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Some topics related to bounding by canonical functions Sean Cox Institute for mathematical logic and foundational research University of M unster (Germany) sean.cox@uni-muenster.de wwwmath.uni-muenster.de/logik/Personen/Cox April 30, 2012

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Some topics related to bounding by canonical functions

Sean Cox

Institute for mathematical logic and foundational research University of M¨ unster (Germany) sean.cox@uni-muenster.de wwwmath.uni-muenster.de/logik/Personen/Cox

April 30, 2012

Sean Cox Bounding by canonical functions

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Outline

1

The partial order (κORD, ≤I) and canonical functions

2

Self-generic structures (“antichain catching”)

3

How antichain catching is related to bounding by canonical functions

4

Forcing Axioms vs. nice ideals on ω2

Sean Cox Bounding by canonical functions

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The partial order ≤I on κORD

Let κ be regular, uncountable and I ⊂ ℘(κ) a normal ideal. e.g. I := NSκ; or I := NS ↾ S for some stationary S ⊂ κ. Define ≤I on κORD by: f ≤I g ⇐ ⇒ {α < κ | f (α) ≤ g(α)} ∈ Dual(I)

Sean Cox Bounding by canonical functions

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The partial order ≤I on κORD

Let κ be regular, uncountable and I ⊂ ℘(κ) a normal ideal. e.g. I := NSκ; or I := NS ↾ S for some stationary S ⊂ κ. Define ≤I on κORD by: f ≤I g ⇐ ⇒ {α < κ | f (α) ≤ g(α)} ∈ Dual(I) ≤I is wellfounded

Sean Cox Bounding by canonical functions

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Canonical functions on κ

Definition (Canonical functions on κ) By recursion: hν :≃ the ≤NSκ-least upper bound of hµ | µ < ν (if such a l.u.b. exists) View each hν as an equivalence class in κORD/ =NSκ.

Sean Cox Bounding by canonical functions

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Canonical functions on κ

Definition (Canonical functions on κ) By recursion: hν :≃ the ≤NSκ-least upper bound of hµ | µ < ν (if such a l.u.b. exists) View each hν as an equivalence class in κORD/ =NSκ. The “first few” (i.e. for ν < κ+); these all map into κ: h0 : α → 0 hν+1 : α → hν(α) + 1 For limit ν < κ+: hν can be defined from earlier ones using sups or diagonal sups

Sean Cox Bounding by canonical functions

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Canonical functions on κ

Definition (Canonical functions on κ) By recursion: hν :≃ the ≤NSκ-least upper bound of hµ | µ < ν (if such a l.u.b. exists) View each hν as an equivalence class in κORD/ =NSκ. The “first few” (i.e. for ν < κ+); these all map into κ: h0 : α → 0 hν+1 : α → hν(α) + 1 For limit ν < κ+: hν can be defined from earlier ones using sups or diagonal sups Theorem (Jech-Shelah; Hajnal) Existence of hκ+ is independent of ZFC.

Sean Cox Bounding by canonical functions

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Canonical functions and ultrapowers

κ κ+V V

Sean Cox Bounding by canonical functions

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Canonical functions and ultrapowers

κ κ+V V Let U ⊂ P(κ) be normal w.r.t. V

Sean Cox Bounding by canonical functions

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Canonical functions and ultrapowers

κ κ+V V Let U ⊂ P(κ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω1 and U is any V -generic for (℘(ω1)/NSω1, ⊂NSω

Sean Cox Bounding by canonical functions

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Canonical functions and ultrapowers

κ κ+V V Let U ⊂ P(κ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω1 and U is any V -generic for (℘(ω1)/NSω1, ⊂NSω Or possibly U ∈ V ; e.g. if κ is a measurable cardinal in V

Sean Cox Bounding by canonical functions

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Canonical functions and ultrapowers

κ κ+V V Let U ⊂ P(κ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω1 and U is any V -generic for (℘(ω1)/NSω1, ⊂NSω Or possibly U ∈ V ; e.g. if κ is a measurable cardinal in V ? ? ult(V , U)

Sean Cox Bounding by canonical functions

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Canonical functions and ultrapowers

κ κ+V V Let U ⊂ P(κ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω1 and U is any V -generic for (℘(ω1)/NSω1, ⊂NSω Or possibly U ∈ V ; e.g. if κ is a measurable cardinal in V ? ? ult(V , U) ν [hν]U

Sean Cox Bounding by canonical functions

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Other characterizations of the first κ+ canonical functions

Could have equivalently used ≤I for any normal ideal I ⊂ ℘(κ)

Sean Cox Bounding by canonical functions

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Other characterizations of the first κ+ canonical functions

Could have equivalently used ≤I for any normal ideal I ⊂ ℘(κ) Non-recursive characterizations of hν (for ν < κ+): “the” function which represents ν in any generic ultrapower by a normal ideal on κ Fix any surjection gν : κ → ν and set hν(α) := otp(g′′

ν α)

Fix any wellorder ∆ of Hκ+ and set hν(α) :≃ otp(M ∩ ν) for any M ≺ (Hκ+, ∈, ∆, {ν}) such that α = M ∩ κ

Sean Cox Bounding by canonical functions

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Bounding by canonical functions

Definition For a normal ideal I ⊂ ℘(κ), Bound(I) means that {hν | ν < κ+} is cofinal in (κκ, ≤I).

Sean Cox Bounding by canonical functions

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Bounding by canonical functions

Definition For a normal ideal I ⊂ ℘(κ), Bound(I) means that {hν | ν < κ+} is cofinal in (κκ, ≤I). Lemma Suppose κ is a successor cardinal. Bound(I) implies that if U is an ultrafilter on V ∩ ℘(κ) such that: U is normal w.r.t. sequences from V U extends the dual of I and j : V →U ult(V , U) is the ultrapower embedding, then j(κ) = κ+V .

Sean Cox Bounding by canonical functions

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Bounding by canonical functions

Definition For a normal ideal I ⊂ ℘(κ), Bound(I) means that {hν | ν < κ+} is cofinal in (κκ, ≤I). Lemma Suppose κ is a successor cardinal. Bound(I) implies that if U is an ultrafilter on V ∩ ℘(κ) such that: U is normal w.r.t. sequences from V U extends the dual of I and j : V →U ult(V , U) is the ultrapower embedding, then j(κ) = κ+V . One can always obtain such a U (even if κ is a successor cardinal) by forcing with PI := (P(κ)/I, ⊆I).

Sean Cox Bounding by canonical functions

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Assuming κ is successor, Bound(I), and U ⊃ Dual(I):

κ κ+V V κ+V ult(V , U)

Sean Cox Bounding by canonical functions

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Saturation implies bounding

Definition Let I be a normal ideal on κ. I is saturated iff PI := (℘(κ)/I, ⊆I) has the κ+-cc. Lemma (folklore) If I is saturated then Bound(I) holds.

Sean Cox Bounding by canonical functions

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Saturation implies bounding

κ+-cc of PI (and that κ is a successor cardinal) implies PI j ˙

G(κ) = κ+V

Then for every f : κ → κ: Df := {S ∈ I+ | ∃ν < κ+ f < hν on S} is dense in PI For each S ∈ Df pick a νS < κ+ such that f < hνS on S Let Af ⊂ Df be a maximal antichain. Set µ := sup{νS | S ∈ Af }; µ < κ+ by κ+-cc of PI. Maximality of Af implies that f ≤I hµ.

Sean Cox Bounding by canonical functions

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♦ implies failure of Bounding

Lemma (folklore?) ♦κ = ⇒ ¬Bound(NSκ)

Sean Cox Bounding by canonical functions

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♦ implies failure of Bounding

Lemma (folklore?) ♦κ = ⇒ ¬Bound(NSκ) Suppose Aα | α < κ is a ♦κ sequence, p : κ × κ ↔bij κ, and f (α) :=

  • tp(Aα)

if Aα codes a wellorder (via p ↾ (α × α))

  • therwise

Fix ν < κ+. Fix b ⊂ κ coding ν. b ∩ α = Aα for stationarily many α

  • tp(b ∩ α) = hν(α) for club-many α

So f (α) = hν(α) for stationarily many α. So f ≮NS hν

Sean Cox Bounding by canonical functions

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Chang’s Conjecture and bounding

Lemma (κ+, κ) ։ (κ, < κ) implies a weak variation of Bound(NSκ).

Sean Cox Bounding by canonical functions

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Bound(NSω1) is well-understood

Theorem (Larson-Shelah; Deiser-Donder) The following are equiconsistent: ZFC + Bound(NSω1) ZFC + there is an inaccessible limit of measurable cardinals Moreover, saturation of NSω1 (which implies Bound(NSω1)) is known to be consistent relative to a Woodin cardinal (Shelah).

Sean Cox Bounding by canonical functions

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What about Bound(NSω2)?

NOTATION: Sm

n := ωm ∩ cof (ωn)

Theorem (Shelah) Suppose I is a normal ideal on ω2 such that S2

0 ∈ I+. Then I is

not saturated. In particular, NSω2 is never saturated. Theorem (Woodin; building on work of Kunen and Magidor) It is consistent relative to an almost huge cardinal that there is some stationary S ⊆ S2

1 such that NSω2 ↾ S is saturated.

(Recall this implies Bound(NSω2 ↾ S)) Question (Well-known open problems)

1 Can NSω2 ↾ S2

1 be saturated?

2 Can Bound(NSω2) hold? What about Bound(NSω2 ↾ S2

1)?

Sean Cox Bounding by canonical functions

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Big gap in known consistency bounds

Question What is the consistency strength of: “Bound(I) holds for some normal ideal I ⊂ ℘(ω2)”? Best known upper bound: almost huge cardinal (Kunen, Magidor, Woodin) Best known lower bound (even assuming that F = NSω2): inaccessible limit of measurables ! (Deiser-Donder)

Sean Cox Bounding by canonical functions

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Big gap in known consistency bounds

Question What is the consistency strength of: “Bound(I) holds for some normal ideal I ⊂ ℘(ω2)”? Best known upper bound: almost huge cardinal (Kunen, Magidor, Woodin) Best known lower bound (even assuming that F = NSω2): inaccessible limit of measurables ! (Deiser-Donder) Lower bound for Bound(ω2) hasn’t even escaped “easy” inner model theory.

Sean Cox Bounding by canonical functions

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Outline

1

The partial order (κORD, ≤I) and canonical functions

2

Self-generic structures (“antichain catching”)

3

How antichain catching is related to bounding by canonical functions

4

Forcing Axioms vs. nice ideals on ω2

Sean Cox Bounding by canonical functions

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Derived ultrapowers

κ = λ+ λ θ

Sean Cox Bounding by canonical functions

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Derived ultrapowers

κ = λ+ λ θ α := M ∩ κ ∈ κ M ≺ (Hθ, ∈, {κ})

Sean Cox Bounding by canonical functions

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Derived ultrapowers

κ = λ+ λ θ α := M ∩ κ ∈ κ M ≺ (Hθ, ∈, {κ}) σM HM crit(σM) = λ+HM λ

Sean Cox Bounding by canonical functions

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Derived ultrapowers

κ = λ+ λ θ α := M ∩ κ ∈ κ M ≺ (Hθ, ∈, {κ}) σM HM crit(σM) = λ+HM λ UM := {s ∈ HM ∩ P(α) | α ∈ σM(s)}

Sean Cox Bounding by canonical functions

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Derived ultrapowers

κ = λ+ λ θ α := M ∩ κ ∈ κ M ≺ (Hθ, ∈, {κ}) σM HM crit(σM) = λ+HM λ UM := {s ∈ HM ∩ P(α) | α ∈ σM(s)} ult(HM, UM)

Sean Cox Bounding by canonical functions

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Derived ultrapowers

κ = λ+ λ θ α := M ∩ κ ∈ κ M ≺ (Hθ, ∈, {κ}) σM HM crit(σM) = λ+HM λ UM := {s ∈ HM ∩ P(α) | α ∈ σM(s)} ult(HM, UM) What if UM is generic over HM for some P ∈ HM?

Sean Cox Bounding by canonical functions

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Self-generic structures

Suppose: I is normal ideal on a successor cardinal κ. M ≺ (Hθ, ∈, {I}, ...) with M ∩ κ ∈ κ σM : HM → Hθ and UM are as on the previous slide P := (℘(κ)/I, ⊆I) and PM := σ−1

M (P).

Definition M is called self-generic for I iff UM is PM-generic over HM.

Sean Cox Bounding by canonical functions

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Relation to saturation and precipitousness

SSelfGen

I

:= {M ≺ H(2κ)+ | M is self-generic for I}

Sean Cox Bounding by canonical functions

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Relation to saturation and precipitousness

SSelfGen

I

:= {M ≺ H(2κ)+ | M is self-generic for I} SSelfGen

I

is stationary

  • I has a

precipitous restriction SSelfGen

I

is I-projective stationary

I is precipitous

SSelfGen

I

contains a “club”

  • I is saturated

(Foreman?)

Sean Cox Bounding by canonical functions

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Relation to saturation and precipitousness

SSelfGen

I

:= {M ≺ H(2κ)+ | M is self-generic for I} SSelfGen

I

is stationary

  • I has a

precipitous restriction Converse holds if κ = ω1 (Schindler) SSelfGen

I

is I-projective stationary

I is precipitous

Converse holds if κ = ω1 (Schindler) SSelfGen

I

contains a “club”

  • I is saturated

(Foreman?)

Sean Cox Bounding by canonical functions

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I-projective stationarity

A set R ⊂ ℘κ(Hθ) is I-projective stationary iff for every S ∈ I+: R ց S := {M ∈ R | M ∩ κ ∈ S} is stationary in ℘κ(Hθ).

Sean Cox Bounding by canonical functions

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I-projective stationarity

A set R ⊂ ℘κ(Hθ) is I-projective stationary iff for every S ∈ I+: R ց S := {M ∈ R | M ∩ κ ∈ S} is stationary in ℘κ(Hθ). Special case of Ralf’s observation: Theorem (Schindler) NSω1 is precipitous ⇐ ⇒ SSelfGen

NSω1

is projective stationary. (in original Feng-Jech sense of “projective stationary”)

Sean Cox Bounding by canonical functions

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For I on ω1, precipitousness implies SSelfGen

I

is large

ω1 V ult(V , G)

Sean Cox Bounding by canonical functions

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For I on ω1, precipitousness implies SSelfGen

I

is large

ω1 V ult(V , G) H := HV

θ

V [G] sees that j′′H is j(I)-self-generic

Sean Cox Bounding by canonical functions

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For I on ω1, precipitousness implies SSelfGen

I

is large

ω1 V ult(V , G) H := HV

θ

V [G] sees that j′′H is j(I)-self-generic

Sean Cox Bounding by canonical functions

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For I on ω1, precipitousness implies SSelfGen

I

is large

ω1 V ult(V , G) H := HV

θ

V [G] sees that j′′H is j(I)-self-generic So V [G] has a branch through the‘tree of finite attempts to build a (countable) j(I)-self-generic object whose intersection with j(ω1) is ωV

1 .

Sean Cox Bounding by canonical functions

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For I on ω1, precipitousness implies SSelfGen

I

is large

ω1 V ult(V , G) H := HV

θ

V [G] sees that j′′H is j(I)-self-generic So V [G] has a branch through the‘tree of finite attempts to build a (countable) j(I)-self-generic object whose intersection with j(ω1) is ωV

1 .

Then use wellfoundedness of ult(V , G).

Sean Cox Bounding by canonical functions

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StatCatch, ProjectiveCatch, and ClubCatch

Definition StatCatch(I) holds iff SSelfGen

I

is stationary ProjectiveCatch(I) holds iff SSelfGen

I

is I-projective stationary ClubCatch(I) holds iff SSelfGen

I

contains a club (relative to “conditional club filter of I”)

Sean Cox Bounding by canonical functions

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Theorem (C.-Zeman) If StatCatch(I) holds for an ideal whose dual concentrates on S2

1,

then there is an inner model with a Woodin cardinal.

Sean Cox Bounding by canonical functions

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Theorem (C.-Zeman) If StatCatch(I) holds for an ideal whose dual concentrates on S2

1,

then there is an inner model with a Woodin cardinal. Unlike for ideals on ω1, StatCatch(I) is MUCH higher in consistency strength that precipitousness.

Sean Cox Bounding by canonical functions

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Theorem (C.-Zeman) If StatCatch(I) holds for an ideal whose dual concentrates on S2

1,

then there is an inner model with a Woodin cardinal. Unlike for ideals on ω1, StatCatch(I) is MUCH higher in consistency strength that precipitousness. Note: ProjectiveCatch(I) does NOT imply that generic ultrapowers by I have strong closure properties.

Sean Cox Bounding by canonical functions

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Outline

1

The partial order (κORD, ≤I) and canonical functions

2

Self-generic structures (“antichain catching”)

3

How antichain catching is related to bounding by canonical functions

4

Forcing Axioms vs. nice ideals on ω2

Sean Cox Bounding by canonical functions

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M → otp(M ∩ θ) resembles a canonical function

κ V θ Suppose U ⊂ ℘(Pκ(Hθ)) is normal w.r.t. V

Sean Cox Bounding by canonical functions

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M → otp(M ∩ θ) resembles a canonical function

κ V θ Suppose U ⊂ ℘(Pκ(Hθ)) is normal w.r.t. V e.g. κ = ω1 and U is generic for ℘(Pω1(Hθ))/NS

Sean Cox Bounding by canonical functions

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M → otp(M ∩ θ) resembles a canonical function

κ V θ Suppose U ⊂ ℘(Pκ(Hθ)) is normal w.r.t. V e.g. κ = ω1 and U is generic for ℘(Pω1(Hθ))/NS ? ? ult(V , U) = Pκ(Hθ)V /U

Sean Cox Bounding by canonical functions

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M → otp(M ∩ θ) resembles a canonical function

κ V θ Suppose U ⊂ ℘(Pκ(Hθ)) is normal w.r.t. V e.g. κ = ω1 and U is generic for ℘(Pω1(Hθ))/NS ? ? ult(V , U) = Pκ(Hθ)V /U [M → otp(M ∩ θ)]U

Sean Cox Bounding by canonical functions

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ProjectiveCatch(I) implies weak version of Bound(I)

Observation (C.) Let θ = (2κ)+. StatCatch(I) implies: for every f : κ → κ there are stationarily many M ∈ ℘κ(Hθ) such that:

  • tp(M ∩ θ) > f (M ∩ κ)

Sean Cox Bounding by canonical functions

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Ideals that bound their completeness

Definition (C.) Suppose J is a normal ideal over ℘κ(Hθ) with completeness κ. We say J bounds its completeness iff for every f : κ → κ: Sf := {M ∈ ℘κ(H(2κ)+) | otp(M) > f (M ∩ κ)} is in the dual of J . Lemma (C.) It is consistent for κ to be supercompact, yet no normal measures on any ℘κ(Hθ) bound their completeness If κ is almost huge, many normal measures that bound completeness. If T is a presaturated tower of ideals with critical point κ, then a tail end of the ideals in the tower bound their completeness.

Sean Cox Bounding by canonical functions

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ProjectiveCatch and bounding

Recall from earlier: Theorem (C.-Zeman) ProjectiveCatch(I) (for I on ω2) gives inner model with Woodin cardinal

Sean Cox Bounding by canonical functions

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ProjectiveCatch and bounding

Recall from earlier: Theorem (C.-Zeman) ProjectiveCatch(I) (for I on ω2) gives inner model with Woodin cardinal Lemma (C.) Suppose I is a normal ideal on κ and ProjectiveCatch(I) holds. Set J := NS ↾ SSelfGen

I

. Then J bounds its completeness (which is κ). Conjecture The consistency strength of “there is an ideal concentrating on IUω1 which bounds its completeness, where the completeness is ω2” is strictly between a supercompact and almost huge cardinal.

Sean Cox Bounding by canonical functions

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ProjectiveCatch and bounding

Recall from earlier: Theorem (C.-Zeman) ProjectiveCatch(I) (for I on ω2) gives inner model with Woodin cardinal Lemma (C.) Suppose I is a normal ideal on κ and ProjectiveCatch(I) holds. Set J := NS ↾ SSelfGen

I

. Then J bounds its completeness (which is κ). Conjecture The consistency strength of “there is an ideal concentrating on IUω1 which bounds its completeness, where the completeness is ω2” is strictly between a supercompact and almost huge cardinal. Note: Bound(I) implies existence of a J which bounds its completeness.

Sean Cox Bounding by canonical functions

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Outline

1

The partial order (κORD, ≤I) and canonical functions

2

Self-generic structures (“antichain catching”)

3

How antichain catching is related to bounding by canonical functions

4

Forcing Axioms vs. nice ideals on ω2

Sean Cox Bounding by canonical functions

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SLIDE 62

Conflict between forcing axioms and nice ideals on ω2

MA: Martin’s Axiom (MAω1) PFA: Proper Forcing Axiom MM: Martin’s Maximum

Sean Cox Bounding by canonical functions

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SLIDE 63

Conflict between forcing axioms and nice ideals on ω2

MA: Martin’s Axiom (MAω1) PFA: Proper Forcing Axiom MM: Martin’s Maximum Theorem (Foreman-Magidor) PFA = ⇒ there is no presaturated ideal on ω2 PFA = ⇒ failure of (ω3, ω2) ։ (ω2, ω1) MM = ⇒ there is no presaturated tower which has completeness ω2 and concentrates on IA.

Sean Cox Bounding by canonical functions

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Conflict between forcing axioms and nice ideals on ω2

MA: Martin’s Axiom (MAω1) PFA: Proper Forcing Axiom MM: Martin’s Maximum Theorem (Foreman-Magidor) PFA = ⇒ there is no presaturated ideal on ω2 PFA = ⇒ failure of (ω3, ω2) ։ (ω2, ω1) MM = ⇒ there is no presaturated tower which has completeness ω2 and concentrates on IA. Theorem (C.) MM is consistent with weakened versions (e.g. (θ, ω2) ։ (ω2, ω1); instances of ProjectiveCatch for ideals with completeness ω2)

Sean Cox Bounding by canonical functions

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Some related results

Theorem (C.-Viale) WRP([ω2]ω) = ⇒ there is no ideal which bounds its completeness and concentrates on the class GICω1 (ω1-guessing, internally club sets). sat(NSω1) + TP(ω2) yields stronger result (with GISω1 in place of GICω1). (WRP and SRP follow from PFA+ and MM, respectively) Corollary PFA+ (resp. MM) implies there is no presaturated tower that concentrates on GICω1 (resp. GISω1).

Sean Cox Bounding by canonical functions

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SLIDE 66

Bounding completeness and trees

Define a partial order on ℘κ(Hθ) by: M ≤r M′ ⇐ ⇒ ∃β < θ M = M′ ∩ Vβ For each α < κ set: T ℘κ(Hθ)

α

:= {M ∈ ℘κ(Hθ) | M ∩ κ = α} (T ℘κ(Hθ)

α

, ≤r) is a tree of height ≤ κ.

Sean Cox Bounding by canonical functions

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SLIDE 67

Tree of models at α

κ = λ+ λ θ sup(M ∩ θ) α := M ∩ κ ∈ κ M ≺ (Hθ, ∈, {κ}) σM HM crit(σM) = λ+HM λ HM

Sean Cox Bounding by canonical functions

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SLIDE 68

Bounding completeness and trees

Observation height(T ℘κ(Hθ)

α

) ≤ κ (for every α < κ)

Sean Cox Bounding by canonical functions

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SLIDE 69

Bounding completeness and trees

Observation height(T ℘κ(Hθ)

α

) ≤ κ (for every α < κ) Lemma Suppose J is a normal ideal on ℘κ(Hθ) with completeness κ. Let I be the projection of J to a normal ideal on κ. If J bounds its completeness, then height(T ℘κ(Hθ)

α

) = κ for I-measure one many α < κ.

Sean Cox Bounding by canonical functions

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SLIDE 70

Bounding completeness and trees

Observation height(T ℘κ(Hθ)

α

) ≤ κ (for every α < κ) Lemma Suppose J is a normal ideal on ℘κ(Hθ) with completeness κ. Let I be the projection of J to a normal ideal on κ. If J bounds its completeness, then height(T ℘κ(Hθ)

α

) = κ for I-measure one many α < κ. Resembles “Strong Chang’s Conjecture”.

Sean Cox Bounding by canonical functions

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SLIDE 71

Useful result of Gitik

Theorem (Gitik) For any club D ⊂ [ω2]ω and any x ∈ R, there are a, b, c ∈ D such that x ∈ Lω2[a, b, c]. Corollary If W is a transitive ZF − model of height ω2 and R − W = ∅, then [ω2]ω − W is stationary. Velickovic strengthened Gitik’s Theorem in a way that shows: [ω2]ω − W is in fact projective stationary.

Sean Cox Bounding by canonical functions

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Some corollaries

Corollary WRP([ω2]ω) (resp. SRP([ω2]ω) = ⇒ if W is a transitive ZF − model of height ω2 and every proper initial segment of W is internally club (resp. internally stationary), then R ⊂ W .

Sean Cox Bounding by canonical functions

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Yet another corollary of Gitik’s Theorem

Observation (C.) Neeman’s and Friedman’s recent models of PFA are not models of WRP([ω2]ω); in particular, they’re not models of PFA+. Fundamentally different from Baumgartner’s classic model of PFA: If κ is supercompact P is any countable support iteration of proper posets which has the κ-cc Then V P | = WRP([κ]ω)

Sean Cox Bounding by canonical functions

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SLIDE 74

Ongoing work and questions

Recall that Bound(NSω1) is well-understood.

1 Is Bound(NSω2) consistent? 2 Find better lower bounds for consistency strength

even need to escape “easy” inner model theory I suspect that our proof that obtains a Woodin cardinal from StatCatch(I) will help

3 Can ProjectiveCatch(NS ↾ S2

1) hold? Can NS ↾ S2 1 be

saturated?

4 Exactly how much can Forcing Axioms tolerate nice

ideals/towers on ω2?

Some partial results with Viale, Weiss (using ideas from Neeman’s PFA forcing)

Sean Cox Bounding by canonical functions

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Note: Bound(NSω2) together with precipitousness of NSω2 has very high consistency strength.

Sean Cox Bounding by canonical functions