[PPT] - Large cardinals and pcf theory in topology and infinite PowerPoint Presentation

SLIDE 1

Large cardinals and pcf theory in topology and infinite combinatorics

Lajos Soukup

Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences

RIMS Set Theory Workshop 2010

Soukup, L (HAS) RIMS 2010 1 / 23

SLIDE 2

Content

Pcf theory and cardinal invariants of the reals

Soukup, L (HAS) RIMS 2010 2 / 23

SLIDE 3

Content

Pcf theory and cardinal invariants of the reals Families of sets

Soukup, L (HAS) RIMS 2010 2 / 23

SLIDE 4

Content

Pcf theory and cardinal invariants of the reals Families of sets Weak Freeze Nation property and Noetherian types of bases

Soukup, L (HAS) RIMS 2010 2 / 23

SLIDE 5

Content

Pcf theory and cardinal invariants of the reals Families of sets Weak Freeze Nation property and Noetherian types of bases Resolvability of monotonically normal spaces

Soukup, L (HAS) RIMS 2010 2 / 23

SLIDE 6

Content

Pcf theory and cardinal invariants of the reals Families of sets Weak Freeze Nation property and Noetherian types of bases Resolvability of monotonically normal spaces Club quessing in a topological construction

Soukup, L (HAS) RIMS 2010 2 / 23

SLIDE 7

Pcf theory and cardinal invariants of the reals

Soukup, L (HAS) RIMS 2010 3 / 23

SLIDE 8

The beginning

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 9

The beginning

Is every set definable?

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 10

The beginning

Is every set definable? Is every set definable in some generic extension?

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 11

The beginning

Is every set definable? Is every set definable in some generic extension? Hamkin’s question: Is every set X definable in some cardinal preserving extension of the ground model?

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 12

The beginning

Is every set definable? Is every set definable in some generic extension? Hamkin’s question: Is every set X definable in some cardinal preserving extension of the ground model? What about X ⊂ ω?

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 13

The beginning

Is every set definable? Is every set definable in some generic extension? Hamkin’s question: Is every set X definable in some cardinal preserving extension of the ground model? What about X ⊂ ω? X = {n ∈ ω : 2ℵn = ℵn+1}.

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 14

The beginning

Is every set definable? Is every set definable in some generic extension? Hamkin’s question: Is every set X definable in some cardinal preserving extension of the ground model? What about X ⊂ ω? X = {n ∈ ω : 2ℵn = ℵn+1}. X = {n ∈ ω : 2ℵ2n = 2ℵ2n+1}.

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 15

The beginning

Is every set definable? Is every set definable in some generic extension? Hamkin’s question: Is every set X definable in some cardinal preserving extension of the ground model? What about X ⊂ ω? X = {n ∈ ω : 2ℵn = ℵn+1}. X = {n ∈ ω : 2ℵ2n = 2ℵ2n+1}. Collapse cardinals

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 16

The beginning

Is every set definable? Is every set definable in some generic extension? Hamkin’s question: Is every set X definable in some cardinal preserving extension of the ground model? What about X ⊂ ω? X = {n ∈ ω : 2ℵn = ℵn+1}. X = {n ∈ ω : 2ℵ2n = 2ℵ2n+1}. Collapse cardinals Cardinal exponentiation can not help

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 17

The beginning

Is every set definable? Is every set definable in some generic extension? Hamkin’s question: Is every set X definable in some cardinal preserving extension of the ground model? What about X ⊂ ω? X = {n ∈ ω : 2ℵn = ℵn+1}. X = {n ∈ ω : 2ℵ2n = 2ℵ2n+1}. Collapse cardinals Cardinal exponentiation can not help Use the spectrum of some cardinal invariant to code!

Soukup, L (HAS) RIMS 2010 4 / 23

SLIDE 18

Coding by MADness

Soukup, L (HAS) RIMS 2010 5 / 23

SLIDE 19

Coding by MADness

a = min{|A| : A ⊂

ω

ω is MAD}

Soukup, L (HAS) RIMS 2010 5 / 23

SLIDE 20

Coding by MADness

a = min{|A| : A ⊂

ω

ω is MAD} spectrum(a)= {|A| : A ⊂

ω

ω is MAD}

Soukup, L (HAS) RIMS 2010 5 / 23

SLIDE 21

Coding by MADness

a = min{|A| : A ⊂

ω

ω is MAD} spectrum(a)= {|A| : A ⊂

ω

ω is MAD} Let E ⊂ ω.

Soukup, L (HAS) RIMS 2010 5 / 23

SLIDE 22

Coding by MADness

a = min{|A| : A ⊂

ω

ω is MAD} spectrum(a)= {|A| : A ⊂

ω

ω is MAD} Let E ⊂ ω. (i) If CH holds then there is a c.c.c poset P such that V P | = “ n ∈ E iff ℵn+1 ∈ spectrum(a)”.

Soukup, L (HAS) RIMS 2010 5 / 23

SLIDE 23

Coding by MADness

a = min{|A| : A ⊂

ω

ω is MAD} spectrum(a)= {|A| : A ⊂

ω

ω is MAD} Let E ⊂ ω. (i) If CH holds then there is a c.c.c poset P such that V P | = “ n ∈ E iff ℵn+1 ∈ spectrum(a)”. (ii) There is a c.c.c poset P such that V P | = “there is a definable cardinal κ s. t. n ∈ E iff κ+n ∈ spectrum(a).

Soukup, L (HAS) RIMS 2010 5 / 23

SLIDE 24

Coding by MADness

Code more than one set!

Soukup, L (HAS) RIMS 2010 6 / 23

SLIDE 25

Coding by MADness

Code more than one set! Code just two sets! Under CH!

Soukup, L (HAS) RIMS 2010 6 / 23

SLIDE 26

Coding by MADness

Code more than one set! Code just two sets! Under CH! Question: Let E, F ⊂ ω. Is there a c.c.c poset P such that V P | = “n ∈ E iff ℵn+1 ∈ spectrum(a), m ∈ F iff ℵω+m+1 ∈ spectrum(a)”.

Soukup, L (HAS) RIMS 2010 6 / 23

SLIDE 27

Coding by MADness

Code more than one set! Code just two sets! Under CH! Question: Let E, F ⊂ ω. Is there a c.c.c poset P such that V P | = “n ∈ E iff ℵn+1 ∈ spectrum(a), m ∈ F iff ℵω+m+1 ∈ spectrum(a)”. No problem with {ℵ1, ℵ2, . . . } and {ℵω+2, ℵω+3, . . . }

Soukup, L (HAS) RIMS 2010 6 / 23

SLIDE 28

Coding by MADness

Code more than one set! Code just two sets! Under CH! Question: Let E, F ⊂ ω. Is there a c.c.c poset P such that V P | = “n ∈ E iff ℵn+1 ∈ spectrum(a), m ∈ F iff ℵω+m+1 ∈ spectrum(a)”. No problem with {ℵ1, ℵ2, . . . } and {ℵω+2, ℵω+3, . . . } Special case: under (G)CH find c.c.c poset P s.t. V P | = {ℵn : 1 ≤ n < ω} ⊂ spectrum(a) ∧ ℵω+1 / ∈ spectrum(a)

Soukup, L (HAS) RIMS 2010 6 / 23

SLIDE 29

Coding by MADness

Code more than one set! Code just two sets! Under CH! Question: Let E, F ⊂ ω. Is there a c.c.c poset P such that V P | = “n ∈ E iff ℵn+1 ∈ spectrum(a), m ∈ F iff ℵω+m+1 ∈ spectrum(a)”. No problem with {ℵ1, ℵ2, . . . } and {ℵω+2, ℵω+3, . . . } Special case: under (G)CH find c.c.c poset P s.t. V P | = {ℵn : 1 ≤ n < ω} ⊂ spectrum(a) ∧ ℵω+1 / ∈ spectrum(a)

Theorem (Fuchino, ∼)

There is a c.c.c poset P such that in V P for each X ⊂ ω there is an

rdinal α such that X = {n ∈ ω : ℵα+n ∈ spectrum(a)}.

Soukup, L (HAS) RIMS 2010 6 / 23

SLIDE 30

Spectrum of a cardinal invariant

Soukup, L (HAS) RIMS 2010 7 / 23

SLIDE 31

Spectrum of a cardinal invariant

x cardinal invariant (e.g. a, b)

Soukup, L (HAS) RIMS 2010 7 / 23

SLIDE 32

Spectrum of a cardinal invariant

x cardinal invariant (e.g. a, b) X = {X : ϕr(X)}

Soukup, L (HAS) RIMS 2010 7 / 23

SLIDE 33

Spectrum of a cardinal invariant

x cardinal invariant (e.g. a, b) X = {X : ϕr(X)} x = min{|X| : X ∈ X} or x = sup{|X| : X ∈ X}

Soukup, L (HAS) RIMS 2010 7 / 23

SLIDE 34

Spectrum of a cardinal invariant

x cardinal invariant (e.g. a, b) X = {X : ϕr(X)} x = min{|X| : X ∈ X} or x = sup{|X| : X ∈ X} spectrum (x)= {|X| : X ∈ X}

Soukup, L (HAS) RIMS 2010 7 / 23

SLIDE 35

Spectrum of a cardinal invariant

x cardinal invariant (e.g. a, b) X = {X : ϕr(X)} x = min{|X| : X ∈ X} or x = sup{|X| : X ∈ X} spectrum (x)= {|X| : X ∈ X} Characterize spectrum (x) for different cardinal invariants!

Soukup, L (HAS) RIMS 2010 7 / 23

SLIDE 36

Cofinality spectrum of groups

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 37

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 38

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows:

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 39

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups.

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 40

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups. Shelah and Thomas: (1) if {κn : n < ω} ∈

(Sym(ω))

ω increasing then pcf({κn : n < ω}) ⊂ CF(Sym(ω))

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 41

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups. Shelah and Thomas: (1) if {κn : n < ω} ∈

(Sym(ω))

ω increasing then pcf({κn : n < ω}) ⊂ CF(Sym(ω)) (2) IF GCH holds and K ⊂ Reg s.t

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 42

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups. Shelah and Thomas: (1) if {κn : n < ω} ∈

(Sym(ω))

ω increasing then pcf({κn : n < ω}) ⊂ CF(Sym(ω)) (2) IF GCH holds and K ⊂ Reg s.t

(i) K has maximal elements,

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 43

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups. Shelah and Thomas: (1) if {κn : n < ω} ∈

(Sym(ω))

ω increasing then pcf({κn : n < ω}) ⊂ CF(Sym(ω)) (2) IF GCH holds and K ⊂ Reg s.t

(i) K has maximal elements, (ii) if µ is singular, sup(K ∩ µ) = µ then µ+ ∈ K

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 44

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups. Shelah and Thomas: (1) if {κn : n < ω} ∈

(Sym(ω))

ω increasing then pcf({κn : n < ω}) ⊂ CF(Sym(ω)) (2) IF GCH holds and K ⊂ Reg s.t

(i) K has maximal elements, (ii) if µ is singular, sup(K ∩ µ) = µ then µ+ ∈ K (iii) if µ is inaccessible, sup(K ∩ µ) = µ then µ ∈ K,

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 45

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups. Shelah and Thomas: (1) if {κn : n < ω} ∈

(Sym(ω))

ω increasing then pcf({κn : n < ω}) ⊂ CF(Sym(ω)) (2) IF GCH holds and K ⊂ Reg s.t

(i) K has maximal elements, (ii) if µ is singular, sup(K ∩ µ) = µ then µ+ ∈ K (iii) if µ is inaccessible, sup(K ∩ µ) = µ then µ ∈ K,

THEN CF(Sym(ω)) = K in some c.c.c generic extension

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 46

Cofinality spectrum of groups

Sym(ω) the group of all permutation of natural numbers Define the cofinality spectrum of Sym(ω) as follows: λ ∈ CF(Sym(ω)) iff Sym(ω) is the union of an increasing chain of λ proper subgroups. Shelah and Thomas: (1) if {κn : n < ω} ∈

(Sym(ω))

ω increasing then pcf({κn : n < ω}) ⊂ CF(Sym(ω)) (2) IF GCH holds and K ⊂ Reg s.t

(i) K has maximal elements, (ii) if µ is singular, sup(K ∩ µ) = µ then µ+ ∈ K (iii) if µ is inaccessible, sup(K ∩ µ) = µ then µ ∈ K,

THEN CF(Sym(ω)) = K in some c.c.c generic extension Problem: Full characterization of CF(Sym(ω))

Soukup, L (HAS) RIMS 2010 8 / 23

SLIDE 47

Additivity spectrum of ideals

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 48

Additivity spectrum of ideals

I ideal

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 49

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I.

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 50

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I. κ ∈ ADD(I) iff there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I.

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 51

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I. κ ∈ ADD(I) iff there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. M meager ideal; N null ideal

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 52

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I. κ ∈ ADD(I) iff there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. M meager ideal; N null ideal add(M) = min(ADD(M)), add(N) = min(ADD(N))

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 53

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I. κ ∈ ADD(I) iff there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. M meager ideal; N null ideal add(M) = min(ADD(M)), add(N) = min(ADD(N)) B the σ-ideal generated by the compact subsets of the irrationals.

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 54

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I. κ ∈ ADD(I) iff there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. M meager ideal; N null ideal add(M) = min(ADD(M)), add(N) = min(ADD(N)) B the σ-ideal generated by the compact subsets of the irrationals. (R \ Q) ≈ ωω

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 55

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I. κ ∈ ADD(I) iff there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. M meager ideal; N null ideal add(M) = min(ADD(M)), add(N) = min(ADD(N)) B the σ-ideal generated by the compact subsets of the irrationals. (R \ Q) ≈ ωω A ⊂ R \ Q is compact iff A is ≤-bounded in ωω, ≤.

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 56

Additivity spectrum of ideals

I ideal ADD(I) : the additivity spectrum of I. κ ∈ ADD(I) iff there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. M meager ideal; N null ideal add(M) = min(ADD(M)), add(N) = min(ADD(N)) B the σ-ideal generated by the compact subsets of the irrationals. (R \ Q) ≈ ωω A ⊂ R \ Q is compact iff A is ≤-bounded in ωω, ≤. B = {B ⊂ ωω : B is ≤∗-bounded in ωω, ≤∗}

Soukup, L (HAS) RIMS 2010 9 / 23

SLIDE 57

Additivity spectrum of ideals

Soukup, L (HAS) RIMS 2010 10 / 23

SLIDE 58

Additivity spectrum of ideals

ADD(I, A)={κ ∈ Reg : ∃ increasing {Aα : α < κ} ⊂ I s.t. ∪α<κAα = A}

Soukup, L (HAS) RIMS 2010 10 / 23

SLIDE 59

Additivity spectrum of ideals

ADD(I, A)={κ ∈ Reg : ∃ increasing {Aα : α < κ} ⊂ I s.t. ∪α<κAα = A} ADD(I)= ∪{ADD(I, A) : A ∈ I+}.

Soukup, L (HAS) RIMS 2010 10 / 23

SLIDE 60

Additivity spectrum of ideals

ADD(I, A)={κ ∈ Reg : ∃ increasing {Aα : α < κ} ⊂ I s.t. ∪α<κAα = A} ADD(I)= ∪{ADD(I, A) : A ∈ I+}.

Theorem

Assume that I ⊂ P(I) is a σ-complete ideal, Y ∈ I+, and A ⊂ ADD(I, Y) is countable. Then pcf(A) ⊂ ADD(I, Y).

Soukup, L (HAS) RIMS 2010 10 / 23

SLIDE 61

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 62

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 63

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ Let {gα : α < κ} ⊂ A be ≤U-increasing, ≤U-cofinal sequence.

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 64

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ Let {gα : α < κ} ⊂ A be ≤U-increasing, ≤U-cofinal sequence. For g ∈ A let U(g)=

x ∈ I : {a ∈ A : x ∈ F a

g(a)} ∈ U

.

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 65

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ Let {gα : α < κ} ⊂ A be ≤U-increasing, ≤U-cofinal sequence. For g ∈ A let U(g)=

x ∈ I : {a ∈ A : x ∈ F a

g(a)} ∈ U

.

U(gα) : α < κ witnesses that κ ∈ ADD(I, Y)

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 66

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ Let {gα : α < κ} ⊂ A be ≤U-increasing, ≤U-cofinal sequence. For g ∈ A let U(g)=

x ∈ I : {a ∈ A : x ∈ F a

g(a)} ∈ U

.

U(gα) : α < κ witnesses that κ ∈ ADD(I, Y) (1) U(g) ∈ I for each g ∈ A

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 67

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ Let {gα : α < κ} ⊂ A be ≤U-increasing, ≤U-cofinal sequence. For g ∈ A let U(g)=

x ∈ I : {a ∈ A : x ∈ F a

g(a)} ∈ U

.

U(gα) : α < κ witnesses that κ ∈ ADD(I, Y) (1) U(g) ∈ I for each g ∈ A (2) If g1 ≤I g2 then U(g1) ⊂ U(g2).

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 68

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ Let {gα : α < κ} ⊂ A be ≤U-increasing, ≤U-cofinal sequence. For g ∈ A let U(g)=

x ∈ I : {a ∈ A : x ∈ F a

g(a)} ∈ U

.

U(gα) : α < κ witnesses that κ ∈ ADD(I, Y) (1) U(g) ∈ I for each g ∈ A (2) If g1 ≤I g2 then U(g1) ⊂ U(g2). (3) {U(gα) : α < κ} = Y.

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 69

pcf(A) ⊂ ADD(I, Y) for countable A ⊂ ADD(I, Y)

For a ∈ A let Fa = {F a

α : α < a} ⊂ I increasing Fa = Y.

Let κ ∈ pcf(A). Fix an ultrafilter U on A such that cf( A/U) = κ Let {gα : α < κ} ⊂ A be ≤U-increasing, ≤U-cofinal sequence. For g ∈ A let U(g)=

x ∈ I : {a ∈ A : x ∈ F a

g(a)} ∈ U

.

U(gα) : α < κ witnesses that κ ∈ ADD(I, Y) (1) U(g) ∈ I for each g ∈ A (2) If g1 ≤I g2 then U(g1) ⊂ U(g2). (3) {U(gα) : α < κ} = Y. Does A ⊂ ADD(I) imply pcf(A) ⊂ ADD(I)? What happens if |A| = ω?

Soukup, L (HAS) RIMS 2010 11 / 23

SLIDE 70

The ideals B and N: restrictions

Soukup, L (HAS) RIMS 2010 12 / 23

SLIDE 71

The ideals B and N: restrictions

Theorem

If A ⊂ ADD(N) is countable, then pcf(A) ⊂ ADD(N).

Soukup, L (HAS) RIMS 2010 12 / 23

SLIDE 72

The ideals B and N: restrictions

Theorem

If A ⊂ ADD(N) is countable, then pcf(A) ⊂ ADD(N). Problem: What about M?

Soukup, L (HAS) RIMS 2010 12 / 23

SLIDE 73

The ideals B and N: restrictions

Theorem

If A ⊂ ADD(N) is countable, then pcf(A) ⊂ ADD(N). Problem: What about M? If κn ∈ ADD(N, Yn) for n ∈ ω then {κn : n < ω} ⊂ ADD(N, Y) for some Y ∈ N +.

Soukup, L (HAS) RIMS 2010 12 / 23

SLIDE 74

The ideals B and N: restrictions

Theorem

If A ⊂ ADD(N) is countable, then pcf(A) ⊂ ADD(N). Problem: What about M? If κn ∈ ADD(N, Yn) for n ∈ ω then {κn : n < ω} ⊂ ADD(N, Y) for some Y ∈ N +. Problem: Given κn ∈ ADD(M, Yn) for n ∈ ω then find Y ∈ M+ s.t. {κn : n < ω} ⊂ ADD(M, Y).

Soukup, L (HAS) RIMS 2010 12 / 23

SLIDE 75

The ideals B and N: restrictions

Theorem

If A ⊂ ADD(N) is countable, then pcf(A) ⊂ ADD(N). Problem: What about M? If κn ∈ ADD(N, Yn) for n ∈ ω then {κn : n < ω} ⊂ ADD(N, Y) for some Y ∈ N +. Problem: Given κn ∈ ADD(M, Yn) for n ∈ ω then find Y ∈ M+ s.t. {κn : n < ω} ⊂ ADD(M, Y). B is the σ-ideal generated by the compact subsets of the irrationals.

Soukup, L (HAS) RIMS 2010 12 / 23

SLIDE 76

The ideals B and N: restrictions

Theorem

If A ⊂ ADD(N) is countable, then pcf(A) ⊂ ADD(N). Problem: What about M? If κn ∈ ADD(N, Yn) for n ∈ ω then {κn : n < ω} ⊂ ADD(N, Y) for some Y ∈ N +. Problem: Given κn ∈ ADD(M, Yn) for n ∈ ω then find Y ∈ M+ s.t. {κn : n < ω} ⊂ ADD(M, Y). B is the σ-ideal generated by the compact subsets of the irrationals.

Theorem

If A ⊂ ADD(B) is progressive and |A| < h, then pcf(A) ⊂ ADD(B).

Soukup, L (HAS) RIMS 2010 12 / 23

SLIDE 77

The ideals B and N: construction

Soukup, L (HAS) RIMS 2010 13 / 23

SLIDE 78

The ideals B and N: construction

Theorem

Assume that I is one of the ideals B, M and N.

Soukup, L (HAS) RIMS 2010 13 / 23

SLIDE 79

The ideals B and N: construction

Theorem

Assume that I is one of the ideals B, M and N. If A = pcf(A) is a non-empty set of uncountable regular cardinals, |A| < min(A)+n for some n ∈ ω, then A = ADD(I) in some c.c.c generic extension V P.

Soukup, L (HAS) RIMS 2010 13 / 23

SLIDE 80

The ideals B and N: construction

Theorem

Assume that I is one of the ideals B, M and N. If A = pcf(A) is a non-empty set of uncountable regular cardinals, |A| < min(A)+n for some n ∈ ω, then A = ADD(I) in some c.c.c generic extension V P. If ∅ = Y ⊂ pcf({ℵn : 1 ≤ n < ω}) then pcf(Y) = ADD(I) in some c.c.c generic extension V P.

Soukup, L (HAS) RIMS 2010 13 / 23

SLIDE 81

The ideals B and N: construction

Theorem

Assume that I is one of the ideals B, M and N. If A = pcf(A) is a non-empty set of uncountable regular cardinals, |A| < min(A)+n for some n ∈ ω, then A = ADD(I) in some c.c.c generic extension V P. If ∅ = Y ⊂ pcf({ℵn : 1 ≤ n < ω}) then pcf(Y) = ADD(I) in some c.c.c generic extension V P. If ℵω+1 < max pcf({ℵn : 1 ≤ n < ω}) then there is an infinite Y ⊂ {ℵn : 1 ≤ n < ω} such that ADD(I) = Y ∪ {ℵω+2} in some c.c.c generic extension V P.

Soukup, L (HAS) RIMS 2010 13 / 23

SLIDE 82

The ideals B and N: construction

Theorem: Assume that I is one of the ideals B, M and N. If A = pcf(A) is a non-empty set of uncountable regular cardinals, |A| < min(A)+n for some n ∈ ω, then A = ADD(I) in some c.c.c generic extension V P.

Soukup, L (HAS) RIMS 2010 14 / 23

SLIDE 83

The ideals B and N: construction

Theorem: Assume that I is one of the ideals B, M and N. If A = pcf(A) is a non-empty set of uncountable regular cardinals, |A| < min(A)+n for some n ∈ ω, then A = ADD(I) in some c.c.c generic extension V P.

The ideals B, M and N have the Hechler property

Soukup, L (HAS) RIMS 2010 14 / 23

SLIDE 84

The ideals B and N: construction

Theorem: Assume that I is one of the ideals B, M and N. If A = pcf(A) is a non-empty set of uncountable regular cardinals, |A| < min(A)+n for some n ∈ ω, then A = ADD(I) in some c.c.c generic extension V P.

The ideals B, M and N have the Hechler property I has the Hechler property iff given any σ-directed poset Q there is a c.c.c poset P such that V P | = a cofinal subset {Iq : q ∈ Q} of I, ⊂ is isomorphic to Q.

Soukup, L (HAS) RIMS 2010 14 / 23

SLIDE 85

Hechler property

Hechler: B has the Hechler property,

Soukup, L (HAS) RIMS 2010 15 / 23

SLIDE 86

Hechler property

Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property,

Soukup, L (HAS) RIMS 2010 15 / 23

SLIDE 87

Hechler property

Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property.

Soukup, L (HAS) RIMS 2010 15 / 23

SLIDE 88

Hechler property

Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property. Hechler: ωω, ≤∗ has the Hechler property,

Soukup, L (HAS) RIMS 2010 15 / 23

SLIDE 89

Hechler property

Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property. Hechler: ωω, ≤∗ has the Hechler property, Define map Φ : ωω, ≤∗ → B by the formula Φ(b) = {x : x ≤∗ b}

Soukup, L (HAS) RIMS 2010 15 / 23

SLIDE 90

Hechler property

Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property. Hechler: ωω, ≤∗ has the Hechler property, Define map Φ : ωω, ≤∗ → B by the formula Φ(b) = {x : x ≤∗ b} Φ is a natural, cofinal, order preserving embedding.

Soukup, L (HAS) RIMS 2010 15 / 23

SLIDE 91

How to obtain a model of A = ADD(I)?

Soukup, L (HAS) RIMS 2010 16 / 23

SLIDE 92

How to obtain a model of A = ADD(I)?

A = pcf(A), |A| < min(A)+n

Soukup, L (HAS) RIMS 2010 16 / 23

SLIDE 93

How to obtain a model of A = ADD(I)?

A = pcf(A), |A| < min(A)+n Q= A, ≤.

Soukup, L (HAS) RIMS 2010 16 / 23

SLIDE 94

How to obtain a model of A = ADD(I)?

A = pcf(A), |A| < min(A)+n Q= A, ≤. I has the Hechler property: f : Q, ≤ ֒ → I, ⊂

Soukup, L (HAS) RIMS 2010 16 / 23

SLIDE 95

How to obtain a model of A = ADD(I)?

A = pcf(A), |A| < min(A)+n Q= A, ≤. I has the Hechler property: f : Q, ≤ ֒ → I, ⊂ A ⊂ ADD(I) is easy

Soukup, L (HAS) RIMS 2010 16 / 23

SLIDE 96

How to obtain a model of A = ADD(I)?

A = pcf(A), |A| < min(A)+n Q= A, ≤. I has the Hechler property: f : Q, ≤ ֒ → I, ⊂ A ⊂ ADD(I) is easy Need:λ / ∈ A then λ / ∈ ADD(I)

Soukup, L (HAS) RIMS 2010 16 / 23

SLIDE 97

How to obtain a model of A = ADD(I)?

A = pcf(A), |A| < min(A)+n Q= A, ≤. I has the Hechler property: f : Q, ≤ ֒ → I, ⊂ A ⊂ ADD(I) is easy Need:λ / ∈ A then λ / ∈ ADD(I) Key observation: If B = pcf(B) is a progressive set of regular cardinals, λ / ∈ B, then for each {fi : i < λ} ⊂ B there is g ∈ B such that |{i : fi ≤ g}| = λ.

Soukup, L (HAS) RIMS 2010 16 / 23

SLIDE 98

The ideals B and N

Soukup, L (HAS) RIMS 2010 17 / 23

SLIDE 99

The ideals B and N

Theorem

Assume that I = B or I = N. Given a nonempty, countable subset A

f uncountable regular cardinals, T.F

.A.E

Soukup, L (HAS) RIMS 2010 17 / 23

SLIDE 100

The ideals B and N

Theorem

Assume that I = B or I = N. Given a nonempty, countable subset A

f uncountable regular cardinals, T.F

.A.E A = pcf(A)

Soukup, L (HAS) RIMS 2010 17 / 23

SLIDE 101

The ideals B and N

Theorem

Assume that I = B or I = N. Given a nonempty, countable subset A

f uncountable regular cardinals, T.F

.A.E A = pcf(A) A = ADD(I) in some c.c.c generic extension.

Soukup, L (HAS) RIMS 2010 17 / 23

SLIDE 102

Different spectrums of posets

Soukup, L (HAS) RIMS 2010 18 / 23

SLIDE 103

Different spectrums of posets

Let P = P, ≤ be a poset

Soukup, L (HAS) RIMS 2010 18 / 23

SLIDE 104

Different spectrums of posets

Let P = P, ≤ be a poset the chain-spectrum of P, S↑(P)= {κ : ∃ unbounded, increasing chain pα : α < κ ⊂ P}

Soukup, L (HAS) RIMS 2010 18 / 23

SLIDE 105

Different spectrums of posets

Let P = P, ≤ be a poset the chain-spectrum of P, S↑(P)= {κ : ∃ unbounded, increasing chain pα : α < κ ⊂ P} the hereditarily unbounded spectrum of P, Sh(P)= {|A| : ∀A′ ∈

A

|A| A′ is unbounded ∧∀B ∈

A

<|A| B is bounded}

Soukup, L (HAS) RIMS 2010 18 / 23

SLIDE 106

Different spectrums of posets

Let P = P, ≤ be a poset the chain-spectrum of P, S↑(P)= {κ : ∃ unbounded, increasing chain pα : α < κ ⊂ P} the hereditarily unbounded spectrum of P, Sh(P)= {|A| : ∀A′ ∈

A

|A| A′ is unbounded ∧∀B ∈

A

<|A| B is bounded} the (unbounded) spectrum of P, S(P)= {|A| : A is unbounded ∧∀B ∈

A

<|A| B is bounded},

Soukup, L (HAS) RIMS 2010 18 / 23

SLIDE 107

Different spectrums of posets

Let P = P, ≤ be a poset the chain-spectrum of P, S↑(P)= {κ : ∃ unbounded, increasing chain pα : α < κ ⊂ P} the hereditarily unbounded spectrum of P, Sh(P)= {|A| : ∀A′ ∈

A

|A| A′ is unbounded ∧∀B ∈

A

<|A| B is bounded} the (unbounded) spectrum of P, S(P)= {|A| : A is unbounded ∧∀B ∈

A

<|A| B is bounded}, the unbounded subset spectrum of P, Ss(P)= {|A| : A ⊂ P(P) ∧ ∪A is unbounded ∧∀B ∈

A

<|A| ∪ B is bounded}.

Soukup, L (HAS) RIMS 2010 18 / 23

SLIDE 108

Different spectrums of posets

Let P = P, ≤ be a poset the chain-spectrum of P, S↑(P)= {κ : ∃ unbounded, increasing chain pα : α < κ ⊂ P} the hereditarily unbounded spectrum of P, Sh(P)= {|A| : ∀A′ ∈

A

|A| A′ is unbounded ∧∀B ∈

A

<|A| B is bounded} the (unbounded) spectrum of P, S(P)= {|A| : A is unbounded ∧∀B ∈

A

<|A| B is bounded}, the unbounded subset spectrum of P, Ss(P)= {|A| : A ⊂ P(P) ∧ ∪A is unbounded ∧∀B ∈

A

<|A| ∪ B is bounded}. S↑(P) ⊂ Sh(P) ⊂ S(P) ⊂ Ss(P)

Soukup, L (HAS) RIMS 2010 18 / 23

SLIDE 109

The poset ωω, ≤∗

S↑(P) chain spectrum Sh(P) hereditary unbounded spectrum Sh(P) (unbounded) spectrum Ss(P) unbounded subset spectrum

Soukup, L (HAS) RIMS 2010 19 / 23

SLIDE 110

The poset ωω, ≤∗

S↑(P) chain spectrum Sh(P) hereditary unbounded spectrum Sh(P) (unbounded) spectrum Ss(P) unbounded subset spectrum S↑(ωω, ≤∗) ⊆ Sh(ωω, ≤∗) ⊆ S(ωω, ≤∗) ⊆ Ss(ωω, ≤∗)

Soukup, L (HAS) RIMS 2010 19 / 23

SLIDE 111

The poset ωω, ≤∗

S↑(P) chain spectrum Sh(P) hereditary unbounded spectrum Sh(P) (unbounded) spectrum Ss(P) unbounded subset spectrum S↑(ωω, ≤∗) ⊆ Sh(ωω, ≤∗) ⊆ S(ωω, ≤∗) ⊆ Ss(ωω, ≤∗) ADD(B) = Ss(ωω, ≤∗)

Soukup, L (HAS) RIMS 2010 19 / 23

SLIDE 112

The poset ωω, ≤∗

S↑(P) chain spectrum Sh(P) hereditary unbounded spectrum Sh(P) (unbounded) spectrum Ss(P) unbounded subset spectrum S↑(ωω, ≤∗) ⊆ Sh(ωω, ≤∗) ⊆ S(ωω, ≤∗) ⊆ Ss(ωω, ≤∗) ADD(B) = Ss(ωω, ≤∗) b = min S↑(ωω, ≤∗) = min ADD(B)

Soukup, L (HAS) RIMS 2010 19 / 23

SLIDE 113

The poset ωω, ≤∗

S↑(P) chain spectrum Sh(P) hereditary unbounded spectrum Sh(P) (unbounded) spectrum Ss(P) unbounded subset spectrum S↑(ωω, ≤∗) ⊆ Sh(ωω, ≤∗) ⊆ S(ωω, ≤∗) ⊆ Ss(ωω, ≤∗) ADD(B) = Ss(ωω, ≤∗) b = min S↑(ωω, ≤∗) = min ADD(B) Brendle, La Berge ℵ2 ∈ S(ωω, ≤∗) \ Sh(ωω, ≤∗)

Soukup, L (HAS) RIMS 2010 19 / 23

SLIDE 114

The poset ωω, ≤∗

S↑(P) chain spectrum Sh(P) hereditary unbounded spectrum Sh(P) (unbounded) spectrum Ss(P) unbounded subset spectrum S↑(ωω, ≤∗) ⊆ Sh(ωω, ≤∗) ⊆ S(ωω, ≤∗) ⊆ Ss(ωω, ≤∗) ADD(B) = Ss(ωω, ≤∗) b = min S↑(ωω, ≤∗) = min ADD(B) Brendle, La Berge ℵ2 ∈ S(ωω, ≤∗) \ Sh(ωω, ≤∗) Brendle, Fuchino: ℵ2 ∈ Sh(ωω, ≤∗) \ S↑(ωω, ≤∗)

Soukup, L (HAS) RIMS 2010 19 / 23

SLIDE 115

The poset ωω, ≤∗

S↑(P) chain spectrum Sh(P) hereditary unbounded spectrum Sh(P) (unbounded) spectrum Ss(P) unbounded subset spectrum S↑(ωω, ≤∗) ⊆ Sh(ωω, ≤∗) ⊆ S(ωω, ≤∗) ⊆ Ss(ωω, ≤∗) ADD(B) = Ss(ωω, ≤∗) b = min S↑(ωω, ≤∗) = min ADD(B) Brendle, La Berge ℵ2 ∈ S(ωω, ≤∗) \ Sh(ωω, ≤∗) Brendle, Fuchino: ℵ2 ∈ Sh(ωω, ≤∗) \ S↑(ωω, ≤∗) Open: S(ωω, ≤∗) Ss(ωω, ≤∗)

Soukup, L (HAS) RIMS 2010 19 / 23

SLIDE 116

The poset HE

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 117

The poset HE

Let E, <E be a poset. Define HE = HE, ≺HE as follows.

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 118

The poset HE

Let E, <E be a poset. Define HE = HE, ≺HE as follows. p = Fp, np, fp ∈ HE iff

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 119

The poset HE

Let E, <E be a poset. Define HE = HE, ≺HE as follows. p = Fp, np, fp ∈ HE iff

FP ∈

E

<ω, np < ω, fp : Fp × np → ω

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 120

The poset HE

Let E, <E be a poset. Define HE = HE, ≺HE as follows. p = Fp, np, fp ∈ HE iff

FP ∈

E

<ω, np < ω, fp : Fp × np → ω

p ≤ q iff

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 121

The poset HE

Let E, <E be a poset. Define HE = HE, ≺HE as follows. p = Fp, np, fp ∈ HE iff

FP ∈

E

<ω, np < ω, fp : Fp × np → ω

p ≤ q iff

Pp ⊇ Fq, np ≥ nq, fp ⊇ fq,

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 122

The poset HE

Let E, <E be a poset. Define HE = HE, ≺HE as follows. p = Fp, np, fp ∈ HE iff

FP ∈

E

<ω, np < ω, fp : Fp × np → ω

p ≤ q iff

Pp ⊇ Fq, np ≥ nq, fp ⊇ fq, fp(a)(i) ≤ fp(b)(i) for all a <E b in Fq and all i ∈ [nq, np)

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 123

The poset HE

Let E, <E be a poset. Define HE = HE, ≺HE as follows. p = Fp, np, fp ∈ HE iff

FP ∈

E

<ω, np < ω, fp : Fp × np → ω

p ≤ q iff

Pp ⊇ Fq, np ≥ nq, fp ⊇ fq, fp(a)(i) ≤ fp(b)(i) for all a <E b in Fq and all i ∈ [nq, np)

Theorem (Farah)

If κ > ω1 is regular then V HE | = there is a κ-chain in ωω, ≤∗ iff one

f the followings happens

(1) E contains a κ-chain or κ∗-chain (2) adding a single Cohen real introduces a κ-chain.

Soukup, L (HAS) RIMS 2010 20 / 23

SLIDE 124

The poset ωω, ≤∗

Theorem: If κ > ω1 is regular then V HE | = there is a κ-chain in ωω, ≤∗ iff

ne of the followings happens

(1) E contains a κ-chain or κ∗-chain (2) adding a single Cohen real introduces a κ-chain.

Soukup, L (HAS) RIMS 2010 21 / 23

SLIDE 125

The poset ωω, ≤∗

Theorem: If κ > ω1 is regular then V HE | = there is a κ-chain in ωω, ≤∗ iff

ne of the followings happens

(1) E contains a κ-chain or κ∗-chain (2) adding a single Cohen real introduces a κ-chain.

E poset, HE embeds E generically into ωω, ≤∗

Soukup, L (HAS) RIMS 2010 21 / 23

SLIDE 126

The poset ωω, ≤∗

Theorem: If κ > ω1 is regular then V HE | = there is a κ-chain in ωω, ≤∗ iff

ne of the followings happens

(1) E contains a κ-chain or κ∗-chain (2) adding a single Cohen real introduces a κ-chain.

E poset, HE embeds E generically into ωω, ≤∗

Corollary

Assume GCH in the ground model. Given any nonempty set K of uncountable regular cardinals there is a c.c.c poset HEK such that V HEK | = K ⊂ S↑(ωω, ≤∗) ⊂ K ∪ {ℵ1}.

Soukup, L (HAS) RIMS 2010 21 / 23

SLIDE 127

Questions

Corollary: Assume GCH in the ground model. Given any nonempty set K of uncountable regular cardinals there is a c.c.c poset HK such that V HK | = K ⊂ S↑(ωω, ≤∗) ⊂ K ∪ {ℵ1}.

Soukup, L (HAS) RIMS 2010 22 / 23

SLIDE 128

Questions

Corollary: Assume GCH in the ground model. Given any nonempty set K of uncountable regular cardinals there is a c.c.c poset HK such that V HK | = K ⊂ S↑(ωω, ≤∗) ⊂ K ∪ {ℵ1}.

Do we need GCH in the corollary above?

Soukup, L (HAS) RIMS 2010 22 / 23

SLIDE 129

Questions

Corollary: Assume GCH in the ground model. Given any nonempty set K of uncountable regular cardinals there is a c.c.c poset HK such that V HK | = K ⊂ S↑(ωω, ≤∗) ⊂ K ∪ {ℵ1}.

Do we need GCH in the corollary above? Can we obtain K = S↑(ωω, ≤∗)?

Soukup, L (HAS) RIMS 2010 22 / 23

SLIDE 130

Questions

Corollary: Assume GCH in the ground model. Given any nonempty set K of uncountable regular cardinals there is a c.c.c poset HK such that V HK | = K ⊂ S↑(ωω, ≤∗) ⊂ K ∪ {ℵ1}.

Do we need GCH in the corollary above? Can we obtain K = S↑(ωω, ≤∗)? Can we prove (some form of) a strong Hechler theorem?

Soukup, L (HAS) RIMS 2010 22 / 23

SLIDE 131

Questions

Corollary: Assume GCH in the ground model. Given any nonempty set K of uncountable regular cardinals there is a c.c.c poset HK such that V HK | = K ⊂ S↑(ωω, ≤∗) ⊂ K ∪ {ℵ1}.

Do we need GCH in the corollary above? Can we obtain K = S↑(ωω, ≤∗)? Can we prove (some form of) a strong Hechler theorem? If Q is a σ-directed poset then there is a c.c.c poset P s. t. V P | = a cofinal subset of ωω, ≤∗ is order isomorphic to Q

Soukup, L (HAS) RIMS 2010 22 / 23

SLIDE 132

Questions

Corollary: Assume GCH in the ground model. Given any nonempty set K of uncountable regular cardinals there is a c.c.c poset HK such that V HK | = K ⊂ S↑(ωω, ≤∗) ⊂ K ∪ {ℵ1}.

Do we need GCH in the corollary above? Can we obtain K = S↑(ωω, ≤∗)? Can we prove (some form of) a strong Hechler theorem? If Q is a σ-directed poset then there is a c.c.c poset P s. t. V P | = a cofinal subset of ωω, ≤∗ is order isomorphic to Q AND S↑(ωω, ≤∗) = S↑(Q)

Soukup, L (HAS) RIMS 2010 22 / 23

SLIDE 133

Questions

Soukup, L (HAS) RIMS 2010 23 / 23

SLIDE 134

Questions

{ℵn : 1 ≤ n < ω} ⊂ CF(sym(ω)) = ⇒ ℵω+1 ∈ CF(sym(ω)).

Soukup, L (HAS) RIMS 2010 23 / 23

SLIDE 135

Questions

{ℵn : 1 ≤ n < ω} ⊂ CF(sym(ω)) = ⇒ ℵω+1 ∈ CF(sym(ω)). {ℵn : 1 ≤ n < ω} ⊂ ADD(B) = ⇒ ℵω+1 ∈ ADD(B).

Soukup, L (HAS) RIMS 2010 23 / 23

SLIDE 136

Questions

{ℵn : 1 ≤ n < ω} ⊂ CF(sym(ω)) = ⇒ ℵω+1 ∈ CF(sym(ω)). {ℵn : 1 ≤ n < ω} ⊂ ADD(B) = ⇒ ℵω+1 ∈ ADD(B). {ℵn : 1 ≤ n < ω} ⊂ S↑(ωω, ≤∗) = ⇒ ℵω+1 ∈ S↑(ωω, ≤∗)

Soukup, L (HAS) RIMS 2010 23 / 23

SLIDE 137

Questions

{ℵn : 1 ≤ n < ω} ⊂ CF(sym(ω)) = ⇒ ℵω+1 ∈ CF(sym(ω)). {ℵn : 1 ≤ n < ω} ⊂ ADD(B) = ⇒ ℵω+1 ∈ ADD(B). {ℵn : 1 ≤ n < ω} ⊂ S↑(ωω, ≤∗) = ⇒ ℵω+1 ∈ S↑(ωω, ≤∗) {ℵn : 1 ≤ n < ω} ⊂ ADD(N) = ⇒ ℵω+1 ∈ ADD(N).

Soukup, L (HAS) RIMS 2010 23 / 23

SLIDE 138

Questions

{ℵn : 1 ≤ n < ω} ⊂ CF(sym(ω)) = ⇒ ℵω+1 ∈ CF(sym(ω)). {ℵn : 1 ≤ n < ω} ⊂ ADD(B) = ⇒ ℵω+1 ∈ ADD(B). {ℵn : 1 ≤ n < ω} ⊂ S↑(ωω, ≤∗) = ⇒ ℵω+1 ∈ S↑(ωω, ≤∗) {ℵn : 1 ≤ n < ω} ⊂ ADD(N) = ⇒ ℵω+1 ∈ ADD(N). {ℵn : 1 ≤ n < ω} ⊂ ADD(M)

?

= ⇒ ℵω+1 ∈ ADD(M).

Soukup, L (HAS) RIMS 2010 23 / 23

SLIDE 139

Questions

{ℵn : 1 ≤ n < ω} ⊂ CF(sym(ω)) = ⇒ ℵω+1 ∈ CF(sym(ω)). {ℵn : 1 ≤ n < ω} ⊂ ADD(B) = ⇒ ℵω+1 ∈ ADD(B). {ℵn : 1 ≤ n < ω} ⊂ S↑(ωω, ≤∗) = ⇒ ℵω+1 ∈ S↑(ωω, ≤∗) {ℵn : 1 ≤ n < ω} ⊂ ADD(N) = ⇒ ℵω+1 ∈ ADD(N). {ℵn : 1 ≤ n < ω} ⊂ ADD(M)

?

= ⇒ ℵω+1 ∈ ADD(M). {ℵn : 1 ≤ n < ω} ⊂ spectrum(a)

?

= ⇒ ℵω+1 ∈ spectrum(a).

Soukup, L (HAS) RIMS 2010 23 / 23

SLIDE 140

Questions

{ℵn : 1 ≤ n < ω} ⊂ CF(sym(ω)) = ⇒ ℵω+1 ∈ CF(sym(ω)). {ℵn : 1 ≤ n < ω} ⊂ ADD(B) = ⇒ ℵω+1 ∈ ADD(B). {ℵn : 1 ≤ n < ω} ⊂ S↑(ωω, ≤∗) = ⇒ ℵω+1 ∈ S↑(ωω, ≤∗) {ℵn : 1 ≤ n < ω} ⊂ ADD(N) = ⇒ ℵω+1 ∈ ADD(N). {ℵn : 1 ≤ n < ω} ⊂ ADD(M)