Constellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Coherent systems of finite support iterations
Diana Carolina Montoya
Joint work with: Vera Fischer, Sy-David Friedman and Diego
Mejía
Kurt Gödel Research Center. University of Vienna.
RIMS workshop on infinite combinatorics and forcing theory

December 1st, 2016

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

ContentsConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Contents

Constellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Contents

Constellations in Cichoń’s diagram

Preservation of mad familiesCoherent system of finite support iterations

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Section 1
Constellations in Cichoń’s diagram

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

The problem

Given a constellation from cardinals in Cichoń’s diagram, is it
possible to find a model where, in addition a can be decided?

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

The unbounding and dominating numbers, b and d.
Definition
If f , g are functions in ω ω , we say that f <∗ g, if there exists ann ∈ ω such that for all m > n, f (m) < g(m). In this case, we saythat g eventually dominates f .

Definition
Let F be a family of functions in ω ω .
I F is dominating, if for all g ∈ ω ω , there exists an f ∈ F suchthat g <∗ f .
I F is unbounded, if for all g ∈ ω ω , there exists an f ∈ F suchthat f ≮∗ g.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

DefinitionI b = min{|F|: F is an unbounded family of functions in ω ω }.
I d = min{|F|: F is a dominating family of functions in ω ω }.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Cardinal Invariants Associated to an Ideal
Let I be a non-trivial σ-ideal on a set X :Definition
I The additivity number:add(I) = min{|J |: J ⊆ I and[J ∈/ I}.

I The covering number:cov(I) = min{|J |: J ⊆ I and

Diana Carolina Montoya
Coherent systems of finite support iterations[J = X }.

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterationsReferences

Definition
I The cofinality number:cof(I) = min{|J |: J ⊆ I and for all M ∈ I there is aJ ∈ J with M ⊆ J}.
I The uniformity number:non(I) = min{|Y |: Y ⊂ X and Y ∈/ I}.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Cichoń’s Diagram

cov N- non M- cof M- cof N6666bℵ1- add N

Diana Carolina Montoya
Coherent systems of finite support iterations--cd66- add M- cov M- non N

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

The method of matrix iterations I
1. It was introduced by Blass and Shelah, to show thatconsistently u < d, where u is the ultrafilter number and d isthe dominating number.
2. Further developed fron Brendle-Fischer, who introduced theterminology matrix iteration for the first time. They used themethod to show that if κ < λ are arbitrary regular uncountablecardinals then there is a generic extension in whicha = b = κ < s = λ. Here a, b and s denote the almost
disjointness, bounding and splitting numbers respectively.
3. Later, classical preservation properties for matrix iterations
were improved by Mejía to provide several examples of models
where the cardinals in Cichoń’s diagram assume many differentvalues, in particular, a model with 6 different values.
Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterationsReferences

An example

Theorem (Mejía)
Let V be a model of ZFC and fix regular uncountable cardinals
θ0 ≤ θ1 ≤ κ ≤ µ and let λ ≥ µ to be a cardinal. Then there is acardinal preserving generic extension in which: add(N ) = θ0 ,cov(N ) = θ1 , b = non(M) = cov(M) = κ, d = µ andnon(N ) = c = λ.
Idea of the proof: Blackboard

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterationsReferences

To remember:
Full generics and restricted generics.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Section 2
Preservation of mad families

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

The almost disjointness number

Definition
Two sets A and B ∈ P(ω) are called almost disjoint if A ∩ B isfinite. We say that an infinite family of sets A ⊆ P(ω) is almost
disjoint if all its elements are pairwise almost disjoint. A familyA ⊆ [ω]ω is called a maximal almost disjoint (abbreviated mad) if it
is almost disjoint and is not properly included in another such family.

Definitiona = min{|A|: A is a mad family of subsets of ω}.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterationsReferences

Preservation of mad families

Definition (Hechler-type almost disjoint families)
For a set Ω define the poset
HΩ := {p : Fp × np → 2 : Fp ∈ [Ω]<ℵ0 and np < ω}. The order isgiven by q ≤ p if and only if p ⊆ q and, for any i ∈ nq \ np , there is
at most one z ∈ Fp such that q(z, i) = 1.
If G is HΩ -generic over V then A = AG := {az : z ∈ Ω} is an a.d.
family where az ⊆ ω is defined as i ∈ az if and only if p(z, i) = 1for some p ∈ G. Moreover, V [G] = V [A] and, when Ω isuncountable, A is mad in V [G].

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Some background I
1. Kunen: Under CH there is a mad family that remains maximal
upon adding a Cohen real.
2. Steprans: there is a mad family which remains maximal uponadding ℵ1 -many Cohen reals.
3. Zhang proved that, under CH there is a mad family in theground model that stays mad after a finite support iteration ofE.
4. Brendle and Fischer: defined a property that is iterable andused to preserve Hechler style mad families added in the firstcolumn of a matrix iteration of ccc posets.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Definition (Brendle-Fischer)
Let A = haz iz∈Ω ∈ M be a family of infinite subsets of ω anda∗ ∈ [ω]ω (not necessarily in M). Say that a∗ diagonalizes M
outside A if, for all h ∈ M, h : ω × [Ω]<ω → ω and for any m < ω,Sthere are i ≥ m and F ∈ [Ω]<ω such that [i, h(i, F )) \ z∈F az ⊆ a∗ .

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

TheoremLet κ be an uncountable regular cardinal. After forcing with Hκ , any
finite support iteration hPξ , Q̇ξ iξ<π where each iterand is either(i) in {C, E} ∪ R or(ii) a ccc poset of size < κpreserves the mad family added by Hκ .

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Section 3
Coherent system of finite support iterations

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

I The method provides a general framework that generalizes themethod of matrix iterations of ccc posets introduced by Blassand Shelah.
I It was motivated by the necessity of adding a third dimension,in order to control the cardinal invariant a.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Coherent system of finite support iterations I
A coherent system (of finite support iterations) s is composed by
the following objects:(I) a partial ordered set I s and an ordinal π s ,(II) a system of posets hPsi,ξ : i ∈ I s , ξ ≤ π s i such that(i) Psi,0 < ·Psj,0 whenever i ≤ j in I s , and(ii) Psi,η is the direct limit of hPsi,ξ : ξ < ηi for each limit η ≤ π s ,(III) a sequence hQ̇si,ξ : i ∈ I s , ξ < π s i where each Q̇si,ξ is aPsi,ξ -name for a poset, Psi,ξ+1 = Psi,ξ ∗ Q̇si,ξ and Psj,ξ forcesQ̇si,ξ < · Psi,ξ Q̇sj,ξ whenever i ≤ j in I s and Psi,ξ < ·Psj,ξV

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

We consider the following particular cases.
1. When I s is a well-ordered set, we say that s is a 2D-coherentsystem (of finite support iterations).
2. If I s is of the form {i0 , i1 } ordered as i0 < i1 , we say that s is acoherent pair (of finite support iterations).
3. If I s = γ s × δ s where γ s and δ s are ordinals and the order of I sis defined as (α, β) ≤ (α0 , β 0 ) if and only if α ≤ α0 and β ≤ β 0 ,we say that s is a 3D-coherent system (of finite supportiterations).

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Standard coherent system of FS iterations I
A coherent system of FS iterations s is standard if(I) it consists, additionally, of:(i) a partition hS s , C s i of π s ,(ii) a function ∆s : C s → I s so that ∆s (i) is not maximal in I s for
all i ∈ C s ,(iii) a sequence hSsξ : ξ ∈ S s i where each Ssξ is either a Suslin ccc
poset or a random algebra, and(iv) a sequence hQ̇sξ : ξ ∈ C s i such that each Q̇sξ is aPs∆s (ξ),ξ -name of a poset which is forced to be ccc by Psi,ξ forall i ≥ ∆s (ξ) in I s , andDiana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagramPreservation of mad families

Coherent system of finite support iterations

References

Standard coherent system of FS iterations II(II) it is satisfied, for any i ∈ I s and ξ < π s , thatss V
 (Sξ ) i,ξ

Q̇si,ξ =Q̇s ξDiana Carolina Montoya
Coherent systems of finite support iterations

1if ξ ∈ S sif ξ ∈ C s and i ≥ ∆s (ξ),
otherwise.

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

The main theorem

TheoremAssume λ<θ1 = λ. Then, there is a ccc poset forcing add(N ) = θ0 ,cov(N ) = θ1 , b = a = κ, non(M) = cov(M) = µ, d = ν andnon(N ) = c = λ.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

Cichoń’s diagram as in Theorem 3.

cov(N )
bnon(M)

cof(M)

µν

b

θ1bb

cof(N )
bdbbcλκθ0ℵ1bbbbbbadd(N )add(M)

cov(M)

non(N )Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

More constellations...
TheoremAssume λ<θ0 = λ. Then, for any of the statements below, there is a
ccc poset forcing it.
(a) add(N ) = θ0 , b = a = κ, cov(I) = non(I) = µ forI ∈ {M, N }, d = ν and cof(N ) = c = λ.
(b) add(N ) = θ0 , cov(N ) = κ, add(M) = cof(M) = µ,non(N ) = ν and cof(N ) = c = λ.
(c) add(N ) = θ0 , cov(N ) = b = a = κ, non(M) = cov(M) = µ,d = non(N ) = ν and cof(N ) = c = λ.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

An application with well-orders of the reals

Theorem
In L, let θ0 < θ1 < κ < µ < ν < λ be uncountable regular cardinals
and, in addition, λ < ℵω . Then there is a cardinals preserving
forcing extension of the constructible universe, L, in which there is a∆13 -well-order of the reals and in addition add(N ) = θ0 ,cov(N ) = θ1 , b = a = κ, non(M) = cov(M) = µ, d = ν andnon(N ) = c = λ.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

References I[1]Jörg Brendle.
Larger cardinals in Cichoń’s diagram.
J. Symbolic Logic, 56(3):795–810, 1991.

[2]

Jörg Brendle and Vera Fischer.
Mad families, splitting families and large continuum.
J. Symbolic Logic, 76(1):198–208, 2011.

[3]

Vera Fischer and Sy David Friedman.
Cardinal characteristics and projective wellorders.
Ann. Pure Appl. Logic, 161(7):916–922, 2010.

[4]

Vera Fischer, Sy-David Friedman, Diego Mejía, and Diana Carolina Montoya.
Coherent systems of finite support iterations.
In preparation.

[5]

Vera Fischer, Sy David Friedman, and Lyubomyr Zdomskyy.
Projective wellorders and mad families with large continuum.
Ann. Pure Appl. Logic, 162(11):853–862, 2011.

[6]

Vera Fischer, Sy David Friedman, and Lyubomyr Zdomskyy.
Cardinal characteristics, projective wellorders and large continuum.
Ann. Pure Appl. Logic, 164(7-8):763–770, 2013.

[7]

Martin Goldstern, Diego Alejandro Mejía, and Saharon Shelah.
The left side of Cichoń’s diagram.
Proc. Amer. Math. Soc.
Accepted.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research CenterConstellations in Cichoń’s diagram

Preservation of mad families

Coherent system of finite support iterations

References

References II[8]Stephen H. Hechler.
Short complete nested sequences in βN\N and small maximal almost-disjoint families.
General Topology and Appl., 2:139–149, 1972.

[9]Haim Judah and Saharon Shelah.
The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems forforcing).
J. Symbolic Logic, 55(3):909–927, 1990.

[10] Diego Alejandro Mejía.
Matrix iterations and Cichon’s diagram.
Arch. Math. Logic, 52(3-4):261–278, 2013.
[11] Diego Alejandro Mejía.
Models of some cardinal invariants with large continuum.
Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku, (1851):36–48, 2013.
[12] Arnold W. Miller.
Some properties of measure and category.
Trans. Amer. Math. Soc., 266(1):93–114, 1981.

Diana Carolina Montoya
Coherent systems of finite support iterations

Kurt Gödel Research Center