Singular cardinal problems

James Cummings CMU

(Cantor) 2^{κ} > κ.

(König) κ^{cf (κ)} > κ, so in particular cf (2^{λ}) > λ for all λ.

(Easton) If GCH holds and F : REG ⟶ CARD is a class function such that κ < λ ⟹ F (κ) ≤ F (λ), cf (F (κ)) > κ then there is a cardinal and cofinality preserving class generic extension in which 2^{κ} = F (κ) for every regular κ.

(Scott) If U is a normal measure on κ, μ < κ and {α < κ : 2^{α} ≤ α^{+μ} } ∈ U then 2^{κ} ≤ κ^{+μ} . In particular GCH does not first fail at κ.

(Silver) If κ is singular strong limit of uncountable cofinality, μ < κ and {α < κ : 2^{α} ≤ α^{+μ} } is stationary then 2^{κ} ≤ κ^{+μ} . In particular GCH does not first fail at κ.

(Galvin and Hajnal) If κ is singular strong limit of uncountable cofinality, and κ = ℵ_{η} for some η < κ (that is to say κ is not a cardinal fixed point) then 2^{κ} < ℵ_{(2^{|η|})^{+}} .

The Singular Cardinals Hypothesis (SCH): For all singular λ

λ^{cf (λ)} = max {λ^{+} , 2^{cf (λ)}}.

Fact: if SCH holds then cardinal arithmetic is determined by the continuum function on the regulars and the cofinality function, and roughly speaking λ^{μ} has the least value possible. To be more precise if SCH holds then

1. Let κ be singular. If the continuum function is eventually constant below κ then 2^{κ} = 2^{<κ}, otherwise 2^{κ} = (2^{<κ})^{+} .

2. If λ ≤ 2^{μ}, λ^{μ} = 2^{μ}. If 2^{μ} < λ then λ^{μ} = λ if μ < cf (λ), and λ^{μ} = λ^{+} if μ ≥ cf (λ).

(Solovay) If κ is strongly compact and λ > κ is singular, strong limit then 2^{λ} = λ^{+} .

The following are equiconsistent:

1. SCH fails.

2. GCH first fails at ℵ_{ω}.

3. GCH fails at a measurable

4. There exists κ with o(κ) = κ^{++} .

Outline of the lectures.

1) How close can we come to proving SCH? Pcf theory (Shelah)

2) Upper bounds for the failure of SCH. Forcing (Gitik, Magidor, Woodin).

3) Lower bounds for the failure of SCH. Core models (Dodd, Gitik, Jensen, Mitchell)

4) Other combinatorics at singulars. Scales, squares, reflection (Cummings, Foreman, Magidor, Shelah).

Let ℙ be a poset.

1. A ⊆ ℙ is cofinal iff ∀p ∈ ℙ ∃q ∈ A p ≤ q.

2. cf (ℙ) = min {|A| : A is cofinal in ℙ }.

3. ⟨p_{i} : i < λ⟩ is a scale in ℙ iff ∀i < j p_{i} < p_{j}, {p_{i} : i < λ} is cofinal and λ = cf (λ).

4. tcf (ℙ) = λ iff there is a scale of length λ in ℙ.

5. ℙ is λ-directed iff for all A ⊆ ℙ with |A| < λ there is q ∈ A such that ∀p ∈ A p ≤ q.

6. Let p⃗ = ⟨p_{i} : i < λ⟩ be increasing. q is a bound for p⃗ iff ∀i p_{i} ≤ q. q is an least upper bound (lub) iff q ≤ r for all bounds r.

Let I be an ideal on a set X, let F be a filter on X.

1. If f, g ∈ ^{X}ON then

f <_{I} g ⟺ {x : f(x) ≥ g(x)} ∈ I
f ≤_{I} g ⟺ {x : f(x) > g(x)} ∈ I
f <_{F} g ⟺ {x : f(x) < g(x)} ∈ F
f ≤_{F} g ⟺ {x : f(x) ≤ g(x)} ∈ F

We study (^{X}ON, ≤_{I}).

Remark: “f <_{I} g” is (in general) stronger than “f ≤_{I} g and f ≠_{I} g”.

2. Let f⃗ = ⟨f_{i} : i < λ⟩ be a sequence such that ∀i < j f_{i} <_{I} f_{j}. Then f is an exact upper bound (eub) for f⃗ iff f is a bound for f⃗ and

∀g <_{I} f ∃i < λ g <_{I} f_{i}.

Fact: an eub is an lub.

Proof: let f be an eub for f⃗ which is not an lub. Let g be a bound such that A = {x : g(x) < f(x)} ∈ I^{+} . Let h be such that h ↾ A = g ↾ A and h ↾ A^{c} = 0. Then h <_{I} f, so h <_{I} f_{i} for some i and g is not a bound. Contradiction.

The converse is false in general.

Example: let I be the NS ideal on ℵ_{1}, and let f_{α}(i) = α for α < ℵ_{1}. Then f = id is an lub but not an eub for ⟨f_{α} : α < ℵ_{1}⟩.

Proof: f is an lub by Fodor. If we write ℵ_{1} = ⋃_{β<ℵ_{1}} S_{β} with S_{β} stationary, and define g(i) to be the least β such that i ∈ S_{β} if β < i and 0 otherwise, then g <_{I} f but g is above each f_{α} for α < ℵ_{1} on a positive set.

Convention: A is usually a set of regular cardinals with |A|^{+} < min (A) and no largest element. D is usually an ultrafilter on A.

Remark: for any singular μ we may choose such an A with sup (A) = μ. If μ < ℵ_{μ} we may in addition choose A to be an interval of regular cardinals.

pcf (A) = {λ : ∃D cf (∏ A/D) = λ}.

J_{<λ}(A) = {B : ∀D ∋ B cf (∏ A/D) < λ}.

Remark: clearly |pcf (A)| ≤ 2^{2^{|A|}}, sup pcf (A) ≤ | ∏ A|.

Some basic facts.

1. J_{<λ} is an ideal.

2. ∏ A/J_{<λ} is λ-directed.

3. cf (∏ A/D) < λ iff D ∩ J_{<λ} ≠ ∅.

4. λ ∈ pcf (A) iff J_{<λ^{+}} \ J_{<λ} ≠ ∅.

5. For λ a limit cardinal, J_{<λ} = ⋃_{μ<λ} J_{<μ}.

6. pcf (A) has a maximal element, namely the least λ where J_{<λ^{+}} = P (A).

7. |pcf (A)| ≤ 2^{|A|}.

Fact: ∏ A/J_{<λ} is λ-directed.

Proof: let J = J_{<λ}, κ = |A|. Reduce to the case where |A|^{+} < μ = cf (μ) < λ and we need an upper bound for ⟨f_{α} : α < μ⟩ which is <_{J}- increasing. Suppose for a contradiction that no bound exists and build a pointwise strictly increasing sequence ⟨g_{β} : β < κ^{+} ⟩ of elements of ∏ A as follows.

g_{β} is not an upper bound so {i : f_{α}(i) > g_{β}(i)} ∈ J^{+} for all large α, and we can find D such that cf (∏ A/D) ≥ λ and g_{β} <_{D} f_{α} for all large α. Choose g_{β+1} so that f_{α} <_{D} g_{β+1} for all α; for all large α there is i such that

g_{β}(i) < f_{α}(i) < g_{β+1} (i).

Find α so large that this holds for every β < κ^{+} , and then find i which works for two values of β. Contradiction since the g_{β} are pointwise increasing.

More basic facts.

1. If B ∈ J_{<λ^{+}} \ J_{<λ}, then tcf (∏ B/J_{<λ}) = λ.

2. J_{<λ^{+}} is generated from J_{<λ} by a single set: that is to say there is a set B such that J_{<λ^{+}} = J_{<λ} + B = {C : C \ B ∈ J_{<λ}}.

3. Let ⟨B_{λ} : λ ∈ pcf (A)⟩ be such that B_{λ} generates J_{<λ^{+}} over J_{<λ}, and for each λ fix ⟨f_{α}^{λ} : α < λ⟩ a scale in ∏ B_{λ}/J_{<λ}. Then for any D we have

cf (∏ A/D) = min {μ : B_{μ} ∈ D},

and if cf (∏ A/D) = λ then ⟨f_{α}^{λ} : α < λ⟩ is a scale in ∏ B_{λ}/D.

Shelah’s trichotomy theorem.

Theorem: let I be an ideal on κ, let ⟨f_{α} : α < λ⟩ be a <_{I}-increasing sequence from ^{κ}ON with λ = cf (λ) > κ^{+} . Then one of the following holds:

(The Good case) There exists an eub f for f⃗.

(The Bad case) There exists an ultrafilter D on κ with D ∩ I = ∅, and a sequence of sets of ordinals ⟨S_{i} : i ≤ κ⟩ with |S_{i}| ≤ κ, such that

∀α < λ ∃h ∈ _{i}^{∏} S_{i} ∃β < λ f_{α} ≤_{D} h <_{D} f_{β}.

(The Ugly Case) For some g ∈ ^{κ}ON the sequence of sets {i : f_{α}(i) < g(i)} is not eventually constant modulo I.

Remark : if I is prime then the Ugly case cannot occur, and the only D which can occur in the Bad case is the dual of I.

Remark: If cf (λ) > 2^{κ}, easy counting arguments show we are in the good case.

Proof : We assume that we are not in the Good, Bad or Ugly case and work towards contradiction.

Step One: show that since we are not in Ugly case, an lub for f⃗ is automatically an eub.

Proof: let f be an lub which fails to be an eub. Let g <_{I} f be such that

A_{α} = {i : g(i) > f_{α}(i)} ∈ I^{+}

for all α. As we are not in Ugly case the sequence of A_{α} eventually stabilises; fix A which is equal to A_{α} mod I for all large α. Define h so that h ↾ A = g ↾ A and h ↾ A^{c} = f ↾ A^{c}, then h is a bound and below f on a positive set. Contradiction!

Step Two: show there is ⟨g_{β} : β < κ^{+} ⟩ a strictly ≤_{I}-decreasing sequence such that each g_{β} is a bound for f⃗.

Proof: Build g⃗ inductively. By Step One g_{β} is not an lub, so we may choose g_{β+1} suitably. At limit β define

h_{α}^{β}(i) = min ({g_{δ}(i) : δ < β} \ f_{α}(i)).

We claim that h_{α}^{β} stabilises mod I for large α, so that we may continue by choosing g_{β} as a representative of the stable class.

Suppose h_{α}^{β} does not stabilise; this means that for a fixed α, A_{αγ} = {i : h_{α}^{β}(i) < f_{γ}(i)} ∈ I^{+} for all large γ. As we are not in Ugly case we may choose A_{α} such that A_{α} =_{I} A_{αγ} for all large γ. The A_{α} are positive and decreasing mod I, so choose D such that D ∩ I = ∅ and A_{α} ∈ D for all α. By construction

∀α ∃γ f_{α} ≤_{D} h_{α}^{β} <_{D} f_{γ}

so setting S_{i} = {g_{δ}(i) : δ < β} we are in Bad case. Contradiction.

Step Three: no sequence as constructed in Step Two can exist.

Proof: let

h_{α}(i) = min ({g_{δ}(i) : δ < κ^{+} } \ f_{α}(i)).

For all α there exists limit β < κ^{+} such that h_{α} = h_{α}^{β}. Fix β such that h_{α} = h_{α}^{β} for unboundedly many α, then choose α such that h_{α} = h_{α}^{β} =_{I} g_{β}. This is a contradiction since by construction h_{α} ≤_{I} g_{β+1} .

This concludes the proof.

Remark: if Bad and Ugly fail then an eub f exists. f will have the property that cf (f(i)) > κ for almost all i, otherwise we would be Bad.

Definition: Let ⟨f_{α} : α < λ⟩ be <_{I}-increasing. β is a good point for f⃗ iff cf (β) > κ and there is a pointwise strictly increasing sequence ⟨h_{γ} : γ < cf (β)⟩ such that h⃗ is “cofinally interleaved” with f⃗ ↾ β mod I; that is ∀α < β ∃γ < cf (β) f_{α} <_{I} h_{γ} and ∀γ < cf (β) ∃α < β h_{γ} <_{I} f_{α}.

Fact: if f⃗ = ⟨f_{α} : α < λ⟩ is <_{I}-increasing, λ = cf (λ) > κ^{+} , and the set of good points is stationary in λ then there exists an eub for f⃗.

Proof : it is enough to show that we are not in Bad or Ugly case. Suppose D and S⃗ witness that we are in Bad case. Let

C = {γ : ∀α < γ ∃h ∈ _{i}^{∏} S_{i} ∃β < γ f_{α} ≤_{D} h <_{D} f_{β}}.

Let β ∈ lim(C) be a good point, as witnessed by ⟨h_{γ} : γ < cf (β)⟩ . Find ⟨γ_{δ} : δ < cf (β)⟩ increasing and H_{δ} ∈ ∏ S_{i} such that h_{γ_{δ}} ≤_{D} H_{δ} <_{D} h_{γ_{δ+1}} .

Find i < κ such that h_{γ_{δ}}(i) ≤ H_{δ}(i) < h_{γ_{δ+1}} (i) for an unbounded set X of δ < cf (β). Since ⟨h_{γ} : γ < cf (β)⟩ is pointwise strictly increasing the values H_{δ}(i) for δ ∈ X are distinct, a contradiction since cf (β) > κ but |S_{i}| ≤ κ.

The proof for Ugly is similar.

Theorem: Let δ, λ be regular with δ^{+} < λ. Then there exists S ⊆ λ∩cof (δ) stationary and ⟨S_{γ} : γ < λ⟩ such that

1. ot(S_{γ}) ≤ δ, S_{γ} ⊆ γ.

2. ∀β ∈ S_{γ} S_{β} = S_{γ} ∩ β.

3. If γ ∈ S, sup (S_{γ}) = γ. Definition: if S and S⃗ are as above then an <_{I}-increasing sequence ⟨f_{α} : α < λ⟩ is docile iff 1. ∀α ∈ S_{γ} ∀i < κ f_{α}(i) < f_{γ}(i).

2. If γ ∈ S then f_{γ}(i) = sup_{α∈S_{γ}} f_{α}(i). Key point: if δ > κ then an docile f⃗ has a stationary set of good points, hence an eub.

In a typical application of docility we would have κ = |A|, δ = |A|^{+} , λ ∈ pcf (A). Here is such an application.

Theorem: if B ∈ J_{<λ^{+}} \J_{<λ}, tcf (∏ B/J_{<λ}) = λ.

Proof: I = J_{<λ} ∪ {C : tcf (∏ C/J_{<λ}) = λ} is an ideal. If B ∉ I choose D such that B ∈ D and D ∩ I = ∅, and observe cf (∏ B/D) = λ. Build ⟨f_{α} : α < λ⟩ a sequence in ∏ B which is docile, increasing mod J_{<λ} and cofinal mod D. Let g be an eub where WLOG g(b) ≤ b for all b ∈ B, and let C = {b : g(b) = b}.

C ∈ I, and so B \ C ∈ D, so modulo D we have g ∈ ∏ B. Find α with g <_{D} f_{α}. f_{α} <_{I} g, contradiction!

A combinatorial lemma.

Lemma: let ⟨f_{β} : β < κ^{+} ⟩ be pointwise increasing and define C_{βγ} = {i : f_{β}(i) < f_{γ}(i)}. Then there is a club E ⊆ κ^{+} such that C_{β,γ} is constant on [E]^{2}.

Proof: C_{βγ} is constant for all large γ, say C_{βγ} = D_{β}. D_{β} is constant for all large β, say D_{β} = D. Now let

E = {ζ : D_{ζ} = D, ∀η < ζ C_{ηζ} = D_{η}}.

Fact: J_{<λ^{+}} is generated from J_{<λ} by a single set.

Proof: Suppose not. Let κ = |A|, reduce to case where λ = cf (λ) > κ^{++} . Fix stationary S ⊆ λ ∩ cof (κ^{+} ) and ⟨S_{α} : α < λ⟩ as above. Let J = J_{<λ}, K = J_{<λ^{+}} .

We will build a matrix of functions

⟨f_{α}^{β} : β < κ^{+} , α < λ⟩

from ∏ A such that ⟨f_{α}^{β} : α < λ⟩ is docile and <_{J}-increasing, and ⟨f_{α}^{β} : β < κ^{+} ⟩ is pointwise increasing. Let h^{β} be an eub for ⟨f_{α}^{β} : α < λ⟩, where we may assume h^{β}(a) ≤ a for all a; let S_{β} = {a : h^{β}(a) = a}, then it is easy to see that S_{β} ∈ K. S_{β} does not generate K over J.

We find C ∈ K such that C \ S_{β} ∉ J, and then choose D_{β} such that C ∈ D_{β}, S_{β} ∉ D_{β}, cf (∏ A/D_{β}) = λ. Now we choose ⟨f_{α}^{β+1} : α < λ⟩ cofinal mod D_{β} where ∀ζ ∀η f_{ζ}^{β} <_{D_{β}} f_{η}^{β+1} .

If β < γ, E(β, γ) = {η : ∀ζ < η f_{ζ}^{γ+1} <_{D_{β}} f_{η}^{β+1} }. If η ∈ S ∩ E(β, γ) then f_{η}^{β+1} =_{D_{β}} f_{η}^{γ+1} . To see this observe that by obedience there is ζ < η such that

{i : f_{η}^{β+1} (i) < f_{η}^{γ+1} (i)} = {i : f_{η}^{β+1} (i) < f_{ζ}^{γ+1} (i)},

which is a set of D_{β}-measure zero.

Now choose η ∈ S ∩ (⋂_{β<γ<κ^{+}} E(β, γ)), and define C_{βγ} = {i : f_{η}^{β+1} (i) < f_{η}^{γ+1} (i)}. Find β_{1} < β_{2} < β_{3} such that C_{β_{1}β_{2}} = C_{β_{2}β_{3}} = C_{β_{1}β_{3}}. This is impossible because by construction

f_{η}^{β_{1}+1} <_{D_{β2}} f_{η}^{β_{2}+1} =_{D_{β2}} f_{η}^{β_{3}+1} .

A technical lemma. Lemma: Suppose sup A < μ = cf (μ) < ν. Let J be an ideal on A such that ∏ A/J is ν-directed, and A ∩ γ ∈ J for all γ < sup A. Then there is a <_{J}-increasing sequence f⃗ = ⟨f_{α} : α < μ⟩ such that

1. f⃗ has an eub g ∈ ∏ A.

2. For all γ < sup A, {a : cf (g(a)) < γ} ∈ J. Proof: Fix ⟨C_{α} : α < μ⟩ such that

1. C_{α} is a family of clubs in α, |C_{α}| ≤ μ.

2. There is C ∈ C_{α} with ot(C) = cf (α).

3. ∀D ∈ C_{α} ∀β ∈ lim(D) D ∩ β ∈ C_{β}.

At every limit stage α, compute for each E ∈ C_{α} with ot(E) < sup A the function g_{E}^{α} : a ⟼ sup_{γ∈E} f_{γ}(a). Choose f_{α} to dominate all such g_{E}^{α}.

Claim 1: Let σ < sup A. There cannot exist an ultrafilter D with D ∩ J = ∅, and a sequence of sets of ordinals ⟨S_{i} : i ≤ κ⟩ with |S_{i}| ≤ σ, such that

∀α < μ ∃h ∈ _{i}^{∏} S_{i} ∃β < μ f_{α} ≤_{D} h <_{D} f_{β}.

Proof: let E be the club of γ < μ such that ∀α < μ ∃h ∈ ∏_{i} S_{i} ∃β < μ f_{α} ≤_{D} h <_{D} f_{β}. Let β ∈ lim(E) ∩ cof (σ^{+} ) and let F ∈ C_{β}. Choose ⟨γ_{j} : j < σ^{+} ⟩ increasing and h_{j} ∈ ∏_{i} S_{i} such that γ_{j} ∈ E ∩ lim(F) and f_{γ_{j}} ≤_{D} h_{j} <_{D} f_{γ_{j+1}} . By construction f_{γ_{j}} >_{D} g_{F ∩γ_{j}}^{γ_{j}}, so choose a_{j} such that

g_{F ∩γ_{j}}^{γ_{j}}(a_{j}) < f_{γ_{j}}(a_{j}) ≤_{D} h_{j}(a_{j}) <_{D} f_{γ_{j+1}} (a_{j}). Find X unbounded such that ∀j ∈ X a_{j} = a, then by construction {h_{j}(a) : j ∈ X} has size σ^{+} , contradiction.

Claim 2: We are not in the Ugly case.

Proof: Similar to Claim 1.

Conclusion: Claim 1 implies we are not in the Bad case, so by trichotomy there is a suitable g. Claim 1 implies that the cofinality of g(a) tends to sup A modulo J.

Let λ be a singular cardinal, κ = cf (λ).

Definition (non standard): PP (λ) is the set of regular μ such that for some uf D on κ and some sequence ⟨λ_{i} : i < κ⟩ of regular cardinals we have lim_{D} λ_{i} = λ, μ = cf (∏_{i} λ_{i}/D).

Definition: pp(λ) is the sup of PP (λ).

Theorem: PP (λ) is an interval of regular cardinals.

Proof: let λ < μ < ν ∈ PP (λ), say ν = cf (∏_{i} λ_{i}/D). Using the technical lemma, find ⟨f_{α} : α < μ⟩ in ∏_{i} λ_{i} increasing mod D with an eub g such that lim_{D} cf (g(i)) = λ. Now cf (∏_{i} cf (g(i))/D) = μ.

Theorem: if A is an interval of regular cardinals then so is pcf (A).

Proof: similar.

Convention: θ is some very large regular cardinal, and <_{θ} is a fixed well ordering of H_{θ}. We will form substructures of (H_{θ}, ∈, <_{θ}). If X ⊆ H_{θ} and Hull(X) is the set of points definable from parameters in X, then Hull(X) is the least substructure containing the set X.

IA chains: A chain of substructures X⃗ is internally approachable (IA) iff X⃗ is increasing and continuous, and X⃗ ↾ (β + 1) ∈ X_{β+1} for all β. It is easy to see that α ⊆ X_{α} and that α < β ⟹ X_{α} ∈ X_{β}.

Characteristic function of a structure: if B is a set of regular cardinals and |X| < min (B) then we may define χ_{X} ∈ ∏ B by χ_{X}(b) = sup (X ∩ b) for all b ∈ B.

Fact: Let X⃗ be an IA chain, let X = X_{α} where cf (α) > ω. Suppose B ⊆ REG with |X| < min (B). Suppose B ⊆ X_{0}. Then for all b ∈ B, X ∩ b contains a club in χ_{X}(b).

Proof: X_{i}, b ∈ X_{i+1} and |X_{i}| < b, so χ_{X_{i}}(b) ∈ X_{i+1} ∩ b. So ⟨χ_{X_{i}}(b) : i < α⟩ is continuous, increasing and cofinal in χ_{X}(b).

Fact: Suppose also that B is an interval of regular cardinals, min (B) = |X|^{+} and |X| ⊆ X. If Z ⊆ X is such that Z is unbounded in χ_{X}(b) for all b ∈ B, then Hull(Z) ∩ sup B = X ∩ sup B.

Proof: Show by induction on b ∈ Z that

Hull(Z) ∩ b = X ∩ b.

Step: let α ∈ Z ∩ b^{+} and fix f ∈ Z such that f : α ≃ b. Then f : Hull(Z) ∩ α ≃ Hull(Z) ∩ b = X∩b, so Hull(Z)∩α = X∩α. As Z is unbounded in X ∩ b^{+} , we are done.

Fact: In the situation above we can reconstruct X ∩ sup B from χ_{X}.

Proof: Intersect Hull(Z) ∩ sup B for all the Z ⊆ H_{θ} such that, for all b ∈ B, Z ∩ χ_{X}(b) contains a club in χ_{X}(b).

1. For any such Z we have that Hull(Z) ∩ sup B ⊇ Hull(Z ∩ X) ∩ sup B = X ∩ sup B.

2. Setting Z = X gives Hull(Z) ∩ sup B = X ∩ sup B.

So the intersection is precisely X ∩ sup B.

Smoothing: we may as well assume that for every limit β of cofinality |A|^{+} we have f_{β}^{λ}(i) = min {sup_{γ∈C} f_{γ}^{λ}(i) : C club in β }. Note we can fix C club in β such that for any club D ⊆ C we have f_{β}^{λ} = sup_{γ∈D} f_{γ}^{λ}.

Theorem: Let A ⊆ REG be an interval of regular cardinals, where min (A) = δ^{+} for some δ and max {|A|^{+} , |pcf (A)|} < min (A), Fix a sequence of generators ⟨B_{λ} : λ ∈ pcf (A)⟩ such that B_{max pcf (A)} = A. Fix ⟨f_{α}^{λ} : α < λ, λ ∈ pcf (A)⟩ such that ⟨f_{α}^{λ} : α < λ⟩ is cofinal in ∏ B_{λ}/J_{<λ}.

Let ⟨X_{α} : α ≤ |A|^{+} ⟩ be an IA chain such that A, B⃗, f⃗ ∈ X_{0}, δ ⊆ X_{0}, |X_{α}| = δ for all α. Let χ_{α} be the characteristic function of X_{α}, X = X_{|A|_{+}} , χ = χ_{|A|_{+}} . Then χ can be obtained by taking a pointwise sup of finitely many of the f_{α}^{λ}.

Proof: let λ ∈ pcf (A), μ = sup (X ∩ λ). We claim

1) ∀a ∈ B_{λ} f_{μ}^{λ}(a) ≤ χ(a)

2) {a ∈ B_{λ} : f_{μ}^{λ}(a) < χ(a)} ∈ J_{<λ}.

For the first claim, observe that by smoothing and the construction of X⃗ there is D ⊆ X ∩ μ such that D is club in μ and f_{μ}^{λ} = sup_{γ∈D} f_{γ}^{λ}. Each f_{γ}^{λ} ∈ X, so f_{μ}^{λ} is pointwise dominated by χ.

For the second claim, choose α < |A|^{+} such that {a ∈ B_{λ} : f_{μ}^{λ}(a) < χ(a)} = {a ∈ B_{λ} : f_{μ}^{λ}(a) < χ_{α}(a)} Now χ_{α} ∈ X so that by elementarity χ_{α} <_{J<λ} f_{μ}^{λ}.

We now define inductively a decreasing sequence of points λ_{i} ∈ pcf (A), along with F_{i} ∈ J_{<λ_{i}}. We let μ_{i} = sup (X ∩ λ_{i}) for all i. λ_{0} = max pcf (A), F_{0} = {a : f_{μ_{0}}^{λ_{0}}(a) < χ(a)}. If F_{i} is not empty then let λ_{i+1} be such that F_{i} ∈ J_{<λ_{i+1}^{+}} \ J_{<λ_{i+1}} , or to put it another way λ_{i+1} = max pcf (F_{i}). F_{i} \ B_{λ_{i+1}} ∈ J_{<λ_{i+1}} as the B⃗ are generators.

F_{i+1} = {a ∈ F_{i} : a ∉ B_{λ_{i+1}} ∨ f_{μ_{i+1}}^{λ_{i+1}} (a) < χ(a)}.

Eventually we reach n such that F_{n} = ∅. By construction, for every a ∈ A there is i < n such that f_{μ_{i}}^{λ_{i}}(a) = χ(a). Since χ dominates each f_{μ_{i}}^{λ_{i}} pointwise, we have that χ is the pointwise supremum of {f_{μ_{i}}^{λ_{i}} : i < n}.

Remark: these ideas are also relevant to Shelah’s “Strong covering” theorems.

A sample application: bounding ℵ_{ω}^{ℵ_{0}} .

Theorem: If 2^{ℵ_{0}} < ℵ_{ω} then ℵ_{ω}^{ℵ_{0}} = pp(ℵ_{ω}).

Proof: Let A = {ℵ_{k} : k < ω, 2^{ℵ_{0}} < ℵ_{k}}. Then pp(ℵ_{ω}) = max pcf (A), and clearly max pcf (A) ≤ ∏ A = ℵ_{ω}^{ℵ_{0}} .

Notice that max {|A|^{+} , |pcf (A)|} < min (A) = (2^{ℵ_{0}})^{+} . If a ∈ [ℵ_{ω}]^{ℵ_{0}} then we can build an IA chain ⟨X_{α} : α ≤ 2^{ℵ_{0}}⟩ such that a∪2^{ℵ_{0}} ⊆ X_{0} and each structure has size 2^{ℵ_{0}}.

There are at most pp(ℵ_{ω}) possibilities for the characteristic function of X_{2^{ℵ0}} , so there are at most pp(ℵ_{ω}) possibilities for X_{2^{ℵ0}} ∩ ℵ_{ω}. This means there are at most pp(ℵ_{ω}) × (2^{ℵ_{0}})^{ℵ_{0}} = pp(ℵ_{ω}) possibilities for a.

Remark: pp(ℵ_{ω}) = max pcf (A) < ℵ_{(2^{ℵ0})^{+}} .

Generalisation: if A is an interval of regular cardinals, min (A)^{|A|} < sup (A) then sup (A)^{|A|} = max pcf (A).

Let μ be a singular strong limit cardinal which is not a cardinal fixed point, say μ = ℵ_{η} for some η < μ. Let A = μ ∩ REG. Then 2^{μ} = max pcf (A).

Since pcf (A) is an interval of regular cardinals and |pcf (A)| ≤ 2^{|A|} = 2^{|η|}, 2^{μ} < ℵ_{(2^{|η|})^{+}} . This generalises the Galvin-Hajnal result.

Theorem: if λ is singular with κ = cf (λ) and J_{κ}^{bd} is the bounded ideal on κ then there is ⟨λ_{i} : i < κ⟩ increasing and cofinal in λ such that tcf (∏_{i} λ_{i}/J_{κ}^{bd} ) = λ^{+} .

Proof: as PP (λ) is an interval we may choose A ⊆ λ with ot(A) = κ such that λ^{+} ∈ pcf (A). Let B ∈ J_{<λ^{++}} \ J_{<λ^{+}} , then tcf (∏ B/J_{<λ^{+}} ) = λ^{+} . Now it is easy to see that J_{<λ^{+}} = J_{<λ} and every element of J_{<λ} is a bounded subset of A, so if ⟨λ_{i} : i < κ⟩ enumerates B then tcf (∏_{i} λ_{i}/J_{κ}^{bd} ) = λ^{+} as required.

Theorem: |pcf (A)| < min (A) ⇒ pcf (pcf (A)) = pcf (A).

Proof: Let A^{∗} = pcf (A), where clearly A^{∗} ⊆ pcf (A^{∗}). For each λ ∈ A^{∗} choose D_{λ} with cf (∏ A/D_{λ}) = λ, and then ⟨f_{α}^{λ} : α < λ⟩ cofinal in cf (∏ A/D_{λ}). Now let μ ∈ pcf (A^{∗}), and let ⟨g_{β} : β < μ⟩ be cofinal in ∏ A^{∗}/D for some D.

Define D^{∗} = {X ⊆ A : {λ : X ∈ D_{λ}} ∈ D}, and let h_{β}(a) = sup_{λ∈A^{∗}} f_{g_{β}(λ)}^{λ}(a).

Claim: if h ∈ ∏ A then h <_{D^{∗}} h_{β} for all large β.

Proof: fix g ∈ ∏ A^{∗} such that h <_{D_{λ}} f_{g(λ)}^{λ} for all λ, and then β such that g <_{D} g_{β}. For each of the D-many λ such that g(λ) < g_{β}(λ), we have h <_{D_{λ}} h_{β}, and so by definition h <_{D^{∗}} h_{β}.

Using the claim we can thin out ⟨h_{β} : β < μ⟩ to a μ-sequence which is increasing and cofinal in ∏(A/D^{∗}). So μ ∈ pcf (A) = A^{∗}, and we have proved that A^{∗} = pcf (A^{∗}).

Theorem: let λ be singular with κ = cf (λ) > ω. Let ⟨λ_{i} : i < κ⟩ be an increasing and continuous sequence of singular cardinals with limit λ. Let μ_{i} = λ_{i}^{+} , μ = λ^{+} . If A = {μ_{i} : i < κ} and B_{μ} is such that J_{<μ^{+}} = J_{<μ} + B_{μ} then {i : μ_{i} ∈ B_{μ}} contains a club. In particular μ ∈ pcf (A) and μ = tcf (∏_{i∈C} μ_{i}/J_{κ}^{bd} ) for some club C.

Proof: Let B ∈ J ⟺ {i : μ_{i} ∈ B \ B_{μ}} ∈ NS_{κ}. Suppose {i : μ_{i} ∉ B_{μ}} is stationary, so that J is not trivial.

Since J_{<μ^{+}} ⊆ J, ∏ A/J is μ^{+} -directed. We may therefore apply the technical lemma to build a <_{J}-increasing ⟨f_{α} : α < μ⟩, which has an eub g ∈ ∏ A such that cf (g(μ_{i})) tends to λ mod J. Now cf (g(μ_{i})) < λ_{i} for all i, so by Fodor there is j < κ such that cf (g(μ_{i})) < λ_{j} for stationarily many i with μ_{i} ∉ B_{μ}; this contradicts the stated property of g.

Localisation.

Theorem: if |pcf (A)| < min (A) (so in particular pcf (pcf (A)) = pcf (A)) then for all B ⊆ pcf (A) and all λ ∈ pcf (B) there exists B_{0} ∈ [B]^{|A|} such that λ ∈ pcf (B_{0}).

Proof: Does anyone know a proof of this that would fit on two slides?

Remark: if ℵ_{ω} is strong limit and 2^{ℵ_{ω}} > ℵ_{ω_{1}} then there are sets ⟨A_{α} : α < ℵ_{1}⟩ with A_{α} ⊆ REG such that α < β ⇒ sup (A_{α}) < min (A_{β}) and ℵ_{ω_{1}+1} ∈ pcf (A_{α}) for all α.

Roughly speaking this is what makes the problem of forcing “ℵ_{ω} is strong limit and 2^{ℵ_{ω}} > ℵ_{ω_{1}}” so hard. It is also crucial in the proof that this statement implies inner models for substantial large cardinals.

Club guessing:

Theorem: if κ, λ are regular and uncountable with κ^{+} < λ then there exists ⟨S_{α} : α ∈ λ ∩ cof (κ)⟩ such that

1. S_{α} is club in α, ot(C_{α}) = κ.

2. For every E club in λ, {α : S_{α} ⊆ E ∩ α} is stationary.

Proof: Start with any choice of S_{α}. Every time you see a bad club E, replace S_{α} by S_{α} ∩ E. Repeat for κ^{+} steps and argue that for some α, S_{α} shrank κ^{+} times.

Sample bound on pp. Theorem: pp(ℵ_{ω}) < ℵ_{ω_{4}}. Proof: If not we can manufacture a topology τ on ℵ_{4} + 1 such that for any Y ≠ ∅, γ.

1. cl(Y) has a maximal element.

2. ∀x ∈ cl(Y) ∃Y_{0} ∈ [Y]^{ℵ_{0}} x ∈ cl(Y_{0}).

3. If cf (γ) > ω then there is C club in γ such that γ = max (cl(C))

We claim no such topology τ can exist. To see this fix ⟨S_{α} : α ∈ ℵ_{3} ∩ cof (ℵ_{1})⟩ which guesses clubs in ℵ_{3}. Build an IA chain ⟨X_{α} : α ≤ ℵ_{3}⟩ of structures of size ℵ_{3} such that ℵ_{3} ∪{τ, S⃗} ∈ X_{0}. If γ_{α} = X_{α} ∩ ℵ_{4} then ⟨γ_{α} : α ≤ ℵ_{3}⟩ is increasing and continuous; let γ = γ_{ℵ_{3}}.

By the hypotheses on τ, fix D ⊆ γ club with γ = max cl(D). Fix α such that S_{α} ⊆ {β : γ_{β} ∈ D}, and let S^{∗} = {γ_{β} : β ∈ S_{α}}. Now let δ = max cl(S^{∗}), so that γ_{α} ≤ δ < ℵ_{4}. As cf (α) = ℵ_{1} we may find ᾱ < α such that δ = max cl{γ_{β} : β ∈ S_{α} ∩ ᾱ}.

Now S_{α} ∈ M_{0} and ⟨γ_{β} : β < ᾱ⟩ ∈ M_{ᾱ+1} , so that δ ∈ M_{ᾱ+1} ∩ ℵ_{4} = γ_{ᾱ+1} . Contradiction!

Remark: a similar method shows that if λ is singular, λ = ℵ_{α+δ} and δ < ℵ_{α} then pp(λ) < ℵ_{α+|δ|^{+4}} .

Remark: there is a theory of “pcf structures”.

More interesting facts: cf ([ℵ_{ω}]^{ℵ_{0}}, ⊆) = pp(ℵ_{ω}). cf (∏ A, <) = max pcf (A). If |pcf (A)| < min (A) then we can choose generators ⟨B_{λ} : λ ∈ pcf (A)⟩ such that ∀λ ∈ B_{μ} B_{λ} ⊆ B_{μ} and pcf (B_{λ}) = B_{λ}. Analogues of Silver’s theorem and the Galvin-Hajnal theorem hold for pp. If δ < ℵ_{4} and cf (δ) = ω then pp(ℵ_{δ}) < ℵ_{ω_{4}}. If λ is the least counterexample to SCH then λ > 2^{ℵ_{0}}, cf (λ) = ℵ_{0}, μ^{ℵ_{0}} ≤ μ^{+} for 2^{ℵ_{0}} < μ < λ, and there exists ⟨λ_{n} : n < ω⟩ increasing and cofinal in λ such that tcf (∏_{n} λ_{n}/J_{ω}^{bd} ) = λ^{++} .

Some hypotheses proposed by Shelah: STRONG: for all singular λ, pp(λ) = λ^{+} . MEDIUM: |pcf (A)| = |A|. WEAK: For singular λ {μ < λ : pp(μ) ≤ λ, cf (μ) = ω} is countable. {μ < λ : pp(μ) ≤ λ, cf (μ) > ω} is finite.

Review of large cardinal notions:

κ is measurable iff there exists a κ-complete non-principal ultrafilter on κ iff there is j : V ⟶ M such that crit(j) = κ and ^{κ}M ⊆ M.

The Mitchell order: if U, V are normal measures on κ then U < V iff U is in the ultrapower by V. < is a well founded partial ordering of height at most (2^{κ})^{+} . o(κ) is the height of <.

κ is λ-strong iff there is j : V ⟶ M such that j(κ) > λ, V_{λ} ⊆ M and ^{κ}M ⊆ M.

Remark: if κ is (κ + 2)-strong then o(α) = (2^{α})^{+} for many α < κ.

κ is λ-supercompact iff there is j : V ⟶ M such that j(κ) > λ and ^{λ}M ⊆ M.

Some forcing results relevant to SCH:

(Silver: supercompact) κ supercompact with 2^{κ} > κ^{+} .

(Prikry) A measurable κ can be made to have cofinality ω by a cardinal preserving forcing.

(Magidor) A cardinal of Mitchell order at least λ = cf (λ) < κ can be made to have cofinality λ in a cardinal preserving extension.

SCH can fail at large singular strong limit cardinals.

(Magidor: supercompact) Supercompact Prikry forcing.

(Magidor: supercompact) ℵ_{ω} strong limit, 2^{ℵ_{ω}} = ℵ_{ω+2} .

(Magidor: huge) 2^{ℵ_{n}} = ℵ_{n+1} for all n, 2^{ℵ_{ω}} = ℵ_{ω+2} .

(Shelah: supercompact) ℵ_{ω} strong limit and 2^{ℵ_{ω}} = ℵ_{α+1} , for α < ℵ_{1}.

(Magidor: huge) 2^{ℵ_{n}} = ℵ_{n+1} for all n and 2^{ℵ_{ω}} = ℵ_{α+1} , for α < ℵ_{1}.

(Shelah: supercompact) μ the least cardinal fixed point of order ω, μ strong limit with 2^{μ} arbitrarily large.

(Radin) Adding a club of V -regulars to large cardinal κ, preserving some large cardinal properties of κ.

(Foreman and Woodin: supercompact) Supercompact Radin forcing.

(Foreman and Woodin: supercompact) GCH fails everywhere, in fact 2^{κ} weakly inaccessible for all κ. For fixed n < ω, can build the model to contain many λ which are ℶ_{n}(λ)- supercompact.

(Woodin: supercompact) 2^{κ} = κ^{+n} for all κ.

Woodin: reduction of hypotheses to level of hypermeasurability.

(∗) GCH + there is j : V ⟶ M and f : κ ⟶ κ such that crit(j) = κ, j(f)(κ) = κ^{++} and κM ⊆ M.

Remark: (∗) follows from GCH + “κ is (κ+2)- strong”, and if j witnesses the strength we may take f(α) = α^{++} .

(Woodin: (∗)) κ measurable with 2^{κ} = κ^{++} .

(Woodin: (∗)) 2^{ℵ_{n}} = ℵ_{n+1} for all n, 2^{ℵ_{ω}} = ℵ_{ω+2} .

(Cummings: strong) GCH holds at every successor, fails at every limit.

(Gitik) (∗) can be forced starting from o(κ) = κ^{++} .

The paradigm shift: up to now powerset of κ was blown up keeping κ large, and then κ was made singular. Problem: GCH will fail at many points on the Prikry sequence, and collapsing to restore GCH tends to collapse at κ also.

(Gitik-Magidor: strong) adding many cofinal ω-sequences to κ without adding bounded subsets.

Example: from κ which is (κ + ω_{1})-strong, 2^{ℵ_{n}} = ℵ_{n+1} for all n and 2^{ℵ_{ω}} = ℵ_{α+1} , for α < ℵ_{1}.

(Segal:strong) Gitik-Magidor for uncountable cofinalities.

(Gitik and Merimovich:strong) ℵ_{ω} strong limit, 2^{ℵ_{ω}} = ℵ_{ω+m} for m < ω, complete freedom below ℵ_{ω}. o(κ) = κ^{++} is an upper bound for the consistency strength of the failure of SCH. More: we can build a model where ℵ_{ω} is strong limit (or even GCH holds below ℵ_{ω}) and 2^{ℵ_{ω}} = ℵ_{ω+2} starting from this hypothesis.

Two routes are available:

1. Gitik showed that forcing over a suitable model with o(κ) = κ^{++} we can produce a model of (∗). Work of Woodin then gives models as required. We also get a model where GCH fails at a measurable.

2. Gitik showed that forcing over a suitable model with o(κ) = κ^{++} gives a suitable ground model for a version of the Gitik-Magidor construction, which will produce a model where GCH holds up to ℵ_{ω} and 2^{ℵ_{ω}} = ℵ_{ω+2} .

Prikry forcing: if κ is measurable, can define a κ^{+} -c.c. forcing which adds no bounded subsets of κ and makes cf (κ) = ω.

U normal measure on κ. Conditions in ℙ_{U} are (s, A) where a ∈ [κ]^{<ω} and A ∈ U. Intuition: s is an initial segment of the ω-sequence and the rest is inside A. Accordingly (s, A) ⊆ (t, B) iff t extends s, A ⊆ B and s \ t ⊆ B. (s, A) ≤^{∗} (t, B) iff s = t and A ⊆ B.

Some attractive properties:

Strongly κ^{+} -c.c. as a union of κ filters.

(Prikry) For every sentence ϕ and condition p there is q ≤^{∗} p deciding ϕ. ≤^{∗} is κ-closed (as U is κ-complete), so we add no bounded subsets of κ.

(Mathias) x⃗ is generic iff every A ∈ U contains a tail of x⃗.

(Kunen) Generic sequence can be obtained by iterating ultrapowers.

Tree version: V a κ-complete ultrafilter on κ. Conditions are trees T ⊆ ^{<ω}κ, which have some finite stem and V -large branching above the stem. T ≤^{∗} U iff T ⊆ U and they have the same stem.

More generally: there is a family V_{s} of ultrafilters, and the tree has V_{s}-large branching at every s above the stem.

Abstractly: (ℙ, ≤, ≤^{∗}) is “Prikry-like” if ≤^{∗} is stronger than ≤, and all questions about the generic extension by (ℙ, ≤) can be decided by strengthening in ≤^{∗}.

(Magidor) Iterated Prikry forcing: finite support on s-parts, full support on A-parts.

Gitik’s iteration technique. A is a set of inaccessibles. For γ ∈ A have a ℙ_{γ}-name for a Prikry-like (ℚ_{γ}, ≤_{γ}, ≤_{γ}^{∗}), which is forced to have size less than min (A \ (γ + 1)).

ℙ_{α} is the set of α-sequences p such that supp (p) is an Easton subset of A (bounded in every inaccessible), p ↾ γ ∈ ℙ_{γ} and p ↾ γ ⊩ p(γ) ∈ ℚ_{γ} for all γ ∈ supp (p).

p ≤ q iff supp (q) ⊆ supp (p) , p ↾ γ ⊩ p(γ) ≤ q(γ) for all γ ∈ dom (q), p ↾ γ ⊩ p(γ) ≤^{∗} q(γ) for all but finitely many γ ∈ dom (q).

p ≤^{∗} q iff supp (q) ⊆ supp (p) , p ↾ γ ⊩ p(γ) ≤^{∗} q(γ) for all γ ∈ dom (q).

Iteration? ℙ_{γ+1} ≃ ℙ_{γ} ∗ ℚ_{γ} and p ⟼ p ↾ β is a projection from ℙ_{α} to ℙ_{β}.

Theorem (Gitik): (ℙ_{α}, ≤, ≤^{∗}) is Prikry-like.

Gitik: from o(κ) = κ^{++} to (∗).

By work of Mitchell, we may assume that our ground model has nice L-like properties (GCH, squares) and there is ⟨U_{α} : α < κ^{++} ⟩ increasing in the Mitchell ordering.

Aim: force to get a model with ⟨Ū_{α} : α < κ^{++} ⟩ a sequence of ultrafilters on κ which is increasing in the Rudin-Keisler ordering. Also we want ⟨π_{αβ} : α < β < κ^{++} ⟩ such that X ∈ Ū_{α} ⟺ π_{αβ}^{−1}[X] ∈ Ū_{β} and π_{αγ} =Ū_{γ} π_{αβ} ∘ π_{βγ}.

Now if j_{α} : V ⟶ M_{α} = Ult(V, Ū_{α}) is the ultrapower map, then π_{αβ} induces a map from M_{α} to M_{β} by π_{αβ}^{∗} : [f]Ū_{α} ⟼ [f ∘ π_{αβ}]Ū_{β}. These maps commute with the j_{α} and with each other, that is j_{β} = j_{α} ∘ π_{αβ}^{∗} and π_{αγ}^{∗} = π_{βγ}^{∗} ∘ π_{αβ}^{∗}.

Given this we can form a limit ultrapower j : V ⟶ M such that j(κ) > κ^{++} and ^{κ}M ⊆ M, and it is then possible to force the existence of a function f such that j(f)(κ) = κ^{++} .

A motivating example (Mitchell).

Assume GCH. Let κ be minimal with o(κ) = 2, A = {α < κ : o(α) = 1}. Fix ⟨U_{α} : α ∈ A⟩ measures of order zero on α ∈ A. Let U, V be measures on κ where U has order zero, V has order 1 and U = [α ⟼ U_{α}]_{V}. Notice that U_{α} and U concentrate on A^{c}, V concentrates on A.

Iterate Prikry forcing at α ∈ A. At stage α observe that j_{U_{α}}(ℙ_{α}) is a Gitik iteration whose support does not contain α. Use GCH and the Prikry lemma to build (canonically) a decreasing ⟨q_{γ} : γ < α^{+} ⟩ in j(ℙ_{α})/G_{α}, such that for every Ȧ a ℙ_{α}-name for a subset of α there is r ∈ G_{α} and γ < α^{+} such that r ⌢ q_{γ} decides α ∈ j_{U_{α}}(Ȧ).

Ū_{α} = {Ȧ^{G_{α}} : ∃r ∈ G_{α} ∃γ r ⌢ q_{γ} ⊩ α ∈ j_{U_{α}}(Ȧ)}.

ℚ_{α} is Prikry forcing defined from the normal measure Ū_{α}.

In V [G_{κ}], extend U to Ū in the same way. Now consider j_{V} (ℙ_{κ}): this is a Gitik iteration of Prikry forcing where the support does contain κ, and in fact the measure used at stage κ in this iteration is Ū.

Building a suitable sequence ⟨q_{γ}^{∗} : γ < κ^{+} ⟩ we define in V [G_{κ}] Ȧ^{G_{κ}} ∈ V̄ iff

∃r ∈ G_{κ} ∃B ∈ Ū r ⌢ (⟨⟩, B) ⌢ q_{γ}^{∗} ⊩ κ ∈ j_{V} (Ȧ).

Ū is normal and concentrates on A^{c}. V̄ concentrates on A, and so is not normal since A consists of cofinality ω ordinals. Define π_{01} with domain A by π_{01} (δ) = min (b_{δ}), then we claim that π_{01} projects V̄ to Ū.

If not then find E ∈ Ū, F ∈ V̄ such that π_{01} [F]∩ E = ∅. Let F = Ḟ^{G_{κ}}, E = Ė^{G_{κ}} and find r ∈ G_{κ}, E^{∗} ∈ Ū and γ such that r ⊩ π_{01} [Ḟ] ∩ Ė = ∅ and r ⌢ (⟨⟩, E^{∗}) ⌢ q_{γ}^{∗} ⊩ κ ∈ j_{V} (Ḟ). Now the condition r ⌢ (⟨⟩, E ∩ E^{∗}) ⌢ q_{γ}^{∗} forces that min (b_{κ}) ∈ E, contradiction.

Woodin: from (∗) to a measurable cardinal where GCH fails. Fix j : V ⟶ M and f such that crit(j) = κ, ^{κ}M ⊆ M and j(f)(κ) = κ^{++} .

Stage 1: Iterate Coll(α^{+} , < f(α)) for α < κ and then do Coll(κ^{+} , < κ^{++} ). Show that in the extension we may lift j and get a situation in which (∗) holds and κ^{++} = κ_{M}^{++} , so that we may take f(α) = α^{++} .

Stage 2: Iterate Add(α, α^{++} ) for α ≤ κ, and argue that j can be lifted so that the measurability of κ is preserved. Less closure than in Silver’s argument from a supercompact, so harder to build generic filters and master conditions: these problems are resolved by forcing, transferring and rearranging of generic filters.

Remark: if you just want failure of GCH at a measurable, you can start with (∗) and iterate Add(α, f(α)) for α < κ followed by Add(κ, κ^{++} ). In this two stage approach you get a bonus; in the final model we can arrange

(∗∗): κ is measurable, 2^{κ} = κ^{++} and for some normal measure U on κ there exists F which is generic over M = Ult(V, U) for the forcing Coll(κ_{M}^{+5} , < j_{U}(κ))_{M}.

Woodin: given a model of (∗∗) we can force using “Prikry forcing with interleaved collapses” to make κ = ℵ_{ω} while preserving κ^{+} . Magidor did supercompact Prikry forcing with interleaved collapses and then went to a certain inner model, analysis of this inner model done by Foreman and Woodin for the GCH fails everywhere result motivated the construction from (∗∗).

How to prove the Prikry lemma for Prikry forcing. U normal, U = {X : κ ∈ j_{U}(X)}. ℙ = ℙ_{U}.

Diagonal intersection: Given a family of conditions (s, A_{s}) let

A = {δ : ∀s max (s) < δ ⟹ δ ∈ A_{s}}.

Then A ∈ U and (s, A) ≤ (s, A_{s}) for all s.

Now fix a statement ϕ.

Stage 1: find A such that if (s, B) decides ϕ for any B, then (s, A) decides ϕ.

Stage 2: Consider (s ⌢ κ, j_{U}(A)) and j_{U}(ϕ). Find A_{s} ∈ U such that all (s ⌢ δ, A) with δ ∈ A_{s} behave the same way wrt ϕ (all force ϕ, all force ¬ϕ or all fail to decide ϕ). Take a diagonal intersection to get A^{∗}.

Stage 3: Given (s, A^{∗}) take an extension of minimal length which decides ϕ. Wlog extension is (t, A^{∗}). If t ≠ s then t = t_{0} ⌢ δ for some δ ∈ A_{t_{0}}, but now by construction (t_{0}, A^{∗}) decides ϕ.

(∗∗): κ is measurable, 2^{κ} = κ^{++} and for some normal measure U on κ there exists F which is generic over M = Ult(V, U) for the forcing Coll(κ_{M}^{+5} , < j_{U}(κ))_{M}.

Define generalised Prikry forcing ℙ as follows: conditions have the form

p = (κ_{0}, p_{0}, κ_{1}, p_{1}, … κ_{n−1}, p_{n−1}, A, H)

where κ_{0} < … < κ_{n−1} < κ, dom (H) = A ∈ U, p_{j} ∈ Coll(κ_{j}^{+5} , < κ_{j+1} ), H(δ) ∈ Coll(δ^{+5} , < κ), [H]_{U} ∈ F.

A typical extension is

q = (κ_{0}, p_{0}^{∗}, κ_{1}, p_{1}^{∗}, … κ_{m−1}, p_{m−1}^{∗}, A^{∗}, H^{∗})

where p_{i}^{∗} ≤ p_{i} for i < n, A^{∗} ⊆ A, H^{∗}(δ) ≤ H(δ) for δ ∈ A^{∗}.

q ≤^{∗} p iff m = n.

Theorem: (ℙ, ≤, ≤^{∗}) is Prikry-like and κ^{+} -c.c.

Corollary: in the extension by (ℙ, ≤) we have κ = κ_{0}^{+ω} , κ strong limit, 2^{κ} = κ^{++} . Now we can do Coll(ℵ_{0}, κ_{0}) to get κ = ℵ_{ω}.

Proof of theorem: Conditions have form p = x∩(A, H) where x ∈ V_{κ} (we call x the lower part of p). x∩(A, H) and x∩(A^{∗}, H^{∗}) are compatible, so ℙ is κ^{+} -c.c.

Proof of Prikry Lemma is just like for Prikry forcing. First show the analogue of the diagonal intersection theorem. Then

Step One: find (A, H) such that for any x, if there exists (B, G) such that x ⌢ (B, G) decides ϕ then x ⌢ (A, H) decides ϕ.

Step Two: consider x ⌢ (κ, p) ⌢ (j_{U}(A), j_{U}(H)). Use genericity to find p = [H^{∗}] ∈ F such that for all x if there is q ≤ p such that x ⌢ (κ, q) ⌢ (j_{U}(A), j_{U}(H)) decides j_{U}(ϕ) then x ⌢ (κ, p) ⌢ (j_{U}(A), j_{U}(H)) decides j_{U}(ϕ).

Step Three: Find A^{∗} ⊆ A such that for all x and all δ ∈ A^{∗}, if there is q ≤ H^{∗}(δ) such that x ⌢ (δ, q) ⌢ (A, H) decides ϕ then x ⌢ (δ, H^{∗}(δ)) ⌢ (A, H) decides ϕ

Step Four: for each x find A_{x}^{∗} ⊆ A^{∗} such that all conditions x ⌢ (δ, H^{∗}(δ)) ⌢ (A, H) behave the same wrt ϕ. Diagonally intersect to get A^{∗∗}.

Now consider an extension of x ⌢ (A^{∗∗}, H^{∗}) which has minimal length and decides ϕ. If it is not direct then wlog it has form y_{0} ⌢ (δ, q) ⌢ (A^{∗∗}, H^{∗}) where y_{0} extends x, δ ∈ A_{y_{0}}^{∗} and q ≤ H^{∗}(δ). By construction y_{0} ⌢ (δ, H^{∗}(δ)) ⌢ (A^{∗∗}, H^{∗}) decides ϕ, but then y_{0} ⌢ (A^{∗∗}, H^{∗}) decides ϕ. Contradiction.

A more elaborate argument gives GCH up to ℵ_{ω} and 2^{ℵ_{ω}} = ℵ_{ω+2} by similar means. Idea: interleave Coll(κ_{i}^{+5} , < κ_{i+1} ) × Coll(κ_{i+1} , κ_{i+1}^{+} ) between successive points κ_{i} and κ_{i+1} . Need a more elaborate mechanism of constraint, functions of two variables representing a generic filter over the second iterated ultrapower for Coll(κ^{+5} , < j(κ))×Coll(j(κ), j(κ^{+} )). Pcf arguments show this idea will not generalise easily to get GCH up to ℵ_{ω} and 2^{ℵ_{ω}} = ℵ_{ω+3} .

Gitik-Magidor method.

Idea: want to add many Prikry sequences to large cardinal κ, without adding bounded subsets of κ. Problem: new reals may be coded by (say) the relationship between two Prikry sequences. Solution: arrange that if x⃗, y⃗ are two Prikry sequences then there is a sequence z⃗ and π_{x}, π_{y} ∈ V such that up to finite perturbation π_{x} ∘ z⃗ agrees with x⃗ and π_{y} ∘ z⃗ agrees with y⃗

To be a little more precise we are given a κ^{+} - directed poset(A, ≺) together with ⟨U_{α} : α ∈ A⟩ a system of measures on κ and ⟨π_{αβ} : α ≺ β⟩ a system of commuting projection maps. These have to satisfy some technical conditions. A will have a minimal element 0 with U_{0} normal.

A condition p then prescribes

1. A set g_{p} ∈ [A]^{κ}, where there is γ_{p} ∈ g_{p} such that ∀β ∈ g_{p} β ≺ γ_{p}.

2. A set p^{β} ∈ [κ]^{<ω} for each β ∈ g_{p}.

3. A tree with stem p^{γ_{p}} and U_{γ_{p}}-large branching.

There are two special kinds of extension.

Direct: q ≤^{∗} p. g_{q} ⊇ g_{p} (maybe with a larger maximal element), and q^{γ} = p^{γ} for all γ ∈ g_{p}.

“Projective”: q ≤^{0} p. g^{q} = g^{p} and q^{β} \ p^{β} is obtained (roughly) by projecting q^{γ} \ p^{γ} along π_{βγ}.

Extension = projective followed by direct. This is transitive because the system π⃗ is “commutative enough”.

The key facts:

1. (ℙ, ≤) is κ^{++} -c.c. and (ℙ, ≤^{∗}) is κ-closed.

2. (ℙ, ≤, ≤^{∗}) is Prikry-like.

3. ℙ adds ⟨x_{α} : α ∈ A⟩, distinct ω-sequences cofinal in κ.

4. ℙ preserves κ^{+} .

How to get a suitable family of measures?

κ is λ-strong, λ successor ordinal or a cardinal with cf (λ) > κ, and GCH holds. Can build a family of size κ^{+λ} .

o(κ) = κ^{+λ} is enough if λ is successor or κ < cf (λ) < λ.

GCH up to ℵ_{ω} and 2^{ℵ_{ω}} = ℵ_{ω+ζ+1} , ζ countable.

Start with GCH and κ which is (κ + ζ + 1)- strong.

To bring κ down to ℵ_{0}, interleave with collapses along the Prikry sequence for U_{0}. It is comparatively easy to get a suitably generic constraint filter because GCH holds.

If ζ is finite the construction is essentially as before.

If ζ is infinite then let ζ + 1 = ∪_{n}D_{n} with D_{n} finite. Build ℙ so that if κ⃗ is the Prikry sequence for U_{0} then (roughly) the cardinals {κ_{n}^{+i} : i ∈ D_{n}} survive in [κ_{n}, κ_{n+1} ).

ℙ is now only κ^{+ζ+2} -c.c. but for all δ ≤ ζ it embeds in a forcing that preserves κ^{+δ+1} .

From a strong cardinal: no bound on 2^{κ} where κ is the least fixed point of order ω.

(Segal) Versions of the Gitik-Magidor construction which make κ singular of uncountable cofinality.

Example: 2^{ℵ_{α}} = ℵ_{α+3} for α ≤ ω_{1} limit, GCH holds except at limits and their successors up to ℵ_{ω_{1}}.

(Gitik and Merimovich) Fine control.

Example: From κ which is (κ + m)-strong, 2^{ℵ_{ω}} = ℵ_{ω+m} and complete freedom below ℵ_{ω}.

Lower bounds: core models, covering lemmas.

Core models: on the hypothesis “there is no X cardinal”, build a model K_{X} and

1. Prove that K_{X} has a nice internal structure. GCH, diamonds, squares, morasses etc.

2. Prove that there is some resemblance between K_{X} and V.

In general for M ⊆ V:

Strong covering: every uncountable set of ordinals is contained in a set of the same V - cardinality which lies in M.

Weak covering: M computes successors of singulars correctly.

If M is a model of GCH and strong covering holds then SCH is true in V.

X = 0^{♯}, K_{X} = L.

(Jensen) If 0^{♯} does not exist, then strong covering holds between L and V.

Once measurable cardinals are allowed into K then we can no longer ask for strong covering by K itself, because of Prikry forcing. Weak covering goes a long way.

X = 0^{†}, K_{X} has maximal form L[μ].

(Dodd-Jensen) If L[μ] exists and 0^{†} does not then EITHER strong covering holds between L[μ] and V OR there is C ∈ V a maximal Prikry generic sequence over L[μ] such that strong covering holds between L[μ, C] and V.

Still a misleadingly simple example. When K contains many measures then there is no longer a uniform set of indiscernibles which works to cover all sets in V, and it is hard to see the set of indiscernibles as a generic object for some forcing.

Typical covering lemma for K a large core model:

x ⊆ ON covered by a set h“(ρ∪I) where h ∈ K, ρ ∈ ON, I ⊆ ON.

I arises from iterating some small “L[E⃗] model” up to a model in K. The analysis gets harder as K gets larger.

Lower bounds for failure of SCH.

(Mitchell) If there is κ singular strong limit with 2^{κ} > κ^{+} then in a suitable core model K, ∀α < κ ∃ν < κ o(ν) ≥ α. If in addition cf (κ) > ω o(κ) = κ^{++} in K.

(Shelah) If λ is the least counterexample to SCH then λ > 2^{ℵ_{0}}, cf (λ) = ℵ_{0}, μ^{ℵ_{0}} ≤ μ^{+} for 2^{ℵ_{0}} < μ < λ, and pp(λ) ≥ λ^{++} .

(Gitik) If λ > 2^{ℵ_{0}} is singular and pp(λ) ≥ λ^{++} then there is an inner model for ∃κ o(κ) = κ^{++} .

These results can be extended: need to understand “o(κ)” as a measure of how many extenders there are at κ.

Gitik and Mitchell: further results.

1) κ strong limit, cf (κ) = δ < κ, 2^{κ} ≥ λ > κ^{+} where λ is not ρ^{+} for cf (ρ) ≤ κ.

1. If δ > ω_{1}, there is an inner model where o(κ) ≥ λ + δ (Optimal by work of Woodin, or by work of Segal).

2. If δ = ω_{1}, there is an inner model where o(κ) ≥ λ (and λ + ω_{1} suffices).

3. If δ = ω then there is an inner model where either o(κ) ≥ λ or {α < κ : o(α) ≥ α^{+n} } is unbounded for all n < ω. The latter case can actually occur, by work of Gitik.

2) If n > 0, cf (κ) = ω, GCH holds below κ and 2^{κ} ≥ κ^{+n+2} then there is an inner model where either o(κ) ≥ κ^{+n+2} + 1 or {α < κ : o(α) ≥ α^{+m} } is unbounded for all m < ω.

3) If ℵ_{ω} is strong limit and 2^{ℵ_{ω}} > ℵ_{ω_{1}} then there is an inner model for a Woodin cardinal (assuming that we can build the Steel core model).

Gitik: more on failure of GCH at a measurable.

1) κ measurable and 2^{κ} = κ^{+α} for α ≥ 2. Then there is an inner model where o(κ) ≥ κ^{+α} or some τ is strong up to the next measurable.

2) κ measurable and 2^{κ} = κ^{+β+1} for β singular, cf (β) ≤ κ. Then there is an inner model where o(κ) ≥ κ^{+β+1} +1 or some τ is strong up to the next measurable.

Complementary consistency results: If α = β + 1 for β a successor or cf (β) > κ, or α is limit with cf (α) > κ then o(κ) = κ^{+α} suffices to make κ measurable with 2^{κ} = κ^{+α} .

Work of Woodin shows o(κ) = κ^{α} + 1 always suffices.

Combinatorial problems about regular cardinals tend to be hardest at successors of singulars. Examples:

1. Is there a κ-Aronszajn tree?

2. Is there a non-trivial non-reflecting stationary subset of κ?

3. κ^{<κ} = κ? (=SCH, essentially)

4. Is there a Jonsson algebra on κ?

Some ways to resolve problems about singular cardinals and their successors.

1. Reflection principles (typically obtained from large cardinals by forcing).

2. Squares (typically obtained by forcing or from inner models).

3. Pcf theory. Q: How are they related?

Definition: Let κ be an uncountable cardinal. A □_{κ}-sequence is a sequence ⟨C_{α} : α < κ^{+} , lim(α)⟩ such that for all α < κ^{+}

1. C_{α} is closed and unbounded in α.

2. If cf (α) < κ, then ot(C_{α}) < κ.

3. For all β ∈ lim(C_{α}), C_{β} = C_{α} ∩ β. We say that □_{κ} holds iff there exists a □_{κ}- sequence. CFM = Cummings, Foreman and Magidor. CS = Cummings and Schimmerling. FM = Foreman and Magidor.

Approachability (Shelah) Definition: ⟨C_{α} : α < κ^{+} ⟩ is an AP_{κ}-sequence iff for a club of α

1. C_{α} is club in α, ot(C_{α}) = cf (α).

2. ∀β < α ∃γ < α C_{α} ∩ β = C_{γ}. Here we could just demand that C_{α} was unbounded and get an equivalent definition.

Very weak square (Foreman and Magidor)

Definition: ⟨C_{α} : α < κ^{+} ⟩ is an VWS _{κ}-sequence iff for a club of α < κ^{+}

1. sup (C_{α}) = α.

2. For every bounded x ∈ [C_{α}]^{ℵ_{0}}, there is β < α with x = C_{β}.

Here demanding that C_{α} be club in α gives a stronger property, the Not So Very Weak Square or NSVWS _{κ}.

By scale we will mean scale of length ℵ_{ω+1} in a product of the form ∏_{n∈A} ℵ_{n} ordered by eventual domination.

(FM) VWS _{ℵ_{ω}} implies that in any scale the set of non-good points of cofinality ℵ_{1} is nonstationary.

(ℵ_{ω+1} , ℵ_{ω}) ↠ (ℵ_{1}, ℵ_{0}) implies that in any scale the set of non-good points of cofinality ℵ_{1} is stationary.

AP_{ℵ_{ω}} implies that in any scale, the set of nongood points of uncountable cofinality is nonstationary.

Relations with large cardinals etc

(FM) If GCH holds and κ is supercompact then there is a class generic extension in which κ is supercompact, cardinals and cofinalities are preserved and VWS _{λ} holds for all singular λ. We can also preserve hugeness.

NSVWS _{λ} fails if λ is the supremum of ω supercompact cardinals.

Definition: Let κ be an uncountable regular cardinal. Let S be a stationary subset of κ.

1. S reflects at α iff α < κ, cf (α) > ω and S ∩α is stationary in α.

2. Refl (S) holds iff every stationary subset of S reflects at some α.

3. S is non-reflecting iff S does not reflect at any α.

Fact: Let □_{κ} hold and let S be a stationary subset of κ^{+} . Then Refl (S) fails.

Definition: Let κ be a cardinal. A □_{κ,λ}-sequence is a sequence ⟨C_{α} : α < κ^{+} , lim(α)⟩ such that

1. C_{α} ⊆ P (α), 1 ≤ |C_{α}| ≤ λ, and C_{α} is a set of closed and unbounded subsets of α.

2. If cf (α) < κ then ∀C ∈ C_{α} ot(C) < κ.

3. ∀C ∈ C_{α} ∀β ∈ lim(C) C ∩ β ∈ C_{β}.

A □_{κ,<λ}-sequence is defined similarly, only we demand that 1 ≤ |C_{α}| < λ.

□_{κ} = □_{κ,1}, □_{κ}^{∗} = □_{κ,κ}.

Fact: Let κ^{<λ} = κ and let □_{κ,<λ} hold. If S ⊆ κ^{+} is stationary then there exists a stationary set T ⊆ S such that T does not reflect at any α with cf (α) ≥ λ.

Theorem (CFM): Let κ be supercompact, and suppose 2^{κ^{+ω}} = κ^{+ω+1} . Let μ, ν be two cardinals (one or both can be finite) such that 1 ≤ μ < ν < ℵ_{ω}. Then there is a generic extension in which

1. All cardinals less than or equal to ν are preserved.

2. ℵ_{ω} = κ_{V}^{+ω} .

3. □_{ℵ_{ω},ν} holds.

4. □_{ℵ_{ω},μ} fails.

Failure of □_{κ} for κ singular requires substantial large cardinals. More information in Ernest Schimmerling’s talk.

Fact (Solovay): If κ is supercompact and λ is a cardinal with κ < λ then □_{λ} fails.

Fact (Shelah) If κ is supercompact and cf (λ) < κ < λ then □_{λ}^{∗} fails.

Fact (Burke and Kanamori): If κ is supercompact and κ, cf (λ) < λ then □_{λ,<cf (λ)} fails.

Fact (Magidor): PFA implies that □_{κ,ℵ_{1}} fails for κ ≥ ℵ_{1}, while “PFA +∀κ ≥ ℵ_{2} □_{κ,ℵ_{2}}” is consistent.

Definition: A □_{λ,cf (λ)}^{ind}-sequence is a matrix of sets ⟨C_{α,i} : α < λ^{+} , i(α) ≤ i < cf (λ)⟩ such that for some increasing sequence ⟨λ_{i} : i < cf (λ)⟩ of regular cardinals with limit λ

1. i(α) < cf (λ) for all α < λ^{+} .

2. ot(C_{α,i}) < λ_{i} for all α.

3. C_{α,i} is club in α.

4. If i(α) ≤ i < j < cf (λ) then C_{α,i} ⊆ C_{α,j}.

5. If i(β) ≤ i < cf (λ) and α ∈ lim(C_{β,i}) then i(α) ≤ i and C_{α,i} = C_{β,i} ∩ α.

6. If α and β are limit ordinals with α < β < λ^{+} then α ∈ lim(C_{β,i}) for all sufficiently large i < μ.

□_{λ,cf (λ)}^{ind} implies □_{λ,cf (λ)} and the transfer principle (ℵ_{1}, ℵ_{0}) ⟶ (λ^{+} , λ).

Theorem (CFM): Let λ be a singular cardinal. Then there exists a forcing poset ℙ such that

1. ℙ is cf (λ)-directed closed.

2. ℙ is < λ-strategically closed.

3. ⊩ _{ℙ}“□_{λ,cf (λ)}^{ind} holds”

Corollary: Let κ be a Laver indestructible supercompact cardinal and let κ ≤ cf (λ) < λ. Then there is a forcing extension in which κ is still supercompact, cardinals and cofinalities up to λ^{+} are preserved, and □_{λ,cf (λ)}^{ind} holds.

Definition: (κ⃗, f⃗) is a very good scale for κ iff

1. κ⃗ = ⟨κ_{i} : i < cf (κ)⟩ is an increasing sequence of regular cardinals cofinal in κ.

2. f⃗ = ⟨f_{α} : α < κ^{+} ⟩ is a scale in ∏_{i} κ_{i}/J_{cf (κ)}^{bd}.

3. For every point α < κ^{+} such that cf (α) > cf (κ) there exists a closed and unbounded set C ⊆ α and i < cf (κ) such that ∀β, γ ∈ C ∀j > i (β < γ ⟹ f_{β}(j) < f_{γ}(j)).

VGS_{κ} holds iff there exists a very good scale for κ.

Theorem (CFM): Let κ be singular, let λ < κ. Then □_{κ,λ} implies VGS_{κ}.

Theorem (CFM): Let κ be singular, and let VGS_{κ} hold. Then for every stationary T ⊆ κ^{+} there are stationary ⟨T_{i} : i < cf (κ)⟩ such that T_{i} ⊆ T and the T_{i} do not reflect simultaneously at any point of cofinality greater than cf (κ).

Theorem (CFM): Let ⟨κ_{n} : n < ω⟩ be an increasing sequence of supercompact cardinals. Let κ = sup_{n} κ_{n}. Then there is a generic extension in which

1. κ = ℵ_{ω}, and κ^{+} is preserved.

2. □_{ℵ_{ω},ω}^{ind} holds.

3. For every finite set f of stationary subsets of ℵ_{ω} there exists N < ω such that if N ≤ n < ω then there exists α of cofinality ℵ_{n} such that all sets in f reflect at α.

Theorem (CFM): Let ℙ be Prikry forcing at κ. In V let S_{0} =_{def} {α < κ^{+} : cf (α) < κ} and let S_{1} = {α < κ^{+} : cf (α) = κ}. Then in V^{ℙ}

1. S_{1} is a non-reflecting stationary set of cofinality ω ordinals.

2. If κ is κ^{+} -supercompact, then any finite set of stationary subsets of S_{0} reflect simultaneously.

3. There are ω subsets of S_{0} which do not reflect simultaneously.

Theorem (CS): □_{κ,ω} holds in V^{ℙ}.

Definition: ADS _{κ} holds iff there exists ⟨A_{α} : α < κ^{+} ⟩ such that

1. A_{α} is unbounded in κ, ot(A_{α}) = cf (κ).

2. For all β < κ^{+} there exists g : β ⟶ κ such that the sequence ⟨A_{α} \ g(α) : α < β⟩ consists of pairwise disjoint sets.

Theorem (CFM): If VGS_{κ} or □_{κ}^{∗} holds then ADS _{κ} holds.

Theorem (CFM): Let κ be singular of cofinality ω. If ADS _{κ} holds then there is stationary S ⊆ Refl ^{∗}([κ^{+} ]^{ℵ_{0}}) which does not reflect to any X with |X| = cf (X) = ℵ_{1}.

Theorem (CFM): Let ⟨κ_{n} : n < ω⟩ be an increasing sequence of supercompact cardinals. Let κ = sup_{n} κ_{n}, and assume that GCH holds above κ.

Then there is a forcing ℙ such that in V^{ℙ}

1. κ = ℵ_{ω} and GCH holds.

2. □_{ℵ_{ω}}^{∗} holds.

3. For all n, any ℵ_{n} stationary subsets of the set ℵ_{ω+1} ∩cof (< ℵ_{n}) reflect simultaneously at some point of cofinality ℵ_{n}.

Theorem (CFM): Let κ be supercompact. Then there is a generic extension W such that

1. κ = ℵ_{2}^{W}.

2. □_{ℵ_{ω}}^{∗} fails in W.

3. If H is Coll(ω, ω_{1})-generic over W then □_{ℵ_{ω}} holds in W [H].

A similar argument shows that for λ regular we can create a situation in which □_{λ} fails, and forcing with Coll(ω, ω_{1}) makes □_{λ} hold. However, in general it may not be possible to force □_{λ} with mild forcing if we demand that □_{λ}^{∗} should fail in the ground model.

Theorem (CFM): Let 1 ≤ n < ω and let λ = ℵ_{n}. Let ℙ be λ-c.c. and suppose that ⊩ _{ℙ}“□_{λ} holds” . Then □_{λ}^{∗} holds in V.

Theorem (CFM): Let MM^{+} hold. Let κ be regular and uncountable. Then there is a forcing extension in which

• There are two stationary subsets of κ which do not reflect simultaneously.

• Stationary subsets of [λ]^{ℵ_{0}} reflect for all λ ≥ ℵ_{2}.