The Semi-Weak Square Principle

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1 The Semi-Weak Square Principle Maxwell Levine Universität Wien Kurt Gödel Research Center for Mathematical Logic Währinger Straße Wien Austria Abstract Cummings, Foreman, and Magidor proved that for any λ < κ, κ,λ implies that κ carries a PCF-theoretic object called a very good scale, but that κ is consistent with the absence of a very good scale at κ. They asked whether κ,<κ is enough to imply the existence of a very good scale for κ, and we resolve this question in the negative. Furthermore, we improve a theorem of Cummings and Schimmerling and show that κ,<κ implies the failure of simultaneous stationary reflection at κ + for any singular κ. Keywords: Set Theory, Forcing, Large Cardinals 1. Some Background The square principle, denoted κ if it holds at a cardinal κ, was distilled by Jensen in order to study Gödel s constructible universe L [6]. It characterizes the minimility of L in a combinatorial sense in particular, square fails above supercompact cardinals but it can hold in larger models. Jensen also defined a weak square principle, denoted κ, and later on Schimmerling introduced a hierarchy of intermediate principles κ,λ for 1 λ κ [7], where κ is equivalent to κ,1 and κ is equivalent to κ,κ. In their paper, Squares, Scales, and Stationary Reflection, Cummings, Foreman, and Magidor demonstrate a complex interplay between generalized square principles, the reflection properties of large cardinals, and the scales used in the PCF theory of Shelah [3]. In particular, they prove that for a singular κ and λ < κ, κ,λ implies the existence of a so-called very good scale, and they show that a very good scale implies the failure of simultaneous stationary reflection at κ +. On the other hand, they construct a model (using countably many supercompact cardinals) in which both κ and simultaneous reflection at κ + hold, and hence there is no very good scale at κ. They ask whether the semi-weak square, κ,<κ, implies the existence of a very good scale, and we will prove here that it does not assuming the existence of a supercompact Preprint submitted to Elsevier February 9, 2018

2 cardinal. Furthermore, we show that κ,<κ does nonetheless imply the failure of simultaneous stationary reflection. It is necessary for us to use large cardinals for our main result, because the absence of a very good scale at any singular cardinal implies consistency of some large cardinal axioms. If κ is a singular cardinal that does not carry a very good scale, then κ fails, hence 0 # exists by Jensen s Covering Lemma, and 0 # implies the consistency of, for example, the existence of a weakly compact cardinal [5]. An exact lower bound for the non-existence of a very good scale is unknown. The conceptual context for this paper is contained entirely in Cummings, Foreman, and Magidor s original study [3]. We aim to give a reasonably selfcontained exposition for anyone with a working knowledge of forcing theory and supercompact cardinals. The essential definitions for this paper are as follows: Definition 1. The semi-weak square principle, κ,<κ, holds if there is a κ,<κ - sequence, which is a sequence C α : α lim(κ + ) such that for all α lim(κ + ): C α consists of clubs C α of order-type less than or equal to κ; C C α, β lim C, C β C β ; 1 C α < κ. Definition 2. Let κ be singular and let κ i : i < cf κ be a sequence of regular cardinals converging to κ. Consider the ordering of eventual domination, where f < g if there is some j < cf κ such that for all i j, f(i) < g(i). A scale at κ is a sequence f α : α < κ + of functions with domain cf κ such that: α, i < cf κ, f α (i) < κ i ; α < β < κ +, f α < f β ; g : cf κ ON such that i < cf κ, g(i) < κ i, α < κ +, g < f α. In other words, a scale is a sequence of functions of length κ + in a given product i<cf κ κ i that is increasing and cofinal in the ordering of eventual domination. Definition 3. For a singular cardinal κ and a scale f = f α : α < κ + in a product i<cf κ κ i, a point α < κ + is very good if cf α > cf κ and there is a club C α and an index j < cf κ such that for all i j, f β (i) : β C is strictly increasing. A scale f = f α : α < κ + is very good if all points α such that cf α > cf κ are very good. Note that a very good scale exists at κ if and only there is a scale f = f α : α < κ + and a club D κ + such that every point α D of cofinality greater than cf κ is very good. There are, naturally, good scales, and also better scales, but their study falls outside the scope of this paper. 2

3 Definition 4. Consider a regular cardinal τ and a stationary subset S τ. S reflects at α < τ if cf α > ω and S α is stationary as a subset of α. If S i : i < γ is a sequence of stationary subsets of τ, then S i : i < γ reflects simultaneously at α < τ if S i α is stationary for all i < γ. If κ is singular, we say that simultaneous reflection holds at κ + if for every regular µ < κ and every sequence S i : i < cf κ of stationary subsets of κ + cof(< µ), there is some α < κ + at which S i : i < cf κ reflects simultaneously. Observe that if κ i : i < cf κ is a sequence of regular cardinals converging to κ, then the sequence of stationary sets κ + cof(κ i ) : i < cf κ cannot reflect simultaneously. To say that simultaneous reflection holds at κ + means that every sequence of stationary sets that can plausibly reflect simultaneously will do so. Other treatments use a more flexible definition of simultaneous reflection, but we will commit to this one for the sake of simplicity. Now that we have introduced the objects of study, we will introduce the tools we will use to obtain our results. Supercompact cardinals will allow us to construct a model where singular cardinals can fail to carry very good scales. Definition 5. A cardinal κ is supercompact if for every λ κ there is an elementary embedding j : V M V with critical point κ such that M λ M. The main forcing poset we need comes from a family of posets defined by Jensen for adding generalized square sequences by initial segments. Cummings, Foreman, and Magidor cover the class of κ,<λ -adding posets, but we will provide a variant of their treatment here for completeness. We will only need the version that adds κ,<κ. Definition 6. S is the poset of bounded initial segments of κ,<κ sequences with closed domain, ordered by direct extension. More precisely, S is the set of conditions p such that: dom p = {α δ : α a limit} for some limit δ < κ + ; α dom p, p(α) is a set of clubs C α of order type less than or equal to κ; α dom p, C p(α), β lim C, C β p(β); α dom p, 1 p(α) < κ. For the ordering, p q if p end-extends q, meaning that max p max q and p (max dom q + 1) = q. Proposition 1. S is κ + -distributive. Proof. First, we claim that any p S can be extended to q < p, i.e. q such that max dom q > max dom p: if max dom p = γ, let q = p γ+ω, { γ+n : n < ω }. Second, suppose p α : α < δ with δ κ is a strictly decreasing sequence such that γ α := max dom p α and for all limits β δ, p β (γ β ) = { γ α : α < β }. Then 3

4 p α : α < δ has a lower bound p δ where max dom p δ = γ δ := sup α<δ γ α and p δ (γ δ ) = { γ α : α < δ }. Now suppose p S f : λ ON with λ κ. Define a strictly decreasing sequence p α : α λ such that for all α, p α+1 decides the value of f(α), and if α is a limit then let p α be the appropriate lower bound. If GCH holds, then S has the κ ++ -chain condition, but we will avoid assumptions about te continuum function. The most important property of S follows from the same reasoning as its distributivity: Proposition 2. S κ,<κ. In order to make use of supercompact cardinals, we need to supplement the S poset with the threading poset. Definition 7. Suppose are given an S-generic filter Q and the κ<κ -sequence C = Q. If λ is an uncountable regular cardinal less than κ, let Tλ be the poset of closed bounded sets c κ + of order-type less than λ such that α lim c, c α C α. The ordering on T λ is end-extension: d c if max d max c and d (max c + 1) = c. The significance of the threading poset lies in the following: Proposition 3. The subset D(S Ṫλ) := {(p, ċ) : d, p ċ = ď, max dom p = max c} S Ṫλ is λ-closed and dense. Proof. To show density: given any (p, ċ), p can be extended to p forcing ċ = ď for some d using κ + -distibutivity, then we extend p to q with max dom q > max d, and so we find (q, d {max dom q}) D. If (p ξ, d ξ ) : ξ < η is a sequence in D(S Ṫλ) with η < λ, then it has a lower bound (p, d ) where max dom p = δ := sup ξ<η p ξ, p(δ ) = { ξ<η d ξ}, and d = ξ<η d ξ {sup ξ<η max d ξ }. It follows that the iteration S Ṫλ is equivalent to a λ-closed forcing. Put a different way, T λ is a quotient of a λ-closed forcing with S. 2. Semi-weak square does not imply existence of a very good scale The threading poset is poorly behaved in general it is not even countably closed so our pivotal lemma is about the fact that the threading poset can be made to preserve stationarity in certain specific cases. Lemma 1. Let G be S-generic and let ν = (κ + ) V. If f : κ + µ is a partition in V [G] for some µ < κ and τ < λ are regular cardinals, then there is some i < µ such that Tλ f 1 (i) cof(τ) is stationary in ν. Proof. Work in V, and suppose that p S f : κ + µ. We can rewrite f as a S Ṫλ-name and instead consider: (p, ) S Tλ f : κ + µ and f V S. We want to show that there is some p p and some i < µ such that (p, ) S Tλ f 1 (i) cof(τ) is stationary. We will inductively define a decreasing sequence 4

5 p ξ : ξ τ in S below p, sequences t i ξ : ξ τ for all i < µ such that (p ξ, t i ξ ) S T λ, and and an increasing sequence of ordinals α ξ : 0 < ξ τ. Zero Step: We will define a sequence of conditions q i : 0 < i µ S below p and a sequence δ i : i µ such that δ i := max dom q i. We can assume that (p, ) f 1 (0) cof(τ) is stationary since otherwise we would be done, and so we find some (q 1, t 0 0) p and a S T λ -name Ċ0 such that (q 1, t 0 0) Ċ0 is a club in ν and Ċ0 f 1 (0) cof(τ) =. Similarly, if q i and t i 0 have been defined, then since we may assume that (q i, ) f 1 (i) cof(τ) is stationary, we we can find q i+1 < q i and t i 0 such that (q i+1, t i 0) Ċi is a club in ν and Ċ i f 1 (i) cof(τ) =. If i is a limit then let δ i = sup j<i δ j and let q i be a condition whose domain maximum is δ i, q i δ j = q j for all j < i, and q i (δ) = { δ j : j < i }. This ensures that there will be a lower bound at every limit step. Finally, let p 0 = q µ. Successor Step: Suppose that p ξ : ξ η, t i ξ : 0 < ξ η for i < µ, and 0 < α ξ : ξ η have been defined already. We will define a sequence of conditions q i : i µ S below p η ; a set of closed bounded sets {s i : i < µ} of order-type less than λ; and a collection of ordinals β i : i < µ above α η such that (q i, s j ) β j Ċj. Furthermore, let δ i := max dom q i. If q i has been defined, then we can find (q i+1, s i+1 ) (q i, t i η) such that δ i+1 > δ i, max s i+1 and β i > α η such that (q i+1, s i+1 ) β i Ċi. This works because Ċi is forced to be a club. Again, if i is a limit then let δ i = sup j<i δ j and let q i be a condition whose maximal element is δ i, q i δ j = q j for all j < i, and q i (δ) = { δ j : j < i }. Now let p η+1 = q µ. Also let t i η+1 = s i {δ } and let α η+1 = sup i<µ β i. Observe that (p η+1, t i η+1) Ċi [α η, α η+1 ) for all i < µ. Limit Step: For i < µ, let t i η = ξ<η ti ξ {sup ξ<η max t i ξ }, and let α η = sup ξ<η α ξ. Let p η be the condition such that p dom p ξ = p ξ for ξ < η, max dom p η is defined as γ := sup ξ<η max dom p ξ, and p η (γ ) = {t i η : i < µ}. Note that p η is in fact a condition in S because µ < κ. Since η τ < λ, (p η, t i η) is a condition in S Ṫλ. Also, since (p η, t i η) Ċi [α ξ, α ξ+1 ) for all i < µ, it follows that (p η, t i η) α η Ċi for all i < µ. The completes the construction. Choose p p τ deciding a value for f(α τ ). If p f(α τ ) = i, then this contradicts the fact that (p, t i τ ) α τ Ċi and Ċ i f 1 (i) cof(τ) =. We must employ some technical theorems for the construction of a model with κ,<κ and no very good scale at κ. They are as follows: Fact 1. [2] Let j : V M be an elementary embedding and let P be a forcing poset. If G is a P-generic filter over V and H is a j(p)-generic filter over M such that j[g] H, then j can be extended to an elementary embedding j : V [G] M[H] given by j (ẋ G ) := j(ẋ) H where j (G) = H. Fact 2. [2] Suppose κ is an inaccessible cardinal, λ < κ is regular, and P is a λ-closed separative poset such that P < κ. Then forcing with Col(λ < κ) is 5

6 equivalent to forcing with Col(λ, < κ)/p. Moreover, if Col(λ, A) where sup A = κ is used in place of Col(λ, < κ), then the conclusion still holds. Fact 3. [1] If τ is a regular uncountable cardinal, S τ cof(ω) is a stationary subset of τ, and P is a countably closed poset, then S remains stationary in any forcing extension by P. Now we are in a position to prove our main result. Theorem 1. If µ is supercompact in V and κ > µ is a singular cardinal such that cf(κ) < µ, then there is a forcing extension in which κ,<κ holds and κ does not carry a very good scale. If κ = µ +δ, then κ = ℵ δ in the forcing extension. In particular, if κ = µ +ω, we can obtain a model where ℵω,<ℵ ω ℵ ω does not carry a very good scale. holds but Proof. Let G be Col(ω 1, < µ)-generic over V. Let S be defined in V [G] and let H be an S-generic filter over V [G]. Then V [G H] is our intended model. Suppose f = f α : α < κ + is a scale at κ in some product i<cf κ τ i and is contained in the model V [G H]. Consider the maps g i : α f α (i) < τ i for each i < cf κ. Apply Lemma 1 to get stationary sets S i κ + cof(ω) for each i such that g i is constant on S i, and moreover such that V [G H] T ω1 S i is stationary in ν where ν = (κ + ) V. Now take a max{ν, S }-supercompact embedding j : V M with critical point µ. We will make repeated use of the fact that if ρ := sup j[ν], then the facts that M = j(ν) is regular and M ν M imply that ρ < j(ν). Claim 1. There exists an extension V [G H L] in which S i is stationary in ν for all i < cf κ and where j can be extended to an elementary embedding j + : V [G H] M[j(G H)]. Claim 2. In V [G H L], the S i s reflect simultaneously. If γ < ν is the point of reflection from Claim 2, then γ cannot be a very good point for the scale f. If it were, it would be witnessed by a club C γ and an index j < cf µ such that f β (i) : β C is increasing for i j. However, β f β (j) is constant on S j γ. Since C S j is stationary, and thus unbounded, this is not possible. Proof of Claim 1. Let I be (Ṫω 1 ) H -generic over V [G H]. We get preservation of stationarity of the S i s due to Lemma 1. Now consider j(col(ω 1, < µ)) = Col(ω 1, < j(µ)) = Col(ω 1, < µ) R where R has the definition α [µ,j(µ)) Col(ω 1, α) in M. Let J be R/(G H I)- generic over V [G H I]. Note that S Ṫω 1 is equivalent to a countably closed poset and of size less than j(µ). By Fact 2, forcing with the quotient R/(Col(µ, < κ) S T ω1 )) is equivalent in M to forcing with R itself, and since R is countably closed and M is closed under countable sequences it follows that the S i s are stationary in V [G H I J] by Fact 3. We also have that 6

7 j[col(ω 1, < µ)] Col(ω 1, < µ) R, so we can apply the Fact 1 to get a partial lift j : V [G] M[G H I J] = M[j (G)]. Now consider j (S). Observe that j (S) is countably closed in V [j (G)] because any countable decreasing sequence of conditions p n : n < ω j (S) is bounded below by p where σ := max dom p = sup n<ω max dom p n and p(σ) = {A} for some sequence of order-type ω unbounded in σ. (So it does not matter if M[j (G)] does not accurately define j (S) because j (ν) has cofinality larger than ω and the other relevant notions are absolute.) Therefore, the S i s are again still stationary if we force with j (S) by Fact 3. Let C = H, write C = C α : α lim(ν), and write j(c) as C α : α lim(j (ν)). Furthermore, let T = I. Consider β lim j [T ], so that there is some α < ν such that j (α) = β. Since j is continuous on sequences of ordinals of countable cofinality and T has order-type ω 1, j [T ] β = j (T α) j (C α ) = C β. It follows that s := C α : α lim(ρ) ρ, {j [T ]} is a master condition for j [S], meaning that for all p S, s j (p). Hence we can force with a j (S)-generic K containing s to apply Fact 1 and get the lift j + : V [G H] M[j + (G H)]. We now have L = I J K, which gives us the claim. Proof of Claim 2. Recall that we assumed that cf κ < µ, and so j + (cf κ) = cf κ because j + ON = j ON. We again use ρ = sup j[ν], where ρ < j(ν). We show that for every i < cf κ, j + (S i ) ρ is stationary in ρ. Consider a club C ρ in M[j + (G H)], and let E = {α < ν : j + (α) D}. Then E will be unbounded in ν and closed under countable sequences: if α n : n < ω E is a sequence with supremum α, then sup n<ω j + (α n ) = j + (α ) D, so α E; if β < ν then define β n : n < ω ν above β and γ n : n < ω ρ so that for all n, j + (β n ) < γ n < j + (β n+1 ), so β < sup β n E. Therefore there will be some α E S i, so j + (α) D j + (S i ). We have demonstrated that, M[j + (G H)] = α < j + (ν), i < cf κ, j + (S i ) α is stationary in α. It follows by elementarity that, V [G H] = α < ν, i < cf κ, S i α is stationary. This completes the proof of our theorem. 3. Semi-weak square and failure of simultaneous stationary reflection Up to now, the extent that semi-weak square impacts stationary reflection was given by the following theorem: 7

8 Theorem 2 (Cummings, Schimmerling). [4] If κ is a singular strong limit cardinal and κ,<κ holds, then for every stationary set S κ +, there is a µ < κ and a sequence of stationary subsets S i : i < cf κ of κ + such that if S i α is stationary for all i < cf κ, then cf α < µ. It turns out that we can weaken the hypotheses and strengthen the conclusion. We can show that even though semi-weak square does not imply the existence of a very good scale, its impact on stationary reflection is the same as that of a very good scale namely, simultaneous stationary reflection fails. Theorem 3. If κ is singular and κ,<κ holds, then for every stationary S κ +, there is a sequence S i : i < cf κ of subsets of S that do not reflect simultaneously. Proof. Fix a stationary set S κ + and a κ,<κ -sequence C α : α lim(κ + ). By κ + -completeness of the club filter, there is some stationary S S and a cardinal µ < κ such that C α < µ for all α S. Let λ := cf κ and choose a strictly increasing sequence κ i : i < λ of regular cardinals converging to κ such that µ < κ 0. We define a scale-like sequence of functions f α : α S in the product i<λ κ i. If α = min S, then let f α (i) = 0 for all i < λ. For all other α S, let j be the least ordinal such that there is some C C α with ot C < κ j, let f α (i) = 0 for i < j, and for i j let, { } f α (i) = sup sup β C S f β (i) : C C α, ot C < κ i + 1. For every i < λ use κ + -completeness of the club filter to find S i S and δ i < κ i such that for all α S i, f α (i) = δ i. We claim that the sequence S i : i < λ does not simultaneously reflect. Suppose for contradiction that α < κ + is a point of simultaneous reflection for this sequence. Pick C C α (it does not matter whether α S ) and choose i such that ot C < κ i. Using the assertion that S i α is stationary, pick β, γ lim C S i such that β < γ. Then C γ C γ, ot(c γ) < ot(c) < κ i, and of course β C S, so it follows by construction that f γ (i) > f β (i). This contradicts the fact that f γ (i) = δ i = f β (i). Acknowledgements I would like to thank Monroe Eskiw, who patiently listened to early versions of these theorems, and who helped me realize that I had proved the full extent of Theorem 3. [1] James Cummings. Notes on singular cardinal combinatorics. Notre Dame Journal of Formal Logic, 46(3), [2] James Cummings. Iterated forcing and elementary embeddings. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of Set Theory, pages Springer,

9 [3] James Cummings, Matthew Foreman, and Menachem Magidor. Squares, scales, and stationary reflection. Journal of Mathematical Logic, 1:35 98, [4] James Cummings and Ernest Schimmerling. Indexed squares. Israel Journal of Mathematics, 131:61 99, [5] Keith Devlin. Constructibility, volume 6 of Perspectives in Mathematical Logic. Springer-Verlag, Berlin, [6] Ronald Jensen. The fine structure of the constructible hierarchy. Annals of Mathematical Logic, 4: , [7] Ernest Schimmerling. Combinatorial principles in the core model for one Woodin cardinal. Annals of Pure and Applied Logic, 74: ,