the result is clearly true for 푛 = 0. 휛푛 Let 푛 ≥ 1 and 퐷 ⊆ P푛 be a directed set of size <휎푛. By deﬁnition, 푝 0푝 1푝 2푝 휛푛[퐷] = {⟨(휌푘, ℎ푘 , ℎ푘 , ℎ푘 ) | 푘 < 푛⟩}, for some (all) 푝 ∈ 퐷. By taking intersection of the measure one sets and unions on the other components of the conditions of 퐷 one can easily form ⃗휛 a condition 푞 which is a ≤ -lower bound fo 퐷. Finally, the proof of the next is identical to [?, Corollary 10.2.53]. + Corollary 4.24. 1lP P 휇ˇ = 휅 .

31See Definition 3.7. 32 ˚ In particular, taking P푛 := P푛 Clause (9) follows. SIGMA-PRIKRY FORCING III 29

Combining all the previous lemmas we ﬁnally arrive at the desired result:

Corollary 4.25. (P, ℓ, 푐, ⃗휛) is a (Σ,⃗S)-Prikry forcing that has property 풟. Moreover, the sequence ⃗휛 is coherent. 4.3. EBPFC is suitable for reflection. In this section we show that (P푛, S푛, 휛푛) is suitable for reﬂection with respect to a relevant sequence of cardinals. Our setup will be the same as the one from page 19 and we will also rely on the notation established in page 22. The main result of the section is Corollary 4.30, which will be preceded by a series of technical lemmas. The ﬁrst one is essentially due to A. Sharon:

* Lemma 4.26 ([Sha05]). For each 푛 < 휔, 푉 Q푛0 |= |휇| = cf(휇) = 휅푛. * Proof. By Lemma 4.14, Q푛0 preserves cofinalities ≤휅푛, and by the Linked0- property [?, Lemma 10.2.41] it preserves cardinals ≥휇+. * * Next we show that Q푛0 collapses 휇 to 휅푛. For each condition 푝 ∈ Q푛0, 푝 푝 푝 * ⋃︀ 푝 denote 푝 := (푎 , 퐴 , 푓 ). Let 퐺 be Q푛0-generic and set 푎 := 푝∈퐺 푎 . By a density argument, otp(푎 ∩ 휇) = 휅푛, and so 휇 is collapsed. Finally, by a * 33 result of Shelah, this implies that 푉 Q푛0 |= cf(|휇|) = cf(휇). + Lemma 4.27. For each 푛 < 휔, 푉 Q푛0 |= |휇| = cf(휇) = 휅푛 = (휎푛) . 0푝 1푝 2푝 Proof. For each 푝 ∈ 푄푛0, 퐹 , 퐹 and 퐹 respectively represent conditions * * * 푀푛 + 푀푛 +3 푀푛 * ∼ in Col(휎푛, <휅푛) , Col(휅푛, 휅 ) and Col(휅 , <푗푛(휅푛)) , where 푀푛 = Ult(푉, 퐸푛 휇). Also, observe that the first of these forcings is nothing but 푉 34 1푝 2푝 Col(휎푛, <휅푛) . Set 퐶푛 := {⟨퐹 , 퐹 ⟩ | 푝 ∈ 푄푛0} and define ⊑ as follows: 1푝 2푝 1푞 2푞 푖푝 푖푞 ⟨퐹 , 퐹 ⟩ ⊑ ⟨퐹 , 퐹 ⟩ iff ∀푖 ∈ {1, 2} 푗푛(퐹 )(훼푝) ⊇ 푗푛(퐹 )(훼푞).

Clearly, ⊑ is transitive, so that C푛 := (퐶푛, ⊑) is a forcing poset. For each condition 푐 in C푛 let us denote by 훼푐 the ordinal 훼푝푐 relative to a condition 푝푐 in Q푛0 witnessing that 푐 ∈ 퐶푛. The following is a routine veriﬁcation:

Claim 4.27.1. C푛 is 휅푛-directed closed. Furthermore, if 퐷 ⊆ C푛 is a directed set of size <휅푛 and 훼 < 휇 is ≤퐸푛 -above all {훼푐 | 푐 ∈ 퐷}. Then, 1 2 1 2 there is ⊑-lower bound ⟨퐹 , 퐹 ⟩ for 퐷 with dom(퐹 ) = dom(퐹 ) ∈ 퐸푛,훼. * * Let 퐺 be a Q푛0-generic filter over 푉 and denote by 퐺 the Q푛0-generic induced by 퐺 and the projection 휚푛 of Lemma 4.14(1). By Lemma 4.26, * 푉 [퐺 ] |= |휇| = cf(휇) = 휅푛, hence it lefts to check that 휅푛 is preserved and + turned into (휎푛) . We prove this in two steps, being the proof of the first one a routine verification. * 푉 * Claim 4.27.2. The map 푒: Q푛0/퐺 → Col(휎푛, <휅푛) ×C푛 defined in 푉 [퐺 ] 0푝 1푝 2푝 by 푝 ↦→ ⟨푗푛(퐹 )(훼푝), ⟨퐹 , 퐹 ⟩⟩ is a dense embedding. + Claim 4.27.3. 푉 [퐺] |= |휇| = cf(휇) = 휅푛 = (휎푛) .

33For Shelah’s theorem, see e.g. [CFM01, Fact 4.5]. 34 * Here we use that 푉휅푛 ⊆ 푀푛. 30 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

* 푉 푉 [퐺*] Proof. Since Q푛0 is 휅푛-directed closed, Col(휎푛, <휅푛) = Col(휎푛, <휅푛) . * Note that C푛 is still 휅푛-directed closed over 푉 [퐺 ] and that in any generic 푉 * + extension by Col(휎푛, <휅푛) over 푉 [퐺 ], “|휇| = 휅푛 = (휎푛) ” holds. Appealing to Easton’s lemma, C푛 is 휅푛-distributive in any extension of * 푉 푉 푉 [퐺 ] by Col(휎푛, <휅푛) . Thus, forcing with Col(휎푛, <휅푛) × C푛 (over * + 푉 [퐺 ]) yields a generic extension where “|휇| = 휅푛 = (휎푛) ” holds. Since (휇+)푉 is preserved, a theorem of Shelah (see [CFM01, Fact 4.5]) yields 푉 “ cf(휇) = cf(|휇|)” in the above generic extension. Thus, Col(휎푛, <휅푛) ×C푛 * + forces (over 푉 [퐺 ]) that “|휇| = cf(휇) = 휅푛 = (휎푛) ” holds. The result now 푉 follows using Claim 4.27.2, as it in particular implies that Col(휎푛, <휅푛) ×C푛 * * and Q푛0/퐺 are forcing equivalent over 푉 [퐺 ]. This completes the proof. ∏︀ Lemma 4.28. For all non-zero 푛 < 휔, 푖<푛 Q푖1 is isomorphic to a product of S푛 with some 휇-directed-closed forcing. 푝 0푝 1푝 2푝 푝 Proof. The map 푝 ↦→ ⟨⟨(휌 푖 , ℎ 푖 , ℎ 푖 , ℎ 푖 ) | 푖 < 푛⟩⟩, ⟨푓 푖 ⟩푖<푛⟩ yields the desired isomorphism. 푛 + Lemma 4.29. For each 푛 < 휔, 푉 P |= |휇| = cf(휇) = 휅푛 = (휎푛) . ∏︀ ∏︀ Proof. Observe that P푛 is a dense subposet of 푖<푛 Q푖1 × 푖≥푛 Q푖0, hence both forcing produce the same generic extension. By virtue of Lemma 4.27 + we have 푉 Q푛0 |= |휇| = cf(휇) = 휅푛 = (휎푛) . Also, Q푛0 is 휎푛-directed-closed, hence Easton’s lemma, Lemma 4.14(3) and Lemma 4.28 combined imply ∏︀ that Q푛0 forces Q푖1 to be a product of a (휅푛−1)-cc forcing times a 휅푛- 푖<푛 ∏︀ distributive forcing. Similarly, Q푛0 forces 푖>푛 Q푖0 to be 휅푛-distributive. ∏︀ ∏︀ Q푛0 Thereby, forcing with 푖<푛 Q푖1 × 푖>푛 Q푖0 over 푉 preserves the above cardinal conﬁguration and thus the result follows. As a consequence of the above we get the main result of the section:

Corollary 4.30. For each 푛 ≥ 2, (P푛, S푛, 휛푛) is suitable for reflection with respect to the sequence ⟨휎푛−1, 휅푛−1, 휅푛, 휇⟩. Proof. We go over the clauses of Deﬁnition 2.10. Clause (1) is obvious. Clause (2) follows from Lemma 4.22 and Lemma 4.23. Clause (4) follows from the comments in Remark 4.12. For Clause (3), note that P푛 forces + “|휇| = cf(휇) = 휅푛 = (휎푛) ” (Lemma 4.29), hence the last paragraph of 휛푛 35 Remark 4.12 implies that S푛 × P푛 forces the same. We conclude this section, establishing two more facts that will be needed for the proof of the Main Theorem in Section 8.

Definition 4.31. For every 푛 < 휔, let T푛 := S푛 × Col(휎푛, <휅푛), and let 휓푛 : P푛 → T푛 be the map deﬁned via

{︃ 0푝푛 ⟨휛푛(푝), 푗푛(퐹 )(훼푝푛 )⟩, if ℓ(푝) > 0; 휓푛(푝) := ⟨1lS푛 , ∅⟩, otherwise. 35 + Recall that 휎푛 := (휅푛−1) (cf. Setup 4). SIGMA-PRIKRY FORCING III 31

Lemma 4.32. Let 푛 < 휔.

(1) T푛 is a 휅푛-cc poset of size 휅푛; (2) 휓푛 defines a nice projection; 휓푛 (3) P푛 is 휅푛-directed-closed; 휓푛 (4) for each 푝 ∈ 푃푛, P푛 ↓ 푝 and (T푛 ↓ 휓푛(푝)) × (P푛 ↓ 푝) isomorphic. In particular, both are forcing equivalent.

Proof. (1) This is obvious.

(2) Let us go over the clauses of Deﬁnition 2.2. Clearly, 휓푛(1lP) = ⟨1lS푛 , ∅⟩, so Clause (1) holds. Likewise, using that 휛푛 is order-preserving it is routine to check that so is 휓푛. Thus, Clause (2) holds, as well.

Let us now prove Clause (3). Let 푝 ∈ 푃푛 and 푡 ≤S푛×Col(휎푛,<휅푛) 휓푛(푝). 0푝푛 Putting 푡 =: ⟨푠, 푐⟩ we have 푠 ⪯푛 휛푛(푝) and 푐 ⊇ 푗푛(퐹 )(훼푝푛 ). On one hand, since 휛푛 is a nice projection, 푞 := 푝 + 푠 is a condition in P푛. On the other hand, there is a function 퐹 and 훽 < 휇 with dom(퐹 ) ∈ 퐸푛,훽 and 36 푞푛 푗푛(퐹 )(훽) = 푐. By possibly enlarging 푎 we may actually assume that 푞푛 훽 = 훼 and also that dom(퐹 ) = 휋 푞푛 [퐴 ]. Let 푟 be the condition in 푞 mc(푎 ),훼푞푛 0푟푛 P푛 with the same entries as 푞 but with 퐹 := 퐹 . Clearly, 푟 ≤ 푞 ≤ 푝. Also, by the way 푟 is deﬁned, 휓푛(푟) = ⟨휛푛(푟), 푐⟩ = ⟨휛푛(푞), 푐⟩ = ⟨푠, 푐⟩ = 푡. Note that if 푢 ∈ 푃푛 is such that 푢 ≤ 푝 and 휓푛(푢) = 푡, then 푢 ≤ 푟. Altogether, 푟 = 푝 + 푡, which yields Clause (3).37 Finally, for Clause (4) one argues in the same lines as in Lemma 4.22.

휓푛 (3) Let 퐷 ⊆ P푛 be a directed set of size < 휅푛. Then, 휓푛[퐷] = {⟨푠, 푐⟩} for 0푝푛 some ⟨푠, 푐⟩ ∈ S푛 × Col(휎푛, <휅푛). Thus, for each 푝 ∈ 퐷, 푗푛(퐹 )(훼푝푛 ) = 푐. <휅푛 Arguing as usual, let 푎 ∈ [휆] be such that both 푎 ∩ 휇 and 푎 have ≤퐸푛 - ⋃︀ 푝푛 greatest elements 훼 and 훽, respectively, and 푎 ⊇ 푝∈퐷 푎 . Then, for each 0푝푛 0푞푛 푝, 푞 ∈ 퐷, 퐵푝,푞 := {휈 < 휅푛 | 퐹 (휋훼,훼 (휈)) = 퐹 (휋훼,훼 (휈))} ∈ 퐸푛,훼 and, ⋂︀푝푛 푞푛 by 휅푛-completedness of 퐸푛,훼, 퐵 := {퐵푝,푞 | 푝, 푞 ∈ 퐷} ∈ 퐸푛,훼. −1 Set 퐴 := 휋훽,훼“퐵. By shrinking 퐴 if necessary, we may further assume 푝 휋 푝 “퐴 ⊆ 퐴 푛 for each 푝 ∈ 퐷. Since 휓 퐷 is constant the map 훽,mc(푎푛) 푛 푝 0푝 1푝 2푝 0 1 2 휛푛 : 푝 ↦→ ⟨(휌푘, ℎ푘 , ℎ푘 , ℎ푘 ) | 푘 < 푛⟩ is so. Let ⟨(휌푘, ℎ푘, ℎ푘, ℎ푘) | 푘 < 푛⟩ be ⋃︀ 푝 0 such constant value. For each 푘 < 휔, set 푓푘 := 푝∈퐷 푓푘 and 퐹 be such 0 0 0푝 that dom(퐹 ) = 퐵 and 퐹 (휈) := 퐹 (휋훼,훼푝푛 (휈)) for some 푝 ∈ 퐷. 1푝 2푝 Observe that {⟨퐹푛 , 퐹푛 ⟩ | 푝 ∈ 퐷} forms a directed subset of C푛 of size < 휅푛 (cf. Lemma 4.27). Using the 휅푛-directed-closedness of C푛 we may let 1 2 ⟨퐹 , 퐹 ⟩ ∈ 퐶푛 be a ⊑-lower bound. Actually, by using the moreover clause 1 2 of Lemma 4.27 we may assume that dom(퐹 ) = dom(퐹 ) ∈ 퐸푛,훼. Thus, by shrinking 퐴 and 퐵 if necessary we may assume dom(퐹 1) = dom(퐹 2) = 퐵.

* 36 푉 푀푛 * ∼ Here we use that Col(휎푛, <휅푛) = Col(휎푛, <휅푛) , where 푀푛 = Ult(푉, 퐸푛 휇). 37 The + here is regarded with respect to the map 휓푛. 32 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

* * Deﬁne 푝 := ⟨푝푘 | 푘 < 휔⟩ as follows: ⎧ 0 1 2 ⎪(푓푘, 휌푘, ℎ푘, ℎ푘, ℎ푘), if 푘 < 푛; * ⎨ 0 1 2 푝푘 := (푎, 퐴, 푓푘, 퐹 , 퐹 , 퐹 ), if 푘 = 푛; ⎪ 0 1 2 ⎩(푎푘, 퐴푘, 푓푘, 퐹푘 , 퐹푘 , 퐹푘 ), if 푘 > 푛, 0 1 2 where (푎푘, 퐴푘, 푓푘, 퐹푘 , 퐹푘 , 퐹푘 ) is constructed as described in Lemma 4.19. * 휓푛 Clearly, 푝 ∈ 푄푛0 and it gives a ≤ -lower bound for 퐷. 휓푛 (4) By Item (2) of this lemma, (T푛 ↓ 휓푛(푝)) × (P푛 ↓ 푝) projects onto 38 P푛 ↓ 푝. Actually both posets are easily seen to be isomorphic. Lemma 4.33. Assume GCH. Let 푛 < 휔. + (1) P푛 is 휇 -Linked; (2) P푛 forces CH휃 for any cardinal 휃 ≥ 휎푛; 휛푛 (3) P푛 preserves the GCH.

Proof. (1) By Deﬁnition 4.10, Lemma 4.19 and the fact that |퐻휇| = 휇. + (2) As P푛 has size ≤ 휇 , Clause (1) together with a counting-of-nice- names argument implies that 2휃 = 휃+ for any cardinal 휃 ≥ 휇+. By Lemma 4.29, + in any generic extension by P푛, |휇| = cf(휇) = 휅푛 = (휎푛) . It thus left to 휃 + verify that P푛 forces 2 = 휃 for 휃 ∈ {휎푛, 휅푛}. I By Clauses (1), (3) and (4) of Lemma 4.32, together with Easton’s lemma, P푛 forces CH휎푛 if and only if T푛 forces CH휎푛 . By Clause (1) of Lemma 4.32, T푛 is a 휅푛-cc poset of size 휅푛, so, the number of T푛-nice names <휅푛 + for subsets of 휎푛 is at most 휅푛 = 휅푛 = 휎푛 , as wanted. + 휇 휅푛 + I The number of nice names for subsets of 휅푛 is ((휇 ) ) = 휇 , and hence CH휅푛 is forced by P푛. 휛푛 (3) By Lemma 4.23, P푛 preserves GCH below 휎푛. By Remark 4.12 and the the fact that S푛 has size < 휎푛, we infer from Clause (2) that GCH holds at cardinals ≥ 휎푛, as well.

5. Nice forking projections In this short section we introduce the notions of nice forking projection, a strengthening of the following key concept from Part I of this series:39 ⃗ Definition 5.1 ([PRS19, S4]). Suppose that (P, ℓP, 푐P, ⃗휛) is a (Σ, S)-Prikry forcing, A = (퐴, E) is a notion of forcing, and ℓA and 푐A are functions with dom(ℓA) = dom(푐A) = 퐴. A pair of functions (t, 휋) is said to be a forking projection from (A, ℓA) to (P, ℓP) iﬀ all of the following hold: (1) 휋 is a projection from A onto P, and ℓA = ℓP ∘ 휋; (2) for all 푎 ∈ 퐴, t(푎) is an order-preserving function from (P ↓ 휋(푎), ≤) to (A ↓ 푎, E);

38See Definition 2.2(4). 39In [PRS19] the following notion is formulated in terms of Σ-Prikry forcings. However the same notion is meaningful in the more general context of (Σ,⃗S)-Prikry forcings. SIGMA-PRIKRY FORCING III 33

(3) for all 푝 ∈ 푃 , {푎 ∈ 퐴 | 휋(푎) = 푝} admits a greatest element, which we denote by ⌈푝⌉A; 푛+푚 (4) for all 푛, 푚 < 휔 and 푏 E 푎, 푚(푎, 푏) exists and satisfies: 푚(푎, 푏) = t(푎)(푚(휋(푎), 휋(푏))); (5) for all 푎 ∈ 퐴 and 푟 ≤ 휋(푎), 휋(t(푎)(푟)) = 푟; (6) for all 푎 ∈ 퐴 and 푟 ≤ 휋(푎), 푎 = ⌈휋(푎)⌉A iff t(푎)(푟) = ⌈푟⌉A; ′ 0 0 ′ ′ (7) for all 푎 ∈ 퐴, 푎 E 푎 and 푟 ≤ 휋(푎 ), t(푎 )(푟) E t(푎)(푟). The pair (t, 휋) is said to be a forking projection from (A, ℓA, 푐A) to (P, ℓP, 푐P) iff, in addition to all of the above, the following holds: ′ ′ ′ (8) for all 푎, 푎 ∈ 퐴, if 푐A(푎) = 푐A(푎 ), then 푐P(휋(푎)) = 푐P(휋(푎 )) and, for ′ 휋(푎) 휋(푎 ) ′ all 푟 ∈ 푃0 ∩ 푃0 , t(푎)(푟) = t(푎 )(푟). Definition 5.2. A pair of functions (t, 휋) is said to be a nice forking projection from (A, ℓA, ⃗휍) to (P, ℓP, ⃗휛) iff all of the following hold: (a)( t, 휋) is a forking projection from (A, ℓA) to (P, ℓP); (b) ⃗휍 = ⃗휛 ∙ 휋, that is, 휍푛 = 휛푛 ∘ 휋 for all 푛 < 휔. Also, for each 푛, 휍푛 is a nice projection from A≥푛 to S푛, and for each 푘 ≥ 푛, 휍푛 A푘 is again a nice projection.

The pair (t, 휋) is said to be a nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓP, 푐P, ⃗휛) if, in addition, Clause (8) of Deﬁnition 5.1 is satisﬁed.

Remark 5.3. If (P, ℓP, 푐P) is a Σ-Prikry forcing then a pair of maps (t, 휋) is a forking projection from (P, ℓP) to (A, ℓA) iﬀ it is a nice forking projection from (P, ℓP, ⃗휛) to (A, ℓA, ⃗휍). Similarly, the same applies to forking projections from (A, ℓA, 푐A, ⃗휍) to (P, ℓP, 푐P, ⃗휛). As we will see, most of the theory of iterations of (Σ,⃗S)-Prikry forcings can be developed starting from the concept of nice forking projection. Nonetheless, to be successful at limit stages, one needs nice forking projections yielding very canonical witnesses for niceness of the 휍푛’s. Roughly ′ speaking, we want that whenever 푝 is a witness for niceness of some 휛푛 ′ ′ then there is a witness 푎 for niceness of 휍푛 which lifts 푝 . This leads to the concept of super nice forking projection that we next introduce:

Definition 5.4. A nice forking projection (t, 휋) from (A, ℓA, ⃗휍) to (P, ℓP, ⃗휛) is called super nice if for each 푛 < 휔 the following property holds: ′ ′ * Let 푎, 푎 ∈ 퐴≥푛 and 푠 ∈ 푆푛 such that 푎 E 푎 + 푠. Then, for each 푝 ∈ 푃≥푛 * 휛푛 ′ * ′ * 휍푛 such that 푝 ≤ 휋(푎) and 휋(푎 ) = 푝 + 휍푛(푎 ), there is 푎 E 푎 with * * ′ * ′ 휋(푎 ) = 푝 and 푎 = 푎 + 휍푛(푎 ).

The notion of super nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓP, 푐P, ⃗휛) is deﬁned similarly. Remark 5.5. The notion of super nice forking projection will not be necessary for the purposes of the current section. Its importance will become ap- parent in Section 7, where we present our iteration scheme (see Claim 7.4.1). 34 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Next we show that if (t, 휋) is a forking projection (not necessarily nice) and ⃗휍 = 휋 ∘ ⃗휛 then 휍푛 satisﬁes Deﬁnition 2.2(3) for each 푛 < 휔.

Lemma 5.6. Let (t, 휋) be a forking projection from (A, ℓA) to (P, ℓP) and suppose that ⃗휍 = 휋 ∘ ⃗휛. Then, for all 푎 ∈ 퐴, 푛 ≤ ℓA(푎) and 푠 ⪯푛 휍푛(푎), 푎 + 푠 = t(푎)(휋(푎) + 푠). Proof. Combining Clauses (2) and (5) of Definition 5.1 with Clause (b) of Definition 5.2 it follows that t(푎)(휋(푎) + 푠) E 푎 and 휍푛(t(푎)(휋(푎) + 푠)) = 푠. 0 Let 푏 ∈ 퐴 such that 푏 E 푎 and 휍푛(푏) ⪯푛 푠. Then, 휋(푏) ≤ 휋(푎) + 푠. By [PRS20, Lemma 2.17], 푏 = t(푏)(휋(푏)), hence Clauses (2) and (7) of 0 0 Definition 5.1 yield 푏 E t(푎)(휋(푏)) E t(푎)(휋(푎) + 푠). We are not done yet with establishing that 푎 + 푠 = t(푎)(휋(푎) + 푠), as we 0 have just dealt with 푏 E 푎. However we can further argue as follows. Let ′ 푏 E 푎 be with 휍푛(푏) ⪯푛 푠. Put, 푏 := t(0(푎, 푏))(0(휋(푎), 휋(푏)) + 휍푛(푏)). It is ′ 0 ′ easy to check that 푏 E 푏 E 푎 and that 휍푛(푏 ) = 휍푛(푏) ⪯푛 푠. Hence, applying ′ 0 the previous argument we arrive at 푏 E 푏 E t(푎)(휋(푎) + 푠). In [PRS20, S2], we drew a map of connections between Σ-Prikry forcings and forking projection, demonstrating that this notion is crucial to define a viable iteration scheme for Σ-Prikry posets. However, to be successful in iterating Σ-Prikry forcings, forking projections need to be accompanied with types, which are key to establish the CPP and property 풟 for (A, ℓA). Definition 5.7 ([PRS20, S2]). A type over a forking projection (t, 휋) is a map tp: 퐴 → <휇휔 having the following properties: (1) for each 푎 ∈ 퐴, either dom(tp(푎)) = 훼 + 1 for some 훼 < 휇, in which case we define mtp(푎) := tp(푎)(훼), or tp(푎) is empty, in which case we define mtp(푎) := 0; (2) for all 푎, 푏 ∈ 퐴 with 푏 E 푎, dom(tp(푎)) ≤ dom(tp(푏)) and for each 푖 ∈ dom(tp(푎)), tp(푏)(푖) ≤ tp(푎)(푖); (3) for all 푎 ∈ 퐴 and 푞 ≤ 휋(푎), dom(tp(t(푎)(푞))) = dom(tp(푎)); (4) for all 푎 ∈ 퐴, tp(푎) = ∅ iff 푎 = ⌈휋(푎)⌉A; (5) for all 푎 ∈ 퐴 and 훼 ∈ 휇 ∖ dom(tp(푎)), there exists a stretch of 푎 to 훼, denoted 푎y훼, and satisfying the following: 훼 휋 (a) 푎y E 푎; (b) dom(tp(푎y훼)) = 훼 + 1; (c) tp(푎y훼)(푖) ≤ mtp(푎) whenever dom(tp(푎)) ≤ 푖 ≤ 훼; (6) for all 푎, 푏 ∈ 퐴 with dom(tp(푎)) = dom(tp(푏)), for every 훼 ∈ 휇 ∖ 훼 훼 dom(tp(푎)), if 푏 E 푎, then 푏y E 푎y ; ˚ ˚ ˚ (7) For each 푛 < 휔, the poset A푛 is dense in A푛, where A푛 := (퐴푛, E) and 퐴˚푛 := {푎 ∈ 퐴푛 | 휋(푎) ∈ 푃˚푛 & mtp(푎) = 0}. Remark 5.8. Note that Clauses (2) and (3) imply that for all 푚, 푛 < 휔, ˚ ˚ ˚ 푎 ∈ 퐴푚 and 푞 ≤ 휋(푎), if 푞 ∈ 푃푛 then t(푎)(푞) ∈ 퐴푛. In the more general context of (Σ,⃗S)-Prikry forcings – where the pair (t, 휋) needs to be a nice forking projection – we need to require a bit more: SIGMA-PRIKRY FORCING III 35

Definition 5.9. A nice type over a nice forking projection (t, 휋) is a type over (t, 휋) which moreover satisﬁes the following: ˚휍푛 휍푛 40 (8) For each 푛 < 휔, the poset A푛 is dense in A푛 .

Remark 5.10. If (P, ℓP, 푐P) is Σ-Prikry then any forking projection (t, 휋) is ˚휍푛 ˚ nice and A푛 = A푛 for all 푛 < 휔. In particular, any type over (t, 휋) is nice. We now turn to collect suﬃcient conditions — assuming the existence of a nice forking projection (t, 휋) from (A, ℓA, 푐A, ⃗휍) to (P, ℓP, 푐P, ⃗휛) — for ⃗ (A, ℓA, 푐A, ⃗휍) to be (Σ, S)-Prikry on its own, and then address the problem of ensuring that the A푛’s be suitable for reﬂection. This study will be needed in Section 6, most notably, in the proof of Theorem 6.16. Setup 5. Throughout the rest of this section, we suppose that:

∙ P = (푃, ≤) is a notion of forcing with a greatest element 1lP; ∙ A = (퐴, E) is a notion of forcing with a greatest element 1lA; ∙ Σ = ⟨휎푛 | 푛 < 휔⟩ is a non-decreasing sequence of regular uncountable cardinals, converging to some cardinal 휅, and 휇 is a cardinal such + that 1lP P 휇ˇ =휅 ˇ ; ⃗ ∙ S = ⟨S푛 | 푛 < 휔⟩ is a sequence of notions of forcing, S푛 = (푆푛, ⪯푛), with |푆푛| < 휎푛; ∙ ℓP, 푐P and ⃗휛 = ⟨휛푛 | 푛 < 휔⟩ are witnesses for (P, ℓP, 푐P, ⃗휛) being (Σ,⃗S)-Prikry; ∙ ℓA and 푐A are functions with dom(ℓA) = dom(푐A) = 퐴, and ⃗휍 = ⟨휍푛 | 푛 < 휔⟩ is a sequence of functions. ∙ (t, 휋) is a nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓP, 푐P, ⃗휛).

Theorem 5.11. Under the assumptions of Setup 5, (A, ℓA, 푐A, ⃗휍) satisfies all the clauses of Definition 3.3, with the possible exception of (2), (7) and + (9). Moreover, if 1lP P “ˇ휅 is singular”, then 1lA A 휇ˇ =휅 ˇ . Proof. Clauses (1) and (3) follow respectively from [PRS19, Lemmas 4.3 and 4.7]. Clause (4) holds by virtue of Clause (4) of Deﬁnition 5.1. Clauses (5) and (6) are respectively proved in [PRS19, Lemmas 4.7 and 4.10], and Clause (8) follows from Clause (b) of Deﬁnition 5.2. Finally, Lemma 3.14(3) yields the moreover part. For more details, see [PRS19, Corollary 4.13].

Next, we give suﬃcient conditions in order for (A, ℓA) to satisfy the CPP. In Part II of this series we prove that CPP follows from property 풟 of (A, ℓA):

Lemma 5.12 ([PRS20, Lemma 2.21]). Suppose that (A, ℓA) has property 풟. Then it has the CPP.

In effect, everything amounts to find sufficient conditions for (A, ℓA) to satisfy property 풟. The following concept will be useful on that respect:

40 ˚ Here A푛 is the forcing from Definition 5.7(7). 36 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Definition 5.13 (Weak Mixing property). The forking projection (t, 휋) is said to have the weak mixing property iﬀ it admits a type tp satisfying that for every 푛 < 휔, 푎 ∈ 퐴, ⃗푟, and 푝′ ≤0 휋(푎), and for every function 푔 : 푊푛(휋(푎)) → A ↓ 푎, if there exists an ordinal 휄 such that all of the following hold:

(1) ⃗푟 = ⟨푟휉 | 휉 < 휒⟩ is a good enumeration of 푊푛(휋(푎)); (2) ⟨휋(푔(푟휉)) | 휉 < 휒⟩ is diagonalizable with respect to ⃗푟, as witnessed by 푝′;41 (3) for every 휉 < 휒:42 ∙ if 휉 < 휄, then dom(tp(푔(푟휉)) = 0; ∙ if 휉 = 휄, then dom(tp(푔(푟휉)) ≥ 1; ∙ if 휉 > 휄, then (sup휂<휉 dom(tp(푔(푟휂))) + 1 < dom(tp(푔(푟휉)); (4) for all 휉 ∈ (휄, 휒) and 푖 ∈ [dom(tp(푎)), sup휂<휉 dom(tp(푔(푟휂)))],

tp(푔(푟휉))(푖) ≤ mtp(푎),

(5) sup휉<휒 mtp(푔(푟휉)) < 휔, 0 ′ ′ ′ then there exists 푏 E 푎 with 휋(푏) = 푝 such that, for all 푞 ∈ 푊푛(푝 ), ′ 0 ′ t(푏)(푞 ) E 푔(푤(휋(푎), 푞 )). Remark 5.14. We would like to emphasize that the above notion make sense even when both (t, 휋) and tp are not nice. This is simply because the above ˚휍푛 clauses do not involve the maps 휍푛’s nor the forcings A푛 . As shown in [PRS20, S2], the weak mixing property is the key to ensure that (A, ℓA) has property 풟. In this respect, the following lemma gathers the results proved in Lemma 2.27 and Corollary 2.28 of [PRS20]:

Lemma 5.15. Suppose that (t, 휋) has the weak mixing property and that (P, ℓP) has property 풟. Then (A, ℓA) has property 풟, as well. In particular, if (P, ℓP) has property 풟 and (t, 휋) has the weak mixing property, then (A, ℓA) has both property 풟 and the CPP. We are still need to verify Clause (2) and (9) of Deﬁnition 3.3. Arguing similarly to [PRS20, Lemma 2.29] we can prove the following:

Lemma 5.16. Suppose that (t, 휋) is as in Setup 5 or, just a pair of maps satisfying Clauses (1), (2), (5) and (7) of Definition 5.1. ˚ Let 푛 < 휔. If (t, 휋) admits a type, and A푛 is defined according to the last ˚휋 ˚ clause of Definition 5.7, if A푛 is ℵ1-directed-closed, then so is A푛. Similarly, ˚휋 ˚휍푛 if A푛 is 휎푛-directed-closed, then so is A푛 . If in addition (t, 휋) admits a nice type then (A, ℓA, 푐A, ⃗휍) satisfies Clauses (2) and (9) of Definition 3.3. Additionally, a routine veriﬁcation gives the following:

41Recall Definition 3.15. 42The role of the 휄 is to keep track of the support when we apply the weak mixing lemma in the iteration (see, e.g. [PRS20, Lemma 3.10]). SIGMA-PRIKRY FORCING III 37

Lemma 5.17. Suppose that (t, 휋) is as in Setup 5. Then, if ⃗휛 is a coherent sequence of nice projections then so is ⃗휍. We conclude this section by providing a suﬃcient condition for the posets A푛’s to be suitable for reﬂection with respect to a sequence of cardinals for which the posets P푛’s were so. Lemma 5.18. Let 푛 be a positive integer. Assume:

(i) 휅푛−1, 휅푛 are regular uncountable cardinals with 휅푛−1 ≤ 휎푛 < 휅푛; ⃗ (ii) (A, ℓA, 푐A, ⃗휍) is (Σ, S)-Prikry; (iii) (t, 휋) is a nice forking projection from (A, ℓA, ⃗휍) to (P, ℓP, ⃗휛); (iv) (P푛, S푛, 휛푛) is suitable for reflection with respect to ⟨휎푛−1, 휅푛−1, 휅푛, 휇⟩; 휍푛 ++ (v) S푛 × A푛 forces “|휇| = cf(휇) = 휅푛 = (휅푛−1) ”. Then (A푛, S푛, 휍푛) is suitable for reflection with respect to ⟨휎푛−1, 휅푛−1, 휅푛, 휇⟩. Proof. Clauses (1), (2) and (4) of Definition 2.10 hold by virtue of hypothe- ses, (푖푣), (푖푖)-(푖푖푖)-(푖푣) and (푖푣), respectively. Now let us address Clause (3). Given hypothesis (푣), we are left with ver- ++ ifying that A푛 forces “|휇| = cf(휇) = 휅푛 = (휅푛−1) ”. By Definition 2.2(4), 휍푛 for every 푎 ∈ 퐴푛,(S푛 ↓ 휍푛(푎)) × (A푛 ↓ 푎) projects onto A푛 ↓ 푎. In addition, by hypothesis (푖푖푖), A푛 projects onto P푛. Since both ends force ++ “|휇| = cf(휇) = 휅푛 = (휅푛−1) ”, the same is true for A푛. 6. Stationary Reflection and Killing a Fragile Stationary Set In this section, we isolate a natural notion of a fragile set and study two aspects of it. In the first subsection, we prove that, given a (Σ,⃗S)- ⋆ Prikry poset P and an 푟 -fragile stationary set 푇˙ , a tweaked version of Sharon’s functor A(·, ·) from [PRS20, S4] yields a (Σ,⃗S)-Prikry poset A(P, 푇˙ ) admitting a nice forking projection to P and killing the stationarity of 푇˙ . In the second subsection, we make the connection between fragile stationary sets, suitability for reflection and non-reflecting stationary sets. The two subsections can be read independently of each other. Setup 6. As a setup for the whole section, we assume that (P, ℓ, 푐, ⃗휛) is a given (Σ,⃗S)-Prikry forcing such that (P, ℓ) satisfies property 풟. Denote ⃗ P = (푃, ≤), Σ = ⟨휎푛 | 푛 < 휔⟩, ⃗휛 = ⟨휛푛 | 푛 < 휔⟩, S = ⟨S푛 | 푛 < 휔⟩. Also, define 휅 and 휇 as in Definition 3.3, and assume that 1lP P “ˇ휅 is singular” <휇 ˚ and that 휇 = 휇. For each 푛 < 휔, we denote by P푛 the countably-closed dense suposet of P푛 given by Clause (2) of Definition 3.3. Recall that by ˚휛푛 ˚ virtue of Clause (9), P푛 is a 휎푛-directed-closed dense subforcing of P푛. We ˚ often refer to P푛 as the ring of P푛. In addition, we will assume that ⃗휛 is a coherence sequence of nice projections (see Definition 3.7). The following concept is implicit in the proof of [CFM01, Theorem 11.1]: ⋆ Definition 6.1. Suppose 푟 ∈ 푃 forces that 푇˙ is a P-name for a stationary subset 푇 of 휇. We say that 푇˙ is 푟⋆-fragile if, looking for each 푛 < 휔 at 38 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

˙ ˙ ⋆ 푇푛 := {(ˇ훼, 푝) | (훼, 푝) ∈ 휇 × 푃푛 & 푝 P 훼ˇ ∈ 푇 }, then, for every 푞 ≤ 푟 , ˙ 푞 Pℓ(푞) “푇ℓ(푞) is nonstationary”. ⋆ ⋆ 6.1. Killing one fragile set. Let 푟 ∈ 푃 and 푇˙ be a P-name for an 푟 - fragile stationary subset of 휇. Let 퐼 := 휔 ∖ ℓ(푟⋆). By Definition 6.1, for all ⋆ ˙ 푞 ≤ 푟 with ℓ(푞) ∈ 퐼, 푞 Pℓ(푞) “푇ℓ(푞) is nonstationary”. Thus, for each 푛 ∈ 퐼, ˙ ⋆ we may pick a P푛-name 퐶푛 for a club subset of 휇 such that, for all 푞 ≤ 푟 , ˙ ˙ 푞 Pℓ(푞) 푇ℓ(푞) ∩ 퐶ℓ(푞) = ∅. Consider the following binary relation: ⋆ ˙ 푅 := {(훼, 푞) ∈ 휇 × 푃 | 푞 ≤ 푟 & ∀푟 ≤ 푞[ℓ(푟) ∈ 퐼 → 푟 Pℓ(푟) 훼ˇ ∈ 퐶ℓ(푟)]}, ˙ + 휇 푉 ˙ and define 푇 := {(ˇ훼, 푝) | (훼, 푝) ∈ (퐸휔) × 푃 & 푝 Pℓ(푝) 훼ˇ ∈ / 퐶ℓ(푝)}. ⋆ ˙ ˙ + It is immediate that for all 푞 ≤ 푟 with 푛 := ℓ(푞) in 퐼, 푞 P 푇 ⊆ 푇 . ˙ + ′ Note that, for all (훼, 푞) ∈ 푅, 푞 P 훼ˇ ∈ / 푇 . Also, if (훼, 푞) ∈ 푅 and 푞 ≤ 푞 then (훼, 푞′) ∈ 푅, as well In this section we will aim to kill the stationarity of the bigger set 푇˙ + in place of 푇 . The whole point for this is that 푇˙ + has the following additional ⋆ ˙ + ˙ property: 푞 ≤ 푟 with 푛 := ℓ(푞) in 퐼, 푞 P푛 푇푛 =휇 ˇ ∖ 퐶푛. This was crucially used in the proof of [PRS20, Lemma 4.24] when we verified the density of ˚ + the ring poset (P훿)푛 in (P훿)푛, for 훿 ∈ acc(휇 + 1). The next proof mimics the argument of [PRS19, Claim 5.6.1]: Lemma 6.2. For all 훾 < 휇 and 푝 ≤ 푟⋆, there is an ordinal 훾¯ ∈ (훾, 휇) and 푝¯ ≤⃗휛 푝, such that (¯훾, 푝¯) ∈ 푅. Proof. We begin proving the following auxiliary claim: Claim 6.2.1. For all 훾 < 휇 and 푝 ≤ 푟⋆ there is an ordinal 훾¯ ∈ (훾, 휇) and ⃗휛 ˙ ˇ ˇ 푝¯ ≤ 푝, such that for all 푞 ≤ 푝¯, 푞 Pℓ(푞) 퐶ℓ(푞) ∩ (ˇ훾, 훾¯) ̸= ∅.

Proof. Let 훾 and 푝 be as above. Set ℓ := ℓ(푝), 푠 := 휛ℓ(푝) and put ′ ′ ˙ 퐷푝,훾 := {푞 ∈ 푃 | 푞 ≤ 푝 & ∃훾 > 훾 (푞 Pℓ(푞) 훾ˇ ∈ 퐶ℓ(푞))}.

Clearly, 퐷푝,훾 is 0-open, hence appealing to Clause (2) of Lemma 3.13 we obtain a condition푝 ¯ ≤⃗휛 푝 with the property that the set 푝¯+푡 푝¯+푡 푈퐷푝,훾 := {푡 ⪯ℓ 푠 | ∀푛 < 휔 (푃푛 ⊆ 퐷푝,훾 or 푃푛 ∩ 퐷푝,훾 = ∅)} 푝¯+푡 is dense in Sℓ ↓ 푠. Note that 푈퐷푝,훾 = {푡 ⪯ℓ 푠 | ∀푛 < 휔 푃푛 ⊆ 퐷푝,훾}. In eﬀect, 푊 (¯푝 + 푡) ⊆ 퐷푝,훾 for all 푡 ∈ 푈퐷푝,훾 . For each 푟 ∈ 푊 (¯푝 + 푡) pick some ordinal 훾푟 ∈ (훾, 휇) witnessing that 푟 ∈ 퐷푝,훾, and put

훾¯ := sup{훾푟 | 푟 ∈ 푊 (¯푝 + 푡)& 푡 ⪯ℓ 푠} + 1. Combining Clauses (훽) and (5) of Definition 3.3 we infer that훾 ¯ < 휇. We claim that푝 ¯ is as desired. Otherwise, let 푞 ≤ 푝¯ forcing the negation of the claim. By virtue of Clause (8) of Definition 3.3, 휛ℓ is a nice projection SIGMA-PRIKRY FORCING III 39 from P≥ℓ to Sℓ, hence Definition 2.2(4) applied to this map yields 푞 = 휛 푞¯ + 휛ℓ(푞), for some푞 ¯ ≤ ℓ 푝¯. Putting 푡 := 휛ℓ(푞), it is clear that 푡 ⪯ℓ 푠. By extending 푡 if necessary, we may freely assume that 푡 ∈ 푈퐷푝,훾 . On the other hand, 푞 ≤0 푤(¯푝, 푞), hence Lemma 3.9 and Clause (3) of Lemma 3.8 yield 푞 ≤0 푤(¯푝, 푞¯ + 푡) + 푡 = 푤(¯푝, 푞¯) + 푡 ∈ 푊 (¯푝 + 푡). This clearly yields a contradiction with our choice of 푞. Now we will take advantage of the previous claim to prove the lemma. So, let 훾 < 휇 and 푝 ≤ 푟⋆. Applying the above claim inductively, we find a ⃗휛 ≤ -decreasing sequence ⟨푝푛 | 푛 < 휔⟩ and an increasing sequence of ordinals below 휇, ⟨훾푛 | 푛 < 휔⟩, such that 푝0 := 푝, 훾0 := 훾, and such that for every 푛 < 휔, the pair (푝푛+1, 훾푛+1) witnesses together the conclusion of Claim 6.2.1 when putting (푝, 훾) := (푝푛, 훾푛). Moreover, Clause (9) of Definition 3.3 allows ˚휛ℓ us to assume that the 푝푛 are taken from Pℓ ↓ 푝, hence we may find푝 ¯ be a ⃗휛 ≤ -lower bound. Setting훾 ¯ := sup푛<휔 훾푛 we have that (¯훾, 푝¯) ∈ 푅. 6.1.1. Definition of the functor and basic properties.

Definition 6.3. Let 푝 be a condition in P.A labeled ⟨푝,⃗S⟩-tree is a function 푆 : dom(푆) → [휇]<휇, where

dom(푆) = {(푞, 푡) | 푞 ∈ 푊 (푝)& 푡 ∈ ∪ℓ(푝)≤푛≤ℓ(푞) S푛 ↓ 휛푛(푞)}, and such that for all (푞, 푡) ∈ dom(푆) the following hold: (1) 푆(푞, 푡) is a closed bounded subset of 휇; (2) 푆(푞′, 푡′) ⊇ 푆(푞, 푡) whenever 푞′ + 푡′ ≤ 푞 + 푡; ˙ + (3) 푞 + 푡 P 푆(푞, 푡) ∩ 푇 = ∅; (4) there is 푚 < 휔 such that for any 푞 ∈ 푊 (푝) and (푞′, 푡′) ∈ dom(푆) with 푞′ ≤ 푞, if 푆(푞′, 푡′) ̸= ∅ and ℓ(푞) ≥ ℓ(푝) + 푚, then (max(푆(푞′, 푡′)), 푞) ∈ 푅. The least such 푚 is denoted by 푚(푆). Remark 6.4. For any pairs (푞′, 푡′), (푞, 푡) of elements of dom(푆) with 푞′ + 푡′ ≤ 푞 + 푡, if if 푞 is incompatible with 푟⋆, then 푆(푞′, 푡′) = ∅. ⃗ ⃗ Definition 6.5. For 푝 ∈ 푃 , we say that 푆 = ⟨푆푖 | 푖 ≤ 훼⟩ is a ⟨푝, S⟩-strategy iff all of the following hold: (1) 훼 < 휇; ⃗ (2) for all 푖 ≤ 훼, 푆푖 is a labeled ⟨푝, S⟩-tree; (3) for every 푖 < 훼 and (푞, 푡) ∈ dom(푆푖), 푆푖(푞, 푡) ⊑ 푆푖+1(푞, 푡); ′ ′ ′ ′ (4) for every 푖 < 훼 and pairs (푞, 푡), (푞 , 푡 ) in dom(푆푖) with 푞 + 푡 ≤ 푞 + 푡, ′ ′ ′ ′ 푆푖+1(푞, 푡) ∖ 푆푖(푞, 푡) ⊑ 푆푖+1(푞 , 푡 ) ∖ 푆푖(푞 , 푡 ); (5) for every limit 푖 ≤ 훼 and (푞, 푡) ∈ dom(푆푖), 푆푖(푞, 푡) is the ordinal ⋃︀ closure of ( 푗<푖 푆푗(푞, 푡)). ⃗ ˙ Definition 6.6. Let A(P, S, 푇 ) be the notion of forcing A := (퐴, E), where: (1)( 푝, 푆⃗) ∈ 퐴 iff 푝 ∈ 푃 and either 푆⃗ = ∅ or 푆⃗ is a ⟨푝,⃗S⟩-strategy; ′ ⃗′ ⃗ (2)( 푝 , 푆 ) E (푝, 푆) iff: (a) 푝′ ≤ 푝; 40 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

(b) dom(푆⃗′) ≥ dom(푆⃗); ⃗ ′ (c) for each 푖 ∈ dom(푆) and (푞, 푡) ∈ dom(푆푖), ′ 푆푖(푞, 푡) = 푆푖(푤(푝, 푞), 푡푞), 43 where 푡푞 := 휛ℓ(푞)(푞 + 푡). For all 푝 ∈ 푃 , denote ⌈푝⌉A := (푝, ∅). Definition 6.7 (Projections and Pitchfork).

(1) Let ℓA := ℓ ∘ 휋 and ⃗휍 := ⃗휛 ∙ 휋, where 휋 : A → P is deﬁned via 휋(푝, 푆⃗) := 푝; ⃗ (2) Deﬁne 푐A : 퐴 → 퐻휇, by letting, for all (푝, 푆) ∈ 퐴, ⃗ ⃗ 푐A(푝, 푆) := (푐(푝), {(푖, 푐(푞), 푆푖(푞, ·)) | 푖 ∈ dom(푆), 푞 ∈ 푊 (푝)}),

where 푆푖(푞, ·) denotes the map 푡 ↦→ 푆푖(푞, 푡); ⃗ (3) Let 푎 = (푝, 푆) ∈ 퐴. The map t(푎): P ↓ 푝 → 퐴 is deﬁned by letting ′ ′ ⃗′ ⃗′ ⃗′ t(푎)(푝 ) := (푝 , 푆 ), where 푆 is a sequence such that dom(푆 ) = dom(푆⃗), and for all 푖 ∈ dom(푆⃗′) the following are true: ′ ′ (a) dom(푆푖) = {(푟, 푡) | 푟 ∈ 푊 (푝 )& 푡 ∈ ∪ℓ(푝′)≤푛≤ℓ(푟) S푛 ↓ 휛푛(푟)}, ′ (b) for all (푞.푡) ∈ dom(푆푖), ′ 44 (*) 푆푖(푞, 푡) = 푆푖(푤(푝, 푞), 푡푞). Remark 6.8. If (P, ℓ, 푐) is Σ-Prikry then we recuperate the corresponding notions from [PRS20, S4].

We next show that (t, 휋) defines a super nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓ, 푐, ⃗휛). The next lemma takes care partially of this task by showing that (t, 휋) is a forking projection from (A, ℓA, 푐A) to (P, ℓ, 푐). Lemma 6.9 (Forking projection). The pair (t, 휋) is a forking projection between (A, ℓA, 푐A) and (P, ℓ, 푐). Proof. We just give some details for the verification of Clauses (2). The rest can be proved arguing similarly to [PRS19, Lemma 6.13]. Here it goes: ⃗ ′ (2) Let 푎 = (푝, 푆) and 푝 ≤ 휋(푎). We just prove that t(푎) is well- defined. The argument for t(푎) being order-preserving is very similar to the one from [PRS19, Lemma 6.13(2)]. If 푆⃗ = ∅, then Definition 6.7(*) ′ ′ ⃗ yields t(푎)(푝 ) = (푝 , ∅) ∈ 퐴. So, suppose that dom(푆) = 훼 + 1. Let ′ ⃗′ ′ ′ ′ ⃗ (푝 , 푆 ) := t(푎)(푝 ) and 푖 ≤ 훼. We shall first verify that 푆푖 is a ⟨푝 , S⟩-labeled ′ tree. Let (푞, 푡) ∈ dom(푆푖) and let us go over the clauses of Definition 6.3. Since the verification of Clauses (3) and (4) are on the same lines of that of (2) we just give details for the latter.

43 ′ Note that 푡푞 ⪯푛 휛푛(푤(푝, 푞)) for some 푛 ∈ [ℓ(푝 ), ℓ(푞)]. Thus, (푤(푝, 푞), 푡푞) ∈ dom(푆푖). And if 푡 ⪯ℓ(푞) 휛ℓ(푞)(푞), then 푡 = 푡푞. 44 Here 푡푞 is as in Definition 6.6((c)). SIGMA-PRIKRY FORCING III 41

′ ′ ′ ′ ′ (2): Let (푞 , 푡 ) ∈ dom(푆푖) be such that 푞 + 푡 ≤ 푞 + 푡. By Clause (6) of Definition 3.3, 푤(푝, 푞′ + 푡′) ≤ 푤(푝, 푞 + 푡). Also, combining [PRS19, Lemma 2.9] and Lemma 3.9 we have the following chain of equalities: 푤(푝, 푞′ + 푡′) = 푤(푝, 푤(푝′, 푞′ + 푡′)) = 푤(푝, 푤(푝′, 푞′)) = 푤(푝, 푞′). Similarly one shows 푤(푝, 푞 + 푡) = 푤(푝, 푞). Thus, 푤(푝, 푞′) ≤ 푤(푝, 푞). Combining Clause (2) of Definition 3.7 with 푞′ + 푡′ ≤ 푞 + 푡 we have ′ ′ ′ ′ 휛ℓ(푞)(푤(푝, 푞 ) + 푡푞′ ) = 휛ℓ(푞)(푞 + 푡 ) ⪯ℓ(푞) 휛ℓ(푞)(푞 + 푡) = 푡푞. ′ ′ ⃗ Thus, 푤(푝, 푞 ) + 푡푞′ ≤ 푤(푝, 푞) + 푡푞. Now use Clause (2) for the labeled ⟨푝, S⟩- ′ ′ ′ ′ ′ ′ tree 푆푖 to get that 푆푖(푞 , 푡 ) = 푆푖(푤(푝, 푞 ), 푡푞′ ) ⊇ 푆푖(푤(푝, 푞), 푡푞) = 푆푖(푞, 푡). To prove that (푝′, 푆⃗′) ∈ 퐴 it remains to argue that 푆⃗′ fulfils the requirements described in Clauses (3), (4) and (5) of Definition 6.5. Indeed, each of these clauses follow from the corresponding ones for 푆⃗. There is just one delicate point in Clause (4), where one needs to argue that ′ ′ 푤(푝, 푞 ) + 푡푞′ ≤ 푤(푝, 푞) + 푡푞. This is done as in Clause (2) above. ′ ′ ⃗′ ⃗ Finally, it is clear that t(푎)(푝 ) = (푝 , 푆 ) E (푝, 푆) (see Definition 6.6). This concludes the verification of Clause (2).

Lemma 6.10. For each 푛 < 휔, 휍푛 is a nice projection from A≥푛 to S푛, and for each 푘 ≥ 푛, 휍푛 A푘 is again a nice projection. Proof. We go over the clauses of Deﬁnition 2.2. Clauses (1) and (2) follow from the fact that 휍푛 is the composition of the projections 휛푛 and 휋 and Clause (3) follows from Lemma 6.9 and Lemma 5.6.

′ ′ Claim 6.10.1. Let 푎, 푎 ∈ 퐴≥푛 and 푠 ⪯푛 휍푛(푎) with 푎 E 푎 + 푠. Then, * * 휛푛 ′ * ′ for each 푝 ∈ 푃≥푛 such that 푝 ≤ 휋(푎) and 휋(푎 ) = 푝 + 휍푛(푎 ) there is * * 휍푛 * * ′ * ′ 푎 ∈ 퐴≥푛 such that 푎 E 푎 with 휋(푎 ) = 푝 and 푎 = 푎 + 휍푛(푎 ). In particular, 휍푛 satisfies Clause (4).

′ ′ ′ Proof. Let 푎 = (푝, 푆⃗), 푎 = (푝 , 푆⃗ ) and 푠 ⪯푛 휍푛(푎) be as above. ′ ′ By Lemma 5.6, 푎 E 푎 + 푠 = t(푎)(푝 + 푠), hence 푝 ≤ 푝 + 푠. Since 휛푛 is a nice projection from P≥푛 to S푛, Definition 2.2(4) yields the existence of a * * 휛푛 ′ * ′ * condition 푝 ∈ 푃≥푛 such that 푝 ≤ 푝 and 푝 = 푝 + 휍푛(푎 ). So, let 푝 be ′ ′ some such condition and set 푡 := 휛푛(푝 ). We have that 휍푛(푎 ) = 푡. Our aim ⃗* * * ⃗* is to find a sequence 푆 such that 푎 := (푝 , 푆 ) is a condition in A≥푛 with * 휍 * ′ the property that 푎 E 푛 푎 and 푎 + 푡 = 푎 . * * As 푛 ≤ ℓ(푝 ), Definition 2 yields 휛푛“푊 (푝 ) = {휛푛(푝)}, hence 푞 + 푡 is well-defined for all 푞 ∈ 푊 (푝*). Moreover, by virtue of Lemma 3.8(3), 푞 + 푡 ∈ 푊 (푝* + 푡) = 푊 (푝′) for every 푞 ∈ 푊 (푝*). ⃗ ⃗′ ′ ⃗* Put 푆 := ⟨푆푖 | 푖 ≤ 훼⟩ and 푆 := ⟨푆푖 | 푖 ≤ 훽⟩. Let 푆 := ⟨푆푖 | 푖 ≤ 훽⟩ * be the sequence where for each 푖 ≤ 훽, 푆푖 is the function with domain * {(푞, 푢) | 푞 ∈ 푊 (푝 )& 푢 ∈ ∪ℓ(푝*)≤푚≤ℓ(푞) S푚 ↓ 휛푚(푞)} defined according to the following casuistic: 42 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

(a) If 푆⃗ is the empty sequence, then {︃ 푆′(푞 + 푡, 푢 ), if 푞 + 푢 ≤ 푞 + 푡; 푆*(푞, 푢) := 푖 푞 푖 ∅, otherwise.

(b) If 푆⃗ is non-empty then there are two more cases to consider: (1) If 훼 < 푖 ≤ 훽, then

{︃ ′ * 푆푖(푞 + 푡, 푢푞), if 푞 + 푢 ≤ 푞 + 푡; 푆푖 (푞, 푢) := 푆훼(푤(푝, 푞), 푢푞), otherwise. * (2) Otherwise, 푆푖 (푞, 푢) := 푆푖(푤(푝, 푞), 푢푞). * By Lemma 3.8(4), 푞 + 푢 = 푞 + 푢푞 for all (푞, 푢) ∈ dom(푆푖 ) and 푖 ≤ 훽. * * We now show that 푆⃗ is a ⟨푝 ,⃗S⟩-strategy by going over the clauses of Deﬁnition 6.5. Clause (1) is indeed obvious, so we begin with (2).

Subclaim 6.10.1.1. Clause (2) holds for 푆⃗*. Proof. Fix some 푖 ≤ 훽 and let us go over the clauses of Definition 6.3. (1): This is obvious. ′ ′ * ′ ′ (2): Let (푞 , 푢 ), (푞, 푢) ∈ dom(푆푖 ) such that 푞 + 푢 ≤ 푞 + 푢. Case (a): We need to distinguish two subcases: * * * ′ ′ I If 푞 + 푢 푞 + 푡, then 푆푖 (푞, 푢) = ∅ and so 푆푖 (푞, 푢) ⊆ 푆푖 (푞 , 푢 ). * ′ I Otherwise, 푞 + 푢 ≤ 푞 + 푡 and so 푆푖 (푞, 푢) = 푆푖(푞 + 푡, 푢푞). On the other ′ ′ ′ ′ hand, since 푞 +푢 ≤ 푞 +푢 ≤ 푞 +푡, we have that 휛푛(푞 +푢 ) ⪯푛 휛푛(푞 +푡) = 푡. ′ ′ ′ * ′ ′ ′ ′ ′ In particular, 푞 + 푢 ≤ 푞 + 푡 and so 푆푖 (푞 , 푢 ) = 푆푖(푞 + 푡, 푢푞′ ). Now, it is routine to check that (푞+푡)+푢푞 = 푞+푢푞 = 푞+푢. Similarly, the same applies ′ ′ ′ * * ′ ′ to 푞 and 푢푞′ . Appealing to Clause (2) for 푆푖 we get 푆푖 (푞, 푢) ⊆ 푆푖 (푞 , 푢 ). Case (b)(1): There are several cases to consider: * I Assume 푞 + 푢 푞 + 푡. Then 푆푖 (푞, 푢) = 푆훼(푤(푝, 푞), 푢푞). ′ ′ * ′ ′ ′ ′ II Suppose 푞 + 푢 푞 + 푡. Then 푆푖 (푞 , 푢 ) = 푆훼(푤(푝, 푞 ), 푢푞′ ). On one hand, by Clause (6) of Definition 3.3, 푤(푝, 푞′ + 푢′) ≤ 푤(푝, 푞 + 푢). Combining [PRS19, Lemma 2.9] with Lemma 3.9, we have 푤(푝, 푞′ +푢′) = 푤(푝, 푤(푝*, 푞′ + 푢′)) = 푤(푝, 푤(푝*, 푞′)) = 푤(푝, 푞′). Similarly, one shows that 푤(푝, 푞 + 푢) = 푤(푝, 푞). Thus, 푤(푝, 푞′) ≤ 푤(푝, 푞). Also, arguing as in page 41, one can prove ′ ′ that 푤(푝, 푞 ) + 푢푞′ ≤ 푤(푝, 푞) + 푢푞. This finally yields * ′ ′ * ′ ′ 푆푖 (푞, 푢) = 푆훼(푤(푝, 푞), 푢푞) ⊆ 푆훼(푤(푝, 푞 ), 푢푞′ ) = 푆푖 (푞 , 푢 ). ′ ′ * ′ ′ ′ ′ ′ II Otherwise, 푞 + 푢 ≤ 푞 + 푡, and so 푆푖 (푞 , 푢 ) = 푆푖(푞 + 푡, 푢푞′ ). ⃗′ Since 훼 < 푖 and 푏 E 푎, Clauses (3) and (5) of Definition 6.5 for 푆 yield ′ * ′ ′ ′ ′ ′ ′ 푆훼(푤(푝, 푞 + 푡), 푢 ) = 푆훼(푞 + 푡, 푢푞′ ) ⊆ 푆푖(푞 + 푡, 푢푞′ ), * ′ ′ ′ ′ where 푢 := 휛ℓ(푞′)((푞 + 푡) + 푢푞′ ). A routine checking gives (푞 + 푡) + 푢푞′ = ′ ′ * ′ ′ ′ ′ ′ ′ 푞 + 푢푞′ , hence 푢 = 푢푞′ , and thus 푆훼(푤(푝, 푞 + 푡), 푢푞′ ) ⊆ 푆푖(푞 + 푡, 푢푞′ ). SIGMA-PRIKRY FORCING III 43

Arguing as in the previous case, 푤(푝, 푞′ + 푡) = 푤(푝, 푞′). Therefore, ′ ′ ′ ′ ′ * ′ ′ 푆훼(푤(푝, 푞 ), 푢푞′ ) ⊆ 푆푖(푞 + 푡, 푢푞′ ) = 푆푖 (푞 , 푢 ). ′ ′ Once again, 푤(푝, 푞 ) + 푢푞′ ≤ 푤(푝, 푞) + 푢푞. Hence, Clause (2) for 푆훼 yields * ′ ′ * ′ ′ 푆푖 (푞, 푢) = 푆훼(푤(푝, 푞), 푢푞) ⊆ 푆훼(푤(푝, 푞 ), 푢푞′ ) ⊆ 푆푖 (푞 , 푢 ). ′ ′ I Assume 푞 + 푢 ≤ 푞 + 푡. Then 푞 + 푢 ≤ 푞 + 푡, as well. In particular, * ′ ′ ′ ′ * ′ ′ 푆푖 (푞, 푢) = 푆푖(푞 + 푡, 푢푞) ⊆ 푆푖(푞 + 푡, 푢푞′ ) = 푆푖 (푞 , 푢 ), ′ 45 where the above follows from Clause (2) for 푆푖. Case (b)(2): This is clear using Clause (2) for 푆푖. * (3): Let (푞, 푢) ∈ dom(푆푖 ). There are two cases to discuss: Case (a): As before there are two cases depending on whether 푞+푢 푞+푡 * * or not. The first case is obvious, as 푆푖 (푞, 푢) = ∅. Otherwise, 푆푖 (푞, 푢) = ′ ′ * ˙ + 46 푆푖(푞 + 푡, 푢푞), and so Clause (3) for 푆푖 yields 푞 + 푢푞 P 푆푖 (푞, 푢) ∩ 푇 = ∅. Case (b)(1): There are two cases to consider: * I Assume 푞 + 푢 푞 + 푡. Then, 푆푖 (푞, 푢) = 푆훼(푤(푝, 푞), 푢푞). Combining * ˙ + 푞 + 푢푞 ≤ 푤(푝, 푞) + 푢푞 with Clause (3) for 푆훼, 푞 + 푢푞 P 푆푖 (푞, 푢) ∩ 푇 = ∅. * ′ I Otherwise 푞 + 푢 ≤ 푞 + 푡, and so 푆푖 (푞, 푢) = 푆푖(푞 + 푡, 푢푞). As in previous * ˙ + cases we have 푞 + 푢푞 P 푆푖 (푞, 푢푞) ∩ 푇 = ∅. Case (b)(2): This follows using Clause (3) for 푆푖. * ′ ′ * ′ (4): Let 푞 ∈ 푊 (푝 ) and a pair (푞 , 푢 ) ∈ dom(푆푖 ) with 푞 ≤ 푞 and ℓ(푞) ≥ * ′ 47 ℓ(푝 ) + 푚푖, where 푚푖 = max{푚(푆훼), 푚(푆푖), 푚(푆푖)} + 1. * ′ ′ * ′ ′ To avoid trivialities, suppose 푆푖 (푞 , 푢 ) ̸= ∅ and put 훿 := max(푆푖 (푞 , 푢 )). * ′ ′ ′ ′ Case (a): Since 푆푖 (푞 , 푢 ) ̸= ∅ we have 푞 + 푢 ≤ 푞 + 푡. In this case * ′ ′ ′ ′ ′ ′ ′ 푆푖 (푞 , 푢 ) = 푆푖(푞 +푡, 푢푞′ ). Since 푞 +푡 ≤ 푞, Clause (4) for 푆푖 yields (훿, 푞) ∈ 푅. Case (b): The verification of the clause in Case (b)(1) and Case (b)(2) is identical to the previous one. Simply note that we can still invoke Clause (4) ′ for 푆푖, 푆훼 or 푆푖, as 푚푖 is sufficiently large. Subclaim 6.10.1.2. Clause (3) holds for 푆⃗*.

* * * Proof. Let 푖 < 훽 and (푞, 푢) ∈ dom(푆푖 ). We show that 푆푖 (푞, 푢) ⊑ 푆푖+1(푞, 푢). * * Case (a): If 푞 + 푢 푞 + 푡 then 푆푖 (푞, 푢) = 푆푖+1(푞, 푢) = ∅ and we are done. * ′ Otherwise, 푞 + 푢 ≤ 푞 + 푡, and so 푆푗 (푞, 푢) = 푆푗(푞 + 푡, 푢푞), for 푗 ∈ {푖, 푖 + 1}. Now, Clause (4) for 푆⃗′ yields the desired property. * * Case (b)(1): Since 훼 < 푖 < 훽 note that both 푆푖 (푞, 푢) and 푆푖+1(푞, 푢) have * been deﬁned according to Case (b)(1). If 푞 + 푢 푞 + 푡, then 푆푖 (푞, 푢) = * 푆훼(푤(푝, 푞), 푢푞) = 푆푖+1(푞, 푢) and the desired property follows trivially. In

45See also the argument of Case (a) above. 46 Once again, we have used that (푞 + 푡) + 푢푞 = 푞 + 푢푞. 47 If 푆⃗ is empty then we convey that 푚(푆훼) := 0. 44 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

* ′ the opposite case, 푆푗 (푞, 푢) = 푆푗(푞 + 푡, 푢푞), for 푗 ∈ {푖, 푖 + 1}. Now it suffices ⃗′ * * to appeal to Clause (3) for 푆 to infer that 푆푖 (푞, 푢) ⊑ 푆푖+1(푞, 푢). Case (b)(2): In this case 푖 ≤ 훼 and there are two more subcases: * * I If 푖 < 훼, then 푖 + 1 ≤ 훼 and thus both 푆푖 (푞, 푢) and 푆푖+1(푞, 푢) have been defined according to Case (b)(2). Now apply Clause (3) for 푆⃗. * * I Otherwise, 푖 = 훼. Then 푆훼(푞, 푢) and 푆훼+1(푞, 푢) have been defined * according to Case (b)(2) and Case (b)(1), respectively. Namely, 푆훼(푞, 푢) = * 푆훼(푤(푝, 푞), 푢푞) and 푆훼+1(푞, 푢) depends on whether 푞 + 푢 ≤ 푞 + 푡 or not. * * If 푞 + 푢 푞 + 푡, then 푆훼+1(푞, 푢) = 푆훼(푤(푝, 푞), 푢푞) = 푆훼(푞, 푢). In the * ′ ⃗′ opposite case, 푆훼+1(푞, 푢) = 푆훼+1(푞 + 푡, 푢푞). By Clause (3) for 푆 , ′ ′ * 푆훼(푞 + 푡, 푢푞) ⊑ 푆훼+1(푞 + 푡, 푢푞) = 푆훼+1(푞, 푢). ′ ′ Also, since 푎 E 푎, 푆훼(푞 + 푡, 푢푞) = 푆훼(푤(푝, 푞 + 푡), 푢푞). Arguing as in previous ′ cases, one can show that 푤(푝, 푞 + 푡) = 푤(푝, 푞), and thus 푆훼(푞 + 푡, 푢푞) = 푆훼(푤(푝, 푞), 푢푞). Altogether, we arrive the the following chain of inclusions: * ′ ′ * 푆훼(푞, 푢) = 푆훼(푤(푝, 푞), 푢푞) = 푆훼(푞+푡, 푢푞) ⊑ 푆훼+1(푞+푡, 푢푞) = 푆훼+1(푞, 푢). Subclaim 6.10.1.3. Clause (4) holds for 푆⃗*. ′ ′ * ′ ′ Proof. Let 푖 < 훽 and (푞, 푢), (푞 , 푢 ) ∈ dom(푆푖 ) such that 푞 + 푢 ≤ 푞 + 푢. There are two main cases to consider, along with their respective subcases. * * Case (a): The case 푞 + 푢 푞 + 푡 is obvious as both 푆푖 (푞, 푢) and 푆푖+1(푞, 푢) * ′ are empty. So, assume that 푞 + 푢 ≤ 푞 + 푡. Then, 푆푗 (푞, 푢) = 푆푗(푞 + 푡, 푢푞) * ′ ′ ′ ′ ′ and 푆푗 (푞 , 푢 ) = 푆푗(푞 + 푡, 푢푞′ ) for 푗 ∈ {푖, 푖 + 1}. Appealing to Clause (4) of Definition 6.5 for 푆⃗′ the desired property follows. * * ′ ′ Case (b)(1): Since 훼 < 푖 < 훽 then both 푆푖 (푞, 푢) (resp. 푆푖 (푞 , 푢 )) and * * ′ ′ 푆푖+1(푞, 푢) (resp. 푆푖+1(푞 , 푢 )) have been defined according to Case (b)(1). * * I If 푞 + 푢 푞 + 푡, then, 푆푖 (푞, 푢) = 푆훼(푤(푝, 푞), 푢푞) = 푆푖+1(푞, 푢), and so * * 푆푖+1(푞, 푠) ∖ 푆푖 (푞, 푠) = ∅. Thus, the desired property holds. * ′ I Otherwise, 푞 +푢 ≤ 푞 +푡 and so 푆푗 (푞, 푢) = 푆푗(푞 +푡, 푢푞) for 푗 ∈ {푖, 푖+1}. * ′ ′ The same applies to 푆푗 (푞 , 푢 ). Note that (푞 + 푡) + 푢푞 = 푞 + 푢푞 and also that ′ ′ ′ ′ ⃗′ (푞 + 푡) + 푢푞′ = 푞 + 푢푞′ . Invoking Clause (4) for 푆 we get the desired result. * * ′ ′ Case (b)(2): In this case we have 푆푖 (푞, 푢) = 푆푖(푤(푝, 푞), 푢푞) and 푆푖 (푞 , 푢 ) = ′ ′ * * ′ ′ 푆푖(푤(푝, 푞 ), 푢푞′ ). On the contrary, the definition of 푆푖+1(푞, 푢) and 푆푖+1(푞 , 푢 ) depends on whether 푖 + 1 ≤ 훼 or not. If 푖 < 훼 then 푖 + 1 ≤ 훼 and so * * ′ ′ ′ ′ 푆푖+1(푞, 푢) = 푆푖+1(푤(푝, 푞), 푢푞) and 푆푖+1(푞 , 푢 ) = 푆푖+1(푤(푝, 푞 ), 푢푞′ ). Using Clause (4) for the sequence 푆⃗ the result follows as usual. * * ′ ′ So, let us assume that 푖 = 훼. Then both 푆훼+1(푞, 푢) and 푆훼+1(푞 , 푢 ) have been defined according to Case (b)(1). There are two more subcases: ′ ′ I Assume that 푞 + 푢 ≤ 푞 + 푡. In this case, 푞 + 푢 ≤ 푞 + 푡 and so * ′ * ′ ′ ′ ′ ′ 푆훼+1(푞, 푢) = 푆훼+1(푞 + 푡, 푢푞) and 푆훼+1(푞 , 푢 ) = 푆훼+1(푞 + 푡, 푢푞′ ). Appealing to Clause (4) for 푆⃗′ it follows that * * * ′ ′ * ′ ′ 푆훼+1(푞, 푢) ∖ 푆훼(푞, 푢) ⊑ 푆훼+1(푞 , 푢 ) ∖ 푆훼(푞 , 푢 ). SIGMA-PRIKRY FORCING III 45

* * I Otherwise, 푆훼+1(푞, 푢) = 푆훼(푤(푝, 푞), 푢푞) = 푆훼(푞, 푢). This yields the desired property. Finally, one verifies that Clause (5) holds for 푆⃗* by appealing to the corresponding clauses for 푆⃗ and 푆⃗′. Combining the previous subclaims it follows that 푎* = (푝*, 푆⃗*) is a con- * dition in A≥푛. We now check that 푎 has the required properties: * 휍 Subclaim 6.10.1.4. 푎 E 푛 푎. Proof. Let us go over the clauses of Definition 6.6. By our choice, 푝* ≤ 푝 and dom(푆⃗*) ≥ dom(푆⃗), so that both Clauses (a) and (b) are true. Now ⃗ ⃗ * assume that 푆 is non-empty and let 푖 ∈ dom(푆) and (푞, 푢) ∈ dom(푆푖 ). ⃗* * Then by definition of 푆 , 푆푖 (푞, 푢) = 푆푖(푤(푝, 푞), 푢푞), so that Clause (c) holds. * * 휛 * 휍 Altogether, 푎 E 푎. Finally, since 푝 ≤ 푛 푝 it follows that 푎 E 푛 푎. Subclaim 6.10.1.5. 푎* + 푡 = 푎′. * * * * ′ Proof. By Lemma 5.6, 푎 + 푡 = t(푎 )(푝 + 푡). Also, since 푝 + 푡 = 푝 , * * ′ * ′ ′ 푎 + 푡 = t(푎 )(푝 ). Thus, we are left with showing that t(푎 )(푝 ) = 푎 . * ′ ′ ⃗ Put t(푎 )(푝 ) = (푝 , 푄). Let 푖 ≤ 훽 and (푞, 푢) ∈ dom(푄푖). By virtue of ′ Definition 6.7(3) we have that 푞 ≤ 푝 and 푢 ⪯푚 휛푚(푞), hence 푞 + 푢 ≤ 푞 + 푡. Case (a): In this case we have the following chain of equalities: * * ′ * ′ ′ 푄푖(푞, 푢) = 푆푖 (푤(푝 , 푞), 푢푞) = 푆푖(푤(푝 , 푞) + 푡, 푢푞) = 푆푖(푞, 푢푞) = 푆푖(푞, 푢). The first equality follows from Definition 6.7(3)(*), the third from Lemma 3.8(1) and the right-most one from Definition 6.3(2) and 푞 + 푢푞 = 푞 + 푢. ′ Case (b): If 훼 < 푖 ≤ 훽 then arguing as before 푄푖(푞, 푢) = 푆푖(푞, 푢). Otherwise, 푖 ≤ 훼 and we have the following chain of equalities * * ′ ′ 푄푖(푞, 푢) = 푆푖 (푤(푝 , 푞), 푢푞) = 푆푖(푤(푝, 푞), 푢푞) = 푆푖(푞, 푢푞) = 푆푖(푞, 푢), ′ For the third equality we used that 푎 E 푎 and 푢푞 = 휛ℓ(푞)(푞 + 푢푞). The above subclaims yield the proof of the claim. The above claim finishes the proof of the lemma. Corollary 6.11. The pair (t, 휋) is a super nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓ, 푐, ⃗휛).

Proof. First, (t, 휋) is a forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓ, 푐, ⃗휛) by virtue of Lemma 6.9. Second, ⃗휍 = ⃗휛 ∘ 휋 by our choice in Definition 6.7. Besides, Lemma 6.10 shows that for each 푛 < 휔, 휍푛 is a nice projection from A≥푛 and S푛. Moreover, Claim 6.10.1 actually shows that (t, 휋) fulfils the requirement appearing in Definition 5.4. This completes the proof. Next, we introduce a map tp and we will latter prove that it defines a nice type over (t, 휋). Afterwards, we will also show that tp witness that the pair (t, 휋) has the weak mixing property. 46 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Definition 6.12. Deﬁne a map tp : 퐴 → <휇휔, as follows. Given 푎 = (푝, 푆⃗) in 퐴, write 푆⃗ as ⟨푆푖 | 푖 < 훽⟩, and then let

tp(푎) := ⟨푚(푆푖) | 푖 < 훽⟩. We shall soon verify that tp is a nice type, but will use the mtp notation of Definition 5.7 from the outset. In particular, for each 푛 < 휔, we will have ˚ ˚ ˚ ˚ A푛 := (퐴푛, E), with 퐴푛 := {푎 ∈ 퐴 | 휋(푎) ∈ 푃푛 & mtp(푎) = 0}. We will often refer to the 푚(푆푖)’s as the delays of the strategy 푆⃗. Arguing on the lines of [PRS19, Lemma 6.15] one can prove the following: ˚휋 Fact 6.13. For each 푛 < 휔, A푛 is a 휇-directed closed forcing. Lemma 6.14. The map tp is a nice type over (t, 휋). Proof. The verification of Clauses (1)–(6) of Definition 5.9 is essentially the same as in [PRS20, Lemma 4.15]. A moment’s reflection makes clear that it suffices to prove Clause (8) to complete the lemma. Let 푎 = (푝, 푆⃗) ∈ 퐴 and to avoid trivialities, let us assume that 푆⃗ ̸= ∅. ⋆ I Suppose that 푝 is incompatible with 푟 . Then, by Remark 6.4, for all 푖 < dom(tp(푎)) and all (푞, 푡) dom(푆푖), 푆푖(푞, 푡) = ∅. Therefore, mtp(푎) = 0. ′ ⃗휛 ˚ ′ Using Definition 3.3(9), let 푝 ≤ 푝 be in 푃ℓ(푝) and set 푏 := t(푎)(푝 ). Com- bining Clauses (2) and (3) of Definition 5.7 with mtp(푎) = 0 it is immediate ′ ˚ ˚ ⃗휍 that mtp(푏) = 0. Also, 휋(푏) = 푝 ∈ 푃ℓ(푝). Thus, 푏 ∈ 퐴ℓ (푎) and 푏 E 푎. ⋆ A I Suppose 푝 ≤ 푟 . Appealing to Clause (5) of Definition 3.3 let 훾 < 휇 be ⃗ above sup푖

dom(푇푖) := {(푞, 푢) | 푞 ∈ 푊 (¯푝)& 푢 ∈ ∪ℓ(푝*)≤푚≤ℓ(푞) S푚 ↓ 휛푚(푞)}, as {︃ 푆푖(푤(푝, 푞), 푢푞), if 푖 < dom(푆⃗), 푇푖(푞, 푢) := 푆max(dom(푆⃗))(푤(푝, 푞), 푢푞) ∪ {훾¯}, otherwise. ⃗ ˚ Arguing as in Claim 6.10.1 one shows that (¯푝, 푇 ) is a condition in Aℓ(푝). ⃗휍 ˚휍푛 휍푛 Clearly 푏 E 푎. Therefore, A푛 is dense in A푛 . We now check that the pair (t, 휋) has the weak mixing property, as witnessed by the type tp given in Deﬁnition 6.12 (see Deﬁnition 5.13).

Lemma 6.15. The pair (t, 휋) has the weak mixing property as witnessed by the type tp from Definition 6.12. ′ 0 Proof. Let 푎, ⃗푟, 푝 ≤ 휋(푎), 푔 : 푊푛(휋(푎)) → A ↓ 푎 and 휄 be as in the statement of the Weak Mixing Property (see Deﬁnition 5.13). More precisely, ⃗푟 = ⟨푟휉 | 휉 < 휒⟩ is a good enumeration of 푊푛(휋(푎)), ⟨휋(푔(푟휉)) | 휉 < 휒⟩ SIGMA-PRIKRY FORCING III 47 is diagonalizable with respect to ⃗푟 (as witnessed by 푝′) and 푔 is a function witnessing Clauses (3)–(5) of Deﬁnition 5.13 with respect to the type tp. 휉 Put 푎 := (푝, 푆⃗) and for each 휉 < 휒, set (푝휉, 푆⃗ ) := 푔(푟휉). Claim 6.15.1. If 휄 ≥ 휒 then there is a condition 푏 in A as in the conclusion Definition 5.13.

Proof. If 휄 ≥ 휒 then Clause (3) yields dom(tp(푔(푟휉)) = 0 for all 휉 < 휒. A Hence, Clause (4) of Definition 5.7 yields 푔(푟휉) = ⌈푝휉⌉ for all 휉 < 휒. Since A 푔(푟휉) E 푎 this in particular implies that 푎 = ⌈푝⌉ . ′ A ′ 0 ′ ′ Set 푏 := ⌈푝 ⌉ . Clearly, 휋(푏) = 푝 and 푏 E 푎. Let 푞 ∈ 푊푛(푝 ). By ′ 0 ′ Clause (2) of Definition 5.13, 푞 ≤ 푝휉, where 휉 is such that 푟휉 = 푤(푝, 푞 ). ′ ′ A 0 A Finally, Definition 5.1(6) yields t(푏)(푞 ) = ⌈푞 ⌉ E ⌈푝휉⌉ = 푔(푟휉). So, hereafter let us assume that 휄 < 휒. For each 휉 ∈ [휄, 휒), Clause (3) of 휉 Definition 5.13 and Definition 6.12 together imply that dom(푆⃗ ) = 훼휉 +1 for some 훼휉 < 휇. Moreover, Clause (3) yields sup휄≤휂<휉 훼휂 < 훼휉 for all 휉 ∈ (휄, 휒). A 휉 Also, the same clause implies that 푔(푟휉) = ⌈푝휉⌉ , hence 푆⃗ = ∅, for all 휉 < 휄. ′ Let ⟨푠휏 | 휏 < 휃⟩ be the good enumeration of 푊푛(푝 ). By Definition 3.3(5),

휃 < 휇. For each 휏 < 휃, set 푟휉휏 := 푤(푝, 푠휏 ). By Definition 5.13(1), 0 푠휏 ≤ 휋(푔(푤(푝, 푠휏 ))) = 휋(푔(푟휉휏 )) = 푝휉휏 , ′ ⃗ 48 for each 휏 < 휃. Set 훼 := sup휄≤휉<휒 훼휉 and 훼 := sup(dom(푆)). By regularity of 휇 and Definition 5.13(3) it follows that 훼 < 훼′ < 휇. Our ′ goal is to define a sequence 푇⃗ = ⟨푇푖 | 푖 ≤ 훼 ⟩, with dom(푇푖) := {(푞, 푢) | ′ ′ ′ ⃗ 푞 ∈ 푊 (푝 )& 푢 ∈ ∪ℓ(푝′)≤푚≤ℓ(푞) S푚 ↓ 휛푚(푞)} for 푖 ≤ 훼 , such that 푏 := (푝 , 푇 ) is a condition in A satisfying the conclusion of the weak mixing property. 푡ℎ ′ As ⟨푠휏 | 휏 < 휃⟩ is a good enumeration of the 푛 -level of the 푝 -tree 푊 (푝′), Lemma 3.6(2) entails that, for each 푞 ∈ 푊 (푝′), there is a unique ordinal 휏푞 < 휃, such that 푞 is comparable with 푠휏푞 . It thus follows from ′ ′ Lemma 3.6(3) that, for all 푞 ∈ 푊 (푝 ), ℓ(푞) − ℓ(푝 ) ≥ 푛 iff 푞 ∈ 푊 (푠휏푞 ). ′ 0 Moreover, for each 푞 ∈ 푊≥푛(푝 ), 푞 ≤ 푠휏 ≤ 푝 , hence 푤(푝 , 푞) is well- 푞 휉휏푞 휉휏푞 defined. Now, for all 푖 ≤ 훼′ and 푞 ∈ 푊 (푝′), let: ⎧ 휉 푆 휏푞 (푤(푝 , 푞), 푢 ), if 푞 ∈ 푊 (푠 )& 휄 ≤ 휉 ; ⎪ min{푖,훼 } 휉휏푞 푞 휏푞 휏푞 ⎨⎪ 휉휏푞 푇 (푞, 푢) := 푖 푆 (푤(푝, 푞), 푢푞), if 푞∈ / 푊 (푠휏 )& 훼 > 0; ⎪ min{푖,훼} 푞 ⎩⎪∅, otherwise.

′ ′ Claim 6.15.2. Let 푖 ≤ 훼 . Then 푇푖 is a labeled 푝 -tree.

Proof. Fix (푞, 푢) ∈ 푑표푚(푇푖) and let us go over the Clauses of Deﬁnition 6.3. The veriﬁcation of (1), (2) and (3) are similar to that of [PRS19, Claim 6.16.1] and, actually, also to that of Claim 6.10.1 above. The reader is thus referred there for more details. We just elaborate on Clause (4).

48Note that 푎 might be ⌈푝⌉A, so we are allowing 훼 = 0. 48 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

′ For each 푖 < 훼 , set 휉(푖) := min{휉 ∈ [휄, 휒) | 푖 ≤ 훼휉}. Subclaim 6.15.2.1. If 푖 < 훼′, then

푚(푇푖) ≤ 푛 + max{mtp(푎), sup휄≤휂<휉(푖) mtp(푔(푟휂)), tp(푔(푟휉(푖))(푖)}. ′ ′ ′ ′ Proof. Let 푞 ∈ 푊푘(푝 ) and (푞 , 푢 ) be a pair in dom(푇푖) with 푞 ≤ 푞, where

푘 ≥ 푛 + max{mtp(푎), sup휄≤휂<휉(푖) mtp(푔(푟휂)), tp(푔(푟휉(푖))(푖)}. ′ ′ ′ ′ Suppose that 푇푖(푞 , 푢 ) ̸= ∅. Denote 휏 := 휏푞′ and 훿 := max(푇푖(푞 , 푢 )). ′ ′ Since ℓ(푞) ≥ ℓ(푝 ) + 푛, note that 푞, 푞 ∈ 푊 (푠휏 ). Also, 휄 ≤ 휉휏 , as otherwise ′ ′ 푇푖(푞 , 푢 ) = ∅. Thus, we fall into the ﬁrst option of the casuistic getting

′ ′ 휉휏 ′ ′ 푇푖(푞 , 푢 ) = 푆 (푤(푝휉휏 , 푞 ), 푢푞′ ). min{푖,훼휉휏 }

I Assume that 휉휏 < 휉(푖). Then, 훼휉휏 < 푖 and so

′ ′ 휉휏 ′ ′ 푇푖(푞 , 푢 ) = 푆 (푤(푝 , 푞 ), 푢 ′ ). 훼휉휏 휉휏 푞 ′ We have that 푤(푝휉휏 , 푞 ) ≤ 푤(푝휉휏 , 푞) is a pair in 푊푘−푛(푝휉휏 ) and that the set 휉휏 ′ ′ 휉휏 푆훼휉휏 (푤(푝휉휏 , 푞 ), 푢푞′ ) is non-empty. Also, 푘 − 푛 ≥ mtp(푔(푟휉휏 )) = 푚(푆훼휉휏 ). 휉휏 So, by Clause (4) for 푆훼휉휏 , we have that (훿, 푤(푝휉휏 , 푞)) ∈ 푅. Finally, since

푞 ≤ 푤(푝휉휏 , 푞), we have (훿, 푞) ∈ 푅, as desired.

I Assume that 휉(푖) ≤ 휉휏 . Then 푖 ≤ 훼휉(푖) ≤ 훼휉휏 , and thus

′ ′ 휉휏 ′ ′ 푇푖(푞 , 푢 ) = 푆푖 (푤(푝휉휏 , 푞 ), 푢푞′ ).

If dom(tp(푎)) ≤ 푖 ≤ sup휄≤휂<휉(푖) 훼휂, by Clause (4) of Deﬁnition 5.13,

tp(푔(푟휉휏 ))(푖) ≤ mtp(푎).

Otherwise, if sup휄≤휂<휉(푖) 훼휂 < 푖 ≤ 훼휉(푖), again by Deﬁnition 5.13(4)

tp(푔(푟휉휏 ))(푖) ≤ max{mtp(푎), tp(푔(푟휉(푖))(푖)}.

휉휏 In either case, 푤(푝휉휏 , 푞) ∈ 푊푘−푛(푝휉휏 ) and 푘 − 푛 ≥ tp(푔(푟휉휏 ))(푖) = 푚(푆푖 ). 휉휏 By Clause (4) of 푆푖 we get that (훿, 푤(푝휉휏 , 푞)) ∈ 푅, hence (훿, 푞) ∈ 푅.

Subclaim 6.15.2.2. 푚(푇훼′ ) ≤ 푛 + sup휄≤휉<휒 mtp(푔(푟휉)). ′ ′ ′ ′ Proof. Let 푞 ∈ 푊푘(푝 ) and (푞 , 푢 ) ∈ dom(푇푖) with 푞 ≤ 푞, where 푘 ≥ 푛 + ′ ′ sup휄≤휉<휒 mtp(푔(푟휉)). Suppose that 푇훼′ (푞 , 푢 ) ̸= ∅ and denote 휏 := 휏푞′ and ′ ′ ′ 훿 := max(푇훼′ (푞 , 푢 )). Since 푘 ≥ 푛, 푞, 푞 ∈ 푊 (푠휏 ). Also, 휄 ≤ 휉휏 , as otherwise ′ ′ ′ ′ 휉휏 ′ ′ ′ ′ ′ 푇훼 (푞 , 푢 ) = ∅. So, 푇훼 (푞 , 푢 ) = 푆훼휉휏 (푤(푝휉휏 , 푞 ), 푢푞′ ). Then 푤(푝휉휏 , 푞 ) ≤ 휉휏 푤(푝휉휏 , 푞) is a pair in 푊푘−푛(푝휉휏 ) with 푘 − 푛 ≥ mtp(푔(푟휉휏 )) = 푚(푆훼휉휏 ). So, 휉휏 Deﬁnition 6.3(4) regarded with respect to 푆훼휉휏 yields (훿, 푤(푝휉휏 , 푞)) ∈ 푅. Once again, it follows that (훿, 푞) ∈ 푅, as wanted.

The combination of the above subclaims yield Clause (4) for 푇푖. Claim 6.15.3. The sequence 푇⃗ is a 푝′-strategy. SIGMA-PRIKRY FORCING III 49

Proof. We need to go over the clauses of Deﬁnition 6.5. However, Clause (1) is trivial, Clause (2) is established in the preceding claim, and Clauses (3) and (5) follow from the corresponding ones of 푆⃗ and the 푆⃗푟휏 ’s. Finally, Clause (4) can be proved as in [PRS19, Claim 6.16.2]. Indeed, for this latter veriﬁcation it is convenient to bear in mind that 훼 > 0 yields 휄 = 0. ′ Thus, we have established that 푏 := (푝 , 푇⃗) is a legitimate condition in A.

′ Claim 6.15.4. Let 휏 < 휃. For each 푞 ∈ 푊푛(푠휏 ), 푤(푝 , 푞) = 푤(푠휏 , 푞) = 푞. Proof. The first equality can be proved exactly as in [PRS19, Claim 6.16.4]. For the second, notice that 푞 and 푤(푠휏 , 푞) are conditions in 푊 (푠휏 ) with the same length. Hence, Lemma 3.6(2) yields 푞 = 푤(푠휏 , 푞), as wanted. ′ 0 Claim 6.15.5. 휋(푏) = 푝 and 푏 E 푎. Proof. The verification is routine. For details we refer the reader to [PRS19, Claim 6.16.3], where a similar statement is proved. 0 49 Claim 6.15.6. For each 휏 < 휃, t(푏)(푠휏 ) E 푔(푟휉휏 ). ⃗ Proof. Let 휏 < 휃 and and denote t(푏)(푠휏 ) = (푠휏 , 푇휏 ). By Lemma 6.9(5) we 0 have that 휋(t(푏)(푠휏 )) = 푠휏 ≤ 푝휉휏 , so Clause (a) of Definition 6.6 holds. 0 A If 휉휏 < 휄, then t(푏)(푠휏 ) E ⌈푝휉휏 ⌉ = 푔(푟휉휏 ), and we are done. So, let us assume that 휄 ≤ 휉휏 . Let 푖 ≤ 훼휉휏 and 푞 ∈ 푊 (푠휏 ). By Definition 6.7(*), 휏 ′ ′ 푇푖 (푞, 푢) = 푇푖(푤(푝 , 푞), 푢푞) and by Claim 6.15.4, 푤(푝 , 푞) = 푤(푠휏 , 푞) = 푞, 휏 50 hence 푇푖 (푞, 푢) = 푇푖(푞, 푢푞) = 푇푖(푞, 푢). Also 푟휉휏푞 = 푤(푝, 푠휏푞 ) = 푤(푝, 푠휏 ) =

푟휉휏 , where the second equality follows from 푞 ∈ 푊 (푠휏 ). Therefore,

휏 휉휏 휉휏 푇푖 (푞, 푢) = 푆 (푤(푝휉휏 , 푞), 푢푞) = 푆 (푤(푝휉휏 , 푞), 푢푞). min{푖,훼휉휏 } 푖 0 Altogether, t(푏)(푠휏 ) E 푔(푟휉휏 ), as wanted. The combination of the above claims yield the proof of the lemma. Let us sum up what we have shown so far:

Corollary 6.16. (t, 휋) is a super nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓ, 푐, ⃗휛) having the weak mixing property. ⃗ In particular, (A, ℓA, 푐A, ⃗휍) is a (Σ, S)-Prikry, (A, ℓA) has property 풟, + 1lA A 휇 =휅 ˇ and ⃗휍 is a coherent sequence of nice projections. Proof. The ﬁrst part follows from Corollary 6.11 and Lemma 6.15. Likewise, (A, ℓA) has property 풟 by virtue of Lemma 5.15, and ⃗휍 is coherent by virtue of Lemma 5.17 (see also Setup 6). Thus, we are left with arguing that ⃗ + (A, ℓA, 푐A, ⃗휍) is (Σ, S)-Prikry and that 1lA A 휇 =휅 ˇ . All the Clauses of Deﬁnition 3.3 with the possible exception of (2), (7) and (9) follow from Theorem 5.11. Also, from this result and the assumptions in Setup 6 it

49 ′ Recall that ⟨푠휏 | 휏 < 휃⟩ was a good enumeration of 푊푛(푝 ). 50 For the second equality we use Definition 6.3(2) for 푇푖 and 푞 + 푢 = 푞 + 푢푞. 50 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

+ follows that 1lA A 휇 =휅 ˇ . Clauses (2) and (9) follow from Lemma 5.16, Fact 6.13 and Lemma 6.14. Finally, Clause (7) follows from Lemma 5.15. + Our next task is to show that after forcing with A the P-name 푇˙ ceases to be stationary. In this respect recall the blanket assumptions of the section displayed in page 38. ⋆ A ˙ + Lemma 6.17. ⌈푟 ⌉ A “푇 is nonstationary”. ⋆ Proof. Let 퐺 be A-generic over 푉 , with ⌈푟 ⌉A ∈ 퐺. Work in 푉 [퐺]. Let 퐺¯ (resp. 퐻푛) denote the generic ﬁlter for P (resp. S푛) induced by 휋 (resp. 휍푛) and 퐺. For all 푎 = (푝, 푆⃗) ∈ 퐺 and 푖 ∈ dom(푆⃗) write 푖 ⋃︁ ¯ 푑푎 := {푆푖(푞, 푡) | 푞 ∈ 퐺∩푊 (푝)& ∃푛 ∈ [ℓ(푝), ℓ(푞)] (푡 ⪯푛 휛푛(푞푛) ∧ 푡 ∈ 퐻푛)}, where ⟨푞푛 | 푛 ≥ ℓ(푝)⟩ is the increasing enumeration of 퐺¯∩푊 (푝) (see Lemma 3.6). Then, let {︃ max(dom(푆⃗)) 푑푎 , if 푆⃗ ̸= ∅; 푑푎 := ∅, otherwise.

Claim 6.17.1. Suppose that 푎 = (푝, 푆⃗) ∈ 퐺. In 푉 [퐺¯], for all 푖 ∈ dom(푆⃗), 푖 푖 ˙ + the ordinal closure cl(푑푎) of 푑푎 is disjoint from (푇 )퐺. Proof. To avoid trivialities we shall assume that 푆⃗ ̸= ∅. We prove the claim by induction on 푖 ∈ dom(푆⃗). The base case 푖 = 0 is trivial, as 푆0 : 푊 (푝) → {∅} (see Deﬁnition 6.5(5)). So, let us assume by induction that 푗 + cl(푑푎) is disjoint from (푇˙ )퐺 for every 0 ≤ 푗 < 푖. 푖 ⋃︀ 푗 Let 훾 ∈ cl(푑푎) ∖ 푗<푖 cl(푑푎). By virtue of Clause (2) of Deﬁnition 6.3 푖 applied to 푆푖, we may further assume that 훾∈ / 푑푎. Succesor case: Suppose that 푖 = 푗 + 1. There are two cases to discuss: ¯ I Assume cf(훾) = 휔. Working in 푉 [퐺], we have 훾 = sup푛<휔 훾푛, where for each 푛 < 휔, there is (푞푛, 푡푛), such that 푞푛 ∈ 퐺¯ ∩ 푊 (푝), 푡푛 ⪯푘 휛푘(푞푛) with 푡푛 ∈ 퐻푘 for some 푘 ∈ [ℓ(푝), ℓ(푞푛)], and 훾푛 ∈ 푆푗+1(푞푛, 푡푛) ∖ 푆푗(푞푛, 푡푛). 51 Strengthening if necessary, we may assume 푞푛 + 푡푛 ≤ 푞푚 + 푡푚 for 푚 ≤ 푛. For each 푛 < 휔 set 훿푛 := max(푆푗+1(푝푛, 푡푛)). Clearly, 훾 ≤ sup푛<휔 훿푛. We claim that 훾 = sup푛<휔 훿푛: Assume to the contrary that this is not the case. Then, there is 푛0 < 휔 such that 훾푚 < 훿푛0 for all 푚 < 휔.

Let 푚 ≥ 푛0. Then, 훾푚 ∈ 푆푗+1(푞푚, 푡푚) ∖ 푆푗(푞푚, 푡푚). Also, since 훾푛0 ∈/ ⃗ 푆푗(푞푛0 , 푡푛0 ), hence 푆푗+1(푞푛0 , 푡푛0 ) ̸= 푆푗(푞푛0 , 푡푛0 ), Deﬁnition 6.5(3) for 푆 yields

훿0 ∈ 푆푗+1(푞푛0 , 푡푛0 ) ∖ 푆푗(푞푛0 , 푡푛0 ). By virtue of Clause (4) of Deﬁnition 6.5,

푆푗+1(푞푛0 , 푡푛0 ) ∖ 푆푗(푞푛0 , 푡푛0 ) ⊑ 푆푗+1(푞푚, 푡푚) ∖ 푆푗(푞푚, 푡푚) ∋ 훾푚.

Thus, as 훾푚 < 훿0, we have that 훾푚 belongs to the left-hand-side set. 푖 Since 푚 above was arbitrary we get 훾 ∈ 푆푗+1(푞푛0 , 푡푛0 ) ⊆ 푑푎. This yields 푖 a contradiction with our original assumption that 훾∈ / 푑푎.

51For this we use Definition 6.3(2). SIGMA-PRIKRY FORCING III 51

⋆ So, 훾 = sup푛<휔 훿푛. Now, let 푛 < 휔 such that ℓ(푝푛⋆ ) ≥ ℓ(푝) + 푚(푆푖+1). ⋆ Then, for all 푛 ≥ 푛 , Clause (4) of Deﬁnition 6.3 yields (훿푛, 푞푛⋆ ) ∈ 푅. In ˙ + particular, (훾, 푞푛* ) ∈ 푅 and thus 푞푛* P 훾ˇ ∈ / 푇 (see page 38). Finally, since + 푞푛* ∈ 퐺¯, we conclude that 훾∈ / (푇˙ )퐺, as desired. ¯ I Assume cf(훾) ≥ 휔1. Working in 푉 [퐺], we have 훾 = sup훼

By Clause (5) of Deﬁnition 6.5, 훿푛0 = sup푘<휔 max(푆푘(푞푛0 , 푡푛0 ), hence there is some 푘0 < 휔 such that 훾 < max(푆푘0 (푞푛0 , 푡푛0 )).

Fix 푚 ≥ 푛0. Since 푞푚 + 푡푚 ≤ 푞푛0 + 푡푛0 , Clause (2) of Deﬁnition 6.3 yields

훾 < max(푆푘0 (푞푛0 , 푡푛0 )) ≤ max(푆푘0 (푞푚, 푡푚)). Also, Clause (3) of Deﬁnition 6.5 implies that

푆푘0 (푞푚, 푡푚) ⊑ 푆휔(푞푚, 푡푚) ∋ 훾푚,

푘0 so that, 훾푚 ∈ 푆푘0 (푞푚, 푡푚). Since 푚 was arbitrary, we infer that 훾 ∈ cl(푑푎 ), which yields a contradiction with our original assumption. ˙ + So, 훾 = sup푛<휔 훿푛. Arguing as in previous cases conclude that 훾∈ / (푇 )퐺. 52 Note that this is the case because 푊 (푝) is a tree with height 휔 and cf(훾) ≥ 휔1. 53 Note that increasing 푞 would only increase 푆푗+1(푞, 푡훼). 52 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA ¯ I Assume cf(훾) ≥ 휔1. Working in 푉 [퐺], we have 훾 = sup훼<푖 훾훼, where for each 훼 < 푖, there is 푡훼 ∈ 퐻푛 with 푡훼 ⪯푛 휛푛(푞) such that 훾훼 ∈ 푆푖(푞, 푡훼).

As in previous cases, here both 푞 and 푛 are fixed and 푞 ∈ 푊≥푚(푆푖)(푝). For each 훼 < 푖, set 훿훼 := max(푆푖(푞, 푡훼)). Once again, we aim to show * that 훾 = sup훼<푖 훿훼. Suppose that this is not the case, and let 훼 < 푖 such that 훾훼 < 훿훼* for all 훼 < 푖. As before, 훾 ̸= 훿훼* , so there is some훼 ¯ < 푖 such that 훾 < max(푆훼¯(푞, 푡훼* )) Now, let 훼 < 푖 be arbitrary and find 푠훼 ⪯푛 푡훼* , 푡훼 in 퐻푛. Then, 훾훼 ∈ 푆푖(푞, 푡훼) ⊆ 푆푖(푞, 푠훼). Also, 푆훼¯(푞, 푡훼* ) ⊆ 푆훼¯(푞, 푠훼) and so max(푆훼¯(푞, 푠훼)) > 훾. By Clause (3) of Definition 6.5 we have that 푆훼¯(푞, 푠훼) ⊑ 푆푖(푞, 푠훼), hence 훾훼 ∈ 푆훼¯(푞, 푠훼). 훼¯ The above shows that 훾 ∈ cl(푑푎 ), which is a contradiction. So, 훾 = sup훼<푖 훿훼. Now proceed as in previous cases, invoking Clause (4) ˙ + of Definition 6.3, and infer that 훾∈ / (푇 )퐺. Claim 6.17.2. Suppose 푎 = (푝, 푆⃗) ∈ 퐴, where 푝 ≤ 푟⋆. For every 훾 < 휇, ⃗ ⃗ ⃗ there exists 훾¯ ∈ (훾, 휇) and (¯푝, 푇 ) E (푝, 푆), such that max(dom(푇 )) = 훼 and for all (푞, 푡) ∈ dom(푇훼), max(푇훼(푞, 푡)) = 훼. Proof. This is indeed what the argument of Lemma 6.14 shows. Working in 푉 [퐺], the above claim yields an unbounded set 퐼 ⊆ 휇 such 훾 훾 that for each 훾 ∈ 퐼 there is 푎훾 = (푝훾, 푆⃗ ) ∈ 퐺 with max(dom(푆⃗ )) = 훾 and 훾 훾 max(푆훾 (푞, 푡)) = 훾 for all (푞, 푡) ∈ dom(푆훾 ). For each 훾 ∈ 퐼, set 퐷훾 := cl(푑푎훾 ). ′ ′ Claim 6.17.3. For each 훾 < 훾 both in 퐼 , 퐷훾 ⊑ 퐷훾′ . ′ ′ Proof. Let 훾 < 훾 be in 퐼 . It is enough to show that 푑푎훾 ⊑ 푑푎훾′ . Namely, we

⃗ ′ will show that 푑푎훾 = 푑푎훾′ ∩훾 +1. Let 푏 = (푟, 푅) ∈ 퐺 be such that 푏E푎훾, 푎훾 . 훾 For the ﬁrst direction, suppose that 훿 ∈ 푑푎훾 and let (푞, 푡) ∈ dom(푆훾 ) be a pair witnessing this. By strengthening 푞 and 푡 if necessary, we may further 54 assume that ℓ(푞) ≥ ℓ(푟) and 푡 ∈ 퐻푛, 푡 ⪯푛 휛푛(푞), where 푛 := ℓ(푞). Let 푟′ ∈ 푊 (푟) ∩ 퐺¯ be the unique condition with ℓ(푟′) = 푛. Also, let ′ ′ ′ ′ ′ ′ ′ 푡 ∈ 퐻푛 be such that 푡 ⪯푛 휛푛(푟 ), 푡. Then 푤(푝훾, 푟 ) = 푞, 푡 = 휛푛(푟 + 푡 ) ′ and 푞 + 푡 ≤ 푞 + 푡. So, by 푏 E 푎훾 and 푏 E 푎훾′ , we get: 훾 훾 ′ ′ ′ 훾′ ′ ′ 훿 ∈ 푆훾 (푞, 푡) ⊆ 푆훾 (푞, 푡 ) = 푅훾(푟 , 푡 ) = 푆훾 (푤(푝훾, 푟 ), 푡 ) ⊆ 푑푎훾′ .

For the other direction, suppose that 훿 ∈ 푑푎훾′ ∩ (훾 + 1) and let (푞, 푡) be a 훾′ pair in dom(푆훾′ ) witnessing this. Again, by strengthening 푞, we may assume that ℓ(푞) ≥ ℓ(푟) and 푡 ∈ 퐻푛, 푡 ⪯푛 휛푛(푞푛), where 푛 := ℓ(푞). Similarly as ′ ′ ′ above, let 푟 ∈ 푊 (푟) ∩ 퐺¯ be with ℓ(푟 ) = 푛, and 푡 ∈ 퐻푛 be such that ′ ′ ′ ′ ′ ′ ′ 푡 ⪯푛 휛푛(푟 ), 푡. Then 푤(푝훾′ , 푟 ) = 푞, 푡 = 휛푛(푟 + 푡 ), 푞 + 푡 ≤ 푞 + 푡 and: ′ ′ 훾′ ′ (1) 푅훾′ (푟 , 푡 ) = 푆훾′ (푞, 푡 ), since 푏 E 푎훾′ ;

54 ′ Suppose that (푞, 푡) is the pair we are originally given and that 푞 ∈ 푊푛(푝) ∩ 퐺¯, where ′ ′ ′ ′ 훾 ′ ′ 푛 ≥ ℓ(푟). Setting 푡 := 휛푛(푞 + 푡) it is immediate that 푞 + 푡 ≤ 푞 + 푡, hence 훿 ∈ 푆훾 (푞 , 푡 ). ′ ′ ′ ′ Also, it is not hard to check that 푞 + 푡 ∈ 퐺, hence 푡 ⪯푛 휛푛(푞 ) and 푡 ∈ 퐻푛. SIGMA-PRIKRY FORCING III 53

′ ′ 훾 ′ ′ ′ ′ (2) 푅훾(푟 , 푡 ) = 푆훾 (푤(푝훾, 푟 ), 푡 ), and so 훾 = max(푅훾(푟 , 푡 )); ′ ′ ′ ′ ⃗ (3) 푅훾(푟 , 푡 ) ⊑ 푅훾′ (푟 , 푡 ), by Clause (3) of Deﬁnition 6.5 for 푅. Combining all three, we get that 훾′ 훾′ ′ 훿 ∈ 푆훾′ (푞, 푡) ∩ (훾 + 1) ⊆ 푆훾′ (푞, 푡 ) ∩ (훾 + 1) = ′ ′ ′ ′ 훾 ′ ′ ′ 푅훾 (푟 , 푡 ) ∩ (훾 + 1) = 푅훾(푟 , 푡 ) = 푆훾 (푤(푝훾, 푟 ), 푡 ) ⊆ 푑푎훾 , as desired. ⋃︀ + Let 퐷 := 훾∈퐼 퐷훾. By Claims 6.17.1 and 6.17.3, 퐷 is disjoint from (푇 )퐺. Additionally, Claim 6.17.3 implies that 퐷 is closed and, since 퐼′ ⊆ 퐷, it is ˙ + also unbounded. So, (푇 )퐺 is nonstationary in 푉 [퐺]. ⋆ ˙ ˙ + Remark 6.18. Note that Lemma 6.17 together with 푟 P 푇 ⊆ 푇 (see page 38) ⋆ A ˙ imply that ⌈푟 ⌉ A “푇 is nonstationary”. The next corollary sums up the content of Subsection 6.1:

Corollary 6.19. Suppose that (Σ,⃗S)-Prikry quadruple (P, ℓ, 푐, ⃗휛) such that, P = (푃, ≤) is a subset of 퐻휇+ , (P, ℓ) has property 풟, ⃗휛 is a coherent sequence + of nice projections, 1lP P 휇ˇ =휅 ˇ and 1lP P “휅 is singular”. ⋆ ⋆ For every 푟 ∈ 푃 and a P-name 푧 for an 푟 -fragile stationary subset of ⃗ 휇, there are a (Σ, S)-Prikry quadruple (A, ℓA, 푐A, ⃗휍) having property 풟, and a pair of maps (t, 휋) such that all the following hold: (a) (t, 휋) is a super nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓ, 푐, ⃗휛) that has the weak mixing property; (b) ⃗휍 is a coherent sequence of nice projections; + (c) 1lA A 휇ˇ =휅 ˇ ; (d) A = (퐴, E) is a subset of 퐻휇+ ; ˚휋 (e) For every 푛 < 휔, A푛 is a 휇-directed-closed; (f) ⌈푟⋆⌉A forces that 푧 is nonstationary. Proof. Since all the assumptions of Setup 6 are valid we obtain from Defi- nitions 6.6 and 6.7, a notion of forcing A = (퐴, E) together with maps ℓA and 푐A, and a sequence ⃗휍 such that, by Corollary 6.16, (A, ℓA, 푐A, ⃗휍) is a (Σ,⃗S)-Prikry quadruple having property 풟 and Clauses (a)–(c) above hold. Clause (d) easily follows from the definition of A = (퐴, E) (see, e.g. [PRS19, Lemma 6.6]), Clause (e) is Fact 6.13 and Clause (f) is Lemma 6.17 together with Remark 6.18. 6.2. Fragile sets vs non-reflecting stationary sets. For every 푛 < 휔, 푉 denote Γ푛 := {훼 < 휇 | cf (훼) < 휎푛−2}, where, by convention, we define 휎−2 and 휎−1 to be ℵ0. The next lemma is an analogue of [PRS19, Lemma 6.1] and will be crucial for the proof of reflection in the model of the Main Theorem. Lemma 6.20. Suppose that: P푛 휇 휇 (i) for every 푛 < 휔, 푉 |= Refl(퐸<휎푛−2 , 퐸<휎푛 ); 54 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

⋆ (ii) 푟 is a condition in P; ˙ (iii) 푇 is a nice P-name for a subset of Γℓ(푟⋆); ⋆ (iv) 푟 P-forces that 푇˙ is a non-reflecting stationary set. Then 푇˙ is 푟⋆-fragile. Proof. Suppose that 푇˙ is not 푟⋆-fragile (see Definition 6.1), and let 푞 be an extension of 푟⋆ witnessing that. Set 푛 := ℓ(푞), so that ˙ 푞 P푛 “푇푛 is stationary”. ˙ Since 푇 is a nice P-name for a subset of Γℓ(푟⋆), it altogether follows that 푞 ˙ 휇 P푛-forces that 푇푛 is a stationary subset of 퐸<휎푛−2 . Let 퐺푛 be P푛-generic containing 푞. By Clause (푖), we have that 푇푛 := ˙ (푇푛)퐺푛 reflects at some ordinal 훾 of cofinality < 휎푛. Since 휛푛 is a nice 휛푛 55 projection, we have that P푛 × S푛 projects to P푛. Then by |푆푛| < 휎푛 and 휛푛 the fact that P푛 contains a 휎푛-directed-closed dense subset, it follows that 푉 휃 := cf (훾) is < 휎푛. In 푉 , fix a club 퐶 ⊆ 훾 of order-type 휃. Work in 푉 [퐺푛]. Set 퐴 := 푇푛 ∩ 퐶, and note that 퐴 is a stationary subset of 훾 of size ≤ 휃. Let 퐻푛 be the S푛-generic filter induced from 퐺푛 by 휛푛. 휛푛 Again, since P푛 contains a 휎푛-directed-closed dense subset, it cannot have added 퐴. So, 퐴 ∈ 푉 [퐻푛]. Let ⟨훼푖 | 푖 < 휃⟩ be some enumeration (possibly with repetitions) of 퐴, and let ⟨훼˙ 푖 | 푖 < 휃⟩ be a sequence of S푛- ˙ ˙ name for it. Pick a condition 푟 in P푛/퐻푛 such that 푟 P푛 퐴 ⊆ 푇푛 ∩ 훾 and ˙ such that 휛푛(푟) S푛 퐴 = {훼˙ 푖 | 푖 < 휃}. Denote 푠 := 휛푛(푟) and note that 푠 ∈ 퐻푛. We now go back and work in 푉 .

′ 휛푛 ′ ′ Claim 6.20.1. Let 푖 < 휃 and 훼 < 훾. For all 푟 ≤ 푟 and 푠 ⪯푛 푠, if 푠 S푛 ′′ 휛푛 ′ ′′ ′ ′′ ′′ ˙ 훼˙ 푖 =훼 ˇ, then there are 푟 ≤ 푟 and 푠 ⪯푛 푠 such that 푟 + 푠 P 훼ˇ ∈ 푇 . Proof. Suppose 푟′, 푠′ are as above. As 푟′ extends 푟 and 푠′ extends 푠, it follows ′ ′ ˙ ′ ˙ that 푟 + 푠 P푛 훼ˇ ∈ 푇푛 and 푠 S푛 훼ˇ ∈ 퐴. So, by the deﬁnition of the name ˙ 0 ′ ′ ˙ 푇푛, there is some 푝 ≤ 푟 + 푠 such that 푝 P 훼ˇ ∈ 푇푛. By Deﬁnition 2.2(4), ′′ ′ ′′ 휛푛 ′ ′′ ′′ ′′ ′′ ˙ let 푠 ⪯푛 푠 and 푟 ≤ 푟 be such that 푟 + 푠 = 푝. So 푟 + 푠 P 훼ˇ ∈ 푇 , as desired.

Fix an injective enumeration ⟨(푖휉, 푠휉) | 휉 < 휒⟩ of 휃 × (S푛 ↓ 푠). Note that 휛푛 56 휛푛 휒 < 휎푛. Using that P푛 is 휎푛-strategically-closed, build a ≤ -decreasing ⃗휛 sequence of conditions ⟨푟휉 | 휉 ≤ 휒⟩, such that, for every 휉 < 휒, 푟휉 ≤ 푟, 휉 and, for any 훼 < 훾, if 푠휉 S푛 훼˙ 푖휉 =훼 ˇ, then there is 푠 ⪯푛 푠휉 such that 휉 ˙ * * 푟휉 + 푠 P 훼ˇ ∈ 푇 . Finally, let 푟 := 푟휒. Note that 휛푛(푟 ) = 휛푛(푟) = 푠, and * hence 휛푛(푟 ) ∈ 퐻푛. * ˙ Claim 6.20.2. 푟 P/퐻푛 퐴 ⊆ 푇 ∩ 훾ˇ.

55 휛푛 More precisely, (P푛 ↓ 푞) × (S푛 ↓ 휛푛(푞)) projects onto P푛 ↓ 푞. 56This is a consequence of Clause (2) of Definition 3.3. SIGMA-PRIKRY FORCING III 55

′ ′ Proof. For each 푖 < 휃, by density, there is some 푠 ⪯푛 푠 in 퐻푛 such that 푠 ′ decides훼 ˙ 푖 to be some ordinal 훼 < 훾. Fix 휉 < 휒 such that (푖휉, 푠휉) = (푖, 푠 ). 휉 ˙ * 휉 ˙ By the construction, 푟휉 + 푠 P 훼ˇ ∈ 푇 , hence 푟 + 푠 P 훼ˇ ∈ 푇 ∩ 훾ˇ.

Finally, since (P, ℓ, 푐, ⃗휛) is (Σ,⃗S)-Prikry, Lemma 3.14(1) implies that P/퐻푛 does not add any new subsets of 휃 and so no new subsets of 퐶, hence P/퐻푛 preserves the stationarity of 퐴, hence the stationarity of 푇 ∩ 훾. This contradicts hypothesis (푖푣).

7. Iteration scheme In this section, we deﬁne an iteration scheme for (Σ,⃗S)-Prikry forcings, following closely and expanding the work from [PRS20, S3]. Setup 7. The blanket assumptions for this section are as follows: <휇 ∙ 휇 is some cardinal satisfying 휇 = 휇, so that |퐻휇| = 휇; * ∙⟨ (휎푛, 휎푛) | 푛 < 휔⟩ is a sequence of pairs of regular uncountable * cardinals, such that, for every 푛 < 휔, 휎푛 ≤ 휎푛 ≤ 휇 and 휎푛 ≤ 휎푛+1; ⃗ ∙ S = ⟨S푛 | 푛 < 휔⟩ is a sequence of notions of forcing, S푛 = (푆푛, ⪯푛), with |푆푛| < 휎푛; ∙ Σ := ⟨휎푛 | 푛 < 휔⟩ and 휅 := sup푛<휔 휎푛. The following convention will be applied hereafter: Convention 7.1. For a pair of ordinals 훾 ≤ 훼 ≤ 휇+:

(1) ∅훼 := 훼 × {∅} denotes the 훼-sequence with constant value ∅; (2) For a 훾-sequence 푝 and an 훼-sequence 푞, 푝 * 푞 denotes the unique 훼-sequence satisfying that for all 훽 < 훼: {︃ 푞(훽), if 훾 ≤ 훽 < 훼; (푝 * 푞)(훽) = 푝(훽), otherwise.

(3) Let P훼 := (푃훼, ≤훼) and P훾 := (푃훾, ≤훾) be forcing posets such that 훼 훾 푃훼 ⊆ 퐻휇+ and 푃훾 ⊆ 퐻휇+ . Also, assume 푝 ↦→ 푝 훾 deﬁnes a 훼 P훾 P훼 projection between P훼 and P훾. We denote by 푖훾 : 푉 → 푉 the map deﬁned by recursion over the rank of each P훾-name 휎 as follows: 훼 훼 푖훾 (휎) := {(푖훾 (휏), 푝 * ∅훼) | (휏, 푝) ∈ 휎}. Our iteration scheme requires three building blocks:

Building Block I. We are given a (Σ,⃗S)-Prikry forcing (Q, ℓ, 푐, ⃗휛) such that (Q, ℓ) satisﬁes property 풟. We moreover assume that Q = (푄, ≤푄) + is a subset of 퐻휇+ , 1lQ Q “ˇ휇 =휅 ˇ & 휅 is singular” and ⃗휛 is a coherent sequence. To streamline the matter, we also require that 1lQ be equal to ∅. ⃗ Building Block II. Suppose that (P, ℓP, 푐P, ⃗휛) is a (Σ, S)-Prikry quadruple having property 풟 such that P = (푃, ≤) is a subset of 퐻휇+ , ⃗휛 is a coherent + sequence of nice projections, 1lP P “ˇ휇 =휅 ˇ ” and 1lP P “ˇ휅 is singular”. 56 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

⋆ For every 푟 ∈ 푃 , and a P-name 푧 ∈ 퐻휇+ , we are given a corresponding ⃗ (Σ, S)-Prikry quadruple (A, ℓA, 푐A, ⃗휍) having property 풟 such that: (a) there is a super nice forking projection (t, 휋) from (A, ℓA, 푐A, ⃗휍) to (P, ℓP, 푐P, ⃗휛) that has the weak mixing property; (b) ⃗휍 is a coherent sequence of nice projections; ˚휋 * 57 (c) for every 푛 < 휔, A푛 is 휎푛-directed-closed; + (d) 1lA A 휇ˇ =휅 ˇ ; (e) A = (퐴, E) is a subset of 퐻휇+ ; By [PRS20, Lemma 2.18], we may also require that: (f) each element of 퐴 is a pair (푥, 푦) with 휋(푥, 푦) = 푥; (g) for every 푎 ∈ 퐴, ⌈휋(푎)⌉A = (휋(푎), ∅); A A (h) for every 푝, 푞 ∈ 푃 , if 푐P(푝) = 푐P(푞), then 푐A(⌈푝⌉ ) = 푐A(⌈푞⌉ ). + Building Block III. We are given a function 휓 : 휇 → 퐻휇+ . Goal 7.2. Our goal is to define a system ⟨(P훼, ℓ훼, 푐훼, ⃗휛훼, ⟨t훼,훾| 훾 ≤ 훼⟩) | 훼 ≤ 휇+⟩ in such a way that for all 훾 ≤ 훼 ≤ 휇+: 훼 (i) P훼 is a poset (푃훼, ≤훼), 푃훼 ⊆ 퐻휇+ , and, for all 푝 ∈ 푃훼, |퐵푝| < 휇, where 퐵푝 := {훽 + 1 | 훽 ∈ dom(푝)& 푝(훽) ̸= ∅}; (ii) The map 휋훼,훾 : 푃훼 → 푃훾 defined by 휋훼,훾(푝) := 푝 훾 forms an projection from P훼 to P훾 and ℓ훼 = ℓ훾 ∘ 휋훼,훾; (iii) P0 is a trivial forcing, P1 is isomorphic to Q given by Building Block I, and P훼+1 is isomorphic to A given by Building Block II when invoked ⋆ with respect to (P훼, ℓ훼, 푐훼, ⃗휛훼) and a pair (푟 , 푧) which is decoded from 휓(훼); ⃗ (iv) If 훼 > 0, then (P훼, ℓ훼, 푐훼, ⃗휛훼) is a (Σ, S)-Prikry notion of forcing satisfying property 풟, whose greatest element is ∅훼, ℓ훼 = ℓ1 ∘ 휋훼,1 + and ∅훼 P훼 휇ˇ =휅 ˇ . Moreover, ⃗휛훼 is a coherent sequence of nice projections such that ⃗휛훼 = ⃗휛훾 ∙ 휋훼,훾 for every 훾 ≤ 훼; + (v) If 0 < 훾 < 훼 ≤ 휇 , then (t훼,훾, 휋훼,훾) is a nice forking projection from + (P훼, ℓ훼, ⃗휛훼) to (P훾, ℓ훾, ⃗휛훾); in case 훼 < 휇 ,(t훼,훾, 휋훼,훾) is furthermore a nice forking projection from (P훼, ℓ훼, 푐훼, ⃗휛훼) to (P훾, ℓ훾, 푐훾, ⃗휛훾), and in case 훼 = 훾 + 1, (t훼,훾, 휋훼,훾) is super nice and has the weak mixing property; (vi) If 0 < 훾 ≤ 훽 ≤ 훼, then, for all 푝 ∈ 푃훼 and 푟 ≤훾 푝훾, t훽,훾(푝훽)(푟) = (t훼,훾(푝)(푟)) 훽. + 7.1. Defining the iteration. For every 훼 < 휇 , fix an injection 휑훼 : 훼 → 휇. As |퐻휇| = 휇, by the Engelking-Karlowicz theorem, we may also fix 푖 + a sequence ⟨푒 | 푖 < 휇⟩ of functions from 휇 to 퐻휇 such that for every + <휇 푖 function 푒 : 퐶 → 퐻휇 with 퐶 ∈ [휇 ] , there is 푖 < 휇 such that 푒 ⊆ 푒 . The upcoming definition is by recursion on 훼 ≤ 휇+, and we continue as long as we are successful.

57 퐴˚푛 is the poset given in Definition 5.7(7) defined with respect to the type map witnessing Clause (a) above. SIGMA-PRIKRY FORCING III 57

I Let P0 := ({∅}, ≤0) be the trivial forcing. Let ℓ0 and 푐0 be the constant function {(∅, ∅)} and ⃗휛0 = ⟨{(∅, 1lS푛 )} | 푛 < 휔⟩. Finally, let t0,0 be the constant function {(∅, {(∅, ∅)})}, so that t0,0(∅) is the identity map. 1 ′ ′ I Let P1 := (푃1, ≤1), where 푃1 := 푄 and 푝 ≤1 푝 iff 푝(0) ≤푄 푝 (0). 휄 Evidently, 푝 ↦→ 푝(0) form an isomorphism between P1 and Q, so we naturally define ℓ1 := ℓ ∘ 휄, 푐1 := 푐 ∘ 휄 and ⃗휛1 := ⃗휛 ∙ 휄. Hereafter, the sequence ⃗휛1 1 is denoted by ⟨휛푛 | 푛 < 휔⟩. For all 푝 ∈ 푃1, let t1,0(푝): {∅} → {푝} be the constant function, and let t1,1(푝) be the identity map. + I Suppose 훼 < 휇 and that ⟨(P훽, ℓ훽, 푐훽, ⃗휛훽, ⟨t훽,훾| 훾 ≤ 훽⟩) | 훽 ≤ 훼⟩ has already been defined. We now define (P훼+1, ℓ훼+1, 푐훼+1, ⃗휛훼+1) and ⟨t훼+1,훾| 훾 ≤ 훼 + 1⟩. II If 휓(훼) happens to be a triple (훽, 푟, 휎), where 훽 < 훼, 푟 ∈ 푃훽 and 휎 is a P훽-name, then we appeal to Building Block II with (P훼, ℓ훼, 푐훼, ⃗휛훼), ⋆ 훼 ⃗ 푟 := 푟 * ∅훼 and 푧 := 푖훽 (휎) to get a corresponding (Σ, S)-Prikry quadruple (A, ℓA, 푐A, ⃗휍). II Otherwise, we obtain (A, ℓA, 푐A, ⃗휍) by appealing to Building Block II ⋆ with (P훼, ℓ훼, 푐훼, ⃗휛훼), 푟 := ∅훼 and 푧 := ∅. In both cases, we obtain a nice forking projection (t, 휋) from (A, ℓA, 푐A, ⃗휍) to (P훼, ℓ훼, 푐훼, ⃗휛훼). Furthermore, each condition in A = (퐴, E) is a pair A (푥, 푦) with 휋(푥, 푦) = 푥, and, for every 푝 ∈ 푃훼, ⌈푝⌉ = (푝, ∅). Now, define P훼+1 := (푃훼+1, ≤훼+1) by letting 푃훼+1 := {푥a⟨푦⟩ | (푥, 푦) ∈ 퐴}, and then ′ ′ ′ letting 푝 ≤훼+1 푝 iff (푝 훼, 푝(훼)) E (푝 훼, 푝 (훼)). Put ℓ훼+1 := ℓ1 ∘ 휋훼+1,1 and define 푐훼+1 : 푃훼+1 → 퐻휇 via 푐훼+1(푝) := 푐A(푝 훼, 푝(훼)). 훼 Let ⃗휛훼 = ⟨휛푛 | 푛 < 휔⟩ be defined in the natural way, i.e., for each 푛 < 휔 a 훼 a and 푥 ⟨푦⟩ ∈ (푃훼)≥푛, we set 휛푛 (푥 ⟨푦⟩) := 휍푛(푥, 푦). Next, let 푝 ∈ 푃훼+1, 훾 ≤ 훼 + 1 and 푟 ≤훾 푝 훾 be arbitrary; we need to define t훼+1,훾(푝)(푟). For 훾 = 훼 + 1, let t훼+1,훾(푝)(푟) := 푟, and for 훾 ≤ 훼, let

a t훼+1,훾(푝)(푟) := 푥 ⟨푦⟩ iff t(푝 훼, 푝(훼))(t훼,훾(푝 훼)(푟)) = (푥, 푦). + I Suppose 훼 ≤ 휇 is a nonzero limit ordinal, and that the sequence ⟨(P훽, ℓ훽, 푐훽, ⃗휛훽, ⟨t훽,훾| 훾 ≤ 훽⟩) | 훽 < 훼⟩ has already been defined according to Goal 7.2. Define P훼 := (푃훼, ≤훼) by letting 푃훼 be all 훼-sequences 푝 such that |퐵푝| < 휇 and ∀훽 < 훼(푝 훽 ∈ 푃훽). Let 푝 ≤훼 푞 iff ∀훽 < 훼(푝 훽 ≤훽 푞 훽). Let ℓ훼 := ℓ1 ∘ 휋훼,1. Next, we define 푐훼 : 푃훼 → 퐻휇, as follows. + II If 훼 < 휇 , then, for every 푝 ∈ 푃훼, let

푐훼(푝) := {(휑훼(훾), 푐훾(푝 훾)) | 훾 ∈ 퐵푝}. + II If 훼 = 휇 , then, given 푝 ∈ 푃훼, ﬁrst let 퐶 := cl(퐵푝), then deﬁne a function 푒 : 퐶 → 퐻휇 by stipulating:

푒(훾) := (휑훾[퐶 ∩ 훾], 푐훾(푝 훾)). 푖 Then, let 푐훼(푝) := 푖 for the least 푖 < 휇 such that 푒 ⊆ 푒 . Set ⃗휛훼 := ⃗휛1 ∙휋훼,1. 58 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Finally, let 푝 ∈ 푃훼, 훾 ≤ 훼 and 푟 ≤훾 푝 훾 be arbitrary; we need to define t훼,훾(푝)(푟). For 훾 = 훼, let t훼,훾(푝)(푟) := 푟, and for 훾 < 훼, let ⋃︀ t훼,훾(푝)(푟) := {t훽,훾(푝 훽)(푟) | 훾 ≤ 훽 < 훼}. 7.2. Verification. Our next task is to verify that for all 훼 ≤ 휇+, the tuple (P훼, ℓ훼, 푐훼, ⃗휛훼, ⟨t훼,훾| 훾 ≤ 훼⟩) fulfills requirements (i)–(vi) of Goal 7.2. It is obvious that Clauses (i) and (iii) hold, so we focus on verifying the rest. The next fact deals with an expanded version of Clause (vi). For the proof we refer the reader to [PRS20, Lemma 3.5]: + Fact 7.3. For all 훾 ≤ 훼 ≤ 휇 , 푝 ∈ 푃훼 and 푟 ∈ 푃훾 with 푟 ≤훾 푝 훾, if we let 푞 := t훼,훾(푝)(푟), then: (1) 푞 훽 = t훽,훾(푝 훽)(푟) for all 훽 ∈ [훾, 훼]; (2) 퐵푞 = 퐵푝 ∪ 퐵푟; (3) 푞 훾 = 푟; (4) If 훾 = 0, then 푞 = 푝; (5) 푝 = (푝 훾) * ∅훼 iff 푞 = 푟 * ∅훼; ′ 0 0 ′ ′ (6) for all 푝 ≤훼 푝, if 푟 ≤훾 푝 훾, then t훼,훾(푝 )(푟) ≤훼 t훼,훾(푝)(푟). We move on to Clause (ii) and Clause (v): Lemma 7.4. Suppose that 훼 ≤ 휇+ is such that for all nonzero 훾 < 훼, ⃗ (P훾, 푐훾, ℓ훾, ⃗휛훾) is (Σ, S)-Prikry. Then: ∙ for all nonzero 훾 ≤ 훼, (t훼,훾, 휋훼,훾) is a nice forking projection from (P훼, ℓ훼, ⃗휛훼) to (P훾, ℓ훾, ⃗휛훾), where 휋훼,훾 is defined as in Goal 7.2(ii); + ∙ if 훼 < 휇 , then (t훼,훾, 휋훼,훾) is furthermore a nice forking projection from (P훼, ℓ훼, 푐훼, ⃗휛훼) to (P훾, ℓ훾, 푐훾, ⃗휛훾) ∙ if 훼 = 훾 + 1, then (t훼,훾, 휋훼,훾) is super nice and has the weak mixing property. Proof. The above items with the exception of niceness can be proved as in [PRS20, Lemma 3.6]. We also acknowledge that the despite the notion of super niceness was not considered in [PRS20] it will automatically follow from niceness and Clause (a) of Building Block II. Thereby, it suffices to prove the following claim in order to complete the argument:

Claim 7.4.1. For all nonzero 훾 ≤ 훼, ⃗휛훼 = ⃗휛훾 ∙ 휋훼,훾. Also, for each 푛, 훼 훼 휛푛 is a nice projection from (P훼)≥푛 to S푛 and for each 푘 ≥ 푛, 휛푛 (P훼)푘 is again a nice projection. Proof. By induction on 훼 ≤ 휇+: I The case 훼 = 1 is trivial, since then, 훾 = 훼 and ⃗휛1 = ⃗휛 ∙ 휄. ′ ′ I Suppose 훼 = 훼 + 1 and the claim holds for 훼 . Recall that P훼 = P훼′+1 was defined by feeding (P훼′ , ℓ훼′ , 푐훼′ , ⃗휛훼′ ) into Building Block II, thus ⃗ obtaining a (Σ, S)-Prikry forcing (A, ℓA, 푐A, ⃗휍) along with the pair (t, 휋). Also, we have that (푥, 푦) ∈ 퐴 iff 푥a⟨푦⟩ ∈ 푃훼. By niceness of (t, 휋) and our recursive definition, ′ ′ 훼 a 훼 훼 a 휛푛 (푥 ⟨푦⟩) = 휍푛(푥, 푦) = 휛푛 (휋(푥, 푦)) = 휛푛 (휋훼,훼′ (푥 ⟨푦⟩)), SIGMA-PRIKRY FORCING III 59 for all 푛 < 휔 and 푥a⟨푦⟩ ∈ (푃훼)≥푛. Hence, ⃗휛훼 = ⃗휛훼′ ∙ 휋훼,훼′ . Using the induction hypothesis for ⃗휛훼′ we arrive at ⃗휛훼 = ⃗휛훾 ∙ 휋훼,훾. Let us now address the second part of the claim. We just show that for 훼 every 푛 < 휔, the map 휛푛 is a nice projection from (P훼)≥푛 to S푛. The 훼 statement that 휛푛 (P훼)푘 is a nice projection can be proved similarly. So, let us go over the clauses of Definition 2.2. Clauses (1) and (2) are evident and Clause (3) follows from Lemma 5.6 applied to (t훼,훼′ , 휋훼,훼′ ). 훼 (4): Let 푝, 푞 ∈ (푃훼)≥푛 and 푠 ⪯푛 휛푛 (푝) be such that 푞 ≤훼 푝 + 푠. Then, ′ ′ ′ ′ (푞 훼 , 푞(훼 )) E (푝 훼 , 푝(훼 )) + 푠. By Clause (a) of Building Block II we have that 휍푛 is a nice projection from A≥푛 to S푛, hence there is (푥, 푦) ∈ 퐴 such 휍푛 ′ ′ ′ ′ ′ ′ that (푥, 푦) (푝 훼 , 푝(훼 )) and (푞 훼 , 푞(훼 )) = (푥, 푦) + 휍푛((푞 훼 , 푞(훼 )). E 훼 ′ ′ 휛푛 Setting 푝 := 푥a⟨푦⟩ it is immediate that 푝 ≤훼 푝 and ′ 훼′ ′ ′ 훼 푞 = 푝 + 휛푛 (푞 훼 ) = 푝 + 휛푛 (푞). + I For 훼 ∈ acc(휇 +1), the first part follows from ⃗휛훼 := ⃗휛1 ∘휋훼,1 and the induction hypothesis. About the verification of the Clauses of Definition 2.2, Clauses (1) and (2) are automatic and Clause (3) follows from Lemma 5.6 applied to (t훼,1, 휋훼,1). About Clause (4) we argue as follows. 훼 Fix 푝, 푞 ∈ (P훼)≥푛 and 푠 ⪯푛 휛푛 (푝) be such that 푞 ≤훼 푝 + 푠. The goal is 훼 ′ ′ 휛푛 ′ 훼 to find a condition 푝 ∈ (푃훼)≥푛 such that 푝 ≤ 푝 and 푞 = 푝 + 휛푛 (푞). 58 Let ⟨훾휏 | 휏 ≤ 휃⟩ be the increasing enumeration of the closure of 퐵푞. For every 휏 ∈ nacc(휃 + 1), 훾휏 is a successor ordinal, so we let 훽휏 denote its predecessor. By recursion on 휏 ≤ 휃, we shall define a sequence of conditions 훾휏 ′ ∏︀ ′ 휛푛 ′ 훾휏 ⟨푝휏 | 휏 ≤ 휃⟩ ∈ 휏≤휃(푃훾휏 ) such that 푝휏 ≤훾휏 푝훾휏 and 푞훾휏 = 푝휏 +휛푛 (푞훾휏 ). In order to be able to continue with the construction at limits stages we ′ ′ shall moreover secure that ⟨푝휏 | 휏 ≤ 휃⟩ is coherent: i.e., 푝휏 훾휏 ′ = 푝휏 ′ for ′ 훾휏 all 휏 ≤ 휏. Also, note that ⟨휛푛 (푞 훾휏 ) | 휏 ≤ 휃⟩ is a constant sequence, so hereafter we denote by 푡 its constant value. To form the first member of the sequence we argue as follows. First note 1 that 푞 1 ≤1 푝 1 + 푠, so that appealing to Definition 2.2(4) for 휛푛 we get 1 ′ ′ 휛푛 ′ a condition 푝−1 ∈ 푃1 such that 푝−1 ≤1 푝 1 and 푞 1 = 푝−1 + 푡. ′ Now, let 푟0 := t훾0,1(푝 훾0)(푝−1). A moment’s reflection makes clear that 푟0 + 푠 is well-defined and also 푞 훾0 ≤훾0 푟0 + 푠. So, appealing to 훾0 훾0 ′ ′ 휛푛 Definition 2.2(4) for 휛푛 we get a condition 푝0 ∈ 푃훾0 such that 푝0 ≤훾0 푟0 1 ′ ′ 휛푛 훾0 1 and 푞 훾0 = 푝0 + 푡. Since 푝−1 ≤1 푝 1 and 휛푛 = 휛푛 ∘ 휋훾0,1 we have 훾0 훾0 휛푛 (푟0) = 휛푛 (푝 훾0). This completes the first step of the induction. ′ Let us suppose that we have already constructed ⟨푝휏 ′ | 휏 < 휏⟩. ′ ′ 휏 is successor: Suppose that 휏 = 휏 +1. Then, set 푟휏 := t훾휏 ,훾휏′ (푝훾휏 )(푝휏 ′ ). Using the induction hypothesis it is easy to see that 푞훾휏 ≤훾휏 푟휏 +푠. Instead 훾휏 of outright invoking the niceness of 휛푛 we would like to use that (t훾휏 ,푏푒푡푎휏 , 휋훾휏 ,훽휏 ) is a super nice forking projection (see Definition 5.4). This will

58 Recall that 퐵푞 := {훽 + 1 | 훽 ∈ dom(푞)& 푞(훽) ̸= ∅}. 60 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

′ ′ secure that the future condition 푝휏 will be coherent with 푝휏 ′ , and therefore with all the conditions constructed so far.

Applying the deﬁnition of t훾휏 ,훾휏′ given at page 57 we have ′ 푟 = (푝 훾 )( (푝 훽 )(푝 ′ )). 휏 t훾휏 ,훽휏 휏 t훽휏 ,훾휏′ 휏 휏

′ Since 푝 훽휏 = 푝 훾휏 * ∅훽휏 , Clause (6) of Fact 7.3 yields ′ ′ (푝 훽 )(푝 ′ ) = 푝 ′ * ∅ . t훽휏 ,훾휏′ 휏 휏 휏 훽휏 ′ So, 푟휏 = t훾휏 ,훽휏 (푝 훾휏 )(푝휏 ′ * ∅훽휏 ). 훽휏 ′ 휛푛 ′ Subclaim 7.4.1.1. 푝 ′ * ∅ ≤ 푝 훽 and 푞 훽 = (푝 ′ * ∅ ) + 푡. 휏 훽휏 훽휏 휏 휏 휏 훽휏 ′ 훾휏 ′ 휛푛 Proof. The first part follows immediately from 푝 ′ ≤ 푝 훾 ′ . For the 휏 훾휏′ 휏 ′ second part note that 푞 훽휏 = 푞 훾휏 * ∅훽휏 , hence Fact 7.4(5) combined with the induction hypothesis yield ′ ′ 푞 훽 = (푞 훽 )(푞 훾 ′ ) = (푞 훽 )(푝 ′ + 푡) = (푝 ′ + 푡) * ∅ . 휏 t훽휏 ,훾휏′ 휏 휏 t훽휏 ,훾휏′ 휏 휏 휏 훽휏 On the other hand, using Lemma 5.6 with respect to ( , 휋 ), t훽휏 ,훾휏′ 훽휏 ,훾휏′ ′ ′ ′ ′ (푝 ′ * ∅ ) + 푡 = (푝 ′ * ∅ )(푝 ′ + 푡) = (푝 ′ + 푡) * ∅ , 휏 훽휏 t훽휏 ,훾휏′ 휏 훽휏 휏 휏 훽휏 where the last equality follows again from Fact 7.4(5). ′ Combining the above expressions we arrive at 푞 훽휏 = (푝휏 ′ * ∅훽휏 ) + 푡. By Clause (f) of Building Block II and the definition of the iteration at successor stage (see page 57), the pair (t훾휏 ,훽휏 , 휋훾휏 ,훽휏 ) is a super nice forking projection from (P훾휏 , ℓ훾휏 , ⃗휛훾휏 ) to (P훽휏 , ℓ훽휏 , ⃗휛훽휏 ). Combining this with the 훾휏 ′ 휛푛 ′ ′ above subclaim we find a condition 푝휏 ≤훾휏 푟휏 such that 푝휏 훽휏 = 푝휏 ′ * ∅훽휏 ′ ′ and 푞 훾휏 = 푝휏 + 푡. Clearly, 푝휏 witnesses the desired property. ′ ⋃︀ ′ 휏 is limit: Put 푝휏 := 휏 ′<휏 푝휏 ′ . Thanks to the induction hypothesis it is 훾휏 ′ 휛푛 evident that 푝휏 ≤훾휏 푝 훾휏 . Also, combining the induction hypothesis with Lemma 5.6 for (t훾휏 ,1, 휋훾휏 ,1) we obtain the following chain of equalities: ⋃︁ ′ ⋃︁ ′ ′ ⋃︁ ′ ′ 59 ′ ′ ′ ′ 푞훾휏 = (푝휏 +푡) = t훾휏 ,1(푝휏 )(푝휏 1+푡) = t훾휏′ ,1(푝휏 )(푝휏 1+푡). 휏 ′<휏 휏 ′<휏 휏 ′<휏 Using the definition of the pitchfork at limit stages (see page 58) we get ⋃︁ ′ ′ ′ ′ ′ ′ t훾휏′ ,1(푝휏 )(푝휏 1 + 푡) = 푝휏 + 푡 = t훾휏 ,1(푝휏 )(푝휏 1 + 푡), 휏 ′<휏 where the last equality follows from Lemma 5.6 for (t훾휏 ,1, 휋훾휏 ,1). 훾휏 ′ 휛푛 ′ Altogether, we have shown hat 푝휏 ≤훾휏 푝 훾휏 and 푞 훾휏 = 푝휏 + 푡. ′ ′ ′ Additionally, 푝휏 훾휏 ′ = 푝휏 ′ for all 휏 < 휏. ′ ′ Finally, putting 푝 := 푝휃 we obtain a condition in (P훼)≥푛 such that 훼 ′ 휛푛 ′ 훼 푝 ≤훼 푝 and 푞 = 푝 + 휛푛 (푞). This completes the argument.

59 ′ ′ ′ Note that for the right-most equality we have used that 푝휏 1 = 푝휏 ′ 1, for all 휏 < 휏. SIGMA-PRIKRY FORCING III 61

This completes the proof of the lemma. Recalling Deﬁnition 3.3(2), for all nonzero 훼 ≤ 휇+ and 푛 < 휔, we need ˚ ˚ to identify a candidate for a dense subposet (P훼)푛 = ((푃훼)푛, ≤훼) of (P훼)푛. + Definition 7.5. For each nonzero 훾 < 휇 , we let tp훾+1 be a type witnessing that (t훾+1,훾, 휋훾+1,훾) has the weak mixing property. 1 60 + Definition 7.6. Let 푛 < 휔. Set 푃˚1푛 := (푄˚푛). Then, for each 훼 ∈ [2, 휇 ], deﬁne 푃˚훼푛 by recursion: {︃ ˚ ˚ {푝 ∈ 푃훼 | 휋훼,훽(푝) ∈ 푃훽푛 & mtp훽+1(푝) = 0}, if 훼 = 훽 + 1; 푃훼푛 := ˚ {푝 ∈ 푃훼 | 휋훼,1(푝) ∈ 푃1푛 & ∀훾 ∈ 퐵푝 mtp훾(휋훼,훾(푝)) = 0}, otherwise. Lemma 7.7. Let 푛 < 휔 and 1 ≤ 훽 < 훼 ≤ 휇+. Then:

(1) 휋훼,훽“푃˚훼푛 ⊆ 푃˚훽푛; (2) For every 푝 ∈ 푃˚훽푛, 푝 * ∅훿 ∈ 푃˚훼푛. Proof. By induction, relying on Clause (4) of Deﬁnition 5.7. We now move to establish Clause (iv) of Goal 7.2. Lemma 7.8. Let 훼 ∈ [2, 휇+]. Then, for all 푛 < 휔 and every directed set 훼 ˚ ˚ 휛푛 * of conditions 퐷 in ( 훼)푛 (resp. ( 훼)푛 ) of size <ℵ1 (resp. <휎푛) there is P P 훼 휛푛 푞 ∈ (푃˚훼)푛 such that 푞 is a ≤훼 (resp. ≤푛 ) lower bound for 퐷. ⋃︀ Moreover, 퐵푞 = 푝∈퐷 퐵푝. Proof. The argument is similar to that of [PRS20, Lemma 3.13]. Remark 7.9. An straightforward modiﬁcation of the lemma shows that for + 휋훼,1 * all 훼 ∈ [2, 휇 ] and 푛 < 휔,(P훼)푛 is 휎푛-directed-closed. + Theorem 7.10. For all nonzero 훼 ≤ 휇 , (P훼, ℓ훼, 푐훼, ⃗휛훼) satisfies all the requirements to be a (Σ,⃗S)-Prikry quadruple, with the possible exceptions of Clause (7) and the density requirements in Clauses (2) and (9).

Additionally, ∅훼 is the greatest condition in P훼, ℓ훼 = ℓ1 ∘ 휋훼,1, ∅훼 P훼 + 휇ˇ = 휅 and ⃗휛훼 is a coherent sequence of nice projections such that

⃗휛훼 = ⃗휛훾 ∙ 휋훼,훾 for every 훾 ≤ 훼. Under the extra hypothesis that for each 훼 ∈ acc(휇+ +1) and every 푛 < 휔, 훼 훼 ˚휛푛 휛푛 + (P훼 )푛 is a dense subposet of (P훼 )푛, we have that for all nonzero 훼 ≤ 휇 , ⃗ (P훼, ℓ훼, 푐훼, ⃗휛훼) is (Σ, S)-Prikry quadruple having property 풟. Proof. We argue by induction on 훼 ≤ 휇+. The base case 훼 = 1 follows from the fact that P1 is isomorphic to Q given by Building Block I. The successor step 훼 = 훽 + 1 follows from the fact that P훽+1 was obtained by invoking Building Block II.

60 Here, 푄˚푛 is obtained from Clause (2) of Definition 3.3 with respect to the triple (Q, ℓ, 푐) given by Building Block I. 62 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Next, suppose that 훼 ∈ acc(휇+ + 1) is such that the conclusion of the lemma holds below 훼. In particular, the hypothesis of Lemma 7.4 are sat- isfied, so that, for all nonzero 훽 ≤ 훾 ≤ 훼,(t훾,훽, 휋훾,훽) is a nice forking projection from (P훾, ℓ훾, ⃗휛훾) to (P훽, ℓ훽, ⃗휛훽). By the very same proof of [PRS20, Lemma 3.14], we have that Clauses (1) and (3)–(6) of Definition 3.3 hold for (P훼, ℓ훼, 푐훼, ⃗휛훼), and that ℓ훼 = ℓ1 ∘ 휋훼,1. Also, Clauses (2) and (9) –without the density requirement– follow from Lemma 7.8. On the other hand, the equality ⃗휛훼 = ⃗휛훾 ∙ 휋훼,훾 follows from Lemma 7.4. + Arguing as in [PRS20, Claim 3.14.2], we also have that 1lP훼 P훼 휇ˇ =휅 ˇ . Finally, since ⃗휛1 is coherent (see Building Block I) and (t훼,1, 휋훼,1) is a nice forking projection, Lemma 5.17 implies that ⃗휛훼 is coherent. 훼 휛푛 To complete the proof let us additionally assume that for every 푛,(˚훼 )푛 훼 P 휛푛 ˚ is a dense subposet of (P훼 )푛. Then, in particular, (P훼)푛 is a dense subposet of (P훼)푛. In effect, the density requirement in Clauses (2) and (9) is automatically fulfilled. About Clause (7), we take advantage of this extra assumption to invoke [PRS20, Corollary 3.12] and conclude that (P훼, ℓ훼) has property 풟. Consequently, Clause (7) for (P훼, ℓ훼, 푐훼, ⃗휛훼) follows by combining this latter fact with Lemma 5.12. 8. A proof of the Main Theorem In this section, we arrive at the primary application of the framework developed thus far. We will be constructing a model where GCH holds ℵ휔 below ℵ휔, 2 = ℵ휔+2 and every stationary subset of ℵ휔+1 reflects. 8.1. Setting up the ground model. We want to obtain a ground model with GCH and 휔-many supercompact cardinals, which are Laver indestructible under GCH-preserving forcing. The first lemma must be well-known, but we could not find it in the literature, so we give an outline of the proof.

Lemma 8.1. Suppose ⃗휅 = ⟨휅푛 | 푛 < 휔⟩ is an increasing sequence of supercompact cardinals. Then there is a generic extension where GCH holds and ⃗휅 remains an increasing sequence of supercompact cardinals. Proof. By preparing the ground model ´ala Laver [Lav78], we may assume that, for each 푛 < 휔, the supercompactness of 휅푛 is indestructible under 휅푛-directed-closed forcing. Claim 8.1.1. There is a generic extension in which ⃗휅 remains an increasing sequence of supercompact cardinals, and Θ := {휃 ∈ CARD | 휃<휃 = 휃} forms a proper class. We now work in the generic extension produced by the preceding claim. For ease of notation, we denote it by 푉 . Let J be Jensen’s iteration to force the GCH. Namely, J is the inverse ˙ limit of the Easton-support iteration ⟨J훼; Q훽 | 훽 ≤ 훼 ∈ Ord⟩ such that, if ˙ ˙ + ˙ 1lJ훼 J훼 “훼 is a cardinal”, then 1lJ훼 J훼 “Q훼 = Add(훼 , 1)” and 1lJ훼 J훼 “Q훼 trivial”, otherwise. Let 퐺 be a J-generic ﬁlter over 푉 . SIGMA-PRIKRY FORCING III 63

Let 푛 < 휔. We claim that J preserves the supercompactness of 휅푛. To this end, let 휃 be an arbitrary cardinal. By possibly enlarging 휃, we may assume that 휃 ∈ Θ. Let 푗 : 푉 → 푀 be an elementary embedding induced by a 휃-supercompact measure over 풫휅푛 (휃). In particular, we are taking 푗 (︀휃 )︀ such that crit(푗) = 휅푛, 푗(휅푛) > 휃, 푀 ∩ 푉 ⊆ 푀 and

푀 = {푗(푓)(푗“휃) | 푓 : 풫휅푛 (휃) → 푉 }. Observe that J can be factored into three forcings: the iteration up to 휅푛, the iteration in the interval [휅푛, 휃) and, finally, the iteration in the interval [휃, Ord). For an interval of ordinals ℐ, let 퐺ℐ denote the Jℐ -generic filter * induced by 퐺. Similarly, we define 퐺ℐ := 퐺ℐ ∩ 푗(J)ℐ . Claim 8.1.2. In 푉 [퐺], there is a lifting 푗 : 푉 [퐺 ] → 푀[퐺* ] of 푗 such 1 휅푛 푗(휅푛) that (︁ )︁ 휃푀[퐺* ] ∩ 푉 [퐺* ] ⊆ 푀[퐺* ]. 푗(휅푛) 푗(휅푛) 푗(휅푛) 푀[퐺* ] 푗(휅푛) Moreover, 퐻휃+ ⊆ 푉 [퐺휃]. * Claim 8.1.3. In 푉 [퐺], there is a lifting 푗2 : 푉 [퐺휃] → 푀[퐺푗(휃)] of 푗1. * ˙ Claim 8.1.4. In 푉 [퐺], there is a lifting 푗3 : 푉 [퐺] → 푀[퐺푗(휃) *퐾] of 푗2. Finally, define 푉 [퐺] 풰 := {푋 ∈ 풫휅푛 (휃) | 푗“휃 ∈ 푗3(푋)}. * ˙ As 푗“휃 ∈ 푀 ⊆ 푀[퐺푗(휃) * 퐾], standard arguments now show that 풰 is a 푉 [퐺] 휃-supercompact measure over 풫휅푛 (휃). In particular, 휅푛 is 휃-supercompact in 푉 [퐺], as wanted.

Note that in the model of the conclusion of the above lemma, the 휅푛’s are no longer indestructible. Our next task is to remedy that, while maintaining GCH. For this, we need the following slight variation of the usual Laver preparation [Lav78]. Lemma 8.2. Suppose that GCH holds, 휒 < 휅 are infinite regular cardinals, and 휅 is supercompact. Then there exists a 휒-directed-closed notion of 휅 forcing L휒 that preserves GCH and makes the supercompactness of 휅 indestructible under 휅-directed-closed forcings that preserve GCH. 휅 Proof. Let 푓 be a Laver function on 휅, as in [Cum10, Theorem 24.1]. Let L휒 ˙ be the direct limit of the Laver-style forcing iteration ⟨R훼; Q훽 | 휒 ≤ 훽 < 훼 < 휅⟩ where, if 훼 is inaccessible, 1lR훼 R훼 GCH, and 푓(훼) encodes an R훼-name R훼 휏 ∈ 퐻훼+ for some 훼-directed-closed forcing that preserves the GCH of 푉 , ˙ ˙ then Q훼 is chosen to be such R훼-name. Otherwise, Q훼 is chosen to be the trivial forcing. As in the proof of [Cum10, Theorem 24.12], we have that after forcing with 휅 L휒, the supercompactness of 휅 becomes indestructible under 휅-directed- closed forcings that preserve GCH. 64 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

휅 We claim that GCH holds in 푉 L휒 . This is clear for cardinals ≥ 휅, since the iteration has size 휅. Now, let 휆 < 휅 and inductively assume GCH<휆. 휅 ∼ ˙ ˙ + Observe that L휒 = R휆+1 * Q, where Q is an R휆+1-name for a 휆 -directed- 휅 closed forcing. In particular, 풫(휆)푉 L휒 = 풫(휆)푉 R휆+1 , and so it is enough to show that 푉 R휆+1 |= CH휆. There are two cases. + ˙ 휆+1 If 휆 is singular, then |R휆| = 휆 , and Q휆 is trivial, so 푉 R |= CH휆. Otherwise, let 훼 be the largest inaccessible, such that 훼 ≤ 휆. Then R휆+1 ˙ is just R훼*Q훼 followed by trivial forcing. Since |R훼| = 훼 and by construction ˙ Q훼 preserves GCH, the result follows.

Corollary 8.3. Suppose that ⃗휅 = ⟨휅푛 | 푛 < 휔⟩ is an increasing sequence of supercompact cardinals. Then, in some forcing extension, all of the following hold: (1) GCH; (2) ⃗휅 is an increasing sequence of supercompact cardinals; (3) For every 푛 < 휔, the supercompactness of 휅푛 is indestructible under notions of forcing that are 휅푛-directed-closed and preserves the GCH. Proof. By Lemma 8.1, we may assume that we are working in a model in which Clauses (1) and (2) already hold. Next, let L be the direct limit of the ˙ iteration ⟨L푛 * Q푛 | 푛 < 휔⟩, where L0 is the trivial forcing and, for each 푛, ˙ if 1l L푛 “휅푛 is supercompact”, then 1l L푛 Q푛 is the (휅푛−1)-directed-closed, GCH-preserving forcing making the supercompactness of 휅푛 indestructible under GCH-preserving 휅푛-directed-closed notions of forcing. (More precisely, ˙ ˙ 휅푛 in the notation of the previous lemma, Q푛 is L휅푛−1 , where, by convention, 휅−1 is ℵ0). Note that, by induction on 푛 < 휔, and Lemma 8.2, we maintain that

1l L푛 “휅푛 is supercompact and GCH holds”. And then when we force with ˙ Q푛 over that model, we make this supercompacness indestructible under GCH -preserving forcing. Then, after forcing with L, GCH holds, and each 휅푛 remains supercompact, indestructible under 휅푛-directed-closed forcings that preserve GCH. 8.2. Connecting the dots. Setup 8. For the rest of this section, we make the following assumptions:

∙ ⃗휅 = ⟨휅푛 | 푛 < 휔⟩ is an increasing sequence of supercompact cardinals. By convention, we set 휅−1 := ℵ0; ∙ For every 푛 < 휔, the supercompactness of 휅푛 is indestructible under notions of forcing that are 휅푛-directed-closed and preserve the GCH; + ++ ∙ 휅 := sup푛<휔 휅푛, 휇 := 휅 and 휆 := 휅 ; ∙ GCH holds below 휆. In particular, 2휅 = 휅+ and 2휇 = 휇+; + 61 ∙ Σ := ⟨휎푛 | 푛 < 휔⟩, where 휎0 := ℵ1 and 휎푛+1 := (휅푛) for all 푛 < 휔.

61 By convention, we set 휎−2 and 휎−1 to be ℵ0. SIGMA-PRIKRY FORCING III 65

∙ ⃗S is as in Deﬁnition 4.11. We now want to appeal to the iteration scheme of the previous section. ⃗ First, observe that 휇, ⟨(휎푛, 휇) | 푛 < 휔⟩, S and Σ respectively fulﬁll all the blanket assumptions of Setup 7. We now introduce our three building blocks of choice:

Building Block I. We let (Q, ℓ, 푐, ⃗휛) be EBPFC as deﬁned in Section 4. By Corollary 4.25, this is a (Σ,⃗S)-Prikry that has property 풟, and ⃗휛 is a coherent sequence of nice projection. Also, Q is a subset of 퐻휇+ and, by + Lemma 4.24, 1lQ Q 휇ˇ =휅 ˇ . In addition, 휅 is singular, so that we have ˚ 1lQ Q “휅 is singular”. Finally, for all 푛 < 휔, Q푛 = Q푛 (see Lemmas 4.18 and 4.23).

Building Block II. Suppose that (P, ℓ, 푐, ⃗휛) is a (Σ,⃗S)-Prikry quadruple having property 풟 such that P = (푃, ≤) is a subset of 퐻휇+ , ⃗휛 is a coherent + sequence of nice projections, 1lP P 휇ˇ =휅 ˇ and 1lP P “휅 is singular”. ⋆ ⋆ For every 푟 ∈ 푃 and a P-name 푧 for an 푟 -fragile stationary subset of 휇, ⃗ there are a (Σ, S)-Prikry quadruple (A, ℓA, 푐A, ⃗휍) having property 풟, and a pair of maps (t, 휋) such that all the following hold:

(a)( t, 휋) is a super nice forking projection from (A, ℓA, 푐A, ⃗휍) to (P, ℓ, 푐, ⃗휛) that has the weak mixing property; (b) ⃗휍 is a coherent sequence of nice projections; + (c) 1lA A 휇ˇ =휅 ˇ ; (d) A = (퐴, E) is a subset of 퐻휇+ ; ˚휋 (e) For every 푛 < 휔, A푛 is 휇-directed-closed; ⋆ ⋆ (f) if 푟 ∈ 푃 and 푧 is a P-name for an 푟 -fragile stationary subset of 휇 then ⋆ A ⌈푟 ⌉ A “푧 is nonstationary”. Remark 8.4. ⋆ ⋆ I If 푟 ∈ 푃 forces that 푧 is a P-name for an 푟 -fragile subset of 휇, we first find some P-name푧 ˜ such that 1lP forces that푧 ˜ is a stationary ⋆ subset of 휇, 푟 P 푧 =푧 ˜ and푧 ˜ is 1lP-fragile. Subsequently, we obtain (A, ℓA, 푐A, ⃗휍) and (t, 휋) by appealing to Corollary 6.19 with ⃗ the (Σ, S)-Prikry triple (P, ℓP, 푐P, ⃗휛), the condition 1lP and the P- A ⋆ A name푧 ˜. In effect, ⌈1lP⌉ forces that푧 ˜ is nonstationary, so that ⌈푟 ⌉ forces that 푧 is nonstationary. ⃗ I Otherwise, we invoke Corollary 6.19 with the (Σ, S)-Prikry forcing (P, ℓP, 푐P, ⃗휛), the condition 1lP and the name 푧 := ∅.

휇 + + Building Block III. As 2 = 휇 , we ﬁx a surjection 휓 : 휇 → 퐻휇+ such that the preimage of any singleton is coﬁnal in 휇+. The next lemma deals with the extra assumption in Theorem 7.10: 66 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Lemma 8.5 (Density of the rings). For each 훼 ∈ acc(휇+ + 1) and every 훼 훼 ˚휛푛 휛푛 62 integer 푛 < 휔, (P훼 )푛 is a dense subposet of (P훼 )푛. Proof. This follows in the same lines of [PRS20, Lemma 4.24], with the only difference that now we use the following: 훼 훼 ˚휛푛 휛푛 (1) At successor stages we can get into the ring (P훼 )푛 by ≤훼 -extending. This is granted by Lemma 6.2. 훾 ˚휛푛 (2) For all 훾 < 훼,(P훾 )푛 is 휎푛-directed-closed. With this property we ˚ take care of the limit stages. (In [PRS20], the full ring (P훾)푛 was 휎푛-closed). We can make this replacement, because of the first item above. Now, we can appeal to the iteration scheme of Section 7 with these build- + ing blocks, and obtain, in return, a sequence ⟨(P훼, ℓ훼, 푐훼, ⃗휛훼) | 1 ≤ 훼 ≤ 휇 ⟩ of (Σ,⃗S)-Prikry quadruples. By Lemma 7.8 and Theorem 7.10 (see also + ˚ 휋훼,1 Remark 7.9), for all nonzero 훼 ≤ 휇 ,(P훼)푛 is 휇-directed-closed and + + 1lP훼 P훼 휇ˇ =휅 ˇ . Note that by the first clause of Goal 7.2, |푃훼| ≤ 휇 for every 훼 ≤ 휇+. + 훼 Lemma 8.6. Let 푛 ∈ 휔∖2 and 훼 ∈ [2, 휇 ). Then ((P훼)푛, S푛, 휛푛 ) is suitable for reflection with respect to ⟨휎푛−2, 휅푛−1, 휅푛, 휇⟩.

Proof. We go over the clauses of Lemma 5.18 with P훼 playing the role of A, 훼 휛푛 playing the role of 휍푛, and P1 playing the role of P. As P1 is given by Building Block I, which is given by Section 4, we simplify the notation here, and — for the scope of this proof — we let P denote the forcing P from Section 4. Clause (i) is part of the assumptions of Setup 8. Clauses (ii) and (iii) are given by our iteration theorem. Clause (iv) is due to Corollary 4.30,63 and the fact that 1 is the Gitik’s EBPFC. Now, we turn to address Clause (v). P 훼 휛푛 That is, we need to prove that in any generic extension by S푛 × (P훼)푛 , ++ |휇| = cf(휇) = 휅푛 = (휅푛−1) . The upcoming discussion assumes the notation of Section 4. By Lemma 4.32, we have: (1) T푛 has the 휅푛-cc and size 휅푛; (2) 휓푛 deﬁnes a nice projection; 휓푛 (3) P푛 is 휅푛-directed-closed; 휓푛 (4) for each 푝 ∈ 푃푛, P푛 ↓ 푝 and (T푛 ↓ 휓푛(푝)) × ((P )푛 ↓ 푝) are forcing equivalent. + ++ By Lemma 4.29, P푛 forces |휇| = cf(휇) = 휅푛 = (휎푛) = (휅푛−1) , and 휋훼,1 by our remark before the statement of this lemma, (P훼)푛 is 휇-directed- closed, hence 휅푛-directed-closed. Combining Clauses (1), (3) and (4) above 62 ˚ (P훼)푛 is as in Definition 7.6. 63 Since (P푛, S푛, 휛푛) is suitable for reflection with respect to ⟨휎푛−1, 휅푛−1, 휅푛, 휇⟩ then so is with respect to ⟨휎푛−2, 휅푛−1, 휅푛, 휇⟩. SIGMA-PRIKRY FORCING III 67

휓푛 휋훼,1 T푛 with Easton’s Lemma, (P )푛 × (P훼)푛 is 휅푛-distributive over 푉 , and so 휋훼,1 ++ 휋훼,1 P푛 × (P훼)푛 forces 휅푛 = (휅푛−1) . Moreover, as P푛 × (P훼)푛 projects to P푛 and the former preserves 휅푛, it also forces |휇| = cf(휇). Altogether, 휋훼,1 ++ P푛×(P훼)푛 forces |휇| = cf(휇) = 휅푛 = (휅푛−1) . To establish that the same 훼 휛푛 configuration is being forced by S푛 × (P훼)푛 , we give a sandwich argument, as follows: 훼 휋훼,1 휛푛 ∙ P푛 × (P훼)푛 projects to S푛 × (P훼)푛 , as witnessed by (푝, 푞) ↦→ (휛푛(푝), 푞); 훼 훼 휛푛 ∙ For any condition 푝 in (P훼)푛,(S푛 ↓ 휛푛 (푝)) × ((P훼)푛 ↓ 푝) projects to (P훼)푛 ↓ 푝, by Definition 2.2(4). ∙ (P훼)푛 projects to P푛 via 휋훼,1. This completes the proof. 훼 + 휛푛 Lemma 8.7. Let 푛 < 휔 and 0 < 훼 < 휇 . Then (P훼)푛 preserves GCH. Proof. The case 훼 = 1 is taken care of by Lemma 4.33. 훼 휛푛 Now, let 훼 ≥ 2. Since (P훼)푛 contains a 휎푛-directed-closed dense subset, it preserves GCH below 휎푛. By the sandwich analysis from the proof of 훼 휛푛 + Lemma 8.6, in any generic extension by ( 훼)푛 , |휇| = cf(휇) = 휅푛 = (휎푛) . 훼 P 휛푛 + So, as (P훼)푛 is a notion of forcing of size ≤ 휇 , collapsing 휇 to 휅푛, it preserves GCH휃 for any cardinal 휃 > 휅푛. 훼 휛푛 휃 + It thus left to verify that (P훼)푛 forces 2 = 휃 for 휃 ∈ {휎푛, 휅푛}. Arguing as in Lemma 8.6, for any condition 푝 in ( 훼)푛,( 푛 ↓ 휓푛(푝)) × I 훼 P T (︀ 휓푛 휋훼,1 )︀ 휛푛 ((P )푛 ↓ 푝) × (P훼)푛 projects onto (P훼)푛 . Recall that the first factor of the product is a 휅푛-cc forcing of size ≤ 휅푛. By Lemma 7.8, the second factor is forcing equivalent to a 휅푛-directed-closed forcing. Thus, by Easton’s lemma, this product preserves CH휎푛 if and only if T푛 ↓ 휓푛(푝) does. And this is indeed the case, as the number of T푛-nice names for subsets of 휎푛 is <휅푛 + at most 휅푛 = 휅푛 = (휎푛) . 휋훼,1 Again, arguing as in Lemma 8.6, ( 1)푛 × ( 훼)푛 projects onto 푛 × I 훼 훼 P P S 휛푛 휛푛 휋훼,1 (P훼)푛 , which projects onto (P훼)푛 . Since (P훼)푛 is forcing equivalent to a

휇-directed-closed, it preserves CH휎푛 . Also, it preserves 휇 and so, by Lemma + + 4.33(1) and the absolutness of the 휇 -Linked property, (P1)푛 is also 휇 - 휋훼,1 Linked in 푉 (P훼)푛 . Once again, counting-of-nice-names arguments implies 휅푛 + + 휋훼,1 that this latter forcing forces 2 ≤ 휇 = (휅푛) . Thus, ( 1)푛 × ( 훼)푛 훼 P P 휛푛 preserves CH휅푛 and so does (P훼)푛 .

Theorem 8.8. In 푉 P휇+ , all of the following hold true: (1) All cardinals ≥ 휅 are preserved; (2) 휅 = ℵ휔, 휇 = ℵ휔+1 and 휆 = ℵ휔+2; ℵ푛 (3) 2 = ℵ푛+1 for all 푛 < 휔; ℵ휔 (4) 2 = ℵ휔+2; (5) Every stationary subset of ℵ휔+1 reflects. 68 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

Proof. (1) We already know that 1l 휇ˇ =휅 ˇ+. By Lemma 3.14(2), 휅 P휇+ P훼 remains strong limit cardinal in 푉 P휇+ . Finally, as Clause (3) of Definition 3.3 + holds for (P휇+ , ℓ휇+ , 푐휇+ , ⃗휛휇+ ), P휇+ has the 휇 -chain-condition, so that all cardinals ≥ 휅++ are preserved. (2) Let 퐺 ⊆ P휇+ be an arbitrary generic over 푉 . By virtue of Clause (1) and Setup 8, it suffices to prove that 푉 [퐺] |= 휅 = ℵ휔. Let 퐺1 the P1-generic filter generated by 퐺 and 휋휇+,1. By Theorem 4.1, 푉 [퐺1] |= 휅 = ℵ휔. Thus, let us prove that 푉 [퐺] and 푉 [퐺1] have the same cardinals ≤ 휅. Of course, 푉 [퐺1] ⊆ 푉 [퐺], and so any 푉 [퐺]-cardinal is also a 푉 [퐺1]- cardinal. Towards a contradiction, suppose that there is a 푉 [퐺1]-cardinal 휃 < 휅 that ceases to be so in 푉 [퐺]. Any surjection witnessing this can be encoded as a bounded subset of 휅, hence as a bounded subset of some 휎푛 for some 푛 < 휔. Thus, Lemma 3.14(1) implies that 휃 is not a cardinal 1 in 푉 [퐻푛], where 퐻푛 is the S푛-generic filter generated by 퐺1 and 휛푛. As 푉 [퐻푛] ⊆ 푉 [퐺1], 휃 is not a cardinal in 푉 [퐺1], which is a contradiction. 푉 P휇+ 푉 S푚 (3) On one hand, by Lemma 3.14(1), 풫(ℵ푛) = 풫(ℵ푛) for some 푚 < 휔. On the other hand, as GCH<휆 holds (cf. Setup 8), Remark 4.12 P휇+ shows that S푚 preserves CHℵ푛 . Altogether, 푉 |= CHℵ푛 . 휅 + (4) By Setup 8, 푉 |= 2 = 휅 . In addition, P휇+ is isomorphic to a notion of forcing lying in 퐻휇+ (see [PRS20, Remark 3.3(1)]) and |퐻휇+ | = 휆. P휇+ 휅 Thus, 푉 |= 2 ≤ 휆. In addition, P휇+ projects to P1, which is isomorphic 휅 to Q, being a poset blowing up 2 to 휆, as seen in Theorem 4.1, so that 푉 P휇+ |= 2휅 ≥ 휆. So, 푉 P휇+ |= 2휅 = 휆. Thus, together with Clause (2), P + ℵ휔 푉 휇 |= 2 = ℵ휔+2. (5) Let 퐺 be P휇+ -generic over 푉 and hereafter work in 푉 [퐺]. Towards a contradiction, suppose that there exists a stationary set 푇 ⊆ 휇 that does not reflect. By shrinking, we may assume the existence of some regular cardinal 휇 * 휃 < 휇 such that 푇 ⊆ 퐸휃 . Fix 푟 ∈ 퐺 and a P휇+ -name 휏 such that 휏퐺 is equal to such a 푇 and such that 푟* forces 휏 to be a stationary subset of 휇 that does not reflect. Since 휇 = 휅+ and 휅 is singular in 푉 , by possibly enlarging * * 푟 , we may assume that 푟 forces 휏 to be a subset of Γℓ(푟*) (see page 53). Furthermore, we may require that 휏 be a nice name, i.e., each element of 휏 ˇ is a pair (휉, 푝) where (휉, 푝) ∈ Γℓ(푟*) × 푃휇+ , and, for each ordinal 휉 ∈ Γℓ(푟*), ˇ the set {푝 ∈ 푃휇+ | (휉, 푝) ∈ 휏} is a maximal antichain. As P휇+ satisfies Clause (3) of Definition 3.3, P휇+ has in particular the 휇+-cc. Consequently, there exists a large enough 훽 < 휇+ such that ⋃︁ 퐵푟* ∪ {퐵푝 | (휉,ˇ 푝) ∈ 휏} ⊆ 훽. * Let 푟 := 푟 훽 and set ˇ ˇ 휎 := {(휉, 푝 훽) | (휉, 푝) ∈ 휏}. From the choice of Building Block III, we may find a large enough 훼 < 휇+ with 훼 > 훽 such that 휓(훼) = (훽, 푟, 휎). As 훽 < 훼, 푟 ∈ 푃훽 and 휎 is a P훽-name, the definition of our iteration at step 훼 + 1 involves appealing to Building SIGMA-PRIKRY FORCING III 69

⋆ 훼 64 Block II with (P훼, ℓ훼, 푐훼, ⃗휛훼), 푟 := 푟*∅훼 and 푧 := 푖훽 (휎). For each ordinal + 휂 < 휇 , denote 퐺휂 := 휋휇+,휂[퐺]. By our choice of 훽 and since 훼 > 훽, we have ˇ ˇ ˇ ˇ 휏 = {(휉, 푝 * ∅휇+ ) | (휉, 푝) ∈ 휎} = {(휉, 푝 * ∅휇+ ) | (휉, 푝) ∈ 푧}, so that, in 푉 [퐺],

푇 = 휏퐺 = 휎퐺훽 = 푧퐺훼 . * ⋆ * ⋆ In addition, 푟 = 푟 * ∅휇+ and so ℓ(푟 ) = ℓ(푟 ). * As 푟 forces that 휏 is a non-reﬂecting stationary subset of Γℓ(푟⋆), it follows ⋆ that 푟 P훼-forces the same about 푧. Claim 8.8.1. 푧 is 푟⋆-fragile. Proof. Recalling Lemma 6.20, it suﬃces to prove that for every 푛 < 휔,

(P훼)푛 휇 휇 푉 |= Refl(퐸<휎푛−2 , 퐸<휎푛 ). This is trivially the case for 푛 ≤ 1. So, let us fix an arbitrary 푛 ≥ 2. 훼 By Lemma 8.6, (( 훼)푛, 푛, 휛푛 ) is suitable for reflection with respect to P S 훼 휛푛 ⟨휎푛−2, 휅푛−1, 휅푛, 휇⟩. Since (P훼)푛 is forcing equivalent to a 휎푛-directed- closed forcing and (by Lemma 8.7) it preserves GCH, 휅푛−1 is a supercompact 훼 휛푛 cardinal indestructible under forcing with ( 훼)푛 . So, recalling Setup 8, 훼 P 휛푛 (P훼)푛 preserves the supercompactness of 휅푛−1. Thus, by Lemma 2.11, 휇 휇 (P훼)푛 푉 |= Refl(퐸<휎푛−2 , 퐸<휎푛 ).

⋆ * ⋆ ⋆ P훼+1 As 푧 is 푟 -fragile and 휋휇+,훼+1(푟 ) = 푟 * ∅훼+1 = ⌈푟 ⌉ ∈ 퐺훼+1, Clause (f) of Building Block II implies that there exists (in 푉 [퐺훼+1]) a club subset of 휇 disjoint from 푇 . In particular, 푇 is nonstationary in 푉 [퐺훼+1] and thus nonstationary in 푉 [퐺]. This contradicts the very choice of 푇 . The result follows from the above discussion and the previous claim. We are now ready to derive the Main Theorem. Theorem 8.9. Suppose that there exist infinitely many supercompact cardinals. Then there exists a forcing extension where all of the following hold: ℵ푛 (1) 2 = ℵ푛+1 for all 푛 < 휔; ℵ휔 (2) 2 = ℵ휔+2; (3) every stationary subset of ℵ휔+1 reflects. Proof. Using Corollary 8.3, we may assume that all the blanket assumptions of Setup 8 are met. Speciﬁcally:

∙ ⃗휅 = ⟨휅푛 | 푛 < 휔⟩ is an increasing sequence of supercompact cardinals that are indestructible under 휅푛-directed-closed notions of forcing that preserve the GCH; + ++ ∙ 휅 := sup푛<휔 휅푛, 휇 := 휅 and 휆 := 휅 ; ∙ GCH holds. Now, appeal to Theorem 8.8. 64Recall Convention 7.1. 70 ALEJANDRO POVEDA, ASSAF RINOT, AND DIMA SINAPOVA

9. Acknowledgments Poveda was partially supported by a postdoctoral fellowship from the Ein- stein Institute of Mathematics of the Hebrew University of Jerusalem. Rinot was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18). Sinapova was partially supported by the National Science Foundation, Career-1454945 and DMS-1954117.

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Einstein Institute of Mathematics, Hebrew University of Jerusalem, Givat- Ram, 91904, Israel. Email address: [email protected]

Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Is- rael. URL: http://www.assafrinot.com Email address: [email protected]

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA URL: https://homepages.math.uic.edu/~sinapova/ Email address: [email protected]