Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration< θ-uf-extendable matrix iterations
Miguel Cardona
Technische Universität Wien
joint work with Jörg Brendle and Diego Mejı́a
RIMS

Axiomatic set and theory and applications
8 November 2018

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background

For f , g ∈ ω ω define

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background

For f , g ∈ ω ω definef ≤∗ g (f is dominated by g) iff ∃m ∈ ω∀n ≥ m(f (n) ≤ g (n)).

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background

For f , g ∈ ω ω definef ≤∗ g (f is dominated by g) iff ∃m ∈ ω∀n ≥ m(f (n) ≤ g (n)).
Considerb = min{|F | F ⊆ ω ω and ¬∃g ∈ ω ω ∀f ∈ F (f ≤∗ g )}.
d = min{|E | E ⊆ ω ω and ∀g ∈ ω ω ∃f ∈ E (g ≤∗ f )}.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundIn ZFC:

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundIn ZFC:M : σ-ideal of meager subsets of 2ω
N : σ-ideal of Lebesgue null sets of 2ω

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundIn ZFC:M : σ-ideal of meager subsets of 2ω
N : σ-ideal of Lebesgue null sets of 2ω
cov(N )bnon(M)
b

b
ℵ1
b

b

cof(M)bcof(N )
b

d
b

b

b

b

b

add(N )

add(M)

cov(M)

non(N )

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsbc

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundIn ZFC:M : σ-ideal of meager subsets of 2ω
N : σ-ideal of Lebesgue null sets of 2ω
cov(N )bnon(M)
b

b
ℵ1
b

b

cof(M)bcof(N )
b

d
b

b

b

b

b

add(N )

add(M)

cov(M)

non(N )

add(M) = min{b, cov(M)}cof(M) = max{d, non(M)}Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsbc

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundFor σ ∈ (2<ω )ω define

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundFor σ ∈ (2<ω )ω define[σ]∞ : = {x ∈ 2ω : ∃∞ n < ω(σ(n) ⊆ x)}\ [=[σ(m)]n<ω m>nMiguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundFor σ ∈ (2<ω )ω define[σ]∞ : = {x ∈ 2ω : ∃∞ n < ω(σ(n) ⊆ x)}\ [=[σ(m)]n<ω m>nand htσ ∈ ω ω byMiguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundFor σ ∈ (2<ω )ω define[σ]∞ : = {x ∈ 2ω : ∃∞ n < ω(σ(n) ⊆ x)}\ [=[σ(m)]n<ω m>nand htσ ∈ ω ω by htσ (i) := |σ(i)| for each i < ω.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundFor σ ∈ (2<ω )ω define[σ]∞ : = {x ∈ 2ω : ∃∞ n < ω(σ(n) ⊆ x)}\ [=[σ(m)]n<ω m>nand htσ ∈ ω ω by htσ (i) := |σ(i)| for each i < ω.
htσ

σ(0) σ(1) σ(2)Miguel Cardona (TU wien). . . σ(n). . .

< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background
For f , g ∈ ω ω define

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background
For f , g ∈ ω ω definef g (f is much less that g ) iff ∀k < ω∃m∀n ≥ m(f (nk ) ≤ g (n)).

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background
For f , g ∈ ω ω definef g (f is much less that g ) iff ∀k < ω∃m∀n ≥ m(f (nk ) ≤ g (n)).
For each f ∈ ω ωMiguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background
For f , g ∈ ω ω definef g (f is much less that g ) iff ∀k < ω∃m∀n ≥ m(f (nk ) ≤ g (n)).
For each f ∈ ω ω define the families
Jf := {X ⊆ 2ω : ∃σ ∈ (2<ω )ω (X ⊆ [σ]∞ and htσ = f )}Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Background
For f , g ∈ ω ω definef g (f is much less that g ) iff ∀k < ω∃m∀n ≥ m(f (nk ) ≤ g (n)).
For each f ∈ ω ω define the families
Jf := {X ⊆ 2ω : ∃σ ∈ (2<ω )ω (X ⊆ [σ]∞ and htσ = f )}
and
If :=

Miguel Cardona (TU wien)

[
g f

Jg .

< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

BackgroundIn ZFC:M : σ-ideal of meager subsets of 2ω
N : σ-ideal of Lebesgue null sets of 2ω
cov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)bbadd(Iid )add(If )
ℵ1
b

b

b

b

b

b
b

d
b

supcofcof(N )

b

b

b

b

b

cof(If )bcof(Iid )bbbadd(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationscBackground
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Some forcing notions• B: Random forcing.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Some forcing notions• B: Random forcing. B adds a real r ∈ 2ω (called random real)that evades all the Borel null sets of 2ω coded in the groundmodel.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Some forcing notions• B: Random forcing. B adds a real r ∈ 2ω (called random real)that evades all the Borel null sets of 2ω coded in the groundmodel.

• E: eventually different real forcing.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Some forcing notions• B: Random forcing. B adds a real r ∈ 2ω (called random real)that evades all the Borel null sets of 2ω coded in the groundmodel.

• E: eventually different real forcing. E adds a real e ∈ ω ω(called eventually different real) that is eventually different
with all real in the ground model.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Some forcing notions• B: Random forcing. B adds a real r ∈ 2ω (called random real)that evades all the Borel null sets of 2ω coded in the groundmodel.

• E: eventually different real forcing. E adds a real e ∈ ω ω(called eventually different real) that is eventually different
with all real in the ground model.

• Ehb : (b, h)-eventually different real forcing.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Some forcing notions• B: Random forcing. B adds a real r ∈ 2ω (called random real)that evades all the Borel null sets of 2ω coded in the groundmodel.

• E: eventually different real forcing. E adds a real e ∈ ω ω(called eventually different real) that is eventually different
with all real in the ground model.

• Ehb : (b, h)-eventually different real forcing. Ehb adds a realQeb ∈ n<ω b(n) (called (b, h)-eventually different real) thatQ
evades all the slaloms in n<ω [b(n)]<h(n) coded in the
ground model.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Examplescov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)bbadd(Iid )add(If )
ℵ1
b

b

b

b

b

b
b

d
b

supcofcof(N )

b

b

b

b

b

cof(If )bcof(Iid )bbbadd(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationscBackground
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Examplescov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)bbadd(Iid )add(If )
ℵ1
b

b

b

b

b

b
b

d
b

supcofcof(N )

b

b

b

b

b

cof(If )bcof(Iid )bbbadd(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )Are these cardinal different?

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationscBackground
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Examplescov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)bbadd(Iid )add(If )
ℵ1
b

b

b

b

b

b
b

d
b

supcofcof(N )

b

b

b

b

b

cof(If )bcof(Iid )bbbadd(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )Are these cardinal different?
For example:• CH implies all these cardinals are equal.
Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationscBackground
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Examplescov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)bbadd(Iid )add(If )
ℵ1
b

b

b

b

b

b
b

d
b

supcofcof(N )

b

b

b

b

b

cof(If )bcof(Iid )bbbadd(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )Are these cardinal different?
For example:• CH implies all these cardinals are equal.
• MA+¬ CH implies all these cardinals are iqual to cMiguel Cardona (TU wien)< θ-uf-extendable matrix iterationscBackground
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

ExamplesTheorem (C. and Mejı́a)
Let µ ≤ ν ≤ κ be uncountable regular cardinals and λ ≥ κ acardinal such that λ<µ = λ. Then there is a function f0 ∈ ω ω anda ccc poset forcing
f ≥∗ f0cov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)bbbadd(If )
ℵ1
b

b

b

supcofcof(N )

b

b

b

b

cof(If )bcof(Iid )νµadd(Iid )

b

b
b

b

d
b

b

λ

κ
b

b

add(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )f ≥∗ f0

Miguel Cardona (TU wien)c< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

ExamplesTheorem(Goldstern and Mejı́a and Shelah 2016)
Let θ ≤ µ ≤ ν = ν ℵ0 ≤ κ = κℵ0 be uncountable regular cardinals
and κ < λ a cardinal such that λ<κ ≤ 2κ and assume b = d = ν.
Then there is a ccc poset forcing

cov(N )bnon(M)
b

b

b

b

d
b

λ

ν

θ
b

cof(N )

b

κ

µ

ℵ1

cof(M)bbbbadd(N )

add(M)

cov(M)

non(N )

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsbc

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Theorem(Mejı́a 2013)
Let θ ≤ µ ≤ ν be uncountable regular cardinals and λ ≥ ν acardinal such that λ<θ = λ. Then there is a ccc poset forcing

cov(N )bnon(M)

cof(M)bbcof(N )
b

λ

µ
b
b

d
b

ν

θ
ℵ1
b

b

b

b

b

add(N )

add(M)

cov(M)

non(N )

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsbc

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Questions
Question
Is it consistent that add(If ) < cov(If ) < non(If ) < cof(If ) for
all increasing function f ?
cov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)bbadd(Iid )add(If )
ℵ1
b

b

b

b

b

b
b

d
b

supcofcof(N )

b

b

b

b

b

cof(If )bcof(Iid )bbbadd(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationscBackground
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesQuestions

Questions
Question
Is it consistent that add(M) < cov(M) < non(M) < cof(M)?

cov(N )bnon(M)
b

b
ℵ1
b

b

cof(M)bcof(N )
b

d
b

b

b

b

b

add(N )

add(M)

cov(M)

non(N )

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsbc

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q,

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

(ii) A set Q ⊆ P is ultrafilter-linked (uf-linked)

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

(ii) A set Q ⊆ P is ultrafilter-linked (uf-linked) if Q is D-linked forany non-principal ultrafilter D on ω.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

(ii) A set Q ⊆ P is ultrafilter-linked (uf-linked) if Q is D-linked forany non-principal ultrafilter D on ω.
(iii) The poset P is θ-uf -linked

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

(ii) A set Q ⊆ P is ultrafilter-linked (uf-linked) if Q is D-linked forany non-principal ultrafilter D on ω.
S(iii) The poset P is θ-uf -linked if P = α<θ Pα for some sequencehPα : α < θi of uf -linked subsets of P.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

(ii) A set Q ⊆ P is ultrafilter-linked (uf-linked) if Q is D-linked forany non-principal ultrafilter D on ω.
S(iii) The poset P is θ-uf -linked if P = α<θ Pα for some sequencehPα : α < θi of uf -linked subsets of P.

When θ = ℵ0 , we write σ-uf-linked. Denote by
Fr := {x ⊆ ω : |ω r x| < ℵ0 } the Frechet filter.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

(ii) A set Q ⊆ P is ultrafilter-linked (uf-linked) if Q is D-linked forany non-principal ultrafilter D on ω.
S(iii) The poset P is θ-uf -linked if P = α<θ Pα for some sequencehPα : α < θi of uf -linked subsets of P.

When θ = ℵ0 , we write σ-uf-linked. Denote by
Fr := {x ⊆ ω : |ω r x| < ℵ0 } the Frechet filter.

(iv) The poset P is θ-Fr -Knaster if any subset of P of size θ

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness
Let P be a poset, F a free filter on ω and let θ be an infinite
cardinal.
(i) A set Q ⊆ P is F -linked if, for any sequence p̄ = hpn : n < ωiin Q, there exists a q ∈ P that forces {n < ω : pn ∈ Ġ } ∈ F + .

(ii) A set Q ⊆ P is ultrafilter-linked (uf-linked) if Q is D-linked forany non-principal ultrafilter D on ω.
S(iii) The poset P is θ-uf -linked if P = α<θ Pα for some sequencehPα : α < θi of uf -linked subsets of P.

When θ = ℵ0 , we write σ-uf-linked. Denote by
Fr := {x ⊆ ω : |ω r x| < ℵ0 } the Frechet filter.

(iv) The poset P is θ-Fr -Knaster if any subset of P of size θ
contains an Fr -linked set of size θ.
Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness forcing

RemarkA set is uf -linked iff it is F -linked for any free filter F .

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesFilter-linkedness forcing

RemarkA set is uf -linked iff it is F -linked for any free filter F .

Theorem(Mejı́a 2019)
If P has p-cc then any subset of P is uf -linked iff it is Fr -linked.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesExamples of Filter-linkedness forcing• Any poset of size ≤ θ is θ-uf-linked. In particular, Cohen

forcing is σ-uf-linked

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesExamples of Filter-linkedness forcing• Any poset of size ≤ θ is θ-uf-linked. In particular, Cohen

forcing is σ-uf-linked• Any complete Boolean algebra that admits a strictly-positive

σ-additive measure is σ-uf-linked. In particular, any random
algebra is σ-uf-linked.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesExamples of Filter-linkedness forcing• Any poset of size ≤ θ is θ-uf-linked. In particular, Cohen

forcing is σ-uf-linked• Any complete Boolean algebra that admits a strictly-positive

σ-additive measure is σ-uf-linked. In particular, any random
algebra is σ-uf-linked.

• E is σ-uf-linked.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

ExamplesExamples of Filter-linkedness forcing• Any poset of size ≤ θ is θ-uf-linked. In particular, Cohen

forcing is σ-uf-linked• Any complete Boolean algebra that admits a strictly-positive

σ-additive measure is σ-uf-linked. In particular, any random
algebra is σ-uf-linked.

• E is σ-uf-linked.

• Ehb is σ-uf-linked.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Preservation of unbounded families
Let θ be a cardinal. A family F ⊆ ω ω is θ-strongly unbounded

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Preservation of unbounded families
Let θ be a cardinal. A family F ⊆ ω ω is θ-strongly unbounded if|F | ≥ θ and |{x ∈ F : x ≤∗ y }| < θ for any y ∈ ω ω .

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Preservation of unbounded families
Let θ be a cardinal. A family F ⊆ ω ω is θ-strongly unbounded if|F | ≥ θ and |{x ∈ F : x ≤∗ y }| < θ for any y ∈ ω ω .

Lemma
If θ is an uncountable regular cardinal and F is a θ-strongly
unbounded family, then b ≤ |F | ≤ d.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Preservation of unbounded families
Let θ be a cardinal. A family F ⊆ ω ω is θ-strongly unbounded if|F | ≥ θ and |{x ∈ F : x ≤∗ y }| < θ for any y ∈ ω ω .

Lemma
If θ is an uncountable regular cardinal and F is a θ-strongly
unbounded family, then b ≤ |F | ≤ d.
For example, Cohen forcing CA adds a θ-strongly unbounded familyof size θ (of Cohen reals) for each uncountable regular θ ≤ |A|.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Preservation of unbounded families
Let θ be a cardinal. A family F ⊆ ω ω is θ-strongly unbounded if|F | ≥ θ and |{x ∈ F : x ≤∗ y }| < θ for any y ∈ ω ω .

Lemma
If θ is an uncountable regular cardinal and F is a θ-strongly
unbounded family, then b ≤ |F | ≤ d.
For example, Cohen forcing CA adds a θ-strongly unbounded familyof size θ (of Cohen reals) for each uncountable regular θ ≤ |A|.

Theorem (Brendle and C. and Mejı́a)
If θ is an uncountable regular cardinal then any θ-Fr-Knaster poset
preserves all the θ-strongly unbounded families from the groundmodel.
Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions< θ-ultrafilter-extendable matrix iteration
For γ, π ordinals, in the ground model V we construct a matrix
iteration P = hPα,ξ , Q̇α,ξ : α ≤ γ, −1 ≤ ξ ≤ πi such that

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions< θ-ultrafilter-extendable matrix iteration
For γ, π ordinals, in the ground model V we construct a matrix
iteration P = hPα,ξ , Q̇α,ξ : α ≤ γ, −1 ≤ ξ ≤ πi such that(i) cof(γ) > ω and γ ≥ θ,

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions< θ-ultrafilter-extendable matrix iteration
For γ, π ordinals, in the ground model V we construct a matrix
iteration P = hPα,ξ , Q̇α,ξ : α ≤ γ, −1 ≤ ξ ≤ πi such that(i) cof(γ) > ω and γ ≥ θ,(ii) Q̇α,−1 = Pα,0 = Cα ,Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions< θ-ultrafilter-extendable matrix iteration
For γ, π ordinals, in the ground model V we construct a matrix
iteration P = hPα,ξ , Q̇α,ξ : α ≤ γ, −1 ≤ ξ ≤ πi such that(i) cof(γ) > ω and γ ≥ θ,(ii) Q̇α,−1 = Pα,0 = Cα ,(iii) for each 0 ≤ ξ < π. Fix ∆(ξ) < γ non-limit and aP∆(ξ),ξ -name Q̇ξ of a θξ -uf-linked poset for some θξ < θ suchthat Pγ, ξ forces it to be ccc.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions< θ-ultrafilter-extendable matrix iteration
For γ, π ordinals, in the ground model V we construct a matrix
iteration P = hPα,ξ , Q̇α,ξ : α ≤ γ, −1 ≤ ξ ≤ πi such that(i) cof(γ) > ω and γ ≥ θ,(ii) Q̇α,−1 = Pα,0 = Cα ,(iii) for each 0 ≤ ξ < π. Fix ∆(ξ) < γ non-limit and aP∆(ξ),ξ -name Q̇ξ of a θξ -uf-linked poset for some θξ < θ suchthat Pγ, ξ forces it to be ccc.
(iv) Pα,ξ+1 = Pα, ξ ∗ Q̇α,ξ ,(Q̇ξQ̇α,ξ :=
1

if α ≥ ∆(ξ),
otherwise.

(v) for ξ limit, Pα,ξ = limdirη,ξ Pα,η .
Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Applications
Theorem(Brendle and C. and Mejı́a)
Let θ0 ≤ θ1 ≤ µ ≤ ν be uncountable regular cardinals and let λ bea cardinal such that ν ≤ λ = λ<θ1 . Then, there is a ccc poset
forcingcov(N )

non(M)bbcof(M)

µ
b

d
b

θ0
b

b

λ
b

ℵ1

cof(N )

b

θ1

ν

b

b

b

b

add(N )

add(M)

cov(M)

non(N )

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsbc

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Sketch of the proof

V Cν = Vν, 0b

V Cα+1 = Vα+1, 0bCV Cα = Vα, 0b

V C1 = V0,1
b

C
V0,0 = Vb

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Sketch of the proofVν, λρ

V Cν = Vν, 0b

BVt(ρ+1), λρb

Vt(ρ)+1, λρbbb

BVt(ρ+1), λρ Vt(ρ)+1, λρ+1bVt(ρ), λρ

1Vt(ρ), λρ+1Vα+1, λρ1Vα+1, λρ+1

Vα,λρ

1

Vα, λρ+1

V1, λρ

1

V1, λρ+1

V0, λρ

1

V0, λρ+1b

V Cβ+1 = Vα+1, 0Vν, λρ+1

b

b

b

C
V Cβ = Vα, 0b

V C1 = V1,0
b

b

b

b

b

C
V = V0,0bbMiguel Cardona (TU wien)b< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Sketch of the proofVν, λρ

V Cν = Vν, 0b

Vt(ρ)+1, λρbQ̇ρξ

BVt(ρ+1), λρ V∆(ξ), ξ

Q̇ρξ

b

b

Vt(ρ), λρ

1V1, λρ

1

V0, λρ

1b

V Cβ+1 = Vα+1, 0Vν, ξBVt(ρ+1), λρ

b

1

b

b

b

b

b

C
V Cβ = Vα, 0b

V C1 = V1,0
b

b

V1, ξ

1

V0, ξ

1

b

b

C
V = V0,0bbbξ = λρ + 1 + ε(ε < λ)Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsb

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Sketch of the proofVν, λρ

V Cν = Vν, 0b

Vt(ρ)+1, λρbVt(ρ), λρ
b

Vν, ξ

Q̇ρξ

BVt(ρ+1), λρ V∆(ξ), ξQ̇ρξ

BVt(ρ+1), λρ

b

b

b

1

Vν, λνµ
b

b

b
b

1
b

b
b

Vα+1, λνµ

V Cβ+1 = Vα+1, 0
b

b

C
b

V C1 = V1,0b

Vα, λνµV Cβ = Vα, 0b

V1, λρ

1

V0, λρ

1

b

V1, ξ

1

V0, ξ

1

b

V1, λνµbbC
V = V0,0bbbξ = λρ + 1 + ε(ε < λ)Miguel Cardona (TU wien)< θ-uf-extendable matrix iterationsb

V0, λνµb

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Matrix iteration

Theorem(Brendle and C. and Mejı́a)
Let θ be an uncountable regular cardinal and Pγ,π a< θ-uf-extendable matrix iteration. Then Pγ,π is θ-uf-Knaster. In
particular, it preserves any θ-strongly unbounded family.

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Applications
Corollary(Brendle and C. and Mejı́a)
Let θ ≤ µ ≤ ν be uncountable regular cardinals and let λ ≥ ν be a
cardinal such that λ<θ = λ. Then, there is a ccc poset forcing
cov(N ) cov(Iid ) cov(If ) supcov non(M) cof(M)
b

b

b

b

b

supcofcof(N )

b

b

b

c
b

µ
add(Iid )add(If )
ℵ1
b

b

b
b

b

d

b

cof(If )bcof(Iid )

λ
b

θ
b

ν
b

add(N ) minadd

b

b

b

b

b

b

add(M) cov(M) minnon non(If ) non(Iid ) non(N )Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Open Problems
Is it consistent with ZFC that(a) add(If ) < non(If ) < cov(If ) < cof(If ) for any increasingf : ω → ω?

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Open Problems
Is it consistent with ZFC that(a) add(If ) < non(If ) < cov(If ) < cof(If ) for any increasingf : ω → ω?

(b) add(N ) < non(N ) < cov(N ) < cof(N )?

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Open Problems
Is it consistent with ZFC that(a) add(If ) < non(If ) < cov(If ) < cof(If ) for any increasingf : ω → ω?

(b) add(N ) < non(N ) < cov(N ) < cof(N )?

(c) add(M) < non(M) < cov(M) < cof(M)?

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Open Problems
Is it consistent with ZFC that(a) add(If ) < non(If ) < cov(If ) < cof(If ) for any increasingf : ω → ω?

(b) add(N ) < non(N ) < cov(N ) < cof(N )?

(c) add(M) < non(M) < cov(M) < cof(M)?
(d) add(SN ) < cov(SN ) < non(SN ) < cof(SN )?

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Open Problems
Is it consistent with ZFC that(a) add(If ) < non(If ) < cov(If ) < cof(If ) for any increasingf : ω → ω?

(b) add(N ) < non(N ) < cov(N ) < cof(N )?

(c) add(M) < non(M) < cov(M) < cof(M)?
(d) add(SN ) < cov(SN ) < non(SN ) < cof(SN )?
(f) add(SN ) < non(SN ) < cov(SN ) < cof(SN )?

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations

Background
Filter-linkedness
Preservation of unbounded families
Matrix iteration

Applications
Open questions

Thank you for your attention!

Miguel Cardona (TU wien)< θ-uf-extendable matrix iterations