On Disjoint Stationary Sequences
Maxwell Levine
University of Freiburg

RIMS Set Theory Workshop
October 26, 2023

Background: Why Disjoint Stationary Sequences?

2/21

Section 1
Background: Why Disjoint Stationary
Sequences?

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Definition (Foreman-Todorčević, 2005)
Given a stationary N ∈ [H(Θ)]κ , we say:

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Definition (Foreman-Todorčević, 2005)
Given a stationary N ∈ [H(Θ)]κ , we say:
I N is internally unbounded if ∀x ∈ Pκ (N), ∃M ∈ N, x ⊆ M,

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Definition (Foreman-Todorčević, 2005)
Given a stationary N ∈ [H(Θ)]κ , we say:
I N is internally unbounded if ∀x ∈ Pκ (N), ∃M ∈ N, x ⊆ M,
I N is internally stationary if Pκ (N) ∩ N is stationary in Pκ (N),Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Definition (Foreman-Todorčević, 2005)
Given a stationary N ∈ [H(Θ)]κ , we say:
I N is internally unbounded if ∀x ∈ Pκ (N), ∃M ∈ N, x ⊆ M,
I N is internally stationary if Pκ (N) ∩ N is stationary in Pκ (N),I N is internally club if Pκ (N) ∩ N is club in Pκ (N),Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Definition (Foreman-Todorčević, 2005)
Given a stationary N ∈ [H(Θ)]κ , we say:
I N is internally unbounded if ∀x ∈ Pκ (N), ∃M ∈ N, x ⊆ M,
I N is internally stationary if Pκ (N) ∩ N is stationary in Pκ (N),I N is internally club if Pκ (N) ∩ N is club in Pκ (N),I N is internally approachable if there is an increasing and continuous
continuous chain hMξ : ξ < κi such that |Mξ | < κSandhMη : η < ξi ∈ Mξ+1 for all ξ < κ such that N = ξ<κ Mξ .

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Definition (Foreman-Todorčević, 2005)
Given a stationary N ∈ [H(Θ)]κ , we say:
I N is internally unbounded if ∀x ∈ Pκ (N), ∃M ∈ N, x ⊆ M,
I N is internally stationary if Pκ (N) ∩ N is stationary in Pκ (N),I N is internally club if Pκ (N) ∩ N is club in Pκ (N),I N is internally approachable if there is an increasing and continuous
continuous chain hMξ : ξ < κi such that |Mξ | < κSandhMη : η < ξi ∈ Mξ+1 for all ξ < κ such that N = ξ<κ Mξ .

Question (Foreman-Todorčević, 2005)
Is it consistent that there is some X ≺ [H(λ)] w/ |X | = ℵ1 that
separates these?

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

3/21

Variants of Internal Approachability
ProblemWhat can we say about the internal structure of elementary submodels?

Definition (Foreman-Todorčević, 2005)
Given a stationary N ∈ [H(Θ)]κ , we say:
I N is internally unbounded if ∀x ∈ Pκ (N), ∃M ∈ N, x ⊆ M,
I N is internally stationary if Pκ (N) ∩ N is stationary in Pκ (N),I N is internally club if Pκ (N) ∩ N is club in Pκ (N),I N is internally approachable if there is an increasing and continuous
continuous chain hMξ : ξ < κi such that |Mξ | < κSandhMη : η < ξi ∈ Mξ+1 for all ξ < κ such that N = ξ<κ Mξ .

Question (Foreman-Todorčević, 2005)
Is it consistent that there is some X ≺ [H(λ)] w/ |X | = ℵ1 that
separates these?

Theorem (Krueger, 2007-9)
Yes, on a stationary subset of [H(λ)]ℵ1 (to all separations).
Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

4/21

Forcing Clubs

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

4/21

Forcing Clubs
ProblemWhat can we say about forcings that add club subsets of stationary sets?

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

4/21

Forcing Clubs
ProblemWhat can we say about forcings that add club subsets of stationary sets?

Theorem (Abraham-Shelah, 1983)
If S ⊂ ℵ1 ∩ cof(ω) is stationary, there is an ℵ1 -distributive forcing that
adds a club through S.

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

4/21

Forcing Clubs
ProblemWhat can we say about forcings that add club subsets of stationary sets?

Theorem (Abraham-Shelah, 1983)
If S ⊂ ℵ1 ∩ cof(ω) is stationary, there is an ℵ1 -distributive forcing that
adds a club through S.

Question (Abraham-Shelah, 1983)
Is it consistent that there is a stationary subset S ⊆ ℵ2 ∩ cof(ℵ1 ) suchthat S ∪ {ℵ2 ∩ cof(ω)} does not contain a club in any ℵ2 -distributive
forcing extension?

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

4/21

Forcing Clubs
ProblemWhat can we say about forcings that add club subsets of stationary sets?

Theorem (Abraham-Shelah, 1983)
If S ⊂ ℵ1 ∩ cof(ω) is stationary, there is an ℵ1 -distributive forcing that
adds a club through S.

Question (Abraham-Shelah, 1983)
Is it consistent that there is a stationary subset S ⊆ ℵ2 ∩ cof(ℵ1 ) suchthat S ∪ {ℵ2 ∩ cof(ω)} does not contain a club in any ℵ2 -distributive
forcing extension?

Theorem (Krueger, 2007)
Assuming Martin’s Maximum, there is stationary subset S ⊆ ℵ2 ∩ cof(ω1 )such that S ∪ {ℵ2 ∩ cof(ω)} does not contain a club in any outer modelwith the same ℵ1 and ℵ2 .
Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

5/21

Disjoint Stationary Sequences

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

5/21

Disjoint Stationary Sequences

Definition (Krueger, 2007)
Let µ be regular. A disjoint stationary sequence on µ+ is a sequencehSα : α ∈ Si such that:Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

5/21

Disjoint Stationary Sequences

Definition (Krueger, 2007)
Let µ be regular. A disjoint stationary sequence on µ+ is a sequencehSα : α ∈ Si such that:I S ⊆ µ+ ∩ cof(µ) is stationary,

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

5/21

Disjoint Stationary Sequences

Definition (Krueger, 2007)
Let µ be regular. A disjoint stationary sequence on µ+ is a sequencehSα : α ∈ Si such that:I S ⊆ µ+ ∩ cof(µ) is stationary,
I Sα is a stationary subset of Pµ (α) for all α ∈ S,

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

5/21

Disjoint Stationary Sequences

Definition (Krueger, 2007)
Let µ be regular. A disjoint stationary sequence on µ+ is a sequencehSα : α ∈ Si such that:I S ⊆ µ+ ∩ cof(µ) is stationary,
I Sα is a stationary subset of Pµ (α) for all α ∈ S,
I Sα ∩ Sβ = ∅ if α 6= β.

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

5/21

Disjoint Stationary Sequences

Definition (Krueger, 2007)
Let µ be regular. A disjoint stationary sequence on µ+ is a sequencehSα : α ∈ Si such that:I S ⊆ µ+ ∩ cof(µ) is stationary,
I Sα is a stationary subset of Pµ (α) for all α ∈ S,
I Sα ∩ Sβ = ∅ if α 6= β.
We write DSS(µ+ ) to say there is a disjoint stationary sequence on µ+ .

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

5/21

Disjoint Stationary Sequences

Definition (Krueger, 2007)
Let µ be regular. A disjoint stationary sequence on µ+ is a sequencehSα : α ∈ Si such that:I S ⊆ µ+ ∩ cof(µ) is stationary,
I Sα is a stationary subset of Pµ (α) for all α ∈ S,
I Sα ∩ Sβ = ∅ if α 6= β.
We write DSS(µ+ ) to say there is a disjoint stationary sequence on µ+ .

Theorem (Friedman-Krueger, 2007)
It is consistent that DSS(µ+ ) holds (actually ∃ disjoint club seq.) forsome regular µ if and only if it is consistent that there is a Mahlo cardinal.

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

6/21

Main Results

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

6/21

Main Results
Question (Krueger, 2009)
Is it consistent that for some regular µ we have DSS(µ+ ) ∧ DSS(µ++ )?

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

6/21

Main Results
Question (Krueger, 2009)
Is it consistent that for some regular µ we have DSS(µ+ ) ∧ DSS(µ++ )?

Question (Krueger, 2009)
Can we separate variants of internal approachability at successive
cardinals?

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

6/21

Main Results
Question (Krueger, 2009)
Is it consistent that for some regular µ we have DSS(µ+ ) ∧ DSS(µ++ )?

Question (Krueger, 2009)
Can we separate variants of internal approachability at successive
cardinals?

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

Maxwell Levine

University of Freiburg

Background: Why Disjoint Stationary Sequences?

6/21

Main Results
Question (Krueger, 2009)
Is it consistent that for some regular µ we have DSS(µ+ ) ∧ DSS(µ++ )?

Question (Krueger, 2009)
Can we separate variants of internal approachability at successive
cardinals?

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which for µ ∈ {ℵ1 , ℵ2 }, there are stationarily manyN ∈ [H(µ+ )]µ that are internally stationary but not internally club.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

7/21

Section 2
Two Disjoint Stationary Sequences

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.
I For µ regular we want a sequence hSα : α ∈ Si such that (1)S ⊆ µ+ ∩ cof(µ) is stationary, (2) Sα is a stationary subset of Pµ (α)
for all α, and (3) Sα ∩ Sβ = ∅ if α 6= β.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.
I For µ regular we want a sequence hSα : α ∈ Si such that (1)S ⊆ µ+ ∩ cof(µ) is stationary, (2) Sα is a stationary subset of Pµ (α)
for all α, and (3) Sα ∩ Sβ = ∅ if α 6= β.
I The set S will be the set of inaccessibles in λ.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.
I For µ regular we want a sequence hSα : α ∈ Si such that (1)S ⊆ µ+ ∩ cof(µ) is stationary, (2) Sα is a stationary subset of Pµ (α)
for all α, and (3) Sα ∩ Sβ = ∅ if α 6= β.
I The set S will be the set of inaccessibles in λ.
I Gitik’s theorem (or Krueger’s analog of it) will be used to generate
the Sα ’s.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.
I For µ regular we want a sequence hSα : α ∈ Si such that (1)S ⊆ µ+ ∩ cof(µ) is stationary, (2) Sα is a stationary subset of Pµ (α)
for all α, and (3) Sα ∩ Sβ = ∅ if α 6= β.
I The set S will be the set of inaccessibles in λ.
I Gitik’s theorem (or Krueger’s analog of it) will be used to generate
the Sα ’s.
I Need an M such that:Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.
I For µ regular we want a sequence hSα : α ∈ Si such that (1)S ⊆ µ+ ∩ cof(µ) is stationary, (2) Sα is a stationary subset of Pµ (α)
for all α, and (3) Sα ∩ Sβ = ∅ if α 6= β.
I The set S will be the set of inaccessibles in λ.
I Gitik’s theorem (or Krueger’s analog of it) will be used to generate
the Sα ’s.
I Need an M such that:I For all δ < λ inaccessible, M ' M  δ ∗ Add(ω) ∗ Q where Q has nicepreservation properties.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.
I For µ regular we want a sequence hSα : α ∈ Si such that (1)S ⊆ µ+ ∩ cof(µ) is stationary, (2) Sα is a stationary subset of Pµ (α)
for all α, and (3) Sα ∩ Sβ = ∅ if α 6= β.
I The set S will be the set of inaccessibles in λ.
I Gitik’s theorem (or Krueger’s analog of it) will be used to generate
the Sα ’s.
I Need an M such that:I For all δ < λ inaccessible, M ' M  δ ∗ Add(ω) ∗ Q where Q has nicepreservation properties.
!!! M itself has nice preservation properties.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

8/21

The Plan (Generally Speaking)
Theorem (Gitik)
If V ⊂ W are models of ZFC with the same ordinals, W \ V contains a
real, and κ is a regular cardinal in W such that (κ+ )W ≤ λ, thenPκW (λ) \ V is stationary.
I Start with V |= “λ Mahlo”.
I For µ regular we want a sequence hSα : α ∈ Si such that (1)S ⊆ µ+ ∩ cof(µ) is stationary, (2) Sα is a stationary subset of Pµ (α)
for all α, and (3) Sα ∩ Sβ = ∅ if α 6= β.
I The set S will be the set of inaccessibles in λ.
I Gitik’s theorem (or Krueger’s analog of it) will be used to generate
the Sα ’s.
I Need an M such that:I For all δ < λ inaccessible, M ' M  δ ∗ Add(ω) ∗ Q where Q has nicepreservation properties.
!!! M itself has nice preservation properties.

I “nice preservation properties” = “Abraham-style projection analysis”
Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

9/21

Review of “Abraham-Style Projection Analysis”

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

9/21

Review of “Abraham-Style Projection Analysis”
Theorem (Mitchell, 1972/3)
Given a weakly compact cardinal λ in V , there is a forcing M such that
V [M] |= “there are no Aronszajn trees on ℵ2 ”.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

9/21

Review of “Abraham-Style Projection Analysis”
Theorem (Mitchell, 1972/3)
Given a weakly compact cardinal λ in V , there is a forcing M such that
V [M] |= “there are no Aronszajn trees on ℵ2 ”.

Theorem (Abraham, 1983)
Given sufficient large cardinals there is a forcing extension in which there
are no Aronszajn trees on ℵ2 or ℵ3 .

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

9/21

Review of “Abraham-Style Projection Analysis”
Theorem (Mitchell, 1972/3)
Given a weakly compact cardinal λ in V , there is a forcing M such that
V [M] |= “there are no Aronszajn trees on ℵ2 ”.

Theorem (Abraham, 1983)
Given sufficient large cardinals there is a forcing extension in which there
are no Aronszajn trees on ℵ2 or ℵ3 .
!!! The point is to use a variant of Mitchell forcing M such that there isa projection
π : Add(τ, λ) × T → M
where T is reasonably closed.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

9/21

Review of “Abraham-Style Projection Analysis”
Theorem (Mitchell, 1972/3)
Given a weakly compact cardinal λ in V , there is a forcing M such that
V [M] |= “there are no Aronszajn trees on ℵ2 ”.

Theorem (Abraham, 1983)
Given sufficient large cardinals there is a forcing extension in which there
are no Aronszajn trees on ℵ2 or ℵ3 .
!!! The point is to use a variant of Mitchell forcing M such that there isa projection
π : Add(τ, λ) × T → M
where T is reasonably closed.
I More generally, the goal here is to make these problems more
tractable.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:
2.1 dom q ⊂ {δ < λ : δ is inaccessible},

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:
2.1 dom q ⊂ {δ < λ : δ is inaccessible},
2.2 | dom q| < µ,Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:
2.1 dom q ⊂ {δ < λ : δ is inaccessible},
2.2 | dom q| < µ,˙
2.3 ∀δ ∈ dom(q), p  ((δ + 1) × τ ) Add∗ (τ,δ+1) “q(δ) ∈ Col(µ,δ)”.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:
2.1 dom q ⊂ {δ < λ : δ is inaccessible},
2.2 | dom q| < µ,˙
2.3 ∀δ ∈ dom(q), p  ((δ + 1) × τ ) Add∗ (τ,δ+1) “q(δ) ∈ Col(µ,δ)”.

We let (p, q) ≤ (p 0 , q 0 ) if and only if:

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:
2.1 dom q ⊂ {δ < λ : δ is inaccessible},
2.2 | dom q| < µ,˙
2.3 ∀δ ∈ dom(q), p  ((δ + 1) × τ ) Add∗ (τ,δ+1) “q(δ) ∈ Col(µ,δ)”.

We let (p, q) ≤ (p 0 , q 0 ) if and only if:(i) p ≤Add∗ (τ,λ) p 0 ,Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:
2.1 dom q ⊂ {δ < λ : δ is inaccessible},
2.2 | dom q| < µ,˙
2.3 ∀δ ∈ dom(q), p  ((δ + 1) × τ ) Add∗ (τ,δ+1) “q(δ) ∈ Col(µ,δ)”.

We let (p, q) ≤ (p 0 , q 0 ) if and only if:(i) p ≤Add∗ (τ,λ) p 0 ,(ii) dom q ⊇ dom q 0 ,Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

10/21

Defining the Forcing
Definition

Given a regular τ and a set of ordinals Y , we let Add∗ (τ, Y ) be the poset
consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
where | dom p| < τ . We let p ≤Add∗ (τ,Y ) q if and only if p ⊇ q.

Definition
Let λ be inaccessible and let τ < µ < λ be regular cardinals such that
τ <τ = τ . We define a forcing M+ (τ, µ, λ) that consists of pairs (p, q)
such that:
1. p ∈ Add∗ (τ, λ),2. q is a function such that:
2.1 dom q ⊂ {δ < λ : δ is inaccessible},
2.2 | dom q| < µ,˙
2.3 ∀δ ∈ dom(q), p  ((δ + 1) × τ ) Add∗ (τ,δ+1) “q(δ) ∈ Col(µ,δ)”.

We let (p, q) ≤ (p 0 , q 0 ) if and only if:(i) p ≤Add∗ (τ,λ) p 0 ,(ii) dom q ⊇ dom q 0 ,(iii) for all δ ∈ dom q 0 , p  ((δ + 1) × τ )Maxwell Levineq 0 (δ)”˙
Add∗ (τ,δ+1) “q(δ) ≤Col(µ,δ)
University of Freiburg

Two Disjoint Stationary Sequences

11/21

The Projection Analysis and Easy Properties

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

11/21

The Projection Analysis and Easy Properties
Definition
Let T = T(M+ (τ, µ, λ)) be the poset consisting of conditions q such that:Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

11/21

The Projection Analysis and Easy Properties
Definition
Let T = T(M+ (τ, µ, λ)) be the poset consisting of conditions q such that:
1. dom q ⊂ λ ∩ {δ < λ : δ is inaccessible},
2. | dom q| < µ,3. ∀δ ∈ dom q,

Maxwell Levine˙Add∗ (τ,δ+1) q(δ) ∈ Col(µ, δ)”.

University of Freiburg

Two Disjoint Stationary Sequences

11/21

The Projection Analysis and Easy Properties
Definition
Let T = T(M+ (τ, µ, λ)) be the poset consisting of conditions q such that:
1. dom q ⊂ λ ∩ {δ < λ : δ is inaccessible},
2. | dom q| < µ,3. ∀δ ∈ dom q,˙Add∗ (τ,δ+1) q(δ) ∈ Col(µ, δ)”.
0

Most importantly, we let q ≤ q if and only if:(i) dom q ⊇ dom q 0 ,(ii) for all δ ∈ dom q,

Maxwell Levine0Add∗ (τ,δ+1) “q(δ) ≤ q (δ)”.

University of Freiburg

Two Disjoint Stationary Sequences

11/21

The Projection Analysis and Easy Properties
Definition
Let T = T(M+ (τ, µ, λ)) be the poset consisting of conditions q such that:
1. dom q ⊂ λ ∩ {δ < λ : δ is inaccessible},
2. | dom q| < µ,3. ∀δ ∈ dom q,˙Add∗ (τ,δ+1) q(δ) ∈ Col(µ, δ)”.
0

Most importantly, we let q ≤ q if and only if:(i) dom q ⊇ dom q 0 ,(ii) for all δ ∈ dom q,0Add∗ (τ,δ+1) “q(δ) ≤ q (δ)”.

Proposition
T is µ-closed and there is a projection Add∗ (τ, λ) × T → M+ (τ, µ, λ).

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

11/21

The Projection Analysis and Easy Properties
Definition
Let T = T(M+ (τ, µ, λ)) be the poset consisting of conditions q such that:
1. dom q ⊂ λ ∩ {δ < λ : δ is inaccessible},
2. | dom q| < µ,3. ∀δ ∈ dom q,˙Add∗ (τ,δ+1) q(δ) ∈ Col(µ, δ)”.
0

Most importantly, we let q ≤ q if and only if:(i) dom q ⊇ dom q 0 ,(ii) for all δ ∈ dom q,0Add∗ (τ,δ+1) “q(δ) ≤ q (δ)”.

Proposition
T is µ-closed and there is a projection Add∗ (τ, λ) × T → M+ (τ, µ, λ).

Proposition
I M+ (τ, µ, λ) is τ -closed and has the λ-chain condition

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

11/21

The Projection Analysis and Easy Properties
Definition
Let T = T(M+ (τ, µ, λ)) be the poset consisting of conditions q such that:
1. dom q ⊂ λ ∩ {δ < λ : δ is inaccessible},
2. | dom q| < µ,3. ∀δ ∈ dom q,˙Add∗ (τ,δ+1) q(δ) ∈ Col(µ, δ)”.
0

Most importantly, we let q ≤ q if and only if:(i) dom q ⊇ dom q 0 ,(ii) for all δ ∈ dom q,0Add∗ (τ,δ+1) “q(δ) ≤ q (δ)”.

Proposition
T is µ-closed and there is a projection Add∗ (τ, λ) × T → M+ (τ, µ, λ).

Proposition
I M+ (τ, µ, λ) is τ -closed and has the λ-chain condition
I M+ (τ, µ, λ) preserves cardinals up to and including µ.
I For all α < λ, M+ (τ, µ, λ) forces |α| = µ.
I M+ (τ, µ, λ) forces λ = µ+ and 2τ = λ.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

12/21

The Factoring

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

12/21

The Factoring
Lemma
If δ0 < λ is inaccessible, then there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ Q
where M+ (τ, µ, δ0 ) ∗ Add(τ ) forces that Q is a projection of a product ofa µ-closed forcing and a τ + -cc forcing.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

12/21

The Factoring
Lemma
If δ0 < λ is inaccessible, then there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ Q
where M+ (τ, µ, δ0 ) ∗ Add(τ ) forces that Q is a projection of a product ofa µ-closed forcing and a τ + -cc forcing.

Sketch.
The outline looks like this:Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

12/21

The Factoring
Lemma
If δ0 < λ is inaccessible, then there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ Q
where M+ (τ, µ, δ0 ) ∗ Add(τ ) forces that Q is a projection of a product ofa µ-closed forcing and a τ + -cc forcing.

Sketch.
The outline looks like this:I we will show that there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ (P × R) where the following
hold in the extension by M+ (τ, µ, δ0 ) ∗ Add(τ ):Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

12/21

The Factoring
Lemma
If δ0 < λ is inaccessible, then there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ Q
where M+ (τ, µ, δ0 ) ∗ Add(τ ) forces that Q is a projection of a product ofa µ-closed forcing and a τ + -cc forcing.

Sketch.
The outline looks like this:I we will show that there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ (P × R) where the following
hold in the extension by M+ (τ, µ, δ0 ) ∗ Add(τ ):I R is a projection of a product of a µ-closed forcing and Add∗ (τ, λ),andMaxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

12/21

The Factoring
Lemma
If δ0 < λ is inaccessible, then there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ Q
where M+ (τ, µ, δ0 ) ∗ Add(τ ) forces that Q is a projection of a product ofa µ-closed forcing and a τ + -cc forcing.

Sketch.
The outline looks like this:I we will show that there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ (P × R) where the following
hold in the extension by M+ (τ, µ, δ0 ) ∗ Add(τ ):I R is a projection of a product of a µ-closed forcing and Add∗ (τ, λ),andI V [M+ (τ, µ, δ0 )][Add(τ, 1)] |= “P is µ-closed”.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

12/21

The Factoring
Lemma
If δ0 < λ is inaccessible, then there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ Q
where M+ (τ, µ, δ0 ) ∗ Add(τ ) forces that Q is a projection of a product ofa µ-closed forcing and a τ + -cc forcing.

Sketch.
The outline looks like this:I we will show that there is a forcing equivalenceM+ (τ, µ, λ) ' M+ (τ, µ, δ0 ) ∗ Add(τ ) ∗ (P × R) where the following
hold in the extension by M+ (τ, µ, δ0 ) ∗ Add(τ ):I R is a projection of a product of a µ-closed forcing and Add∗ (τ, λ),andI V [M+ (τ, µ, δ0 )][Add(τ, 1)] |= “P is µ-closed”.

I The crux is that the forcing P is interpreted the same way in two
different sub-extensions, which makes the argument go through.
Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

13/21

Details to the Factoring

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

13/21

Details to the Factoring
Claim

V [Add∗ (τ, δ + 1)]’s version Col(µ, δ0 ) is still correct inV [M+ (τ, µ, δ0 )][Add(τ, 1)].

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

13/21

Details to the Factoring
Claim

V [Add∗ (τ, δ + 1)]’s version Col(µ, δ0 ) is still correct inV [M+ (τ, µ, δ0 )][Add(τ, 1)].

Proof.
I G is M+ (τ, µ, δ0 )-generic over V ,I r is Add(τ )-generic over V [G ],δ0
I H is the Add∗ (τ, δ0 )-generic induced from G by πAdd,

I K is the generic for the quotient of M+ (τ, µ, δ0 ) by Add∗ (τ, δ0 ), i.e.
the generic such that V [H][K ] = V [G ],I T is the generic forT(M+ (τ, µ, δ0 )), so V [G ] ⊂ V [T ][H].

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

13/21

Details to the Factoring
Claim

V [Add∗ (τ, δ + 1)]’s version Col(µ, δ0 ) is still correct inV [M+ (τ, µ, δ0 )][Add(τ, 1)].

Proof.
I G is M+ (τ, µ, δ0 )-generic over V ,I r is Add(τ )-generic over V [G ],δ0
I H is the Add∗ (τ, δ0 )-generic induced from G by πAdd,

I K is the generic for the quotient of M+ (τ, µ, δ0 ) by Add∗ (τ, δ0 ), i.e.
the generic such that V [H][K ] = V [G ],I T is the generic forT(M+ (τ, µ, δ0 )), so V [G ] ⊂ V [T ][H].
Adjoining G does not change the def. of Add(τ ), and K is def.’d in termsof subsets of τ adjoined by H, so V [G ][r ] = V [H][K ][r ] = V [H][r ][K ].

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

13/21

Details to the Factoring
Claim

V [Add∗ (τ, δ + 1)]’s version Col(µ, δ0 ) is still correct inV [M+ (τ, µ, δ0 )][Add(τ, 1)].

Proof.
I G is M+ (τ, µ, δ0 )-generic over V ,I r is Add(τ )-generic over V [G ],δ0
I H is the Add∗ (τ, δ0 )-generic induced from G by πAdd,

I K is the generic for the quotient of M+ (τ, µ, δ0 ) by Add∗ (τ, δ0 ), i.e.
the generic such that V [H][K ] = V [G ],I T is the generic forT(M+ (τ, µ, δ0 )), so V [G ] ⊂ V [T ][H].
Adjoining G does not change the def. of Add(τ ), and K is def.’d in termsof subsets of τ adjoined by H, so V [G ][r ] = V [H][K ][r ] = V [H][r ][K ].
V [H][r ] ⊂ V [H][r ][K ] = V [H][K ][r ] = V [G ][r ] ⊂ V [T ][H][r ] = V [H][r ][T ],Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

13/21

Details to the Factoring
Claim

V [Add∗ (τ, δ + 1)]’s version Col(µ, δ0 ) is still correct inV [M+ (τ, µ, δ0 )][Add(τ, 1)].

Proof.
I G is M+ (τ, µ, δ0 )-generic over V ,I r is Add(τ )-generic over V [G ],δ0
I H is the Add∗ (τ, δ0 )-generic induced from G by πAdd,

I K is the generic for the quotient of M+ (τ, µ, δ0 ) by Add∗ (τ, δ0 ), i.e.
the generic such that V [H][K ] = V [G ],I T is the generic forT(M+ (τ, µ, δ0 )), so V [G ] ⊂ V [T ][H].
Adjoining G does not change the def. of Add(τ ), and K is def.’d in termsof subsets of τ adjoined by H, so V [G ][r ] = V [H][K ][r ] = V [H][r ][K ].
V [H][r ] ⊂ V [H][r ][K ] = V [H][K ][r ] = V [G ][r ] ⊂ V [T ][H][r ] = V [H][r ][T ],and Easton’s Lemma implies that T does not add new < µ-sequences
over V [H][r ], so therefore K does not add new < µ-sequences over
Maxwell Levine
University of Freiburg
V [H][r ] since it is an intermediate factor of the extension.

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis
Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis
Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

RemarkThis is *not* a lifting argument.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis
Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

RemarkThis is *not* a lifting argument.

Proof.
I V |= “λ1 < λ2 are Mahlo”.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis
Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

RemarkThis is *not* a lifting argument.

Proof.
I V |= “λ1 < λ2 are Mahlo”.
I M1 = M+ (ω, ℵ1 , λ1 ), Ṁ2 be an M1 -name for M+ (ω, λ1 , λ2 ).

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis
Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

RemarkThis is *not* a lifting argument.

Proof.
I V |= “λ1 < λ2 are Mahlo”.
I M1 = M+ (ω, ℵ1 , λ1 ), Ṁ2 be an M1 -name for M+ (ω, λ1 , λ2 ).
I G1 is M1 -generic over V , G2 is Ṁ2 [G1 ]-generic over V [G1 ].

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis
Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

RemarkThis is *not* a lifting argument.

Proof.
I V |= “λ1 < λ2 are Mahlo”.
I M1 = M+ (ω, ℵ1 , λ1 ), Ṁ2 be an M1 -name for M+ (ω, λ1 , λ2 ).
I G1 is M1 -generic over V , G2 is Ṁ2 [G1 ]-generic over V [G1 ].
I The main issue is to make sure that the DSS added by G1 is not
destroyed by adjoining G2 .

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

14/21

Taking Advantage of the Projection Analysis
Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which there are disjoint stationary sequences on ℵ2 and ℵ3 .

RemarkThis is *not* a lifting argument.

Proof.
I V |= “λ1 < λ2 are Mahlo”.
I M1 = M+ (ω, ℵ1 , λ1 ), Ṁ2 be an M1 -name for M+ (ω, λ1 , λ2 ).
I G1 is M1 -generic over V , G2 is Ṁ2 [G1 ]-generic over V [G1 ].
I The main issue is to make sure that the DSS added by G1 is not
destroyed by adjoining G2 .
I This works because of the projection analysis, which preserves
stationarity of the relevant stationary sets.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

15/21

Consecutively Separating Interally Club and Int. Stat.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

15/21

Consecutively Separating Interally Club and Int. Stat.
Theorem (Krueger)
PFA2 implies that for all λ ≥ ℵ2 regular there are stationarily manyN ∈ [λ]ℵ1 which are internally stationary but not internally club.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

15/21

Consecutively Separating Interally Club and Int. Stat.
Theorem (Krueger)
PFA2 implies that for all λ ≥ ℵ2 regular there are stationarily manyN ∈ [λ]ℵ1 which are internally stationary but not internally club.

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which for µ ∈ {ℵ1 , ℵ2 }, there are stationarily manyN ∈ [H(µ+ )]µ that are internally stationary but not internally club.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

15/21

Consecutively Separating Interally Club and Int. Stat.
Theorem (Krueger)
PFA2 implies that for all λ ≥ ℵ2 regular there are stationarily manyN ∈ [λ]ℵ1 which are internally stationary but not internally club.

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which for µ ∈ {ℵ1 , ℵ2 }, there are stationarily manyN ∈ [H(µ+ )]µ that are internally stationary but not internally club.
I Same model as the first theorem.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

15/21

Consecutively Separating Interally Club and Int. Stat.
Theorem (Krueger)
PFA2 implies that for all λ ≥ ℵ2 regular there are stationarily manyN ∈ [λ]ℵ1 which are internally stationary but not internally club.

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which for µ ∈ {ℵ1 , ℵ2 }, there are stationarily manyN ∈ [H(µ+ )]µ that are internally stationary but not internally club.
I Same model as the first theorem.

Proposition
Let P1 have the δ-chain condition, let P2 be ν-closed, and let X be a set
such that |X |δ ≤ ν with δ + ≤ ν. If S ⊆ [X ]δ is stationary and internallystationary but not internally club, then P1 × P2 forces “S is SaISbnIC”.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

15/21

Consecutively Separating Interally Club and Int. Stat.
Theorem (Krueger)
PFA2 implies that for all λ ≥ ℵ2 regular there are stationarily manyN ∈ [λ]ℵ1 which are internally stationary but not internally club.

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which for µ ∈ {ℵ1 , ℵ2 }, there are stationarily manyN ∈ [H(µ+ )]µ that are internally stationary but not internally club.
I Same model as the first theorem.

Proposition
Let P1 have the δ-chain condition, let P2 be ν-closed, and let X be a set
such that |X |δ ≤ ν with δ + ≤ ν. If S ⊆ [X ]δ is stationary and internallystationary but not internally club, then P1 × P2 forces “S is SaISbnIC”.
I i.e. δ = ℵ1 , ν = ℵ2 , S ⊆ [ℵ2 ]ℵ1 , P1 = Add(ω, λ2 ), P2 = ℵ2 -closed
termspace T.

Maxwell Levine

University of Freiburg

Two Disjoint Stationary Sequences

15/21

Consecutively Separating Interally Club and Int. Stat.
Theorem (Krueger)
PFA2 implies that for all λ ≥ ℵ2 regular there are stationarily manyN ∈ [λ]ℵ1 which are internally stationary but not internally club.

Theorem (L., 2023)
Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forcing
extension in which for µ ∈ {ℵ1 , ℵ2 }, there are stationarily manyN ∈ [H(µ+ )]µ that are internally stationary but not internally club.
I Same model as the first theorem.

Proposition
Let P1 have the δ-chain condition, let P2 be ν-closed, and let X be a set
such that |X |δ ≤ ν with δ + ≤ ν. If S ⊆ [X ]δ is stationary and internallystationary but not internally club, then P1 × P2 forces “S is SaISbnIC”.
I i.e. δ = ℵ1 , ν = ℵ2 , S ⊆ [ℵ2 ]ℵ1 , P1 = Add(ω, λ2 ), P2 = ℵ2 -closed
termspace T.
I This uses that we are adding Add(ω)-sets as opposed to Add(ω1 )sets in the second step!
Maxwell Levine

University of Freiburg

Related Results

16/21

Section 3
Related Results

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Question (Authors of “Eightfold Way”, 2018)
Can we get failure of AP(ℵ2 ) (i.e. failure of ℵ2 ∈ I [ℵ2 ]), an ℵ2 -Aronszajn
tree, and stationary reflection for subsets of ℵ2 ∩ cof(ω) from a Mahlo?

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Question (Authors of “Eightfold Way”, 2018)
Can we get failure of AP(ℵ2 ) (i.e. failure of ℵ2 ∈ I [ℵ2 ]), an ℵ2 -Aronszajn
tree, and stationary reflection for subsets of ℵ2 ∩ cof(ω) from a Mahlo?

Theorem (Gilton-Krueger, 2020)
Yes.

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Question (Authors of “Eightfold Way”, 2018)
Can we get failure of AP(ℵ2 ) (i.e. failure of ℵ2 ∈ I [ℵ2 ]), an ℵ2 -Aronszajn
tree, and stationary reflection for subsets of ℵ2 ∩ cof(ω) from a Mahlo?

Theorem (Gilton-Krueger, 2020)
Yes.(Argument uses DSS(ℵ2 ) to get ¬AP(ℵ2 ).)Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Question (Authors of “Eightfold Way”, 2018)
Can we get failure of AP(ℵ2 ) (i.e. failure of ℵ2 ∈ I [ℵ2 ]), an ℵ2 -Aronszajn
tree, and stationary reflection for subsets of ℵ2 ∩ cof(ω) from a Mahlo?

Theorem (Gilton-Krueger, 2020)
Yes.(Argument uses DSS(ℵ2 ) to get ¬AP(ℵ2 ).)Theorem (Cox, 2021)
PFA implies ¬DSS(ℵ2 ) ∧ ¬AP(ℵ2 ).

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Question (Authors of “Eightfold Way”, 2018)
Can we get failure of AP(ℵ2 ) (i.e. failure of ℵ2 ∈ I [ℵ2 ]), an ℵ2 -Aronszajn
tree, and stationary reflection for subsets of ℵ2 ∩ cof(ω) from a Mahlo?

Theorem (Gilton-Krueger, 2020)
Yes.(Argument uses DSS(ℵ2 ) to get ¬AP(ℵ2 ).)Theorem (Cox, 2021)
PFA implies ¬DSS(ℵ2 ) ∧ ¬AP(ℵ2 ).

Question (Gilton, 2019)
What is the exact consistency of this separation?

Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Question (Authors of “Eightfold Way”, 2018)
Can we get failure of AP(ℵ2 ) (i.e. failure of ℵ2 ∈ I [ℵ2 ]), an ℵ2 -Aronszajn
tree, and stationary reflection for subsets of ℵ2 ∩ cof(ω) from a Mahlo?

Theorem (Gilton-Krueger, 2020)
Yes.(Argument uses DSS(ℵ2 ) to get ¬AP(ℵ2 ).)Theorem (Cox, 2021)
PFA implies ¬DSS(ℵ2 ) ∧ ¬AP(ℵ2 ).

Question (Gilton, 2019)
What is the exact consistency of this separation?

Theorem (L., 2023)
A Mahlo cardinal: Con(∃λ
Mahlo)=⇒ Con(¬DSS(ℵ2 ) ∧ ¬AP(ℵ2 )).
Maxwell Levine

University of Freiburg

Related Results

17/21

Failure of Approachability without a Disjoint Stationary
Sequence
Problem
So when are or aren’t there disjoint stationary sequences?
Cited years are from publications or preprints in what follows!

Question (Authors of “Eightfold Way”, 2018)
Can we get failure of AP(ℵ2 ) (i.e. failure of ℵ2 ∈ I [ℵ2 ]), an ℵ2 -Aronszajn
tree, and stationary reflection for subsets of ℵ2 ∩ cof(ω) from a Mahlo?

Theorem (Gilton-Krueger, 2020)
Yes.(Argument uses DSS(ℵ2 ) to get ¬AP(ℵ2 ).)Theorem (Cox, 2021)
PFA implies ¬DSS(ℵ2 ) ∧ ¬AP(ℵ2 ).

Question (Gilton, 2019)
What is the exact consistency of this separation?

Theorem (L., 2023)
A Mahlo cardinal: Con(∃λ
Mahlo)=⇒ Con(¬DSS(ℵ2 ) ∧ ¬AP(ℵ2 )).(Website version!)
Maxwell Levine

University of Freiburg

Related Results

18/21

Disjoint Stationary Sequences on an Interval

Maxwell Levine

University of Freiburg

Related Results

18/21

Disjoint Stationary Sequences on an Interval

Theorem (Jakob)
Assuming the consistency of countably many Mahlo cardinals, there are
disjoint stationary sequences on ℵn for all 0 < n < ω.

Maxwell Levine

University of Freiburg

Concluding Thoughts

19/21

Section 4
Concluding Thoughts

Maxwell Levine

University of Freiburg

Concluding Thoughts

20/21

Questions

Maxwell Levine

University of Freiburg

Concluding Thoughts

20/21

Questions
Fact (Krueger)
If µ is regular and 2µ = µ+ , then DSS(µ+ ) is equivalent to the existenceof a stationary set U ⊆ [H(µ+ )]µ such that every N ∈ U is internallyinternally unbounded but not internally club.

Maxwell Levine

University of Freiburg

Concluding Thoughts

20/21

Questions
Fact (Krueger)
If µ is regular and 2µ = µ+ , then DSS(µ+ ) is equivalent to the existenceof a stationary set U ⊆ [H(µ+ )]µ such that every N ∈ U is internallyinternally unbounded but not internally club.
I So we could not use this to automatically strengthen “successive
disjoints stationary sequences” to “successive separations ofinternally stationary and internally club”.

Maxwell Levine

University of Freiburg

Concluding Thoughts

20/21

Questions
Fact (Krueger)
If µ is regular and 2µ = µ+ , then DSS(µ+ ) is equivalent to the existenceof a stationary set U ⊆ [H(µ+ )]µ such that every N ∈ U is internallyinternally unbounded but not internally club.
I So we could not use this to automatically strengthen “successive
disjoints stationary sequences” to “successive separations ofinternally stationary and internally club”.

Question
Is it consistent for µ regular that exactly one of DSS(µ+ ) and “internallyclub and internally unbounded are distinct for [H(µ+ )]µ ” holds?

Maxwell Levine

University of Freiburg

Concluding Thoughts

20/21

Questions
Fact (Krueger)
If µ is regular and 2µ = µ+ , then DSS(µ+ ) is equivalent to the existenceof a stationary set U ⊆ [H(µ+ )]µ such that every N ∈ U is internallyinternally unbounded but not internally club.
I So we could not use this to automatically strengthen “successive
disjoints stationary sequences” to “successive separations ofinternally stationary and internally club”.

Question
Is it consistent for µ regular that exactly one of DSS(µ+ ) and “internallyclub and internally unbounded are distinct for [H(µ+ )]µ ” holds?

Fact (Folklore)
If 2µ = µ+ , a separation between internally approachable and internallyunbounded implies the consistency of a Mahlo cardinal.

Maxwell Levine

University of Freiburg

Concluding Thoughts

20/21

Questions
Fact (Krueger)
If µ is regular and 2µ = µ+ , then DSS(µ+ ) is equivalent to the existenceof a stationary set U ⊆ [H(µ+ )]µ such that every N ∈ U is internallyinternally unbounded but not internally club.
I So we could not use this to automatically strengthen “successive
disjoints stationary sequences” to “successive separations ofinternally stationary and internally club”.

Question
Is it consistent for µ regular that exactly one of DSS(µ+ ) and “internallyclub and internally unbounded are distinct for [H(µ+ )]µ ” holds?

Fact (Folklore)
If 2µ = µ+ , a separation between internally approachable and internallyunbounded implies the consistency of a Mahlo cardinal.

Question
What is the exact equiconsistency strength of the separation of internallyapproachable and internally unbounded for [H(µ+ )]µ for regular µ?
Maxwell Levine

University of Freiburg

Concluding Thoughts

21/21

Arigatō!

Maxwell Levine

University of Freiburg