A COHERENT SEQUENCE IN Lp GΣ (R)

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1 A COHERENT SEQUENCE IN Lp GΣ (R) NAM TRANG MARTIN ZEMAN 1. INTRODUCTION Suppose (P, Σ) is a hod pair such that Σ has branch condensation (here P is countable). We work under the assumption AD + +Θ = Θ Σ + MC(Σ). Under this assumption, V = Lp GΣ (R), where Lp GΣ (R) is the stack of countably iterable Θ-g-organized premice over R that project to R as defined in [4]. A proof of this fact can be found in [1]. The construction of the coherent sequence here is heavily based on the construction of κ sequences in L[E] models done in [3], though much complexity that appears in [3] due to the existence of pluripotent L[E] levels that gives rise to protomice does not occur in our situation. Before stating our main result, let us fix some notation. Let C = {τ < Θ Lp GΣ (R) τ ZF and there are no R-cardinals in Lp GΣ (R) τ}. 1 Notice that C is a club in Θ by reflection and the fact that Lp GΣ (R) Θ ZF. For each τ C, let N τ be the least segment of Lp GΣ (R) such that there is an n < ω such that ρ Nτ n τ and ρ Nτ n+1 = R. Notice that o(n τ ) > τ and τ = Θ Nτ. Furthermore, τ is a strong cutpoint of N τ, that is τ is not measurable in N τ and there are no extenders on the sequence of N τ that overlap τ. Now, we re ready to state our main result. Theorem 1.1. There is a sequence D = D τ τ C with the following properties: 1. D τ τ C 2. D τ is closed, and if cof(τ) > ω then D τ is unbounded in τ. 3. if τ < τ, then D τ = D τ τ. We say E is a thread through D if E C is club in Θ and for every limit point α of E, E α = D α. We remark that by the construction, if Θ has uncountable cofinality in a parent universe with the same reals, then the sequence D does not admit a thread since any thread through D will produce an M Lp GΣ (R) such that ρ M ω 1 Here ZF is ZF minus Powerset Axiom. = R and o(m) Θ. Contradiction.

2 We note that by the same proof, we obtain the same result if we assumed V = Lp gσ (R), i.e. V is the stack of countably iterable g-organized premice over R that project to R (see [4]). We remark that the construction below goes through for both Jensen indexing and Mitchell-Steel indexing. In the construction of [3], Jensen indexing is used since it relies on the appropriate condensation lemma for that indexing. Here, we simply replace instances where the condensation lemma is used by a simple comparison argument, which does not depend on any particular indexing scheme. Finally, we remark that the sequence D can be turned into a coherent sequence D on Θ in the sense of [2] by purely combinatorial means. The sequence D is similarly nonthreadable in the sense above. 2. THE CONSTRUCTION We fix some more notations. Here are some objects associated to each τ C. The notations follow closely those of [3]. S E,G Σ α (R) α < Θ, J E,G Σ α (R) α < Θ are the Jensen s S and J hierarchies relative to G Σ and E, which is the extender sequence of Lp GΣ (R). n τ is the unique n such that ρ Nτ n τ and ρ Nτ n+1 = R. p τ is the standard parameter for N τ. ρ τ = ρ Nτ n τ. H τ is the n τ -th reduct of N τ. h nτ +1 τ = h N τ is the Σ (nτ ) 1 Skolem function of N τ, so there is a set H τ such that y = h τ (x, p τ ) where x R if and only if z H τ, (z, y, x, p τ ) H τ. For more details and background on fine structure, see [3]. Our construction here resembles that of the square sequence in [3] for the C α α S 0, where S 0 is, roughly speaking, the set over which the Jensen s construction of square in L can be adapted in the L[E] context in a straightforward manner. Here the set S 0 in [3] is C. We first define an approximation B τ τ C to our desired sequence D τ τ C. Definition 2.1. Let τ C, B τ is the set of all τ C τ satisfying: n τ = n τ = n, and N τ, N τ are Θ-g-organized R-premice of the same type. 2 2 In Mitchell-Steel indexing, premice are of types I, II, III; in Jensen indexing, premice are of types A, B, C. 2

3 There is a map σ ττ : N τ N τ which is Σ (nτ ) 0 -preserving (with respect to the language of Θ-g-organized R-premice) and such that 1. τ = cr(σ ττ ), and σ ττ (τ) = τ; 2. σ ττ (p τ ) = p τ ; 3. for each α p τ, there is a generalized witness Q τ (α) for α with respect to N τ and p τ such that Q τ (α) rng(σ ττ ). For a discussion on solidity witnesses and generalized witnesses, see [5] or [3]. The important point is that being a generalized witness (being a Π (nτ ) 1 notion) is preserved downward under the embeddings σ ττ. Notice also that the maps σ ττ are uniquely determined (and the uniqueness of such maps is not influenced by (3) in the definition). To see this, let a N τ, then x R, a = h τ (x, p τ ); this Σ (n) 1 definition is preserved upwards by the Σ (n) 0 map σ ττ and as σ ττ (x) = x for any real x, σ ττ (a) = h τ (x, p τ ). Lastly, the maps σ ττ are not cofinal at the n-th level, and hence not Σ (n) 1 preserving. Otherwise, by soundness of the N τ s, the maps σ ττ are onto, hence N τ = N τ, which is not possible. Lemma 2.2. Let τ C, and τ < τ in B τ. Then rng(σ τ τ) rng(σ ττ ). Proof. First we show sup((σ τ τ) ωρ τ ) < sup((σ ττ ) ωρ τ ) (2.1) Suppose this is false. Pick an x R such that h τ (x, p τ ) = τ, so there is a z H τ such that H τ (z, τ, x, p τ ). Applying σ ττ, we get H τ (σ ττ (z), τ, x, p τ ). Choose ζ < ωρ τ such that z S E,G Σ (R) and let ζ = σ ζ ττ (ζ). By the failure of 2.1, there is a ζ < ωρ τ such that ζ ζ = σ τ τ(ζ ). So the statement ( u n S E,G Σ ζ (R))( δ n < τ) H τ (u n, δ n, x, p τ ) is a Σ (n) 0 formula held in N τ, hence can be pulled back by σ τ τ. So h τ (x, p τ ) is defined and σ τ τ(h τ (x, p τ )) = τ, which contradicts the fact that τ = cr(σ τ τ). This proves can be used to show that x R, h τ (x, p τ ) is defined h τ (x, p τ ) is defined. To see this, we need to see that the relation ( x 0 )(x 0 = h τ (x, p τ )) is uniformly Σ (n) 1. This follows from the fact that h τ is a good Σ (n) 1 -function, so substituting h τ for v 0 in the relation ( x 0 )(x 0 = v 0 ) yields the result. The rest is just as in the previous paragraph. From the discussion in the previous paragraph, we know σ τ τ( h τ (x, p τ )) = σ ττ ( h τ (x, p τ )) whenever h τ (x, p τ ) is defined. By soundness of N τ, we have the desired conclusion. Remark 2.3. In the proof of the above lemma, (3) of Definition 2.1 is never used. 3

4 Lemma 2.4. Let τ C and τ B τ. Then B τ τ = B τ min(b τ ). Proof. Let τ B τ τ. We ll show that τ B τ. By the previous lemma, rng(σ τ τ) rng(σ ττ ), we can define the map σ : N τ N τ by σ = (σ ττ ) 1 σ τ τ. It s easy to see that σ satisfies all requirements in Definition 2.1 except maybe for item 3. But this is also true because we can pull back the generalized witness by a Σ (n) 0 map. Hence σ = σ τ τ and τ B τ. Let τ = min(b τ ) and τ B τ τ. We may assume τ > τ. Define σ = σ ττ σ τ τ. To show that σ = σ τ τ, it suffices to verify (3) of Definition 2.1 as the other conditions are evident. If Q(α) rng(σ τ τ) is a generalized witness for α p τ with respect to N τ and p τ then Q(α) rng(σ) by the previous lemma, so τ B τ. The above lemma tells us that the sequence B τ τ C is almost coherent. Inspecting the proof of the lemma, it s easy to see that condition (3) in the Definition 2.1 is the reason full coherency may fail. We now modify the sequence B τ τ C a bit to get full coherency. For τ C, we let τ(0) = τ τ(i + 1) = min(b τ(i) ) l τ = the least i such that B τ(i) = Notice that for any τ C, l τ is defined and is less than ω. Now for any τ C, D τ = B τ(0) B τ(1)... B τ(lτ 1), and for any τ D τ, σ τ τ = σ τ(1)τ(0) σ τ(2)τ(1)... σ τ τ(j) where j is such that τ B τ(j). Note that the maps σ τ τ are unique with critical point τ, sending τ to τ, and p τ p τ. Lemma 2.5. The sequence D τ τ C is coherent. Proof. Let τ C and τ D τ. First it is easy to see that to min(b τ ) D τ, B τ = D τ [min(b τ ), τ ) (2.2) Now define τ (i) from τ the same way τ(i) is defined from τ. Using (2.2), it is easy to see that D τ must be an initial segment of D τ. It remains to prove for every τ C, D τ is closed and if cof(τ) > ω, D τ is unbounded. 4

5 Lemma 2.6. Suppose τ C and cof(τ) > ω. Then D τ is unbounded in τ. Proof. We first assume N τ is E-active, that is, N τ = J E,G Σ α (R), F τ, where F τ is the (amenable code) for the top extender of N τ. Say cr(f τ ) = κ and ϑ = (κ + ) Nτ. Note that κ > τ; this is because F τ is R-complete over N τ and τ = Θ Nτ, as discussed above, is a strong cutpoint for N τ. We may assume n = 1, where ωρ Nτ n+1 = R, and ωρ Nτ n want to show there is a τ D τ such that τ τ. τ. Let τ < τ. We First, let X = h τ ({τ, P} {p τ }). Then X is countable Σ 1 substructure of N τ. Let σ : N N τ be the inverse of the collapse and σ(κ, ϑ, τ, p, κ) = (κ, ϑ, τ, p τ, κ). Note that τ < κ and ϑ = (κ + ) N. Now let τ = sup(σ τ). Then τ < τ because we put τ in X, and τ < τ because X is countable and cof(τ) > ω. We claim that τ D τ. By [4, Lemma 3.10], N is Θ-g-organized. Suppose N = J E,G Σ α (R), F. Then N is a R-premouse of the same type as N τ as an R-premouse. Let be the ultrapower of N by the extender induced by σ with generators in R and the ultrapower is formed using functions f N; so is isomorphic to {σ(f)(x) f N x R}. Let σ be the ultrapower map and σ : N τ be the natural map. So σ = σ σ. It s not hard to see that Los theorem holds and σ is Σ 0 preserving, hence is wellfounded. From now on, we identify with its transitive collapse. Now, contains all the reals. We need to see that is a Θ-g-organized R-premouse of the same type as N τ. The fact that is Θ-g-organized follows from [4, Lemma 3.10] and the fact that σ embeds into the Θ-g-organized R-premouse N τ. Now σ is Σ 0 and cofinal and cr( F ) τ = cr( σ). This implies the top extender of measures all sets in and hence is a premouse (and not a protomouse). Let x ; so x = σ(f)(a) for some f N and a R. But f = h N (b, p) for some b R, so σ(f) = h(b, p), where p = σ(p) = σ 1 (p τ ). So x = h(b, p)(a). This gives = R, τ ωρn, and p R. Solidity of is guaranteed by the fact that σ : N τ is Σ (n) 0 -preserving. Since rng(σ) rng(σ ), rng(σ ) contains a generalized witness for σ (β) for each β p. This shows p is the standard parameter for and is sound. ωρ n+1 Finally, we show = N τ by showing that N τ. Note that N τ because τ is a cardinal in but τ < τ and hence cannot be a cardinal in N τ. is countably iterable since it is embeddable into N τ. Hence N τ by a simple comparison argument using the fact that both are sound, Θ-g-organized premice over R projecting to R. 3 3 In [3], this is where the index-dependent condensation lemma is used to show the corresponding fact about levels of L[E]. In our case, things are greatly simplified as R is a strong cutpoint of both and N τ ; hence, we can show N τ by directly comparing them. The argument that neither side moves during the comparison is the same as that in the proof that two sound mice projecting to ω are lined up. 5

6 Now suppose N τ is either B-active (that is, the top predicate of N τ codes a branch) or passive. Let N,, σ, σ, τ be defined as above. Again, by [4, Lemma 3.10], is a countably iterable Θ-g-organized, R-sound mouse over R. The same argument as above shows = N τ. This completes the proof of the lemma. Lemma 2.7. Suppose τ C. Then D τ is closed. Proof. Let τ be a limit point of D τ. We need to see that τ D τ. First, note that τ C as C is closed. Now form the direct limit, σ τ τ τ D τ of the direct system N τ, σ ττ τ τ, and τ, τ D τ τ. The direct limit is well-founded as we can define a map σ from into N τ in an obvious manner. So from now on, we identify with its transitive collapse. The following properties are evident: for τ D τ τ, σ τ τ is Σ (n) 0 -preserving; σ σ τ τ = σ τ τ; τ = σ τ τ(τ ); τ = cr(σ) and σ( τ) = τ. To see that is a premouse of the same type as N τ, just notice that the direct limit maps (into ) preserve Π 2 properties upwards on a tail-end (cf. [3, Lemma 3.8]); is also Θ-gorganized by [4, Lemma 3.10] since N τ is such and σ is sufficiently elementary. Now let p = σ τ(p τ τ ) for some τ < τ. Given x, there is an x N τ for some τ D τ τ such that σ τ(x τ ) = x. By R-soundness of N τ, x n+1 = h τ (a, p τ ) for some a R. Since Σ (n) n+1 1 facts are preserved upwards by the direct limit maps, x = h (a, p). This n+1 implies = h (R p), hence ωρn+1 = R and p Rn+1, that is, p is a very good parameter for. Next, we need to see that p is the standard parameter for to conclude that is R- sound. To see this, notice that rng(σ τ τ) rng(σ), σ( p) = p τ, and σ τ τ(p τ ) = p τ for any τ D τ τ. This implies that rng(σ) contains a generalized witness for each α p τ with respect to N τ and p τ. The above facts imply that p is indeed the standard parameter for, and hence is R-sound. Consequently, = N τ and σ = σ ττ by a similar interpolation argument as in Lemma 2.6. This gives us that τ D τ. 6

7 References [1] G. Sargsyan and J. R. Steel. The Mouse Set Conjecture for sets of reals, available at gs481/papers.html To appear in the Journal of Symbolic Logic. [2] Ernest Schimmerling. Coherent sequences and threads. Advances in Mathematics, 216(1):89 117, [3] Ernest Schimmerling and Martin Zeman. Characterization of κ in core models. Journal of Mathematical Logic, 4(01):1 72, [4] F. Schlutzenberg and N. Trang. Scales in hybrid mice over R, available at math.cmu.edu/ namtrang [5] Martin Zeman. Inner models and large cardinals, volume 5. Walter de Gruyter,