Cardinal invariants and the generalized Baire spaces Diana Carolina - - PowerPoint PPT Presentation

Cardinal invariants and the generalized Baire spaces

Diana Carolina Montoya Kurt Gödel Research Center Universität Wien

11th Young set theory workshop Bernoulli Center. Laussane, June 25, 2018

Contents

Classic cardinal invariants and their generalizations A word on Cichón’s diagram The generalized ultrafjlter number The independence number

2

Section 1 Classic cardinal invariants and their generalizations

3

Motivation

“Cardinal invariants are simply the smallest cardinals ≤ 𝔡 for which various results, true for ℵ0, become false...”

Andreas Blass, Combinatorial cardinal characteristics of the continuum, 2010

Classical cardinal invariants of the Baire space 𝜕𝜕 have been extensively studied and understood. In fact, it is possible to directly abstract several defjnitions from 𝜕 to an arbitrary uncountable cardinal 𝜆.

4

The generalized Baire spaces

Let 𝜆 be an uncountable regular cardinal satisfying 𝜆<𝜆 = 𝜆. The generalized Baire space is just the set of functions 𝜆𝜆 endowed with the topology generated by the sets of the form: [𝑡] = {𝑔 ∈ 𝜆𝜆 ∶ 𝑔 ⊇ 𝑡} for 𝑡 ∈ 𝜆<𝜆. Denote NWD𝜆 to be the collection of nowhere dense subsets of 𝜆<𝜆 with respect to this topology, recall that a set 𝐵 ⊆ 𝜆𝜆 is nowhere dense if for every 𝑡 ∈ 𝜆<𝜆 there exists 𝑢 ⊇ 𝑡 such that [𝑢] ∩ 𝐵 = ∅.

5

Then it we defjne the generalized 𝜆-meager sets in 𝜆𝜆 to be 𝜆-unions of elements in NWD𝜆 and denote ℳ𝜆 to be the 𝜆-ideal that 𝜆-meager sets determine (here

𝜆-ideal means an ideal that in addition is closed under unions of size ≤ 𝜆).

It is well known that the Baire category theorem can be lifted to this context, i.e. it holds that the intersection of 𝜆-many open dense sets is open (Friedman, Hyttinen, Kulikov, 2014).

6

“The beginning”

Since 1995, with the paper “Cardinal invariants above the continuum ” from Cum- mings and Shelah, the study of the invariants associated to these spaces and their interactions has been developing. It is also important to mention that the study of these spaces has been also ap- proached from the point of view of Descriptive Set Theory (Calderoni, Friedman, Hyttinen, Kulikov, Moreno, Motto Ros) and Topology (Korch).

7

Some cardinal invariants

Defjnition

If 𝑔, 𝑕 are functions in 𝜆𝜆, we say that 𝑔 <∗ 𝑕, if there exists an 𝛽 < 𝜆 such that for all 𝛾 > 𝛽, 𝑔(𝛾) < 𝑕(𝛾). In this case, we say that 𝑕 eventually dominates 𝑔.

Defjnition

Let 𝔊 be a family of functions from 𝜆 to 𝜆.

▶ 𝔊 is dominating, if for all 𝑕 ∈ 𝜆𝜆, there exists an 𝑔 ∈ 𝔊 such that 𝑕 <∗ 𝑔. ▶ 𝔊 is unbounded, if for all 𝑕 ∈ 𝜆𝜆, there exists an 𝑔 ∈ 𝔊 such that 𝑔 ≮∗ 𝑕.

8

The unbounding and dominating numbers

Defjnition

▶ The unbounding number:

𝔠(𝜆) = min{|𝔊|∶ 𝔊 is an unbounded family of functions in 𝜆𝜆}

▶ The dominating number:

𝔢(𝜆) = min{|𝔊|∶ 𝔊 is a dominating family of functions in 𝜆𝜆}

9

Cardinal invariants associated to an ideal

Let ℐ be a 𝜆-ideal (closed under 𝜆-sized unions) on 𝜆𝜆:

Defjnition

▶ The additivity number:

add(ℐ) = min{|𝒦|∶ 𝒦 ⊆ ℐ and ⋃ 𝒦 ∉ ℐ}.

▶ The covering number:

cov(ℐ) = min{|𝒦|∶ 𝒦 ⊆ ℐ and ⋃ 𝒦 = 𝜆𝜆}.

10

Defjnition

▶ The cofjnality number:

cof(ℐ) = min{|𝒦|∶ 𝒦 ⊆ ℐ and for all 𝑁 ∈ ℐ there is a

𝐾 ∈ 𝒦 with 𝑁 ⊆ 𝐾}.

▶ The uniformity number:

non(ℐ) = min{|𝑍 |∶ 𝑍 ⊂ 𝑌 and 𝑍 ∉ ℐ}.

11

Cichón’s diagram

Cichón’s diagram summarizes the provable ZFC relationships between some car- dinal invariants related to the 𝜏-ideals of meager and null sets (with respect to the standard product measure) on the classical Baire space.

12

Cichoń’s Diagram on the Baire space 𝜕𝜕

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✻ ✻

cof 𝒪 non 𝒪 cof ℳ

𝔢

cov ℳ add ℳ

𝔠

non ℳ cov 𝒪 add 𝒪

ℵ1 𝔡

Figure 1: Cichón’s diagram

13

Hasse’s diagram

Go to independence number

14

15

Why cardinal invariants of these spaces? I

There are some remarkable difgerences between the countable and the uncountable cases that make this study interesting and present new challenges for future

• research. Here some examples:

▶ Expected bounds: Typically, classical invariants take values in the interval

[ℵ1, 𝔡]. However, in the uncountable for instance, the generalization of the

splitting number 𝔱(𝜆) can be ≤ 𝜆, and actually large cardinals are necessary to have the expected inequality 𝔱(𝜆) ≥ 𝜆+ (Suzuki, 1998). Also, Ben-Neria and Gitik found the optimal large cardinal assumption to get 𝔱(𝜆) > 𝜆+,

• 2014. Specifjcally:

16

Why cardinal invariants of these spaces? II

Theorem

Let 𝜆, 𝜇 be regular uncountable cardinals such that 𝜆+ < 𝜇. 𝔱(𝜆) = 𝜇 is equiconsistent to the existence of a measurable cardinal 𝜆 with 𝑝(𝜆) = 𝜇.

▶ New ZFC results: Some examples

▶ Raghavan and Shelah showed that, for uncountable 𝜆, the inequality 𝔱(𝜆) ≤

𝔠(𝜆) holds whereas in the countable case, there are two difgerent forcing

extensions in which inequalities 𝔱 < 𝔠 and 𝔠 < 𝔱 hold respectively.

▶ They also proved that, if 𝜆 > ℶ𝜕 then 𝔢(𝜆) ≤ 𝔰(𝜆). Recently, Fischer and

Soukup showed (among others) that the same conclusion can be obtained under the hypothesis cf(𝔰(𝜆)) ≤ 𝜆.

17

Why cardinal invariants of these spaces? III

▶ Roitman’s problem: It asks whether from 𝔢 = ℵ1 it is possible to prove that

𝔟 = ℵ1 (still open!).

▶ So far, Shelah gave the best approximation to an answer to this problem: he

developed the method of template iteration forcing to give a model in which the inequality 𝔢 < 𝔟 is satisfjed, yet in his model the value of 𝔢 is ℵ2; the question that is still open asks if it is possible to fjnd such a model but in addition having 𝔢 = ℵ1.

▶ In the uncountable in contrast, Blass, Hyttinen and Zhang (2007) proved that,

in ZFC for uncountable regular 𝜆 Roitman’s problem can be solved on the positive, i.e. if 𝔢(𝜆) = 𝜆+, then 𝔟(𝜆) = 𝜆+.

▶ Additionally, Fischer and Soukup have proved that from the assumption 𝔢(𝜆) =

𝜆+ other relatives from 𝔟(𝜆) can be decided to have value 𝜆+. Namely, 𝔟𝑓(𝜆) = 𝔟𝑕(𝜆) = 𝜆+.

18

Why cardinal invariants of these spaces? IV

▶ Global results: Cummings and Shelah used an Easton-like iteration to prove

the following:

Theorem

Assume GCH, if 𝜆 → (𝛾(𝜆), 𝜀(𝜆), 𝜈(𝜆)) is a class function from the class of all regular cardinals to the class of cardinal numbers, with

𝜆+ ≤ 𝛾(𝜆) = cf(𝛾(𝜆)) ≤ cf(𝜀(𝜆)) ≤ 𝜀(𝜆) ≤ 𝜈(𝜆) and cf(𝜈(𝜆)) > 𝜆 for

all 𝜆. Then, there exists a class forcing ℙ, preserving all cardinals and cofjnalities, such that in the generic extension 𝔠(𝜆) = 𝛾(𝜆), 𝔢(𝜆) = 𝜀(𝜆) and 𝜈(𝜆) = 2𝜆.

19

Why cardinal invariants of these spaces? V

▶ More cardinal invariants via combinatorial characterizations: In the count-

able case the following holds:

Theorem (Bastoszyński)

Let 𝑔 and 𝑕 be two functions in 𝜕𝜕. We say that 𝑔 and 𝑕 are eventually difgerent if there is 𝑜 ∈ 𝜕, such that for all 𝑛 ≥ 𝑜 𝑔(𝑛) ≠ 𝑕(𝑛) (and write 𝑔 ≠∗ 𝑕), then: non ℳ = min{|ℱ|∶ (∀𝑕 ∈ 𝜕𝜕)(∃𝑔 ∈ ℱ)¬(𝑔 ≠∗ 𝑕)}. cov ℳ = min{|ℱ|∶ (∀𝑕 ∈ 𝜕𝜕)(∃𝑔 ∈ ℱ)(𝑔 ≠∗ 𝑕)}. Then, if we defjne for arbitrary uncountable regular 𝜆:

▶ nm(𝜆) = min{|ℱ|∶ (∀𝑕 ∈ 𝜆𝜆)(∃𝑔 ∈ ℱ)¬(𝑔 ≠∗ 𝑕)}. ▶ cv(𝜆) = min{|ℱ|∶ (∀𝑕 ∈ 𝜆𝜆)(∃𝑔 ∈ ℱ)(𝑔 ≠∗ 𝑕)}.

20

Why cardinal invariants of these spaces? VI

The following holds:

Proposition

▶ 𝔠(𝜆) ≤ nm(𝜆) ≤ non(ℳ𝜆). ▶ cov(ℳ𝜆) ≤ cv(𝜆) ≤ 𝔢(𝜆).

Moreover, if 𝜆 is strongly inaccessible, the corresponding cardinals coincide.

▶ Club versions:

▶ Cummings and Shelah defjned the ”club” versions of 𝔢(𝜆) and 𝔠(𝜆), namely

given 𝑔, 𝑕 ∈ 𝜆𝜆, we say that 𝑔 <∗

cl 𝑕 (𝑕 club dominates 𝑔), if there exists a

club 𝐷 on 𝜆 so that, for every 𝛽 ∈ 𝐷, 𝑔(𝛽) < 𝑕(𝛽) and defjned 𝔠cl(𝜆) and 𝔢cl(𝜆) accordingly. They proved:

21

Why cardinal invariants of these spaces? VII

Theorem 𝔠cl(𝜆) = 𝔠(𝜆), 𝔢cl(𝜆) ≤ 𝔢(𝜆) and if 𝜆 is regular and > ℶ𝜕, 𝔢cl(𝜆) = 𝔢(𝜆).

▶ The pseudointersection number: 𝔮(𝜆) is defjned as the minimum size of a

family ℱ of subsets of 𝜆 with the strong intersection property (i.e. for every

ℱ′ ⊆ ℱ, |ℱ′|< 𝜆, ⋂ ℱ′ is unbounded) and no pseudointersection of

size 𝜆 (i.e. no set 𝑌 ∈ [𝜆]𝜆 such that 𝑌 ⊆∗ 𝐺, for all 𝐺 ∈ ℱ). Raghavan and Shelah proved also that 𝔱(𝜆) ≤ 𝔮cl(𝜆) ≤ 𝔠(𝜆) where

𝔮cl(𝜆) is the minimum size of a family of clubs without a pseudointersection

• f size 𝜆. Recently, in joint work with Fischer and Soukup, we have proved

that there is a model where 𝔮(𝜆) < 𝔮cl(𝜆).

22

Some results

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻

𝔢(∈∗

𝑞)(𝜆)

cof ℳ(𝜆)

𝔢(𝜆)

cov ℳ(𝜆) add ℳ(𝜆)

𝔠(𝜆)

non ℳ(𝜆)

𝔠(∈∗

𝑞)(𝜆)

𝔠(∈∗)(𝜆) 𝜆+ 𝔢(∈∗)(𝜆) 2𝜆

Cichoń’s Diagram on the uncountable for 𝜆 strongly inaccessible.

23

𝜆-Sacks forcing

Let 𝜆 be strongly inaccessible. Conditions in 𝕋𝜆 are 𝜆-closed sub-trees 𝑈 ⊆ 2<𝜆 such that ∀𝑡 ∈ 𝑈, ∃𝑢 ∈ 𝑈, 𝑡 ⊆ 𝑢 splitting and the limit of splitting nodes is also

• splitting. Also 𝑈 ≤ 𝑇 if 𝑈 ⊆ 𝑇.

▶ It has good fusion properties. ▶ It has the generalized ℎ-Sacks property where ℎ ∈ 𝜆𝜆 is defjned by ℎ(𝛽) =

2|𝛽|, i.e. given 𝑇 ∈ 𝕋𝜆 and ̇ 𝑔 an 𝕋𝜆-name for an element in 𝜆𝜆, there

are 𝑈 ≤ 𝑇 and 𝐺 ∶ 𝜆 → [𝜆]<𝜆 ℎ-slalom (|𝐺(𝛽)|≤ ℎ(𝛽)) such that

𝑈 ⊩ ̇ 𝑔(𝛽) ∈ 𝐺(𝛽) for all 𝛽 < 𝜆.

24

The iteration of 𝕋𝜆 with 𝜆-support of length 𝜆++ has:

▶ Good fusion, so cardinals ≤ 𝜆+ are preserved. ▶ The generalized Sacks property and, as a consequence 𝔢(∈∗)(𝜆) as well

as the other cardinals in the extended diagram are equal to 𝜆+.

25

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻

𝔢(∈∗

𝑞)(𝜆)

cof(ℳ𝜆)

𝔢(𝜆)

cov(ℳ𝜆) add(ℳ𝜆)

𝔠(𝜆)

non(ℳ𝜆)

𝔠(∈∗

𝑞)(𝜆)

𝔠(∈∗)(𝜆) 𝜆+ 𝔢(∈∗)(𝜆) 2𝜆

Figure 7: Efgect of the iteration of 𝜆−Sacks forcing

26

𝜆-Miller Forcing

Let ℱ be a 𝜆-complete fjlter on 𝜆. Defjne 𝕅𝕁𝜆

ℱ to be the following forcing notion:

Conditions in 𝕅𝕁𝜆

ℱ are 𝜆-closed sub-trees 𝑈 of the set of increasing sequences in

𝜆<𝜆, such that every node can be extended to a ℱ-splitting node.

Also we want that if 𝛽 < 𝜆 is limit, 𝑣 ∈ 𝜆𝛽, and for arbitrarily large 𝛾 < 𝛽, 𝑣 ↾ 𝛾

ℱ-splits in 𝑈, then 𝑣 ℱ-splits in 𝑈;

27

Properties of 𝜆-Miller forcing

▶ It has good fusion which implies ≥ 𝜆+ are preserved. ▶ 𝕅𝕁𝜆 𝒟, where 𝒟 is the club fjlter, adds a Cohen subset of 𝜆. ▶ 𝕅𝕁𝜆 ℱ generically adds an unbounded function over 𝜆𝜆 ∩ 𝑊. ▶ The product 𝕅𝕁𝜆 ℱ × 𝕅𝕁𝜆 ℱ adds a 𝜆-Cohen function. ▶ 𝕅𝕁𝜆 𝒱 has the pure decision property when 𝒱 is an ultrafjlter. i.e. if 𝑈 ∈ 𝕅𝕁𝜆 𝒱

and 𝜒 is a formula in the forcing language, there is 𝑇 ≤ 𝑈 with the same stem such that 𝑇 decides 𝜒 i.e. 𝑇 ⊩ 𝜒 or 𝑇 ⊩ ¬𝜒.

28

The generalized ultrafjlter number

Let 𝜆 be an uncountable cardinal.

Defjnition

𝔳(𝜆) = min{|ℬ|∶ ℬ is a base for a uniform ultrafjlter on 𝜆}.

Uniform means that all the sets in the ultrafjlter have size 𝜆. Also, if 𝒱 is an ultrafjlter

• n 𝜆, ℬ ⊆ 𝒱 is a base if given 𝐺 ∈ 𝒱, there is 𝐶 ∈ ℬ such that 𝐶 ⊆∗ 𝐺.

29

Theorem (Brooke-Taylor, Fischer, Friedman, M.)

Suppose 𝜆 is a supercompact cardinal, 𝜆∗ is a regular cardinal with 𝜆 < 𝜆∗ ≤ Γ and Γ is a cardinal that satisfjes Γ𝜆 = Γ. Then there is a forcing extension in which cardinals have not been changed satisfying:

𝜆∗ = 𝔳(𝜆) = 𝔠(𝜆) = 𝔢(𝜆) = 𝔟(𝜆) = 𝔱(𝜆) = 𝔰(𝜆) = cov(ℳ𝜆) = add(ℳ𝜆) = non(ℳ𝜆) = cof(ℳ𝜆) and 2𝜆 = Γ.

If in addition (Γ)<𝜆∗ ≤ Γ then we can also provide that

𝔮(𝜆) = 𝔲(𝜆) = 𝔦𝒳(𝜆) = 𝜆∗ where 𝒳 is a 𝜆-complete ultrafjlter on 𝜆.

Go to fjnal

30

Section 2 The independence number

31

Independent families

Defjnition (Notation)

Let 𝒝 be a family of infjnite subsets of 𝜕:

▶ We denote FF(𝒝) the family of fjnite partial functions from 𝒝 to 2. Given

ℎ ∈ FF(𝒝), 𝒝ℎ = ⋂{𝐵ℎ(𝐵) ∶ 𝐵 ∈ 𝒝 ∩ dom(ℎ)}, where 𝐵ℎ(𝐵) = 𝐵

if ℎ(𝐵) = 0 and 𝐵ℎ(𝐵) = 𝜕 𝐵 otherwise.

▶ We refer to {𝒝ℎ ∶ ℎ ∈ FF(𝒝)} as the family of Boolean combinations of

𝒝 associated to ℎ.

Defjnition

A family 𝒝 ⊆ [𝜕]𝜕 is called independent if for for every ℎ ∈ FF(𝒝), the set 𝒝ℎ is infjnite. An independent family 𝒝 is said to be maximal independent if it is not properly contained in another independent family.

32

An example due to Fichtenholz-Kantorovic

Put 𝒟 = [ℚ]<𝜕 and for any real 𝑠 ∈ ℝ, look at the set:

𝑌𝑠 = {𝐺 ∈ 𝒟 ∶ |𝐺 ∩ (−∞, 𝑠)| is even}

Then, the family {𝑌𝑠 ∶ 𝑠 ∈ ℝ} is independent: Let 𝑠0 < 𝑠2 < … < 𝑠𝑙 and

𝑡0 < 𝑡1 < 𝑡2 < … < 𝑡𝑚 two sets of reals, then the set ⋂

𝑗≤𝑙

𝐹𝑠𝑗 ∩ ⋂

𝑘≤𝑚

(𝜕 𝐹𝑡𝑘)

is infjnite. Why? Let’s look at the following drawing:

33

In the fjgure, the set of rationals {𝑟0, 𝑟1, …} ∈ ⋂𝑗≤𝑙 𝐹𝑠𝑗 ∩ ⋂𝑘≤𝑚(𝜕 𝐹𝑡𝑘)

34

The independence number

Defjnition

𝔧 = min{|𝒝|∶ 𝒝 is a maximal independent family of subsets of 𝜕}. 𝔧 is a cardinal invariant, in the sense that ℵ1 ≤ 𝔧 ≤ 𝔡, some lower bounds for it

are the cardinal invariants 𝔢 and 𝔰.

Go to Hasse’s diagram

35

How to add an independent real?

The following results are due to Brendle:

Lemma

Let 𝒝 be an independent family. Then there is an ideal 𝒦𝒝 on 𝜕 with the following properties:

• 1. 𝒦𝒝 ∩ {𝒝ℎ ∶ ℎ ∈ FF(𝒝)} = ∅.
• 2. For every 𝑌 ∈ [𝜕]𝜕 there is ℎ ∈ FF(𝒝) such that either 𝑌 ∩ 𝒝ℎ or

𝒝ℎ 𝑌 belongs to 𝒦𝒝.

Whenever 𝒝 be an independent family and 𝒦𝒝 is an ideal satisfying properties (1) and (2) of the lemma above, we say that 𝒦𝒝 is an independence diagonalization ideal associated to 𝒝.

36

The forcing

Defjnition

Let 𝒝 be an independent family and let 𝒦𝒝 be an independence diagonalization ideal associated to it. The poset 𝔺(𝒦𝒝) consists of all pairs (𝑡, 𝐹) where

𝑡 ∈ [𝜕]<𝜕, 𝐹 ∈ [𝒦𝒝]<𝜕 with extension relation defjned as follows: (𝑢, 𝐺) ≤ (𝑡, 𝐹) if and only if 𝑢 ⊇ 𝑡, 𝐺 ⊇ 𝐹 and (𝑢\𝑡) ∩ ⋃ 𝐹 = ∅.

This poset 𝔺(𝒦𝒝) is 𝜏-centered, so it preserves cardinals. Additionally it has the following weakly diagonalization property:

37

Lemma

Let 𝐻 be a 𝔺(𝒦𝒝) generic fjlter. Then 𝑦𝐻 ∶= ⋃{𝑡 ∶ ∃𝐺(𝑡, 𝐺) ∈ 𝐻} is an infjnite subset of 𝜕 such that in 𝑊 [𝐻], 𝒝 ∪ {𝑦𝐻} is independent, while for every

𝑍 ∈ ([𝜕]𝜕\𝒝) ∩ 𝑊, the family 𝒝 ∪ {𝑦𝐻, 𝑍 } is not independent.

As a corollary we obtain:

Theorem

(GCH) Let 𝜆 < 𝜇 be regular uncountable cardinals. There is a ccc generic extension in which 𝔧 = 𝔢 = 𝜆 < 𝔡 = 𝜇.

38

A generic maximal independent family

Shelah constructed a maximal independent family, which remains a witness to

𝔧 = ℵ1 in a model of 𝔳 = ℵ2. With Fischer, we showed that, over a model of GCH

for example, his construction naturally gives rise to the existence of a countably closed, ℵ2-cc poset ℙ, which generically adjoins a maximal independent family, which turns to be Sacks indestructible.

39

Lemma

Let 𝒝 be an independent family and let 𝒠(𝑌) to be the set of all functions

ℎ ∈ FF(𝒝) for which 𝑌 ∩ 𝒝ℎ is fjnite, then:

id(𝒝) = {𝑌 ⊆ 𝜕 ∶ ∀ℎ ∈ FF(𝒝)∃ℎ′ ⊇ ℎ(𝒝ℎ′ ∩ 𝑌) is fjnite}

= {𝑌 ⊆ 𝜕 ∶ 𝒠(𝑌) is dense in FF(𝒝)}

is an ideal on 𝜕, to which we refer as the independence density ideal associated to

𝒝. Here when we say “dense” in FF(𝒝), we mean dense respect to the inclusion

relation.

40

The poset

Defjnition

Let ℙ be the poset of all pairs (𝒝, 𝐵) where 𝒝 is a countable independent family,

𝐵 ∈ [𝜕]𝜕 such that for all ℎ ∈ FF(𝒝) the set 𝒝ℎ ∩ 𝐵 is infjnite. The extension

relation on ℙ is given by: (ℬ, 𝐶) ≤ (𝒝, 𝐵) if and only if ℬ ⊇ 𝒝 and 𝐶 ⊆∗ 𝐵.

Proposition

Let 𝐻 be ℙ-generic over 𝑊. Then 𝒝𝐻 = ⋃{𝒝 ∶ ∃𝐵 ∈ [𝜕]𝜕 with (𝒝, 𝐵) ∈ 𝐻} is a maximal independent family.

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Theorem

The generic maximal independent family adjoined by ℙ over a model of CH and

2ℵ0 = ℵ1 remains maximal after the countable support iteration of Sacks forcing 𝕋 of length 𝜕2.

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Comparing the two ideals I

▶ Let 𝒝 be an independent family. Then id(𝒝) ⊆ 𝒦𝒝. ▶ If 𝒝 is an independent family which is not maximal, then id(𝒝) ⊊ 𝒦𝒝.

Defjnition

An independent family 𝒝 is said to be densely maximal if for every 𝑌 ∈ [𝜕]𝜕\𝒝 and every ℎ ∈ FF(𝒝), there is ℎ′ ∈ FF(𝒝) for which either 𝑌 ∩ 𝒝ℎ′ of

𝒝ℎ′\𝑌 is fjnite.

Proposition

If 𝒝 is densely maximal independent, then 𝒦𝒝 ⊆ id(𝒝) and so 𝒦𝒝 = id(𝒝).

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Comparing the two ideals II

Proposition

The maximal family that turned to be Sacks indestructible is densely independent.

Corollary

A densely maximal independent family 𝒝 such that the dual fjlter of its diagonalization ideal id(𝒝) is generated by a Ramsey fjlter and the co-fjnite sets remains maximal after the countable support iteration of Sacks forcing, as well as after the countable support product of Sacks forcing.

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The generalized case

Defjnition

Let 𝒝 be a family of unbounded subsets of 𝜆 of size ≥ 𝜆:

▶ We call BF𝜆(𝒝) the family of functions from 𝒝 to 2 with domain of size < 𝜆. ▶ Given ℎ ∈ BF𝜆(𝒝), 𝒝ℎ = ⋂{𝐵ℎ(𝐵) ∶ 𝐵 ∈ 𝒝 ∩ dom(ℎ)}, where

𝐵ℎ(𝐵) = 𝐵 if ℎ(𝐵) = 0 and 𝐵ℎ(𝐵) = 𝜆 𝐵 otherwise.

Defjnition

A family 𝒝 ⊆ [𝜆]𝜆 such that |𝒝|≥ 𝜆 is called 𝜆-independent if for for every

ℎ ∈ BF𝜆(𝒝), the set 𝒝ℎ is unbounded on 𝜆. A 𝜆-independent family 𝒝 is said

to be 𝜆-maximal independent if it is not properly contained in another

𝜆-independent family.

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A generalization of Brendle’s result

Fischer and Shelah have characterized fjlters associated to an independent family that can be used to diagonalize an independent family of subsets of 𝜕 by using Mathias forcing with respect to such fjlters.

Defjnition

Let 𝒝 be a 𝜆-independent family. A 𝜆-complete fjlter ℱ is called a diagonalization fjlter for the family 𝒝 if the following hold:

• 1. For every 𝐺 ∈ ℱ and ℎ ∈ BF𝜆(𝒝), |𝐺 ∩ 𝒝ℎ|= 𝜆.
• 2. ℱ ∩ {𝒝ℎ ∶ ℎ ∈ BF𝜆(𝒝)} = ∅.

In addition, a maximal diagonalization fjlter is a 𝜆-complete fjlter that is maximal with respect to properties (1) and (2), i.e. there is no 𝜆-complete fjlter ℱ′ ⊃ ℱ satisfying these properties.

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Finally...

In recent work with V. Fischer we have proved the following diagonalization lemma in order to decide the value of 𝔧(𝜆) in the model where 𝔳(𝜆) was small. Specifjcally, it is decided to be 𝜆∗.

Go to model with small u

Lemma

Let 𝒝 be a 𝜆-independent family, ℱ be a diagonalization fjlter and 𝐻 be a

𝕅𝜆

𝒱-generic fjlter over a ground model 𝑊. Put 𝑦𝒱 to be the generic Mathias

function added by this forcing notion, then:

• 1. 𝒝 ∪ {𝑦𝒱} is 𝜆-independent.
• 2. If 𝑧 ∈ ([𝜆]𝜆 ∩ 𝑊 ) 𝒝 is such that 𝒝 ∪ {𝑧} is 𝜆-independent, then

𝒝 ∪ {𝑦𝒱, 𝑧} is not.

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