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Title Toward the Consistency Strength of Stationary Set Reflection on Small Cardinals
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Author Northrup, Cynthia
Publication Date 2015
Peer reviewed|Thesis/dissertation
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Toward the Consistency Strength of Stationary Set Reflection on Small Cardinals
DISSERTATION
submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in Mathematics
by
Cynthia Northrup
Dissertation Committee: Professor Dr. Martin Zeman, Chair Professor Dr. Penelope Maddy Professor Dr. Vladimir Baranovsky
2015 c 2015 Cynthia Northrup DEDICATION
To Scott and Maxwell.
ii TABLE OF CONTENTS
Page
LIST OF FIGURES iv
ACKNOWLEDGMENTS v
CURRICULUM VITAE vi
ABSTRACT OF THE DISSERTATION vii
1 Definitions and Background 1 1.1 Foundations ...... 1 1.2 Large Cardinals ...... 3 1.3 Forcing ...... 6 1.4 Reflection ...... 11
2 Reflection and Square 14 2.1 Current Bounds on Reflection ...... 14 2.2 Square ...... 15
3 Forcing 18 3.1 The Initial Forcing ...... 18 3.2 Building the Generic Filter above Pκ ...... 21 4 Club Shooting 25 4.1 Background ...... 25 4.2 Shooting a Club ...... 26
Bibliography 32
iii LIST OF FIGURES
Page
3.1 ...... 22
iv ACKNOWLEDGMENTS
I would like to thank the ARCS Foundation for their support of my research, specifically the generous donations of Sue and Nicolaos Alexopoulos, and Peggy and Joseph Stemler.
Additionally, I would like to acknowledge the support I received from the GAANN program.
Thank you to Sarah Eichhorn, Chris O’neal, and De Gallow for their support and advice. They are phenomenal teachers and were always there to help me improve my own teaching and prepare for a career in academia.
Many thanks to my advisor, Martin Zeman, for being patient and understanding to the many stresses of graduate school that can distract from research. My committee members, Vladimir Baranovsky for taking the time to venture into the world of set theory, and Penelope Maddy for being so supportive from the time of my advancement.
Special thanks to Jacquelyn Rische and my other friends at UC Irvine, especially Zachary Faubion, Andres Forero Cuervo, and Monroe Eskew, for their willingness to talk logic when- ever the need arose.
Thank you to Donna McConnell for her continued support, advice, and endless help through- out this process.
v CURRICULUM VITAE
Cynthia Northrup
EDUCATION Doctor of Philosophy in Mathematics 2015 University of California, Irvine Master of Science in Mathematics 2010 University of California, Irvine Bachelor of Art in Mathematics 2006 California State University, Northridge
FELLOWSHIPS ARCS Fellow 2013–2014 Achievement Rewards for College Scientists Pedagogical Fellow 2013–2014 Sponsored by the UC Irvine Teaching, Learning, and Technology Center GAANN Fellow 2010–2012 U.S. Department of Education and the UC Irvine Mathematics Department
TEACHING EXPERIENCE at UC Irvine Instructor Pre-Calculus (2 summer sessions, 1 quarter online) 2009, 2011, 2013 Differential Calculus (2 quarters, 2 summer sessions) 2012-2013 Integral Calculus (1 summer session) 2014 Teaching Assistant Pre-Calculus (1 summer session) 2009 Differential Calculus (3 quarters) 2009, 2014 Integral Calculus (3 quarters, 1 summer session) 2008, 2010-2011 Multivariable Calculus (1 quarter) 2013 Infinite Series and Linear Algebra (2 summer sessions) 2010-2011 Linear Algebra (1 summer session) 2013 Mathematics for Economists (2 quarters) 2009-2010
vi ABSTRACT OF THE DISSERTATION
Toward the Consistency Strength of Stationary Set Reflection on Small Cardinals
By
Cynthia Northrup
Doctor of Philosophy in Mathematics
University of California, Irvine, 2015
Professor Dr. Martin Zeman, Chair
A set S ⊆ ω3 is stationary in ω3 if it intersects all closed and unbounded subsets in ω3. We say that S reflects below ω3 if there is some α < ω3 for which S ∩ α is stationary in α.
In my thesis I obtain a model M for which every subset of ω3 focusing on cofinality ω reflects at a point of cofinality ω1. That is, M |= Refl(ω3, ω, ω1). To do this, I adjust the methods of Gitik in order to get set reflection. This argument may help show the way to getting the consistency strength.
Due to existing bounds, our strategy begins with κ of Mitchell order κ which concentrates on
+ α -supercompact cardinals α. We collapse κ to ω3, change the cofinality of many measurables between ω2 and ω3, and shoot clubs through the non-reflecting stationary sets. For any non- reflecting stationary set S, we use the forcing to add a club C for which C ∩ S = ∅, thus rendering this S no longer stationary. We take care that any more stationary sets we add in this process, have a club shot through them and so are no longer stationary in the final
ω3 generic extension. Additionally, S ⊆ Sω sets can only have a club shot through them under certain circumstances. In our situation, we use the fact that the complement of these non- reflecting stationary sets will be fat, i.e. for any club C ⊆ α, E ∩ C contains closed sets of ordinals of arbitrarily large order-type below α, to allow us to shoot a club without collapsing κ.
vii Chapter 1
Definitions and Background
It is assumed that the reader has a general knowledge of set theory and familiarity with the concept of forcing. This chapter contains many of the definitions and topics discussed beyond, for ease of reference. Details on the main topics we use, stationary reflection and the square principle, will be discussed in chapter 2. For additional details and background, please refer to the works of Jech [10] and Kunen [13].
1.1 Foundations
We work in a model of set theory, V , under the assumption of ZFC. That is, the Zermelo- Fraenkel axioms along with the axiom of choice.
Definition 1.1. A set is an ordinal if it is transitive and well ordered by ∈.
Definition 1.2. An ordinal α is called a cardinal if there is no bijection between α and γ for any ordinal γ < α. To be more clear, here is the natural progression of the cardinal numbers.
1 + ++ 0 < 1 < 2 < ... < ω < ω1 < ω2 < ... < κ < κ < κ < ...
Definition 1.3. The cofinality of a cardinal κ, cf(κ), is the smallest cardinal λ such that there exists a sequence hαξ : ξ < λi cofinal in κ.
Definition 1.4. The cardinal κ is regular if cf(κ) = κ and singular otherwise.
Definition 1.5. For κ a regular uncountable cardinal, a set C ⊂ κ is closed if for every sequence hαξ : ξ < γi ⊂ C, where γ < κ, supξ<γ(αξ) ∈ C.
Definition 1.6. For a limit ordinal κ a set C is said to be club if it is closed and unbounded in κ.
Definition 1.7. For κ a cardinal with cf(κ) > ω a set S ⊂ κ is said to be stationary if S ∩ C 6= ∅ for all clubs C in κ.
κ Definition 1.8. Sγ = {α < κ : cf(α) = γ)}
Definition 1.9. A filter on a nonempty set S is a collection F of subsets of S such that
1. S ∈ F and ∅ ∈/ F ,
2. If X ∈ F and Y ∈ F , then X ∩ Y ∈ F ,
3. If X,Y ⊂ S,X ∈ F , and X ⊂ Y , then Y ∈ F .
Definition 1.10. A filter F is principal if F = {X ⊂ S : Xo ⊂ X} for some nonempty
Xo ⊂ S.
Definition 1.11. An filter U on a set S with |S| = κ is uniform if |X| = κ for every X ∈ U.
Definition 1.12. Let hXα|α < κi be a sequence of subsets of κ. The diagonal intersection of Xα, α < κ, is defined as follows:
4 Xα = {ξ < κ|ξ ∈ ∩ Xα}. α<κ α<ξ
2 Definition 1.13. A filter F on a cardinal κ is normal if it is closed under diagonal inter- sections.
Definition 1.14. A filter U on a set S is an ultrafilter if for every X ⊂ S, either X ∈ U or S − X ∈ U.
Definition 1.15. A filter U is κ-complete if for any γ < κ,
\ Xα ∈ U for all α < γ ⇒ Xα ∈ U. α<γ
Definition 1.16. An elementary embedding is a map π : M → N such that for every formula φ and every a1, ..., an ∈ M,
M |= φ[a1, ..., an] if and only if N |= φ[π(a1), ..., π(an)]
1.2 Large Cardinals
Large cardinals are cardinals which not only cannot be proven to exist in a model of ZFC, but cannot be proven to be consistent with ZFC. Definitions for these large cardinals are used to give a measure to how far above ZFC we must assume in order to gain a given result. Their natural hierarchy gives other results their “consistency strength” as a standard of comparison. The large cardinals we use are defined below.
Definition 1.17. An inaccessible cardinal is one which is uncountable, regular, and 2λ < κ for all λ < κ.
Definition 1.18. An inaccessible cardinal κ is called a Mahlo cardinal if the set of all regular cardinals below κ is stationary.
Definition 1.19. An uncountable cardinal κ is measurable if there exists a κ-complete nonprincipal ultrafilter (or κ-measure) U on κ.
3 Definition 1.20. Pκ(A)= {X ⊂ A : |X| < κ}
Definition 1.21. An κ-measure U on Pκ(A) is fine if for any a ∈ A, U extends the filter generated by aˆ = {y ∈ Pκ(A): a ∈ y}.
Definition 1.22. An ordinal function f on a set S is regressive if f(α) < α for all α ∈ S, α > 0.
Definition 1.23. A fine measure U on Pκ(A) is normal if any regressive function
f : Pκ(A) → A is constant on a set in U.
Definition 1.24. An uncountable cardinal κ is supercompact if for every A such that
|A| ≥ κ there exists a fine normal measure on Pκ(A).
Definition 1.25. An uncountable cardinal κ is called λ-supercompact, for λ ≥ κ if ∃ a
normal measure on Pκ(λ).
Definition 1.26. Given a measurable cardinal κ with κ-measure U, we construct UltU (V ), the ultrapower of V by U, as follows. Let V κ be the collection of functions with domain κ.
1. We say f ∼ g ⇐⇒ {ξ < κ : f(ξ) = g(ξ)} ∈ U
2. [f] = {g ∈ V κ : f ∼ g}
3. [f] E [g] ⇐⇒ {ξ < κ : f(ξ) ∈ g(ξ)} ∈ U
This gives us hV κ/U, Ei, where V κ/U = {[f]: f ∈ V κ}
Theorem 1.1. (Lo´s)Let φ(v0, ..., vn) be a formula in the language of set theory and let
κ f0, ..., fn ∈ V . Then we have
κ V /U |= φ([f0], ..., [fn]) ⇐⇒ {ξ < κ : φ(f0(ξ), ..., fn(ξ))} ∈ U
4 κ For each a ∈ V , let d : V → V /U be given by d(a) = [ca], where ca is the constant function with value a. ThenLo´s’sTheorem gives us
κ V |= φ(x0, ..., xn) ⇐⇒ V /U |= φ(d(x0), ..., d(xn))
and so d is an elementary embedding. Since E is well-founded on V κ/U, and satisfies several
κ other properties, we can obtain an isomorphism e : V /U → UltU (V ), where UltU (V ) is the transitive collapse of V κ/U.
Definition 1.27. j := e ◦ d is the canonical ultrapower embedding.
Definition 1.28. The critical point of an elementary embedding is the smallest ordinal which is not mapped to itself.
Theorem 1.2. Given a measurable cardinal κ, we can construct an elementary embedding
j : V → UltU (V ), where UltU (V ) is a transitive ∈-structure, such that crit(j) = κ.
Theorem 1.3. If κ is an uncountable cardinal, and there exists an elementary embedding from V into a transitive ∈-structure M such that crit(j) = κ, then κ is measurable and the set U = {x ⊆ κ : κ ∈ j(x)} is a normal κ-measure on κ.
Definition 1.29. Let κ be a measurable cardinal. The Mitchell order is a well-founded partial order on the collection of normal measures on κ given by
U1 ¡ U2 if and only if U1 ∈ UltU2 (V )
If U is a normal measure on κ , and α is an ordinal, then α = o(U) if and only if there
exists an increasing (in Mitchell order) sequence hUξ : ξ < α + 1i of ultrafilters on κ such
that U = Uα.
Definition 1.30. The Mitchell order of κ, denoted o(κ), is the supremum of lengths of all such chains of ultra filters on κ.
5 - o(κ) = 0 if and only if there exists no ultrafilter U on κ, {α < κ : α is measurable} ∈ U.
- o(κ) ≥ 1 if and only if there exists an ultrafilter U on κ, {α < κ : α is measurable} ∈ U.
- o(κ) ≥ 2 if and only if there exists an ultrafilter U on κ, {α < κ : o(α) ≥ 1} ∈ U.
- For any limit ordinal α, o(κ) ≥ α if and only if o(κ) ≥ β for all β < α.
If the Generalized Continuum Hypothesis holds, then o(κ) ≤ κ++ for every measurable cardinal κ.
1.3 Forcing
Paul Cohen [3][4] introduced forcing and used it to prove the independence of the Continuum Hypothesis and the Axiom of Choice. Forcing is used to extend a transitive model M by adjoining a new set G ⊂ M in order to obtain a larger transitive model M[G] called a generic extension. Note that G ∈ M[G] but G/∈ M for nontrivial cases. Our choice of partial order, or notion of forcing, determines what is true in the generic extension. We include some introductory concepts here, but a more thorough treatment can be found in Kunen’s Set Theory [13], for example.
Definition 1.31. A binary relation ≤ on a set P is a partial ordering of P if it is reflexive and transitive. The direction that the symbol ≤ faces varies by publication, but for this work we will use ≤ to mean stronger than.
Definition 1.32. A set D ⊂ P is dense in P if for any p ∈ P there exists a q ∈ D such that q ≤ p.
Definition 1.33. A set A ⊂ P is open in P if for any p ∈ A, q ≤ p, we have q ∈ A.
Definition 1.34. A filter G is P-generic over M if it intersects all dense sets of P in M.
6 Definition 1.35. Conditions p, q ∈ P are comparable if either p ≤ q or q ≤ p.
Definition 1.36. Conditions p, q ∈ P are compatible if there exists some r ∈ P for which r ≤ p and r ≤ q.
A generic filter consists of conditions which are compatible. This means that adjoining G takes elements which are like approximations and glues them into the actual element that was not in the ground model. Finite conditions result in a countable ∪G, for example.
Definition 1.37. A partial order P is λ-closed if and only if whenever hpξ|ξ < γi, γ < λ, is a decreasing sequence of elements of P, then ∃q ∈ P ∀ξ < γ(q ≤ pξ). That is, there is one condition, q, which holds all the information from the sequence and agrees with any statement forced by some pξ.
Theorem 1.4. If P is λ-closed a partial order in M, where λ is a cardinal in M, then P preserves cardinals ≤ λ.
Definition 1.38. Let hP, ≤i be a partial order. A chain in P is a set C ⊂ P such that ∀p, q ∈ C (p ≤ q ∪ q ≤ p).
Definition 1.39. A ⊂ P is an antichain if ∀p, q ∈ P, p 6= q −→ p and q are not compatible.
Definition 1.40. For a cardinal θ, P satisfies the θ-chain condition (θ-c.c.) if every antichain has size less than θ.
Theorem 1.5. Assume P ∈ M, θ is a regular cardinal of M, and M |= “P has the θ-c.c.” Then P preserves cardinals ≥ θ.
Example 1.1. Cohen Forcing Let M be a countable transitive model of ZFC Let
P = {f : f is a partial function, f : ω1 × ω → {0, 1}, |f| < ω1}
7 ordered by reverse inclusion. Then any generic filter G is a set of compatible partial functions. S Hence G = g : ω1 × ω → {0, 1}. Consider the funtions hgα : α < ω1i given by gα(n) =
ω g(α, n). Then F : ω1 → 2 , with F (α) = gα, can be shown to be a surjection using a density
ω argument. Thus M[G] |= 2 = ω1.
Example 1.2. Collapse Forcing Let M be a countable transitive model of ZFC. Given an uncountable cardinal λ and a regular cardinal κ < λ. Let
P = Col(κ, λ) = {f : f is a partial function, f : κ → λ, |f| < λ}
under reverse inclusion.
Then for any generic filter G, ∪G : κ → λ is a surjection which collapses λ to κ.
This collapse will change cofinalities, but in a uniform way. For example, under Col(ω1, λ), all ordinals of uncountable cofinality in M will have cofinality ω1 in the generic extension.
When looking at the requirements for reflection, we look at various models as examples. One of these is the Core Model, K, a relative notion which depends on the large cardinals V satisfies. Given a large cardinal hypothesis φ, the “core model below φ” refers to the inner model that satisfies our criteria under the assumption that φ does not hold, while keeping K as close to L as possible (meaning that it satisfies weak covering, that is, K computes the successors of singular and weakly compact cardinals correctly). K incorporates all large cardinal hypotheses from V . “There exists no model with a Woodin cardinal” is the strongest anti large cardinal hypothesis under which we can construct K. If there exists an inner model with a Woodin cardinal, then there is no way to construct a model with weak covering.
Using core model theory it follows that
• A lot of ordinals between ω2 and ω3 in V look like inaccessibles in K.
8 • There are a lot of measurables in K.
So V and K disagree on cofinalities. If we want reflection, we have to change a lot of cofinalities and start with large cardinals.
Example 1.3. Namba Forcing
Suppose we would like to change the cofinality of ω2 to be ω, while preserving ω1. Conditions are trees with
• Nodes in Pω(ω2), finite subsets of ω2.
• A finite stem.
• Every node below the stem has ω2 many splittings.
• Ordered by inclusion.
<ω Then for any generic G, ∩G is a branch (an increasing sequence in [ω2] ) called a generic
branch. We can consider ∪ ∩ G, a countable sequence cofinal in ω2. This changes cf(ω2) to
ω, while ω1 is preserved.
The Necessity for Large Cardinals:
Now, taking this a step further, we aim to change the cofinality of ω3 to be ω, while preserving
ω1 and ω2. Not only do our previous methods fall short of this goal, but using core model theory it can be shown that such a model must contain measurables of high Mitchell order and in fact must contain a Prikry sequence. [8]
Definition 1.41. Let us now describe the basic Prikry forcing. Let κ be a measurable cardinal and U a normal ultrafilter on κ. The partial order P consists of pairs hp,Ai such that
9 • p is an increasing sequence of finite length in κ
• A ∈ U
• min(A) > max(p)
Conditions of the Prikry forcing are ordered by ≤ and ≤∗ where hp,Ai ≤ hq,Bi if and only if
• q / p (q is an initial segment of p)
• p − q ⊂ B
• A ⊂ B
and hp,Ai ≤∗ hq,Bi if and only if
•h p,Ai ≤ hq,Bi
• p=q
Definition 1.42. Forcing with hP, ≤, ≤∗i. P adds an ω-sequence, cofinal in κ, called a Prikry sequence.
Thus for any extension V [G] we have cf(κ) = ω, all cardinals are preserved, and no new bounded sets are added to κ.
Prikry forcing allowed us to change cf(κ) to ω. Here we would like to change cf(κ) to an uncountable value. We consider the following.
Theorem 1.6. (Magidor, 1978 [15]) Assume a measurable κ with o(κ) = δ, δ a regular cardinal, δ < κ
10 • Changes cf(κ) to δ
• Adds a closed unbounded subset of κ of ordertype δ
Theorem 1.7. (Mitchell, 1987 [17]) The above assumption is optimal.
Theorem 1.8. (Radin, 1982 [18]) There exists a forcing which additionally adds a closed unbounded subset which consists of cardinals from the ground model. In this way combines a variety of ways of changing cofinalities. This means that we can now change cofinalities in a highly non-uniform way.
Example 1.4. Radin Forcing Conditions are of the form hs, Ai where
• s is a sequence of normal measures.
• A is an element of a ultrafilter on sets of normal measures.
• In the generic extension we have hUn : n ∈ ωi, where Un is a κn-measure for each n ∈ ω.
1.4 Reflection
Stationary set reflection is a measure of whether a property which holds on a large structure also holds on substructures. Sets which reflect are thought of as large, but this is not a precise notion. For definitions 1.43-1.46, let κ be an uncountable regular cardinal, S ⊂ κ stationary.
Definition 1.43. S reflects at γ if and only if S ∩ γ is stationary in γ.
Definition 1.44. S reflects if and only if there is a γ < κ at which S reflects.
11 Definition 1.45. Refl(S) is the statement: Every stationary subset of S reflects.
κ Definition 1.46. Refl(κ, γ, λ) is the statement: Every stationary subset of Sγ reflects at a point of cofinality λ.
Definition 1.47. Refl(θ, S) is the statement: For any collection hSδ|δ < θi of stationary subsets of S, ∃α at which they all reflect. This is referred to as simultaneous reflection.
To begin, let us prove the following.
Lemma 1.1. For any A ⊆ κ in V , j(A) ∩ κ = A in M.
Theorem 1.9. Refl(κ, ω, < ω) holds for κ measurable.
Proof. Since κ is measurable, consider M ∼= Ult(U, V ) and canonical embedding j : V → M.
κ Take any S ⊆ Sω stationary in V . Then (by lemma 1.1) j(S) ∩ κ = S. We claim that S is stationary in M. For any club C ⊆ κ in M, C ⊆ κ club in V by M a subclass of V . Hence, C ∩ S 6= ∅. So, M |= j(S) reflects at some inaccessible below j(κ) (specifically, κ).
Thus V |= S reflects at some inaccessible below κ. ∴ Refl(κ, ω, < κ).
Theorem 1.10. Con(ZFC + ∃ measurable) ⇒ Con(ZFC + Refl(ω2, ω, ω1))
Proof. Given κ measurable, let j : V → M be the canonical embedding. P = col(ω1, < κ). Force with P over V and j(P) over M. Then extend j to j0 : V [G] → M[G][H]. We want to show that V [G] |= Refl(ω2, ω, ω1). Recall, that this means we want V [G] |=“Every stationary
ω2 S ⊂ Sω reflects at a point of cofinality ω1”.
ω2 0 Lemma 1.2. For any S ⊂ Sω stationary in V [G], consider j (S) ∩ κ = S is stationary in M[G][H].
Lemma 1.3. S ∈ M[G].
12 Proof. col(ω1, [κ, j(κ)))) was taken in M and we will show that this is the same as col(ω1, [κ, j(κ)))) taken in M[G]. Then, in M[G], the forcing is ω1-closed.
Recall that col(ω1, [κ, j(κ))) adjoins a surjection from ω1 → α ∀α ∈ [κ, j(κ)).
Lemma 1.4. The forcing notion col(ω1, [κ, j(κ))) preserves the stationarity of S.
Proof. By Lemma 1.5.
λ Lemma 1.5. Assume N |= ZFC, λ is regular, E ⊆ Sω stationary, Q is a poset in N, and
N |= Q is ω1−closed. Then E remains stationary in λ after forcing with Q.
˙ Proof. Let N[G] |= C is club in λ. Then ∃q ∈ G, q C is club in λ.
Pick some large θ so that we may work in Hθ. Take some elementary substructure X of Hθ so that X contains q, C,˙ Q,E, Eˇ (and anything else we need) and λ˜ ∈ E, where by λ regular ˜ ˜ uncountable, λ := sup(X ∩ λ) < λ, cf(λ) = ω. Pick an increasing sequence hλn : n < ωi, ˜ ˜ λn ∈ X (but λ, hλn : n < ωi are not in X) so that supm<ω(λm) = λ.
Construct hqn : n < ωi descending, q0 ≤ q and an ascending sequence of ordinals hγn : n < ωi ˜ ˙ such that for any n < ω, λn < γn < λ and qn λn ∈ C by induction on n.
Take λn, qn, and γn−1 to X. (We actually need to take max{γn−1, λn}, but wlog we use λn.) ˙ In X, qn C is unbounded. ˇ ⇒ ∃qn+1 ≤ qn and γn ∈ ORD ∩ C such that qn+1 λn < γˇn. ˇ Since we are in X, γn < λ ⇒ (by elementarity) Hθ |= γn ∈ C, λn < γn ˜ In N, hγni −→ λ, hqni a descending length ω-chain, and Q is ω1-closed. ˜ ˙ ˙ ⇒ ∃q˜ ≤ qn (∀n < ω). So we have (still in N), hγni −→ λ ∈ E,q ˜ γˇn ∈ C, C closed ˜ˇ ˙ ˙ ˇ ⇒ q˜ λ ∈ C. Thus,q ˜ γˇn ∈ C ∩ E. Therefore, N[G] |= E ∩ C 6= ∅.
13 Chapter 2
Reflection and Square
2.1 Current Bounds on Reflection
We are interested in the bounds on the consistency strength of Refl(ω3, ω, ω1), so let us review the motivations for our choice. First, to understand why we do not discuss reflection
κ+ + on stationary sets of the form Sκ = {α < κ : cf(α) = κ)}, consider the following.
Lemma 2.1. (Jech [11]) For θ a regular uncountable cardinal, α < θ, cf(α) > ω, ∃ club C ⊆ α of ordertype α such that β ∈ C =⇒ cf(β) < cf(α).
κ+ So, consider C ⊆ κ club with elements of cofinality less than κ. Since Sκ contains β with
κ+ κ+ cf(β) = κ, then Sκ ∩ C = ∅, and so Sκ cannot reflect.
Theorem 2.1. (Harington, Shelah [9])
Con(ZFC + Mahlo) −→ Con(ZFC+Refl(ω2, ω, ω1))
M[G] Using Col(ω1, < κ) to get κ = ω2 , they use the fact that κ is Mahlo to ensure that the forcing adds no bounded subsets of κ. Then, for any non-reflecting stationary set S, using
14 the process of “shooting a club”, add a club CS for which CS ∩ S = ∅. This method will not work for our purposes, as the collapse would change cofinalities of inaccessibles to ω2.
The existing bounds on the cardinal strength of Refl(ω3, ω, ω1) are as follows.
Theorem 2.2. (Cummings, Shelah)
Con(ZFC + Supercompact) −→ Con(ZFC+Refl(ω3, ω, ω1))
Theorem 2.3. (Zeman)
Con(ZFC + Subcompact) −→ Con(ZFC+Refl(ω3, ω, ω1) + Simultaneous reflection on ω2)
That is, given ω1- many stationary subsets of ω2 concentrating on cof(ω), there is a γ < ω2 at which they all reflect. Note that subcompact is significantly smaller than any (nontriv- ial) instance of a supercompact and it is known that any variant of reflection at ω3 with simultaneous reflection at ω2 has high consistency strength. If you require reflection without simultaneous reflection below, this should be much weaker in consistency strength. We keep the following in mind, however.
Theorem 2.4. (Cox [5])
Refl(ω3, ω, ω1) requires at least measurable cardinals of high Mitchell order, slightly above order κ.
2.2 Square
Discovered by Jensen in 1972, the combinatorial principle square (κ) is typically used in building objects of size κ+. Square is considered to be a kind of incompactness principle, along with the existence of Aronszajn trees. These incompactness principles are at odds with compactness principles such as Shelah’s singular compactness theorem, the tree property, and stationary set reflection. The tension between these helps us understand them better. Let us give the basic definitions and make this notion more precise.
15 + Definition 2.1. κ: There exists a sequence hCα : α ∈ Lim(κ )i such that
1. Cα is a closed and unbounded subset of α
2. If β ∈ Lim(Cα) then Cβ = Cα ∩ β
3. If cf(α) < κ then |Cα| < κ
As an example of these principles witnessing various forms of incompactness and non- reflection we have the following.
+ Theorem 2.5. (Solovay) Let κ hold and let S be a stationary subset of κ . Then Refl(S) fails.
∗ Jensen’s weak square κ, however, is not strong enough to produce non-refelcting sets.
Definition 2.2. Let κ be an uncountable cardinal, and let ν be a cardinal, 1 ≤ ν ≤ κ.
+ κ,ν: There exists a sequence hCα|α ∈ Lim(κ )i such that
1. Cα is nonempty and |Cα| ≤ ν, and each C ∈ Cα is a closed unbounded subset of α,
2. if C ∈ Cα and β ∈ Lim(C) then C ∩ β ∈ Cβ,
3. if cf(α) < κ then |C| < κ for every C ∈ Cα.
∗ Definition 2.3. Weak Square, denoted κ, is κ,κ, the weakest of the above principles.
Definition 2.4. Global on Singular Cardinals:
There exists hCα|α a singular cardinali such that for each α,
1. Cα is a closed unbounded subset of Card ∩ α of order-type less than α.
2. The limit points of Cα are singular cardinals.
16 3. Cα¯ = Cα ∩ α¯ whenever α¯ is a limit point of Cα.
Definition 2.5. Global : There exists hCα|α a singular ordinali such that for each α,
1. Cα is a closed unbounded subset of α of order-type less than α.
2. The limit points of Cα are singular ordinals.
3. Cα¯ = Cα ∩ α¯ whenever α¯ is a limit point of Cα.
Global is equivalent to the conjunction of on the singular cardinals together with κ for all uncountable cardinals κ.
Let us consider the current lower bounds on the consistency strength of Global .
Theorem 2.6. (Jensen [12]) L |= Global .
Theorem 2.7. (Welch [19]) Global holds in the Dodd-Jensen core model.
Theorem 2.8. (Wylie [20]) Global holds in Jensen’s core model with measures of order zero.
Theorem 2.9. (Zeman) Global holds in the core model below zero-pistol.
Theorem 2.10. (Zeman) Global holds in all extender models which are currently known to exist.
Due to the relationship between Reflection and Square, as a consequence of our Reflection result we get the failure of global on singular cardinals below an inaccessible, κ, which was our measurable in the ground model.
17 Chapter 3
Forcing
Given a measurable cardinal κ of high Mitchell order, we will construct a generic extension where κ is regular and every stationary set S ⊂ κ∩cof(ω) has a reflection point of cofinality
ω1. To do this, we do an iteration up to κ. S is inside our iteration, so we can map it to j(S) and find the generic filter H, for the part above κ, which keeps S stationary.
3.1 The Initial Forcing
Given κ measurable, j : V −→ M the corresponding elementary embedding. Consider the following Easton Support Iteration:
P0 is the trivial forcing. ˙ Pα+1 = Pα ∗ Qα, where Qα = Add(ℵα+1, 1).
Add(ℵα+1, 1) consists of conditions p : a → 2, where a ⊂ ℵα+1, |a| < ℵα+1 ordered by end extension. We take direct limits at strong inaccessibles, inverse limits otherwise.
We continue to Pκ in this way.
18 For definitions 3.1 and 3.2, let Pα be an iteration of length α, where α is a limit ordinal.
Definition 3.1. Pα is a direct limit if for every α-sequence p,
p ∈ Pα iff ∃β < α p β ∈ Pβ and ∀ξ ≥ β p(ξ) = 1
That is, we have no new information and trivially extend all shorter conditions to length α.
Definition 3.2. Pα is an inverse limit if for every α-sequence p,
p ∈ Pα iff ∀β < α p β ∈ Pβ
So every condition is an extension of some shorter condition, but not every such extension is a condition.
D ˙ E Lemma 3.1. (The Factor Lemma). Let Pα+β be a forcing iteration of Qξ|ξ < α + β , (α) Pα ˙ where each Pξ, ξ ≤ α + β is either a direct limit of inverse limit. In V , let Pβ be the D ˙ E (˙α) forcing iteration of Qα+ξ|ξ < β such that for every limit ordinal ξ < β, Pξ is either a
direct or inverse limit, according to whether Pα+ξ is a direct or inverse limit. If Pα+ξ is an
inverse limit for every limit ordinal ξ ≤ β such that cf(ξ) ≤ |Pα|, then Pα+β is isomorphic ˙ (α) to Pα ∗ Pβ .
Lemma 3.2. Add(κ+, 1) is κ+-closed.
+ Proof. Given a chain of conditions in Add(κ , 1) of length κ, hpγ|γ < κi, let p := ∪pγ. We
+ claim that p is a condition in Add(κ , 1) below all pγ. Clearly p ≤ pγ, ∀γ < κ.
+ + + + + p ∈ Add(κ , 1) because each |pγ| < κ , κ < κ , and κ is regular =⇒ |p| = ∪|pγ| < κ .
Lemma 3.3. Add(κ+, 1) is (2κ)+-c.c.
19 Proof. |Add(κ+, 1)| = |[κ+]κ| = (κ+)κ (by κ+ ≥ κ) = 2κ (by 2κ ≥ κ+). So any antichain has size < (2κ)+.
Lemma 3.4. (Baumgartner [16])
Suppose hPα : α ≤ δi, hQα : α < δi an iterated forcing with Pα an inverse limit when α is a ˙ limit ordinal, cf(α) < κ. If α Qα is κ-closed then Pδ is κ-closed.
Lemma 3.5. For each n < ω, Pm is ℵn-closed ∀m < n.
˙ ˙ Proof. Fix n < ω. Pn is ℵn-closed and Pn+1 = Pn ∗ Qn, where Pn “Qn is ℵn-closed” and
so Pn+1 is ℵn-closed. ˙ ˙ Assume Pm is ℵn-closed. Then Pm+1 = Pm ∗ Qm and Pm+1 “Qm is ℵm-closed”. Then Pm+1 is ℵn-closed.
Theorem 3.1. GCH holds below κ in V Pκ .
+ Proof. Successor Steps: From Lemmas 3.2 and 3.3 we have that Add(ℵα , 1) preserves cardi-
V Pα + ℵα + + ℵα + + V Pα+1 nals ≤ ℵα and ≥ (2 ) . We know that Add(ℵα , 1) collapses (2 ) to ℵα = ℵα+2 . So,
Pα+1 + in V (after forcing with Add(ℵα , 1)) we have
V Pα ℵα + ℵα V Pα ℵα+2 = (2 ) > (2 ) ≥ ℵα+1.
+ ℵα V Pα ℵα V Pα+1 Also, Add(ℵα , 1) does not add subsets of ℵα, so (2 ) = (2 ) and so it is still a
Pα+1 ℵα cardinal and thus V |= 2 = ℵα+1.
∴ we have GCH at successor steps.
Limit Steps: ˙ (n) ˙ (n) ∀n < ω, Pω = Pn ∗ Qω . We have Pn is ℵn-closed and Pn “Qω is ℵn-closed”. Then from
Pn Lemma 3.4 in V and the inverse limit at Pω we have our theorem.
20 Consider a stationary set S in κ. In V , we have the canonical embedding j, S = j(S) ∩ κ and S ⊆ ω3∩cof(ω) due to the fact that κ is measurable. We need to have S stationary in
M[H] V [G] M[H] M[H], cf(κ) = ω1, and ω1 = ω1 . Then M[H] |= κ is a reflection point of j(S) of
cofinality ω1. This implies that M[H] |= (∃α < j(κ)), cf(α) = ω1, and α is a reflection point of S. So, by elementarity, the same is true in V [G].
We see that S is stationary in M[H].
If we want the generic to get cf(κ) = ω1, we can’t do it over V [G]. We must force further, but without affecting the stationarity of S.
3.2 Building the Generic Filter above Pκ
0 We first extend j to j so that we have Pκ yielding GCH below κ and j defined in V [G][H] ensuring that κ is measurable in V [G][H]. For any P-name σ in V , define j0(σG) = j(σ)H .
+ From the Factor Lemma, we have that j(Pκ) factors into Pκ ∗ Add(κ , 1) ∗ R. We begin by considering a generic G. We automatically have GCH below κ, since Pκ is the same in each V and M. And so for any G, we take the same G in M. Now we have V[G] and M[G]. We will see that since κ-sequences are the same in both V[G] and M[G], then Add(κ+, 1)V [G] = Add(κ+, 1)M[G] as well. Take generic H ⊂ Add(κ+, 1)V [G] = Add(κ+, 1)M[G].
We will build an F generic in R. To do this, we count the dense sets (in several steps), build a chain of conditions from those dense sets, and then extend that chain into a generic filter F.
We will show that |{D|D ⊂ R}M[G][H]|V [G][H] = κ+. Let D = {D : D ⊂ R dense, D ∈ M[G][H]}. Then we claim that
Lemma 3.6. |D|V [G][H] = κ+.
21 Figure 3.1
− −
− j(α)
Pj(α)/Pα+1
α− −α Pj(α)
Pα+1 V N j
Proof. To begin counting, we see that for each D in M[G][H] that is dense in R, there exists ˙ M ˙ G∗H˙ a name D ∈ Vj(κ)+1 such that D = D . M[G][H] |= j(κ) is inaccessible. ⇒ There exists a bijection b ∈ M[G][H] (and so b ∈ V [G][H] as well, so our construction remains in V [G][H])
M[G][H] M[G][H] ˜ ˜ −1 b : j(κ) −→ Vj(κ) . For any D ∈ D, D ⊂ Vj(κ) ⇒ ∃D ⊂ j(κ), D := b [D]. We want to consider all possible subsets of j(κ) in M[G][H], so we need to look at their ˇ names. Subsets α ⊂ j(κ) have names of the formα ˙ = ∪ξ + antichain in Pκ ∗Add(κ , 1). Since M |= j(κ)-c.c., eachα ˙ has size at most j(κ)×j(κ) ' j(κ). M M M V [G][H] ⇒ Eachα ˙ ⊂ Vj(κ) ⇒ α˙ ∈ Vj(κ)+1. So, we need only determine |Vj(κ)+1| . In M, Vj(κ)+1 is the V construction up to j(κ) + 1. Since κ is inaccessible, |Vκ| = κ in V [G][H]. We have GCH up to 2κ = κ+ in V [G][H]. M If f : κ −→ Vκ+1 then ∀α < κ f(α) ∈ CVκ+1 (α), CVκ+1 = Vj(κ)+1. M If [f] ∈U Vj(κ)+1 then for almost every α < κ f(α) ∈ Cκ+1(α). M κ κ κ κ In V , |Vj(κ)+1| = |{[f]U : f : κ −→ Vκ+1}| = |Vκ+1| = (2 ) = 2 . M + Then, since we have GCH in V [G][H], |Vj(κ)+1| = κ . 22 Now we show that M[G][H] is closed under κ-sequences in V [G][H]. That is, given a sequence in V [G][H] with elements from M[G][H], the sequence itself is in M[G][H]. Lemma 3.7. a ∈ [M[G]]κ ∩ V [G][H] ⇒ a ∈ M[G][H]. G∗H˙ Proof. Let a := haξ|ξ < κi ∈ V [G][H], aξ ∈ M[G][H]. ⇒ ∃a˙ ∈ V ,a ˙ = a In the presence of the Axiom of Choice, every set can be well-ordered [10], so we think of each aξ as an ordinal. a˙ ∈ V Pκ ⇒ a˙ ⊂ κ j(˙a) =a ˙, so a =a ˙ G ∈ M[G] a˙ ⊂ κ ⊂ Vκ. j|Vκ = id|Vκ Lemma 3.8. R is κ+-closed. Proof. R is the continuation of the Easton support Iteration in M[G][H] starting at Add(κ++, 1). We have already shown in Lemma 3.2 that Add(κ++, 1) is κ++ -closed and we can use the Factor Lemma to show that R is κ++-closed. Now, we will build the generic filter F. We will first construct a chain of conditions from the dense sets of R. R is κ+-closed, V [G][H] |= |{D : D ⊂ M[G][H] dense}M[G][H]| = κ+. + Construct a sequence hpα|α < κ i as follows: + + |D| = κ , so enumerate them hDα|α < κ i. Take pα ∈ Dα. ∀pα, Dα+1 dense ⇒ (∃pα+1)pα+1 ∈ Dα+1 and pα+1 ≤ pα. + For limit α, α < κ ⇒ ∃qα ≤ pβ∀β < α (by closure). Then, since Dα is dense, ∃pα ∈ Dα, + pα ≤ qα. Then we have hpα : α < κ i a set that intersects all dense sets. + Note: We have constructed the sequence hpα : α < κ i ∈ V [G][H] but, since M[G][H] is 23 + closed under κ-sequences in V [G][H], we in fact have that hpα : α < κ i ∈ M[G][H]. This is necessary in order to continue building our generic filter for R. + Lemma 3.9. There exists a generic filter F ⊃ hpα : α < κ i. + Proof. Let F = {p ∈ R|(∃α < κ )pα ≤ p} F is a filter on R + • Since pα ∈ F ∀α < κ , F 6= ∅ •∀ p ≤ q, p ∈ F , there exists pα ≤ p ≤ q ⇒ q ∈ F as well. •∀ p, q ∈ F ∃pα ≤ p, pβ ≤ q. Without loss of generality, assume pα ≤ pβ ⇒ pα ≤ p, q. + + F contains hpα : α < κ i: This is clear, since pα ≤ pα∀α < κ . F is generic: ∀Dα dense in R, pα ∈ Dα ∩ F . R will not disturb P(κ) too much. We want j(P)κ to preserve the stationarity of S. If S is stationary in the ground model, V , then we need the iteration and then the κth step glues them into one club. 24 Chapter 4 Club Shooting ω3 To get reflection for every stationary set S ⊆ Sω , we use the method of “shooting a club”. For any stationary set S that does not reflect, we use the forcing to add a club C for which C ∩S = ∅, thus rendering this S no longer stationary. Typically, if we shoot a club through a stationary set at ω1 this is a distributive forcing. We are concerned that we do not add more stationary sets in this process. In fact, we will add many stationary sets in this process, but their stationarity is destroyed by adding clubs in the club shooting iteration. Additionally, ω3 S ⊆ Sω sets can only have a club shot through them under certain circumstances, namely that they need to be “fat”. In our situation, we use the fact that the complements of non-reflecting stationary sets will be fat to allow us to shoot a club without collapsing κ. 4.1 Background The notion of being a stationary set is not absolute, meaning that a stationary set S of a model M can become nonstationary in a generic extension M[G] after forcing. This idea, of shooting a club, was introduced for S ⊆ ℵ1 by Baumgartner, Harrington, and Kleinberg 25 [2]. Abraham and Shelah [1] then generalized this concept, and showed that in order to introduce a club into S without adding new sets of size < α, S must be fat. Definition 4.1. We say that A is a fat subset of α if for any club C ⊆ α, A ∩ C contains closed sets of ordinals of arbitrarily large order-type below α. 4.2 Shooting a Club In [6] Gitik takes time to arrange clubs which he already had. A club set is added in such a way that all of its initial segments are added first. We utilize this forcing under a different assumption, namely that U(κ, 0) concentrates on α+-supercompact cardinals. We begin by describing the forcing notion Pκ as introduced by Gitik [7]. The traditional notation would be Pκ, but we avoid confusion with our earlier forcing. Since we have GCH and κ is a measurable cardinal with Mitchell order o(κ) = κ. Then we take U~ to be the coherent sequence of length lU~ = κ + 1 for which U(κ, 0) concentrates on α+-supercompact cardinals α. That is, U~ is a function with domain {(α, β)|α < lU~ and β < OU~ (α)}. lU~ is the length of U~ , and the function OU~ (α) is the order of U~ at α. For each (α, β) ∈ domU~ , 1. U~ (α, β) is a normal ultrafilter on α, and α α α ~ ~ 2. if jβ : V → Nβ is the canonical embedding then jβ (U) α + 1 = U(α, β). ~ ~ We use dom1U to mean the set of α such that (α, β) ∈ domU for some β. Let ~ + A = {α < κ | α∈ / dom1U, α is an α supercompact cardinal } ∈ U(κ, 0) 26 ~ + Without loss of generality, ∀α ∈ dom1U, A ∩ α ∈ U(κ, 0). For α ∈ A, α is α -supercompact + which guarantees some fine normal measure on Pα(α ). Fix U(α) to be this measure over + ++ Pα(α ) for each α ∈ A so that |[id]U(α)| = α . Our iterated forcing will change the cofinality of every α, α+ to ω, in order to build a generic club through A ∩ α inside of V [G] for a lot of α’s below κ. ~ We define, by induction, the iteration Pα for α ∈ the closure of {β, β + 1 | β ∈ A∪dom1U}. Suppose Pα is defined, α ∈ A. Fix from the beginning some well-ordering, W , of Vλ such that for any inaccessible β < λ, W |Vβ : Vβ ⇐⇒ β, where λ is a cardinal much above ++ all the cardinals we are working with. Let hAβ|β < α i be the W -least enumeration of ++ V + Pα(α ) all canonical Pα-names of subsets of Pα (α ). Let j : V −→ N ' V /U(α) be the canonical embedding. + Let G ⊂ Pα be generic. Since we are working in the case that α is α -supercompact, de- ++ fine an α -sequence of Pα+1-names of elements of Pj(α)/Pα deciding all the statements 00 + ++ 00 j α ∈ j(Aβ)(β < α ) as follows. By definition of Pα, j G = G. Define an Easton- ++ ++ increasing sequence hpγ|γ < α i of elements of Pj(α)/Pα such that for all γ < α , pγ+1 decides α ∈ j(Aγ) and every initial segment of the sequence will lie in N[G]. We utilize two lemmas due to Gitik [7] to guarantee each of the following At successor steps we have the Easton version of the Prikry Property. That is, for any pγ there exists an Easton extension pγ+1 ≤E pγ deciding α ∈ j(Aγ). In other words, either ˙ ˙ pγ+1 α ∈ j(Aγ) or pγ+1 α∈ / j(Aγ). At limit steps γ, we have built hpδ|δ < γi, where pδ2 ≤E pδ1 ∀δ2 > δ1. Since we also have pδ1 (γ + 1) = pδ2 (γ + 1), then there exists p ∈ Pα such that p (γ + 1) = pδ (γ + 1) and p ≤E pδ for every δ < γ. 27 Define the ultrafiler U 0 in V [G] as follows: For A ⊂ κ set A ∈ U 0 if for some γ < α+, and ˙ ˙ 00 some Pα-name A of A in N[G], pγ αˇ ∈ j(A). Since j G = G, this definition does not depend on the particular name for A. It follows that U 0(α) is a fine ultrafilter. Now, even though we began with supercompact cardinals in place of strongly compact ones, 0 our ultrafilter extension U (α) lacks normality. So we let Qα be the strongly compact Prikry forcing, in place of a supercompact one. Conditions in Qα are of the form hQ1, ..., Qn,Bi, where + 1. hQ1, ..., Qni is an increasing sequence of elements of Pα(α ), + 2. B is a tree of increasing sequences of elements of Pα(α ) so that Qn is contained in every element of such a sequence, 0 3. For every η ∈ B, SucB(η) ∈ U (α) Then, Qα is α-weakly closed, has the Prikry property, and changes the cofinalities of α and α+ to ω. ˙ Define Pα+1 = Pα ∗ Qα. ~ ~ For α ∈ dom1U, if α is a limit point of dom1U, define Pα to be a variation on the Magidor Prikry forcing [14], where instead of full support in the second coordinate, only the Easton support is allowed. ~ ~ U~ If α ∈ dom1U is not a limit point of dom1U, then O (α) = 1 and the cardinality of the forcing below α is less than α. So, by Levy-Solovay, U(α, 0) generates the normal ultrafilter in the extension. Fix a generic subset Gκ of Pκ and set Gα = Gκ ∩ Pκ. Then every Gα will be a V -generic subset of Pα. 28 T Let E = β<κ U(κ, β) For γ < κ, set U~ E(γ) = {β ∈ E|O (β) ≥ γ and E ∩ β ∈ ∩σ<γU(β, σ)} For every β ∈ E(γ), bβ, the generic sequence to β, is almost contained in E ∩ β. The ordertype of bβ ≥ γ, and the set E(γ) is stationary in U(κ, γ). Since Pκ satisfies the κ − c.c., Baumgartner [16] give us that every club of κ in V [Gκ] contains a club of κ in V . The same holds if we replace κ with α ∈ A ∩ dom1(U). From Abraham-Shelah [1] we can shoot a club through a fat subset, but constructing such a set requires several steps. Set E0 = {α < κ|α measurable in V and ∀γ < α E(γ) ∩ α a stationary subset of α}. Then 0 T E ∈ β<κ U(κ, β). For any α ∈ E0 ∪ {κ}, we see that E ∩ α is a fat subset of α in V [G] i.e. for any club C ⊆ α, E ∩ C contains closed sets of ordinals of arbitrarily large order-type below α. By induction we define 0 0 E0 = {α ∈ E ∩ E |α ∈ A} = E ∩ E ∩ A. 0 U~ Eβ = {α ∈ E ∩ E |O (α) = β and ∀δ < β, Eδ ∩ α ∈ U(α, δ)} for 1 ≤ β ≤ κ (1) (1) Then each Eβ ∈ U(κ, β). Set E = ∪β<κEβ. E ∈ ∩β<κU(κ, β). Let α ∈ E(1) ∪ {κ}. Define P [E ∩ α] to be the forcing notion in V [Gα] consisting of all closed subsets of E ∩ α ordered by end extension. Definition 4.2. A forcing notion (P, ≤) is (α, ∞)-distributive if the intersection of less than α many dense open subsets of P is dense open. (1) Theorem 4.1. For any α ∈ E , there exists a V [Gα]-generic club subset of P [E ∩ α] in 29 V [Gα+1]. (1) (1) Proof. Recall that E = ∪β<κEβ. For any α ∈ E , we use induction on β < κ for which α ∈ Eβ. If α ∈ E0, then we used the strongly compact Prikry forcing on α. This resulted in + 1. cf(α) = cf(α ) = ω in V [Gα+1]. 2. No new bounded subsets of α in V [Gα+1]. Then, from E ∩ α a fat subset of α the forcing P [E ∩ α] is an (α, ∞)-distributive forcing notion in V [Gα] [1]. The set of all dense subsets of P [E ∩ α] which are in V [Gα] can be written in V [Gα+1] as a union of ω many sets, each of them is in V [Gα] and has cardinality less than α. So, by P [E ∩α](α, ∞)-distributive, we can define a set meeting all dense subsets of P [E ∩ α] of V [Gα]. Assume the theorem is proved for all α ∈ Eγ, where γ < β < κ. If α ∈ Eβ, consider bα, the generic sequence to α. Define b := bα −ξ, where ξ < α is the least such that b ∈ Eδ ∩α for every δ < β. Let hγξ|ξ < τi be the increasing continuous enumeration of b. Define by induction an increasing continuous sequence of closed sets hCξ|ξ < τi so that 1. Cξ ∈ V [Gγξ+1] is a V [G − γξ]-generic club through E ∩ γξ. 2. Cξ+1 is an end extension of Cξ. S 0 3. For limit ξ, Cξ = {Cξ0 |ξ < ξ}. Successor: Define Cξ+1 using the inductive assumption. 0 S 0 Limit: Consider Cξ = {Cξ |ξ < ξ} for limit ξ. We claim that Cξ is V [Gγξ ]-generic. Let D 30 0 be a dense open subset of P [E ∩ γξ] in V [Gγξ ]. Then ∃ξ < ξ, D ∩ P [E ∩ γξ] will belong to 0 V [Gγξ0 ] and it will be a dense subset of P [E ∩ γξ ] in V [Gγξ0 ]. Since Pγξ satisfies the γξ chain 00 00 condition, γξ |ξ < ξ is almost contained in every closed and unbounded set of γξ. Also note that for δ < γξ, the forcing Pγξ /Pδ does not add any new bounded subsets to δ. The Cγξ0 extends to some elements of D ∩ P [E ∩ γξ]. 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