[PPT] - on Rather new topi: Many op en questions Continuum funtion PowerPoint Presentation

SLIDE 1 F

r ing,

Combinato ri s and Denabilit y Summa ry: T

da

y: Denable W ello rders La rge a rdinals F

r ing

axioms Ca rdinal ha ra teristi s (new!) Other

ntexts

T uesda y: Ca rdinal Cha ra teristi s

n κ

Rather new topi : Many

p

en questions Continuum fun tion 2κ Dominating, b

unding

numb ers Conalit y

f

the symmetri group Almost disjointness, splitt i ng numb ers W ednesda y: Mo dels

f

PF A , BPF A

SLIDE 2 Denable W ello rders In ZF , A C is equivalent to: H(κ+) an b e w ello rdered fo r every κ When an w e

btain

a denab le w ello rder

f

H(κ+) ?

Σ

n denab le w ello rder

f

H(κ+) : W ello rder

f

H(κ+) whi h is Σ n denab le

ver

H(κ+) with κ as a pa rameter Rema rks: 1. If n is at least 3, then κ an b e elimi nated, as {κ} is Π 2 denab le 2. If λ is a limi t a rdinal and H(κ+) has a denab le w ello rder fo r

nally

many κ < λ, then H(λ) has a denab le w ello rder

Σ

n denab le w ello rder

f

H(κ+) with pa rameters : W ello rder

f

H(κ+) whi h is Σ n denab le

ver

H(κ+) with a rbitra ry elements

f

H(κ+) as pa rameters

SLIDE 3 Denable W ello rders: La rge a rdinals and H(ω 1) The b est situation: V = L → Ea h H(κ+) = Lκ+ has a Σ 1 denab le w ello rder Denable w ello rders and La rge Ca rdinals H(ω 1)

Σ

n denab le w ello rder

f

H(ω 1) (with pa rameters) ∼

Σ

1 n+ 1 denab le w ello rder

f

the reals (with real pa rameters) Theo rem (Manseld) Σ 1 2 w ello rder

f

the reals → every real b elongs to L. (Ma rtin-Steel) A Σ 1 n+ 2 w ello rder

f

the reals is

nsistent

with n W

din

a rdinals but in onsistent with n W

din

a rdinals and a measurable a rdinal ab

ve

them.

SLIDE 4 Denable W ello rders: La rge Ca rdinals and H(ω 2) H(ω 2) A fo r ing is small i it has size less than the least ina essible. Small fo r ings p reserve all la rge a rdinals. Theo rem (Asp er-F) There is a small fo r ing whi h fo r es CH and a denab le w ello rder

f

H(ω 2) . The ab

ve

w ello rder is not Σ 1 . In fa t: Theo rem (W

din)

Measurable W

din

a rdinal + CH → there is no w ello rder

f

the reals whi h is Σ 1

ver

H(ω 2) . Ho w ever:

SLIDE 5 Denable W ello rders: La rge a rdinals and H(ω 2) Theo rem (A vraham-Shelah) There is a small fo r ing whi h fo r es ∼ CH and a w ello rder

f

the reals whi h is Σ 1

ver

H(ω 2) . Question 1. Is there a small fo r ing whi h fo r es a Σ 2 w ello rder

f

H(ω 2) ?

SLIDE 6 Denable W ello rders: La rge a rdinals and H(ω 2) Ab

ut

the p ro

f
f:

Theo rem (Asp er-F) There is a small fo r ing whi h fo r es CH and a denab le w ello rder

f

H(ω 2) . T w

ingredients:

Canoni al fun tion

ding

Strongly t yp e-guessing

ding

(Asp er)

SLIDE 7 Denable W ello rders: La rge a rdinals and H(ω 2) Canoni al funtion

ding

F

r

ea h α < ω 2 ho

se

fα : ω 1 → α

nto

and dene gα : ω 1 → ω 1 b y: gα(γ) =

rdert

yp e fα[γ]. gα is a anoni al fun tion fo r α . No w

de

A ⊆ ω 2 b y B ⊆ ω 1 as follo ws:

α ∈

A i gα(γ) ∈ B fo r a lub

f γ

Assuming GCH , the fo r ing to do this is ω

strategi all

y losed and

ω

2

.

SLIDE 8 Denable W ello rders: La rge a rdinals and H(ω 2) Asp er

ding

A lub-sequen e in ω 1

f

height τ is a sequen e C = ( Cδ | δ ∈ S) where S ⊆ ω 1 is stationa ry and ea h Cδ is lub in δ

f
rdert

yp e τ .

C

is strongly t yp e-guessing i fo r every lub C ⊆ ω 1 there is a lub D ⊆ ω 1 su h that fo r all δ in D ∩ S ,

rdert

yp e( C ∩ C +

δ ) = τ

, where C +

δ

denotes the set

f

su esso r elements

f

Cδ . An

rdinal γ

is p erfe t i ωγ = γ. Lemma (Asp er) Assume GCH . Let B ⊆ ω 1 . Then there is an

ω

strategi all

y losed, ω 2

fo

r ing that fo r es: γ ∈ B i the γ

th

p erfe t

rdinal

is the height

f

a strongly t yp e-guessing lub sequen e.

SLIDE 9 Denable W ello rders: La rge a rdinals and H(ω 2) T

p

rove: Theo rem (Asp er-F) There is a small fo r ing whi h fo r es CH and a denab le w ello rder

f

H(ω 2) . Assume GCH . W rite H(ω 2) as Lω 2[ A] , A ⊆ ω 2 . Use Canoni al fun tion

ding

to

de

A b y B ⊆ ω 1 . Use Asp er

ding

to

de

B denab ly

ver

H(ω 2) . Problem: B

nly
des

H(ω 2)

f

the ground mo del, not H(ω 2)

f

the extension! Solution: P erfo rm b

th
dings

simult aneously. The fo r ing is a hyb rid fo r ing: halfw a y b et w een iteration and p ro du t.

SLIDE 10 Denable W ello rders: La rge a rdinals and H(κ) H(κ) Theo rem (Asp er-F) There is a lass fo r ing whi h fo r es GCH , p reserves all sup er ompa t a rdinals (as w ell as a p rop er lass

f

n

huge

a rdinals fo r ea h n ) and adds a denab le w ello rder

f

H(κ+) fo r all regula r κ ≥ ω 1 . Co rolla ry There is a lass fo r ing whi h fo r es GCH , p reserves all sup er ompa t a rdinals (as w ell as a p rop er lass

f

n

huge

a rdinals fo r ea h n ) and adds a pa rameter-free denab le w ello rder

f

H(δ) fo r all a rdinals δ ≥ ω 2 whi h a re not su esso rs

f

singula rs. Su esso rs

f

singula rs? Σ 1 denab le w ello rders?

SLIDE 11 Denable W ello rders: La rge a rdinals and H(κ) Su esso rs

f

singula rs: Theo rem (Asp er-F) Supp

se

that there is a j : L( H(λ+)) → L( H(λ+)) xing λ , with riti al p

int < λ.

Then there is no denab le w ello rder

f

H(λ+) with pa rameters. Question 2. Is there a small fo r ing that adds a denab le w ello rder

f

H(ℵω+ 1) with pa rameters?

Σ

1 denab le w ello rders: Theo rem There is a lass fo r ing whi h fo r es GCH , p reserves all sup er ompa t a rdinals (as w ell as a p rop er lass

f

n

huge

a rdinals fo r ea h n ) and adds a Σ 1 denab le w ello rder

f

H(κ+) with pa rameters fo r all regula r κ ≥ ω 1 .

SLIDE 12 Denable W ello rders and F

r ing

Axioms Question 3. Is there a small fo r ing that adds a Σ 1 denab le w ello rder

f

H(ω 3) ? Denable w ello rders and F

r ing

Axioms H(ω 1) Theo rem MA is

nsistent

with a Σ 1 3 w ello rder

f

the reals. (Cai edo-F) BPF A + ω 1 = ω L 1 (whi h is

nsistent

relative to a ree ting a rdinal) implies that there is a Σ 1 3 w ello rder

f

the reals. Theo rem (Hjo rth) Assume ∼ CH and every real has a # . Then there is no

Σ

1 3 w ello rder

f

the reals.

SLIDE 13 Denable W ello rders and F

r ing

Axioms Question 4. Do es BPF A + 0# do es not exist imply that there is a

Σ

1 3 w ello rder

f

the reals? Question 5. Is BMM

nsistent

with a p roje tive w ello rder

f

the reals? PF A is not. Question 6. Is MA

nsistent

with the nonexisten e

f

a p roje tive w ello rder

f

the reals? F

r

H(ω 2) : Theo rem (Cai edo-V eli k

vi )

BPF A + ω 1 = ω L 1 implies that there is a Σ 1 denab le w ello rder

f

H(ω 2) . Theo rem (La rson) Relative to enough sup er ompa ts, there is a mo del

f

MM with a denab le w ello rder

f

H(ω 2) .

SLIDE 14 Denable W ello rders and F

r ing

Axioms F

r

la rger H(κ) : Theo rem MA is

nsistent

with a denab le w ello rder

f

H(κ+) fo r all κ . (Ree ting a rdinal) BSPF A is

nsistent

with a denab le w ello rder

f

H(κ+) fo r all κ . (Enough sup er ompa ts) MM is

nsistent

with a denab le w ello rder

f

H(κ+) fo r all regula r κ ≥ ω 1 .

SLIDE 15 Denable W ello rders and Ca rdinal Cha ra teris ti s New

ntext

fo r denab le w ello rders: Ca rdinal Cha ra teristi s T emplate iteration T : A

untable

supp

rt, ω

2

iteration

whi h adds a Σ 1 3 w ello rder

f

the reals (and a Σ 1 w ello rder

f

H(ω 2) ). It is not p rop er, but is S

p

rop er fo r ertain stationa ry S ⊆ ω 1 . Broad p roje t: Mix the template iteration with a va riet y

f

iterations fo r

ntrolling

a rdinal ha ra teristi s. Theo rem (V.Fis her

F)

Ea h

f

the follo wing is

nsistent

with a Σ 1 3 w ello rder

f

the reals: d < c , b < a = s, b < g.

b =

the b

unding

numb er, a = the almost disjointness numb er,

s =

the splitt i ng numb er, g = the group wise densit y numb er

SLIDE 16 Denable W ello rders The template iteration T is gentle (ωω b

unding)

but also exible (it an b e mixed with any

untable

supp

rt

p rop er iteration

f

p

sets
f

size ω 1 ) One an also ask fo r ni ely denab le witnesses to a rdinal ha ra teristi s. A sample result: Theo rem (F-Zdomskyy) It is

nsistent

that a = ω 2 and there is a Π 1 2 innite maximal almost disjoint family . Question 7. Is it

nsistent

with a = ω 2 that there is a Σ 1 2 innite maximal almost disjoint family?

SLIDE 17 Denable W ello rders in

ther

Contexts Questions. 8. Is it

nsistent

that fo r all innite regula r κ , GCH fails at κ and there is a denab le w ello rder

f

H(κ+) ? 9. Is the tree p rop ert y at ω 2

nsistent

with a p roje tive w ello rder

f

the reals? 10. Is it

nsistent

that the nonstationa ry ideal

n ω

1 is saturated and there is a Σ 1 4 w ello rder

f

the reals? 11. Is it

nsistent

that GCH fails at a measurable a rdinal κ and there is a denab le w ello rder

f

H(κ+) ?

SLIDE 18 Ca rdinal Cha ra teristi s at κ Ca rdinal ha ra teristi s

n ω

is a vast subje t. Examples from Blass' survey:

a, b, d, e, g, h, i, m, p, r, s, t, u

These a re all at most c , the a rdinalit y

f

the

ntinuum.

κ

regula r, un oun tab le. W e

nsider

analogues

f

some

f

the ab

ve

fo r κ

a(κ), b(κ), d(κ) . . .

Why?

SLIDE 19 Ca rdinal Cha ra teristi s at κ F

ur

reasons: 1. Higher iterated fo r ing Ca rd Cha rs ω / Countable supp

rt

iterations ≡ Ca rd Cha rs κ / Higher supp

rt

iterations 2. La rge a rdinal

ntext:

Ca rd Cha rs at a measurable 3. Global b ehaviour as κ va ries, Internal

nsisten y

4. Solve p roblems at κ that a re unsolved at ω Illustr at e with some examples

SLIDE 20 Ca rdinal Cha ra teristi s at κ The Ca rd Cha r 2κ Global b ehaviour Theo rem (Co rolla ry to Easton's Theo rem) It is

nsistent

that 2α = α++ fo r all regula r α . F

r ing

used: Easton p ro du t

f α
Cohen

fo r ings Cohen(α, α++) . Internal

nsisten y

Theo rem (F-Ondrejovi) Assuming that 0# exists, there is an inner mo del in whi h 2α = α++ fo r all regula r α . F

r ing

used: Reverse Easton iteration

f α
Cohen

fo r ings.

SLIDE 21 Ca rdinal Cha ra teristi s at κ La rge a rdinal

ntext

Theo rem (W

din)

Assume that κ is hyp ermeasurable. Then in a fo r ing extension, κ is measurable and 2κ = κ++ . F

r ing

used: Reverse Easton iteration

f α
Cohen

fo r ings, α ≤ κ ,

α

ina essible, follo w ed b y Cohen(κ+, κ++) .

SLIDE 22 Ca rdinal Cha ra teristi s at κ No w lo

k

at The Ca rd Cha r d(κ) Global Behaviour Theo rem (Cummings-Shelah) It is

nsistent

that d(α) = α+ < 2α fo r all regula r α . F

r ings

used: α

Cohen

p ro du t and α

He hler

iteration.

SLIDE 23 Ca rdinal Cha ra teristi s at κ La rge a rdinal

ntext

Theo rem (F-Thompson) Assume that κ is hyp ermeasurable. Then in a generi extension, κ is measurable and d(κ) = κ+ < 2κ . F

r ing

used: Reverse Easton iteration

f α
Sa ks

p ro du ts, α ≤ κ ,

α

ina essible. T w

interesting

p

ints:

i. If y

u

try this with κ

Cohen

and κ

He hler

then y

u

need some sup er ompa tness ii. The p ro

f

is easier than W

din's

p ro

f,

whi h

nly

gives

κ+ <

2κ

SLIDE 24 Ca rdinal Cha ra teristi s at κ La rge Ca rdinal

ntext

together with Global Behaviour Theo rem (F-Thompson) Assume that κ is hyp ermeasurable. Then in a generi extension, κ is measurable and (α) = α+ < 2α fo r all regula r α . F

r ings

used: (Reverse Easton iteration

f

) α

Sa ks

at ina essible

α ≤ κ

, α

Cohen

p ro du t follo w ed b y α

He hler

iteration at su esso rs

f

non-ina essibles, something new at α+ , α ina essible (α+

Cohen

p ro du t follo w ed b y a mixture

f α
Sa ks

p ro du t and

α+

He hler

iteration). Con lusion: Understanding d(κ) in the La rge a rdinal setting requires a a reful hoi e

f

fo r ings; mixing it with the Global Behaviour

f d(α)

requires the invention

f

new fo r ings

SLIDE 25 Ca rdinal Cha ra teristi s at κ Rema rk. (F-Honzik) Easton's Theo rem fo r 2α has b een w

rk

ed

ut

in the la rge a rdinal setting. But: Question 12. What Global Behaviours fo r d(α) a re p

ssible

when there is a measurable a rdinal?

SLIDE 26 Ca rdinal Cha ra teristi s at κ The Ca rd Cha r CofSym(α) Sym(α) = group

f

p ermutations

f α

under

mp
sition.

CofSym(α) = least λ su h that Sym(α) is the union

f

a stri tl y in reasing λ - hain

f

subgroups. Sha rp and Thomas: CofSym(α) an b e anything reasonable. But its Global Behaviour is nontrivial! Theo rem (Sha rp-Thomas) (a) Supp

se

that α < β a re regula r and GCH holds. Then in a

nalit

y-p reserving fo r ing extension, CofSym(α) = β . (b) If CofSym(α) > α+ then CofSym(α+) ≤ CofSym(α) . Question 13. Is it

nsistent

that CofSym(ω) = CofSym(ω 1) = ω 3 ?

SLIDE 27 Ca rdinal Cha ra teristi s at κ CofSym has b een studied in the La rge Ca rdinal setting: Theo rem (F-Zdomskyy) Supp

se

that κ is hyp ermeasurable. Then in a fo r ing extension, κ is measurable and CofSym(κ) = κ++ . F

r ings

used: Iteration

f

Mill er(κ) (with

ntinuous

lub-splitti ng) and a generalisation

f

Sa ks(κ) . The p ro

f

also uses g l(κ) (group wise densit y numb er fo r

ntinuous

pa rtitions).

SLIDE 28 Ca rdinal Cha ra teristi s at κ

a(κ)

and d(κ)

a(κ) =

minimum size

f

a (size at least κ ) maximal almost disjoint family

f

subsets

f κ

An

ld
p

en p roblem: Question 14. Do es d(ω) = ω 1 imply a(ω) = ω 1 ? But this is solved at un oun tab le a rdinals! Theo rem (Blass-Hyttinen-Zhang) F

r

un oun tab le α , d(α) = α+ implies

a(α) = α+

SLIDE 29 Ca rdinal Cha ra teristi s at κ Are there

ther
p

en questions fo r ω whi h an b e solved fo r un oun tab le a rdinals? Question 15. Can p(κ) b e less than t(κ) ? Ma yb e it will help to assume that κ is a la rge a rdinal. Question 16. Can s(κ) b e singula r? Mo re

p

en Questions. 17. (Without la rge a rdinals) Is b(κ) < a(κ)

nsistent

fo r an un oun tab le κ ? 18. Whi h Global Behaviours fo r b(α), d(α) a re internally

nsistent?

Cummings-Shelah answ ered this fo r

rdina

ry

nsisten y

. 19. (Without sup er ompa tness) Can s(κ) b e greater than κ+ ? Zapletal: Need (almost) a hyp ermeasurable. 20. Is it

nsistent

that CofSym(κ) = κ+++ fo r a measurable κ ?

SLIDE 30 Some mo dels

f

PF A , BPF A Let C b e a lass

f

fo r ings F A(C) = F

r ing

Axiom fo r C F

r

P in C , an hit ω 1

many

p redense sets in P with a lter

n

P BF A(C) = Bounded F

r ing

Axiom fo r C F

r

P in C , an hit ω 1

many

p redense sets

f

size ≤ ω 1 in P with a lter

n

P PF A = F A( Prop er) = Prop er F

r ing

Axiom BPF A = BF A( Prop er) = Bounded Prop er F

r ing

Axiom Useful F a t. (Baga ria, Stavi-Vnnen ) BPF A is equivalent to the

Σ

1 elementa rit y

f

H(ω 2) V in H(ω 2) V [ G] fo r p rop er P and P

generi

G

SLIDE 31 Some mo dels

f

PF A , BPF A Theo rem (a) (Baumga rtner) If there is a sup er ompa t then PF A holds in a p rop er fo r ing extension. (b) (Goldstern-Shelah) If there is a ree ting a rdinal (i.e., a regula r κ su h that H(κ) ≺Σ 2 V ) then BPF A holds in a p rop er fo r ing extension.

SLIDE 32 Some mo dels

f

PF A , BPF A Ca rdinal Minimal i t y V is a rdinal minimal i whenev e r M is an inner mo del with the

rre t

a rdinals (i.e., Ca rd M = Ca rd V ) then M = V . Lo al version: κ a a rdinal. V is κ

minimal

i whenev e r M is an inner mo del with the

rre t

a rdinals ≤ κ then H(κ) M = H(κ) . Examples L is triviall y a rdinal minimal . Let x b e κ

Sa ks, κ
Mill

er

r κ
Laver
ver

L . Then L[ x] is not a rdinal minimal . Let f : κ → κ+ b e a minimal

llapse
f κ+

to κ

ver

L . Then L[ f ] is a rdinal minimal . Mo re interesting examples: Co re mo dels

SLIDE 33 Some mo dels

f

PF A , BPF A Theo rem Let K b e the

re

mo del fo r a measurable, hyp ermeasurable, strong

r

W

din

a rdinal. Then K is a rdinal minimal . In fa t, K is

κ

minimal

fo r all κ ≥ ω 2 .

ω

1

minimal

i t y fails fo r

re

mo dels, and in fa t whenev e r 0# exists: Theo rem Supp

se

that 0# exists. Then V is not ω 1

minimal

. In fa t, there is an inner mo del M with the

rre t ω

1 whi h is a fo r ing extension

f

L.

SLIDE 34 Some mo dels

f

PF A , BPF A Another sour e

f

a rdinal minimal i t y: Mo dels

f

fo r ing axioms SPF A = F A( Semip rop er) = Semip rop er F

r ing

Axiom BSPF A = BF A( Semip rop er) = Bounded Semip rop er F

r ing

Axiom Theo rem (V eli k

vi )

Supp

se

that SPF A holds. Then V is ω 2

minimal

. There is a related result fo r BPF A: Theo rem (Cai edo-V eli k

vi )

Supp

se

that BPF A holds. Then V is

ω

2

minimal

with resp e t to inner mo dels satisfyi ng BPF A : If M is an inner mo del satisfyi ng BPF A with the

rre t ω

2 then H(ω 2) M = H(ω 2) .

SLIDE 35 Some mo dels

f

PF A , BPF A Theo rem (a) Supp

se

that there is a sup er ompa t. Then in some fo r ing extension, PF A holds and the universe is not ω 2

minimal

. (b) Supp

se

that there is a ree ting a rdinal. Then in some fo r ing extension, BPF A holds and the universe is not ω 2

minimal

. The p ro

fs

a re based

n:

Lemma (Collapsing to ω 2 with nite

nditions)

Assume GCH . Supp

se

that κ is ina essible and S denotes [κ]ω

f

V . Then there is a fo r ing P su h that: (a) P fo r es κ = ω 2 . (b) P is S

p

rop er, and hen e p reserves ω 1 , in any extension

f

V in whi h S remains stationa ry .

SLIDE 36 Some mo dels

f

PF A , BPF A W e sk et h the p ro

f
f

(a): (a) Supp

se

that there is a sup er ompa t. Then in some fo r ing extension, PF A holds and the universe is not ω 2

minimal

.

κ

sup er ompa t. Collapse κ to ω 2 with nite

nditions,

p ro du ing V [ F] . P erfo rm Baumga rtner's PF A iteration, but at stage α < ω 2 , ho

se

a fo r ing in V [ F ↾ α, Gα] whi h is S

p

rop er there; a rgue that it is also S

p

rop er in V [ F, Gα] . Imp

rtant:

Only use names from V [ F ↾ α, Gα] , to k eep the fo r ing small! Diagonal iteration V erify that PF A (indeed F A(S − Prop er) ) holds in V [ F, G] . As κ = ω 2 b

th

in V [ F] and in V [ F, G] , this sho ws that V [ F, G] is not ω 2

minimal

.

SLIDE 37 Some mo dels

f

PF A , BPF A Ho w to

llapse

an ina essible κ to ω 2 with nite

nditions?

Let # : [κ]ω → κ b e inje tive. P

nsists
f

all pairs p = ( A, S) su h that: 1. A is a nite set

f

disjoint losed intervals [α, β] , α ≤ β < κ ,

f(α) ≤ ω

1 . 2. S is a nite subset

f [κ]ω

(side

nditions).

3. T e hni al. 4. Let F b e the set

f

un oun tab le

nalit

y α fo r [α, β] in A, together with κ . The height

f

x ∈ S is the least element

f

F greater than sup x . Then: i. (Closure under trun ation) x in S , α in F implies x ∩ α in S . ii. (Almost an ∈

hain)

If x, y ∈ S have the same height then

#( x) ∈

y , #( y) ∈ x

r

x = y .

SLIDE 38 Some mo dels

f

PF A , BPF A The fo r ing is κ

and

adds a lub in κ

nsisting
nly
f
rdinals
f
nalit

y ≤ ω 1 . So κ b e omes ω 2 . Questions. 21. Supp

se

that BSPF A holds. Then is V ω 2

minimal

with resp e t to inner mo dels satisfyi ng BSPF A ? 22. Is there a fo r ing whi h

llapses

an ina essible to ω 3 with nite

nditions?