On a fragment of the Strong Reflection
Principle consistent with CH
Gunter Fuchs

RIMS Set Theory Workshop 2023
Oct. 24, 2023

Motivation: Martin’s Axiom

For a class Γ of forcings, FAΓ says that for any ω1 -sized collection ofdense subsets of a forcing in Γ, there is a filter that meets each of
the dense sets in the collection.

1

Motivation: Martin’s Axiom

For a class Γ of forcings, FAΓ says that for any ω1 -sized collection ofdense subsets of a forcing in Γ, there is a filter that meets each of
the dense sets in the collection.
Familiar instances:1

Motivation: Martin’s Axiom

For a class Γ of forcings, FAΓ says that for any ω1 -sized collection ofdense subsets of a forcing in Γ, there is a filter that meets each of
the dense sets in the collection.
Familiar instances:• Martin’s Axiom at ω1 , MAω1 (Γ =the collection of all c.c.c. forcings)• PFA (Γ =the collection of all proper forcings)• MM (Γ =the collection of all stationary set preserving forcings)

1Motivation: Martin’s Axiom

For a class Γ of forcings, FAΓ says that for any ω1 -sized collection ofdense subsets of a forcing in Γ, there is a filter that meets each of
the dense sets in the collection.
Familiar instances:• Martin’s Axiom at ω1 , MAω1 (Γ =the collection of all c.c.c. forcings)• PFA (Γ =the collection of all proper forcings)• MM (Γ =the collection of all stationary set preserving forcings)
The instance of main interest for this talk:• The Subcomplete Forcing Axiom, SCFA (Γ =the collection of all
subcomplete forcings - I’ll write SC for this class)1

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.

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Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.

2Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.
• No nontrivial ccc forcing is subcomplete.

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.
• No nontrivial ccc forcing is subcomplete.
• Examples of subcomplete forcing:2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.
• No nontrivial ccc forcing is subcomplete.
• Examples of subcomplete forcing:• Namba forcing (assuming CH).

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.
• No nontrivial ccc forcing is subcomplete.
• Examples of subcomplete forcing:• Namba forcing (assuming CH).
• Příkrý forcing.

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.
• No nontrivial ccc forcing is subcomplete.
• Examples of subcomplete forcing:• Namba forcing (assuming CH).
• Příkrý forcing.
• Generalized Příkrý forcing (Minden).

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.
• No nontrivial ccc forcing is subcomplete.
• Examples of subcomplete forcing:• Namba forcing (assuming CH).
• Příkrý forcing.
• Generalized Příkrý forcing (Minden).
• Magidor forcing (of length ω1 ) (F.)

2

Properties of subcomplete forcing• Can be iterated with revised countable support (Jensen) or using
nice iterations (Miyamoto, F-Switzer); allows dropping atechnical condition in the definition of subcompleteness.
• Subcomplete forcing adds no new reals.
• preserves:• stationary subsets of ω1 .
• ω1 -Souslin trees.
• The ♢ principle, if it holds.

• Every countably closed forcing is subcomplete.
• No nontrivial ccc forcing is subcomplete.
• Examples of subcomplete forcing:• Namba forcing (assuming CH).
• Příkrý forcing.
• Generalized Příkrý forcing (Minden).
• Magidor forcing (of length ω1 ) (F.)• Every ω2 -distributive forcing (assuming CH, F.)2

Forcing SCFA

• Assuming the existence of a supercompact cardinal κ, one caniterate proper forcings with countable support, with iterands
given by a Laver function for the supercompactness of κ,
producing a model in which PFA + κ = ω2 = 2ω holds.
(Baumgartner)3Forcing SCFA

• Assuming the existence of a supercompact cardinal κ, one caniterate proper forcings with countable support, with iterands
given by a Laver function for the supercompactness of κ,
producing a model in which PFA + κ = ω2 = 2ω holds.
(Baumgartner)• This can be modified to work for SPFA, by iterating semi-proper
forcings with rcs, inserting collapses to ω1 after each step in theiteration. (Foreman-Magidor-Shelah)3Forcing SCFA

• Assuming the existence of a supercompact cardinal κ, one caniterate proper forcings with countable support, with iterands
given by a Laver function for the supercompactness of κ,
producing a model in which PFA + κ = ω2 = 2ω holds.
(Baumgartner)• This can be modified to work for SPFA, by iterating semi-proper
forcings with rcs, inserting collapses to ω1 after each step in theiteration. (Foreman-Magidor-Shelah)• This can be modified to work for SCFA, by iterating subcomplete
forcings.

3During the iteration to force SCFA, ♢ will be forced, and since ♢ ispreserved by subcomplete forcing, it will hold in the final model.
Thus, the natural model of SCFA satisfies ♢. (Jensen)

4During the iteration to force SCFA, ♢ will be forced, and since ♢ ispreserved by subcomplete forcing, it will hold in the final model.
Thus, the natural model of SCFA satisfies ♢. (Jensen)
Recently, H. Sakai showed that SCFA is even consistent with ♢+ .

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During the iteration to force SCFA, ♢ will be forced, and since ♢ ispreserved by subcomplete forcing, it will hold in the final model.
Thus, the natural model of SCFA satisfies ♢. (Jensen)
Recently, H. Sakai showed that SCFA is even consistent with ♢+ .
This looks very different from MM.

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During the iteration to force SCFA, ♢ will be forced, and since ♢ ispreserved by subcomplete forcing, it will hold in the final model.
Thus, the natural model of SCFA satisfies ♢. (Jensen)
Recently, H. Sakai showed that SCFA is even consistent with ♢+ .
This looks very different from MM.
However, many of the known effects of MM carry over to SCFA (orrelated forcing principles), e.g.:• consequences on stationary reflection of sets of ordinals,• the failure of □ principles (extending prior work of Jensen),• existence of well-orders of the reals (to a certain degree).
How to approach SRP?

4

Stationary reflection
Many important consequences of forcing axioms have to do with thereflection of stationary sets (of ordinals).
Definition
Let κ be an uncountable regular cardinal, and let S ⊆ κ be stationary.
An ordinal ρ < κ is said to be a reflection point of S if ρ has
uncountable cofinality and S ∩ ρ is stationary in ρ.

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Stationary reflection
Many important consequences of forcing axioms have to do with thereflection of stationary sets (of ordinals).
Definition
Let κ be an uncountable regular cardinal, and let S ⊆ κ be stationary.
An ordinal ρ < κ is said to be a reflection point of S if ρ has
uncountable cofinality and S ∩ ρ is stationary in ρ.
Reflection may occur as a result of large cardinal properties. A
concrete reason why it occurs can be that a stronger property holds:
Definition
Let κ be an uncountable regular cardinal. Let Sκω be the set ofordinals less than κ that have countable cofinality. Friedman’s
problem at κ, FPκ , says that every stationary set S ⊆ Sκω contains a
closed set of order type ω1 .

5

Stationary reflection
Many important consequences of forcing axioms have to do with thereflection of stationary sets (of ordinals).
Definition
Let κ be an uncountable regular cardinal, and let S ⊆ κ be stationary.
An ordinal ρ < κ is said to be a reflection point of S if ρ has
uncountable cofinality and S ∩ ρ is stationary in ρ.
Reflection may occur as a result of large cardinal properties. A
concrete reason why it occurs can be that a stronger property holds:
Definition
Let κ be an uncountable regular cardinal. Let Sκω be the set ofordinals less than κ that have countable cofinality. Friedman’s
problem at κ, FPκ , says that every stationary set S ⊆ Sκω contains a
closed set of order type ω1 .
Clearly, the supremum of the closed set in the previous definition isa reflection point of S.
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SRP• originally introduced by Todorčević.

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SRP

• originally introduced by Todorčević.
• shown by Feng and Jech to be equivalent to their “projective
stationary reflection principle.”

6

SRP

• originally introduced by Todorčević.
• shown by Feng and Jech to be equivalent to their “projective
stationary reflection principle.”I view the principle as a generalization of Friedman’s problem. The
first concept to generalize is the notion of a closed subset of a given
order type.

6

SRP

• originally introduced by Todorčević.
• shown by Feng and Jech to be equivalent to their “projective
stationary reflection principle.”I view the principle as a generalization of Friedman’s problem. The
first concept to generalize is the notion of a closed subset of a given
order type.
Definition
Let κ be a regular uncountable cardinal, and let S ⊆ [Hκ ]ω bestationary. A continuous ∈-chain through S of length λ is a sequence⟨Xi | i < λ⟩ of members of S, increasing with respect to ∈, such that∪for every limit j < λ, Xj = i<j Xi .

6

The second thing that needs translating is “being a stationary subsetof Sκω .”7

The second thing that needs translating is “being a stationary subsetof Sκω .”Definition
Let κ be an uncountable regular cardinal. S ⊆ [Hκ ]ω is projective
stationary if for every stationary set T ⊆ ω1 , the set{X ∈ S | X ∩ ω1 ∈ T} is stationary.

7The second thing that needs translating is “being a stationary subsetof Sκω .”Definition
Let κ be an uncountable regular cardinal. S ⊆ [Hκ ]ω is projective
stationary if for every stationary set T ⊆ ω1 , the set{X ∈ S | X ∩ ω1 ∈ T} is stationary.
Here is the characterization of SRP, due to Feng and Jech, which I
take as the official definition.

7

The second thing that needs translating is “being a stationary subsetof Sκω .”Definition
Let κ be an uncountable regular cardinal. S ⊆ [Hκ ]ω is projective
stationary if for every stationary set T ⊆ ω1 , the set{X ∈ S | X ∩ ω1 ∈ T} is stationary.
Here is the characterization of SRP, due to Feng and Jech, which I
take as the official definition.
Definition
Let κ ≥ ω2 be regular. Then the strong reflection principle at κ,
denoted SRP(κ), states that whenever S ⊆ [Hκ ]ω is projective
stationary, there is a continuous ∈-chain of length ω1 through S. The
strong reflection principle SRP states that SRP(κ) holds for every
regular κ ≥ ω2 .

7

Consequences of SRP

SRP is a consequence of MM, and it captures many of the major
consequences of MM.
• NSω1 is ω2 -saturated,• 2ω = ω2 ,• Friedman’s problem at regular κ > ω1 ,• τ ω1 = τ for every regular τ > ω1 ,• the failure of □κ , for every uncountable κ,• the singular cardinal hypothesis.

8Consequences of SRP

SRP is a consequence of MM, and it captures many of the major
consequences of MM.
• NSω1 is ω2 -saturated,• 2ω = ω2 ,• Friedman’s problem at regular κ > ω1 ,• τ ω1 = τ for every regular τ > ω1 ,• the failure of □κ , for every uncountable κ,• the singular cardinal hypothesis.
Since SCFA is consistent with ♢, the highlighted consequences of
SRP don’t follow from SCFA.

8

Consequences of SRP

SRP is a consequence of MM, and it captures many of the major
consequences of MM.
• NSω1 is ω2 -saturated,• 2ω = ω2 ,• Friedman’s problem at regular κ > ω1 ,• τ ω1 = τ for every regular τ > ω1 ,• the failure of □κ , for every uncountable κ,• the singular cardinal hypothesis.
Since SCFA is consistent with ♢, the highlighted consequences of
SRP don’t follow from SCFA.

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Question
Is there a version of SRP for SCFA with the following properties:• it is natural,• it follows from SCFA,• it implies the “major” consequences of SCFA?

9The key
Definition
PS is the forcing notion consisting of continuous ∈-chains through S
of countable successor length, ordered by end-extension. For p ∈ PS ,we write p = ⟨Mpi | i ≤ ℓp ⟩.
Fact (Feng & Jech)
Let κ be an uncountable regular cardinal, and let S ⊆ [Hκ ]ω bestationary. Then
1. for every countable ordinal α, the set of conditions p with ℓp ≥ α
is dense in PS ,2. for every a ∈ Hκ , the set of conditions p such that there is an
i < ℓp with a ∈ Mpi is dense in PS ,3. PS is countably distributive.

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The key

Fact (Feng & Jech)
Let κ ≥ ω2 be an uncountable regular cardinal. Then a stationary set
S ⊆ [Hκ ]ω is projective stationary iff PS preserves stationary subsetsof ω1 .

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The key

Fact (Feng & Jech)
Let κ ≥ ω2 be an uncountable regular cardinal. Then a stationary set
S ⊆ [Hκ ]ω is projective stationary iff PS preserves stationary subsetsof ω1 .
So with hindsight, the strong reflection principle can be formulated
as follows.
Whenever κ ≥ ω2 is regular, S ⊆ [Hκ ]ω is stationary, andthe forcing PS is stationary set preserving, then S contains
a continuous ∈-chain of length ω1 .

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Canonical fragments of SRP

Definition
Let Γ be a forcing class. Then a stationary subset S of Hκ , whereκ ≥ ω2 is regular, is Γ-projective stationary iff PS ∈ Γ.

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Canonical fragments of SRP

Definition
Let Γ be a forcing class. Then a stationary subset S of Hκ , whereκ ≥ ω2 is regular, is Γ-projective stationary iff PS ∈ Γ.
Generalizing the above formulation of SRP, we arrive at:Definition
Let Γ be a forcing class. Let κ ≥ ω2 be regular. The Γ-fragment of thestrong reflection principle at κ, denoted Γ-SRP(κ), states thatwhenever S ⊆ [Hκ ]ω is Γ-projective stationary, then S contains acontinuous chain of length ω1 . The Γ-fragment of the strong
reflection principle, Γ-SRP, states that Γ-SRP(κ) holds for every
regular κ ≥ ω2 .

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Clearly then:• SSP-SRP is equivalent to SRP,
• FAΓ =⇒ Γ-SRP.

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Characterizing SC-projective stationarity

Definition
A model N = LAτ of ZFC− is full if there is a transitive model N′ of
ZFC− with N ∈ N′ , such that, letting θ = On ∩ N, θ is regular and′uncountable in N′ and N = HNθ . A well-founded model A of ZFC− isfull if the transitive isomorph of A is full.

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Characterizing SC-projective stationarity

Definition
A model N = LAτ of ZFC− is full if there is a transitive model N′ of
ZFC− with N ∈ N′ , such that, letting θ = On ∩ N, θ is regular and′uncountable in N′ and N = HNθ . A well-founded model A of ZFC− isfull if the transitive isomorph of A is full.
Definition
Let κ be a regular uncountable cardinal. A stationary set S ⊆ [Hκ ]ω isspread out if for all sufficiently large θ, whenever N and X are suchthat Hθ ⊆ N = LAτ |= ZFC− , τ > θ, S, a ∈ X ≺ N, X is countable and N|X
is full, then there is a Y ≺ N such that N|X ∼= N|Y, Y ∩ Hκ ∈ S, and theisomorphism fixes a.

14

Characterizing SC-projective stationarity
TheoremLet κ be an uncountable regular cardinal, and let S ⊆ [Hκ ]ω . The
following are equivalent:• S is spread out• S is SC-projective stationary

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Characterizing SC-projective stationarity
TheoremLet κ be an uncountable regular cardinal, and let S ⊆ [Hκ ]ω . The
following are equivalent:• S is spread out• S is SC-projective stationary
Disclaimer: Literally, in this theorem, SC has to be taken to be theclass of all ∞-subcomplete forcing notions, a simplification of
subcompleteness introduced by F. & Switzer. One can slighty
complicate the definition of “spread out” to “fully spread out” inorder to characterize subcomplete projective stationarity. For
simplicity, I won’t distinguish between the two unless it matters.

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Characterizing SC-projective stationarity
TheoremLet κ be an uncountable regular cardinal, and let S ⊆ [Hκ ]ω . The
following are equivalent:• S is spread out• S is SC-projective stationary
Disclaimer: Literally, in this theorem, SC has to be taken to be theclass of all ∞-subcomplete forcing notions, a simplification of
subcompleteness introduced by F. & Switzer. One can slighty
complicate the definition of “spread out” to “fully spread out” inorder to characterize subcomplete projective stationarity. For
simplicity, I won’t distinguish between the two unless it matters.
Thus, SC-SRP says that every spread out subset of some [Hκ ]ω , whereκ > ω1 is regular, contains a continuous ∈-chain of length ω1 .
15

Playing with spread out sets

1. Let κ be a regular cardinal. Let Hκ ⊆ N = LAτ , where κ < τ and τ
is a cardinal in L[A], and let τ ′ = (τ + )L[A] , N′ = LAτ ′ . Then,
whenever X′ is countable and N′ |X′ ≺ N′ , it follows that
X = X′ ∩ LAτ is full. In particular, the set of X ∩ Hκ as in thedefinition of being spread out contains a club in [Hκ ]ω .

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Playing with spread out sets

1. Let κ be a regular cardinal. Let Hκ ⊆ N = LAτ , where κ < τ and τ
is a cardinal in L[A], and let τ ′ = (τ + )L[A] , N′ = LAτ ′ . Then,
whenever X′ is countable and N′ |X′ ≺ N′ , it follows that
X = X′ ∩ LAτ is full. In particular, the set of X ∩ Hκ as in thedefinition of being spread out contains a club in [Hκ ]ω .
2. If κ ≤ 2ω and in the setting of the definition of “spread out”,
N|X ∼= N|Y, then the isomorphism must be the identity on Hκ . So
X ∩ Hκ = Y ∩ Hκ . Thus, if S ⊆ [Hκ ]ω is spread out, then it contains
a club.

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Playing with spread out sets

1. Let κ be a regular cardinal. Let Hκ ⊆ N = LAτ , where κ < τ and τ
is a cardinal in L[A], and let τ ′ = (τ + )L[A] , N′ = LAτ ′ . Then,
whenever X′ is countable and N′ |X′ ≺ N′ , it follows that
X = X′ ∩ LAτ is full. In particular, the set of X ∩ Hκ as in thedefinition of being spread out contains a club in [Hκ ]ω .
2. If κ ≤ 2ω and in the setting of the definition of “spread out”,
N|X ∼= N|Y, then the isomorphism must be the identity on Hκ . So
X ∩ Hκ = Y ∩ Hκ . Thus, if S ⊆ [Hκ ]ω is spread out, then it contains
a club.
3. Hence, for ω2 ≤ κ ≤ 2ω , SC-SRP(κ) is vacuously true.

16Playing with spread out sets

1. Let κ be a regular cardinal. Let Hκ ⊆ N = LAτ , where κ < τ and τ
is a cardinal in L[A], and let τ ′ = (τ + )L[A] , N′ = LAτ ′ . Then,
whenever X′ is countable and N′ |X′ ≺ N′ , it follows that
X = X′ ∩ LAτ is full. In particular, the set of X ∩ Hκ as in thedefinition of being spread out contains a club in [Hκ ]ω .
2. If κ ≤ 2ω and in the setting of the definition of “spread out”,
N|X ∼= N|Y, then the isomorphism must be the identity on Hκ . So
X ∩ Hκ = Y ∩ Hκ . Thus, if S ⊆ [Hκ ]ω is spread out, then it contains
a club.
3. Hence, for ω2 ≤ κ ≤ 2ω , SC-SRP(κ) is vacuously true.
4. Obviously, every spread out set is projective stationary, since if
PS is subcomplete, then it is also stationary set preserving.

16This creates an interesting situation: in order for SC-SRP to capture a
lot of the consequences of SRP, in particular those of SRP(ω2 ), we
better work with SC-SRP + CH, or else SC-SRP(ω2 ) gives us nothing!

17

This creates an interesting situation: in order for SC-SRP to capture a
lot of the consequences of SRP, in particular those of SRP(ω2 ), we
better work with SC-SRP + CH, or else SC-SRP(ω2 ) gives us nothing!
So in order for SC-SRP(ω2 ) to be similar to SRP(ω2 ), we should
augment it with CH, which contradicts SRP.

17

This creates an interesting situation: in order for SC-SRP to capture a
lot of the consequences of SRP, in particular those of SRP(ω2 ), we
better work with SC-SRP + CH, or else SC-SRP(ω2 ) gives us nothing!
So in order for SC-SRP(ω2 ) to be similar to SRP(ω2 ), we should
augment it with CH, which contradicts SRP.
But how similar is SC-SRP to SRP? We need some spread out sets!

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Main Tool

Lemma (after Jensen)κLet κ > 2ω be a regular cardinal, θ > 22 regular,Hθ ⊆ N = LAτ |= ZFC− , N|X ≺ N countable and full, and κ ∈ X. Letη ∈ κ ∩ X be such that η ω < κ. Let ⟨λi | i < n⟩ be regular cardinals in
N|X, each greater than κ. Let a be some element of X.
Then there is an ω-club of ordinals κ0 < κ for which there are a Y∼
and a π such that N|Y ≺ N, π : N|X ←→ N|Y and1. π fixes a, κ, η, λ0 , . . . , λn−1 .
2. For i < n, sup(X ∩ λi ) = sup(Y ∩ λi ).
3. X ∩ η = Y ∩ η.
4. sup(Y ∩ κ) = κ0 .

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Examples of spread out sets

Write Sκω = {β < κ | cf(β) = ω}.
Lemma
Let κ > 2ω be a regular cardinal, and let B ⊆ Sκω be stationary. Thenthe setS = {X ∈ [Hκ ]ω | sup(X ∩ κ) ∈ B}
is spread out.

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Examples of spread out sets

Write Sκω = {β < κ | cf(β) = ω}.
Lemma
Let κ > 2ω be a regular cardinal, and let B ⊆ Sκω be stationary. Thenthe setS = {X ∈ [Hκ ]ω | sup(X ∩ κ) ∈ B}
is spread out.
Corollary
SC-SRP implies Friedman’s problem at every regular κ > 2ω , andhence the failure of □κ , for every cardinal κ ≥ 2ω .

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Examples of spread out sets
Lemma
Let κ > 2ω be regular. Let ⃗D = ⟨Di | i < ω1 ⟩ be a partition of ω1 into
stationary sets and ⃗A = ⟨Ai | i < ω1 ⟩ a sequence of stationary subsets
of Sκω . Let
S = {X ∈ [Hκ ]ω | ∀i < ω1 (X ∩ ω1 ∈ Di → sup(X ∩ κ) ∈ Ai )}.
Then S is spread out.

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Examples of spread out sets
Lemma
Let κ > 2ω be regular. Let ⃗D = ⟨Di | i < ω1 ⟩ be a partition of ω1 into
stationary sets and ⃗A = ⟨Ai | i < ω1 ⟩ a sequence of stationary subsets
of Sκω . Let
S = {X ∈ [Hκ ]ω | ∀i < ω1 (X ∩ ω1 ∈ Di → sup(X ∩ κ) ∈ Ai )}.
Then S is spread out.
Funny point: Of course, the set S is projective stationary, since it isspread out. But the usual proof for this assumes that the sequence ⃗D
is maximal, in the sense that for every stationary set T ⊆ ω1 , there isan i < ω1 such that T ∩ Di is stationary. So if this assumption is
dropped, then I don’t think it was known that this set is projective
stationary.
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Corollary
SC-SRP implies that for every regular κ > 2ω and every collection Tof up to ω1 many stationary subsets of Sκω , there is an “exact”
reflection point δ for T . That is, δ is a reflection point of each T ∈ T ,∪and ( T ) ∩ δ contains a club.

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Corollary
SC-SRP implies that for every regular κ > 2ω and every collection Tof up to ω1 many stationary subsets of Sκω , there is an “exact”
reflection point δ for T . That is, δ is a reflection point of each T ∈ T ,∪and ( T ) ∩ δ contains a club.
Corollary
SC-SRP implies that for every regular κ > 2ω , κω1 = κ.

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Corollary
SC-SRP implies that for every regular κ > 2ω and every collection Tof up to ω1 many stationary subsets of Sκω , there is an “exact”
reflection point δ for T . That is, δ is a reflection point of each T ∈ T ,∪and ( T ) ∩ δ contains a club.
Corollary
SC-SRP implies that for every regular κ > 2ω , κω1 = κ.
Corollary
SC-SRP implies SCH.

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So SC-SRP does capture the major consequences of SCFA listed in
the beginning.

22

So SC-SRP does capture the major consequences of SCFA listed in
the beginning.
Another example of spread out sets can be formulated in terms of
mutual stationarity:
Definition
Let K be a collection of regular cardinals with supremum δ, and let⃗S = ⟨Sκ | κ ∈ K⟩ be a sequence such that for every κ ∈ K, Sκ is asubset of κ. Then ⃗S is mutually stationary if for every algebra A on δ,
there is an N ≺ A such that for all κ ∈ N ∩ K, sup(N ∩ κ) ∈ Sκ .

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Fact (Foreman & Magidor)
Let K be a set of uncountable regular cardinals, and let ⃗S = ⟨Sκ |κ ∈ K⟩ be a sequence such that for every κ ∈ K, Sκ is a stationary
subset of Sκω . Then, ⃗S is mutually stationary: for any algebra A on
sup K, there is a countable N ≺ A such that for all κ ∈ N ∩ K,
sup(N ∩ κ) ∈ Sκ .

23

Fact (Foreman & Magidor)
Let K be a set of uncountable regular cardinals, and let ⃗S = ⟨Sκ |κ ∈ K⟩ be a sequence such that for every κ ∈ K, Sκ is a stationary
subset of Sκω . Then, ⃗S is mutually stationary: for any algebra A on
sup K, there is a countable N ≺ A such that for all κ ∈ N ∩ K,
sup(N ∩ κ) ∈ Sκ .
Note that this can be equivalently formulated as saying:
Corollary
Let K be a set of regular cardinals with min(K) > ω1 , and let ⃗S = ⟨Sκ |κ ∈ K⟩ be a sequence such that for every κ ∈ K, Sκ is a stationary
subset of Sκω . Let δ = sup K. Then the set
S = {M ∈ [Hδ ]ω | ∀κ ∈ M ∩ K

sup(M ∩ κ) ∈ Sκ }is projective stationary.

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Another example
Theorem
Let K be a set of regular cardinals such that min(K) > 2ω and suchthat whenever κ < λ, κ, λ ∈ K, then κω < λ. Let ⃗S = ⟨Sκ | κ ∈ K⟩ besuch that for every κ ∈ K, Sκ ⊆ Sκω is stationary in κ. Let ρ ≥ sup(K) be
regular. Then the set
S = {M ∈ [Hρ ]ω | ∀κ ∈ M ∩ K (sup(M ∩ κ) ∈ Sκ )}is spread out.

24Another example
Theorem
Let K be a set of regular cardinals such that min(K) > 2ω and suchthat whenever κ < λ, κ, λ ∈ K, then κω < λ. Let ⃗S = ⟨Sκ | κ ∈ K⟩ besuch that for every κ ∈ K, Sκ ⊆ Sκω is stationary in κ. Let ρ ≥ sup(K) be
regular. Then the set
S = {M ∈ [Hρ ]ω | ∀κ ∈ M ∩ K (sup(M ∩ κ) ∈ Sκ )}is spread out.
Corollary (SC − SRP + CH)
Let K be a set of regular cardinals such that min(K) > ω1 . Let⃗S = ⟨Sκ | κ ∈ K⟩ be such that for every κ ∈ K, Sκ ⊆ Sκ is stationary inωκ. Let ρ ≥ sup(K) be regular. Then the set
S = {M ∈ [Hρ ]ω | ∀κ ∈ M ∩ K (sup(M ∩ κ) ∈ Sκ )}is spread out.
24

This can be “diagonalized” as before giving some information about“mutually stationary simultaneous reflection” under SC-SRP.
Definition
Let κ be a regular cardinal, and let ⃗S be a sequence of stationary
subsets of κ. Then eTr(⃗S) is the exact trace of ⃗S, that is, the set ofexact simultaneous reflection points of ⃗S.

25

This can be “diagonalized” as before giving some information about“mutually stationary simultaneous reflection” under SC-SRP.
Definition
Let κ be a regular cardinal, and let ⃗S be a sequence of stationary
subsets of κ. Then eTr(⃗S) is the exact trace of ⃗S, that is, the set ofexact simultaneous reflection points of ⃗S.
TheoremAssume SC-SRP. Let K be a set of regular cardinals with min(K) > 2ω .
Let ⃗S = ⟨Sκ,i | κ ∈ K, i < ω1 ⟩ be such that for every κ ∈ K and i < ω1 ,Sκ,i is a subset of Sκω stationary in κ. For κ ∈ K, let ⃗Sκ = ⟨Sκ,i | i < ω1 ⟩.
Then the sequence⟨eTr(⃗Sκ ) | κ ∈ K⟩
is mutually stationary.

25

Limitations

Typical examples of statements that follow from MM but not from
SRP are principles of simultaneous reflection of stationarity thathave a “diagonal” character.

26

Diagonal Reflection
Definition
Let λ be a regular cardinal, let S ⊆ λ be stationary, and let κ < λ.
The simultaneous reflection principle Refl(κ, S) says that whenever⃗S = ⟨Si | i < κ⟩ is a sequence of stationary subsets of S, then ⃗S has asimultaneous reflection point. (Magidor et al)
The diagonal reflection principle DSR(<κ, S) says that whenever⟨Sα,i | α < λ, i < jα ⟩ is a sequence of stationary subsets of S, wherejα < κ for every α < λ, then there are an ordinal γ < λ of
uncountable cofinality and a club F ⊆ γ such that for every α ∈ F andevery i < jα , Sα,i ∩ γ is stationary in γ. (F.)
The version of the principle in which jα ≤ κ is denoted DSR(κ, S).
If F is only required to be unbounded, then the resulting principle iscalled uDSR(<κ, S), and if it is required to be stationary, then it isdenoted sDSR(<κ, S). (F. & Lambie-Hanson)27

Larson had previously introduced a diagonal reflection principle at2
ω2 , called OSRω2 , which turns out to be equivalent to DSR(ω1 , Sωω ),
ω2and also to uDSR(ω1 , Sω ).

28

Larson had previously introduced a diagonal reflection principle at2
ω2 , called OSRω2 , which turns out to be equivalent to DSR(ω1 , Sωω ),
ω2and also to uDSR(ω1 , Sω ).
Theorem (Larson)
MM implies OSRω2 , but SRP does not.
So this theorem separates SRP from MM, showing that SRP does not
capture all consequences of MM. Note that since SC-SRP is just afragment of SRP, SC-SRP also does not imply OSRω2 .

28

Larson had previously introduced a diagonal reflection principle at2
ω2 , called OSRω2 , which turns out to be equivalent to DSR(ω1 , Sωω ),
ω2and also to uDSR(ω1 , Sω ).
Theorem (Larson)
MM implies OSRω2 , but SRP does not.
So this theorem separates SRP from MM, showing that SRP does not
capture all consequences of MM. Note that since SC-SRP is just afragment of SRP, SC-SRP also does not imply OSRω2 . In fact, this canbe improved:
Lemma
Assume that SRP holds and continues to hold after ω2 -directedclosed forcing. Then there is an ω2 -strategically closed forcing of size
ω2 in whose extensions the following hold:
1. SC-SRP,
2
2. there is a nonreflecting stationary subset of Sωω .
2That is, SC-SRP does not even imply Refl(1, Sω
ω ).

28

In fact, even SCFA + 2ω = ω2 is consistent with this failure ofreflection, and more (Sakai-Switzer).

29

In fact, even SCFA + 2ω = ω2 is consistent with this failure ofreflection, and more (Sakai-Switzer).
Let me recall Larson’s theorem:
Theorem (Larson)
MM implies OSRω2 , but SRP does not.

29

In fact, even SCFA + 2ω = ω2 is consistent with this failure ofreflection, and more (Sakai-Switzer).
Let me recall Larson’s theorem:
Theorem (Larson)
MM implies OSRω2 , but SRP does not.
Note that this theorem does not separate SCFA from SC-SRP:

29

In fact, even SCFA + 2ω = ω2 is consistent with this failure ofreflection, and more (Sakai-Switzer).
Let me recall Larson’s theorem:
Theorem (Larson)
MM implies OSRω2 , but SRP does not.
Note that this theorem does not separate SCFA from SC-SRP:
Sure, SC-SRP does not imply OSRω2 . But SCFA doesn’t either!

29

Larson’s separation at higher cardinals

Theorem (F. & Lambie-Hanson)
Let κ > ω2 be regular, and suppose that SRP holds and continues tohold in any forcing extension obtained by κ-directed closed forcing.
Then there is a κ-strategically closed forcing notion that produces
forcing extensions in which
1. SRP continues to hold, but
2. uDSR(1, Sκω ) fails.

30

Larson’s separation at higher cardinals

Theorem (F. & Lambie-Hanson)
Let κ > ω2 be regular, and suppose that SRP holds and continues tohold in any forcing extension obtained by κ-directed closed forcing.
Then there is a κ-strategically closed forcing notion that produces
forcing extensions in which
1. SRP continues to hold, but
2. uDSR(1, Sκω ) fails.
Accompanying this, I have shown:
Theorem
If SCFA holds and κ > 2ω is regular, then DSR(ω1 , Sκω ) holds.
This was (part of) the original point of introducing the DSR principles.

30

Separation achieved

Putting these last two results together, we have succeeded inseparating SC-SRP from SCFA:
Assuming κ > 2ω is regular and SRP holds and continues to holdafter κ-directed closed forcing, we can produce a model where SRP
holds (hence so does SC-SRP) but SCFA fails, because uDSR(1, Sκω )fails. So, morally, on paper, we have:31Separation achieved

Putting these last two results together, we have succeeded inseparating SC-SRP from SCFA:
Assuming κ > 2ω is regular and SRP holds and continues to holdafter κ-directed closed forcing, we can produce a model where SRP
holds (hence so does SC-SRP) but SCFA fails, because uDSR(1, Sκω )fails. So, morally, on paper, we have:
Fact
SC-SRP does not imply SCFA.

31

But the separation is suboptimal

However, my interpretation of all of this is that the right axioms are
SCFA + CH/SC-SRP + CH.
Hence, I consider the separation result to be suboptimal, in that it
doesn’t separate SCFA + CH from SC-SRP + CH.

32

Still suboptimal, probably

TheoremLet Γ be the class of subcomplete, uncountable cofinality preserving
forcing notions. If ZFC is consistent with BFA(Γ, ≤ω2 ), then ZFC isconsistent with the conjunction of the following:
1. CH,
2. BFA(Γ, ≤ω2 ),3. SC-SRP(ω2 ),4. ¬BSCFA(≤ω2 ).

33

Still suboptimal, probably

TheoremLet Γ be the class of subcomplete, uncountable cofinality preserving
forcing notions. If ZFC is consistent with BFA(Γ, ≤ω2 ), then ZFC isconsistent with the conjunction of the following:
1. CH,
2. BFA(Γ, ≤ω2 ),3. SC-SRP(ω2 ),4. ¬BSCFA(≤ω2 ).
Note: Here, I really mean subcomplete, not ∞-subcomplete.

33

Still suboptimal, probably

TheoremLet Γ be the class of subcomplete, uncountable cofinality preserving
forcing notions. If ZFC is consistent with BFA(Γ, ≤ω2 ), then ZFC isconsistent with the conjunction of the following:
1. CH,
2. BFA(Γ, ≤ω2 ),3. SC-SRP(ω2 ),4. ¬BSCFA(≤ω2 ).
Note: Here, I really mean subcomplete, not ∞-subcomplete.
It would be nice to achieve a true separation.

33

Thank you!

34