Long chains of special guessing models Boban Velickovic IMJ-PRG - - PowerPoint PPT Presentation

Long chains of special guessing models

Boban Velickovic

IMJ-PRG Universit´ e Paris Diderot

Reflections on Set Theoretic Reflection Bagaria 60 Conference Sant Bernat, November 16 2018

Outline

Background and history

This is joint work with my PhD student R. Mohammadpour. Question What are guessing models and why should we care about them? Guessing models: technical notion isolated by Viale following his work with Weiss on two cardinal tree properties they capture the combinatorial essence of supercompactness, but can exist at small cardinals the existence of such models follows from PFA and implies many of its important consequences

Motivation: Get higher cardinal versions of strong forcing axioms. Our higher forcing axioms will imply 2ℵ0 > ℵ2 so they will contradict PFA, yet we want them to retain and extend some of the important consequences

• f PFA. The existence of guessing models does not bound the continuum, so

it is a natural test question. Goal: Formulate a higher cardinal generalization of a guessing model, show that it is consistent to have them, and that this has some desirable consequences.

Definitions

Fix an uncountable cardinal θ. Let Rθ = Hθ (or Vθ). M ≺ Rθ and let M be the transitive collapse of M. Let jM ∶ M → M be the inverse of the collapsing map πM. Let κ = min{α ∈ M ∶ M ∩ α ≠ α}. Let κM be the critical point of jM. So jM(κM) = κ. Fix γ ≤ κ. Definition (Viale) M is a γ-guessing model if for every Z ∈ M and f ∶ Z → 2, if f is γ-approximated in M, i.e. f ↾ C ∈ M, for all C ∈ Pγ(Z) ∩ M, then f is guessed in M, i.e. there is f ∈ M such that f ↾ M = f ↾ M. We are primarily interested in the case κ = ω2 and γ = ω1.

Write P∗

κ(Rθ) for the set of all M ≺ Rθ such that M ∩ κ ∈ κ. For γ ≤ κ we

let Gκ,γ(Rθ) = { M ∈ P∗

κ(Rθ) ∶ M is γ-guessing}.

Definition (Viale) GM(κ,γ,Rθ) is the statement that Gκ,γ(Rθ) is stationary. GM(κ,γ) is the principle: GM(κ,γ,Rθ) holds, for all sufficiently large θ. Remark If M is γ-guessing and γ ≤ γ′ ≤ κ then M is γ′-guessing. If M is ℵ0-guessing then it is 0-guessing.

Lemma (Viale)

1

If M is ℵ0-guessing then κM and κ are inaccessible.

2

M ≺ Vδ is ℵ0-guessing iff M = Vδ, for some δ. The following is a reformulation of Magidor’s characterization of supercompactness in terms of ℵ0-guessing models. Theorem (Magidor) κ is supercompact iff GM(κ,ℵ0) holds. Remark For this reason we use the term Magidor models for ℵ0-guessing models.

Theorem (Weiss) GM(ω2,ω1) implies the failure of ◻(λ), for all regular λ ≥ ω2, and the tree property at ω2, in fact, the two cardinal tree property TP(ω2,λ), for λ ≥ ω2. Theorem (Viale, Weiss) PFA implies GM(ω2,ω1). It is not known if GM(ω2,ω1) implies the Singular Cardinal Hypothesis, but a slight strengthening of it does imply SCH.

Definition Suppose M ≺ Rθ is of size ω1. We say that M is:

1

internally unbounded if Pω1(M) ∩ M is unbounded in Pω1(M),

2

internally stationary if Pω1(M) ∩ M is stationary in Pω1(M),

3

internally club (IC) if Pω1(M) ∩ M contains a club in Pω1(M). Theorem (Viale) Suppose that the set of internally unbounded guessing models is stationary in P∗

ω2(Rθ), for all large enough θ. Then SCH holds.

Remark The proof of the Viale-Weiss theorem shows that the above assumption follows from PFA.

Special guessing models

A guessing model may not remaining guessing in a generic extension of the

• universe. In order to prevent this we can specialize it. This is analogous to

weakly specializing a tree of height ω1. Suppose M is an IC ω1-guessing model. For countable X ∈ M we let F(X) = {(Z,f) ∶ Z ∈ X,f ∈ M ∩ 2Z∩X}. For a sequence ⃗ X = (Xξ)ξ of elements of M, let F( ⃗ X) = ⋃ξ F(Xξ). Definition We say that M is special if there is an increasing continuous sequence ⃗ X = (Xξ)ξ of countable sets in M whose union is M and a function s ∶ F( ⃗ X) → ω such that if ξ < η, Z ∈ M, f ∈ 2Z∩Xξ, g ∈ 2Z∩Xη, f ⊆ g, and s(Z,f) = s(Z,g) then f is guessed in Xξ.

It is easy to see that if M is a special IC-model of size ω1 then M is a guessing model in any ω1-preserving extension of V , i.e. M is an indestructible guessing model. If M is an IC ω1-guessing model there is a natural proper poset that specializes it. Definition Elements of PM are triples p = (Mp,sp,dp) where:

1

Mp is a finite ∈-chain of countable elementary submodels of M,

2

sp is a finite partial specializing map on F(Mp),

3

dp ∶ Mp → [M]<ω is such that if P ∈ Q then dp(P) ∈ Q. We let q ≤ p if Mp ⊆ Mq, sp ⊆ sq and dp(P) ⊆ dq(P), for all P ∈ Mp. The role of dp is to make sure that the generic sequence of countable elementary submodels is continuous.

We let SGM(ω2,ω1) denote the statement that for every large enough θ there are stationary many special IC-guessing models in P∗

ω2(Rθ).

Viale-Weiss proof actually shows that PFA implies SGM(ω2,ω1). Proposition SGM(ω2,ω1) implies:

1

there are no ω1-Souslin trees

2

there are no weak Kurepa trees on ω1. Remark SGM(ω2,ω1) was studied by Cox and Krueger who showed that it is consistent with continuum being arbitrary large.

What kind of guessing models should we expect from our higher forcing axioms? Theorem (Trang) Suppose there is a supercompact cardinal. Then there is a generic extension in which GM(ω3,ω2) holds. However, in Trang’s model CH holds, so this is too weak for what we want. How about GM(ω3,ω1)? It implies the tree property at ω3, but it is not clear if it implies GM(ω2,ω1) so we may lose some of the consequences we already had, such as the tree property at ω2. In order to formulate the right principle we need to look at one more important application of guessing models.

Approachability ideal

Definition Let λ be a regular cardinal and ¯ a = (aξ ∶ ξ < λ) a sequence of bounded subsets of λ. We let B(¯ a) denote the set of all δ < λ such that there is a cofinal c ⊆ δ such that:

1

• tp(c) < δ, in particular δ is singular,

2

for all γ < δ, there is η < δ such that c ∩ γ = aη. Definition (Shelah) Suppose λ is regular. I[λ] is the ideal generated by the sets B(¯ a), for sequences ¯ a as above, and the non stationary ideal NSλ.

Approachability ideal

This ideal was defined by Shelah in the late 1970s. I[λ] and its variations have been extensively studied over the past 40 years. For regular κ < λ we let Sκ

λ = {α < λ ∶ cof(α) = κ}.

Theorem (Shelah) Suppose λ is a regular cardinal.

1

Then S<λ

λ+ ∈ I[λ+].

2

Suppose κ is regular and κ+ < λ. Then there is a stationary subset of Sκ

λ which belongs to I[λ].

The approachability property APκ+ states that κ+ ∈ I[κ+]. For a regular cardinal κ the issue is to understand I[κ+] ↾ Sκ

κ+.

Approachability ideal

We concentrate on the case κ = ω1. Fact Suppose ¯ a = (aξ ∶ ξ < ω2) is a sequence of bounded subsets of ω2. Let M ≺ Hθ be an ω1-guessing model of size ω1 such that ¯ a ∈ M. Then M ∩ ω2 ∉ B(¯ a). Therefore, GM(ω2,ω1) implies that Sω1

ω2 ∉ I[ω2]. However, one can ask a

stronger question. Question (Shelah) Can I[ω2] ↾ Sω1

ω2 consistently be the nonstationary ideal on Sω1 ω2 ?

Approachability ideal

Note that this cannot follow from GM(ω2,ω1) since it requires the continuum to be at least ω3. Theorem (Mitchell) Suppose κ is κ+-Mahlo. Then there is a generic extension in which κ = ω2 and I[ω2] ↾ Sω1

ω2 is the non stationary ideal on Sω1 ω2 .

Remark In Mitchell’s model ω3 ∈ I[ω3]. It is not known if one can have Mitchell’s result for two consecutive cardinals, say ω2 and ω3.

Strong guessing models

Definition Let θ > ω2 be a regular cardinal. We say that M ∈ P∗

ω3(Rθ) is a strong

ω1-guessing model if M can be written as the union of an increasing ω1-continuous chain (Mξ ∶ ξ < ω2) of special ω1-guessing models of size ω1. G+

ω3,ω1(Rθ) = { M ∈ P∗ ω3(Rθ) ∶ M is a strong ω1-guessing model}.

Definition GM+(ω3,ω1) states that G+

ω3,ω1(Rθ) is stationary, for all large enough θ.

Remark GM+(ω3,ω1) obviously implies Mitchell’s result.

Strong guessing models

Theorem GM+(ω3,ω1) implies the following:

1

all ω1-Aronszajn trees are special

2

there are no weak ω1- Kurepa trees

3

the tree property at ω2 and ω3

4

Singular Cardinal Hypothesis

5

I[ω2] ↾ Sω1

ω2 is the non stationary ideal on Sω1 ω2 .

Moreover, GM+

ω3,ω1 has the right structural form, i.e. if a particular instance

can be forced by a poset in the appropriate class then by meeting ω2 dense sets we can ’pull’ this back to V . Question Is GM+(ω3,ω1) consistent?

Fix sufficiently large θ and let Rθ = Hθ (or Vθ). If the collection of IC guessing models of size ω1 is stationary in P∗

ω2(Rθ) then there is a poset

using side conditions of two types forcing an instance of GM+(ω3,ω1). Fix an algebra H ∶ R<ω

θ

→ Rθ. Let C be the collection of countable submodels of (Rθ,∈,H), and U be the collection of IC guessing submodels of (Rθ,∈,H) of size ω1. Definition Let MH be the collection of finite ∈-chains closed under intersections of models from C ∪ U. The order is reverse inclusion. MH is C ∪ U-strongly proper and will add an ω1-continuous ∈-chain of length ω2 consisting of models in U. We can simultaneously specialize each

• f these models.

Suppose M ∈ MH and N ∈ M ∩ U. We let M(N) = {N ∩ M ∶ M ∈ M ∩ C,N ∈ M}. Definition Let QH be the set of triples p = (Mp,sp,dp) where:

1

Mp ∈ MH,

2

sp, dp are functions on Up = Mp ∩ U,

3

p(N) = (Mp(N),sp(N),dp(N)) ∈ PN, for all N ∈ Up. We let: q ≤ p if Mp ⊆ Mq and q(N) ≤ p(N), for all N ∈ Up. QH is C-proper and U-strongly proper and creates a strong ω1-guessing model of size ω2 that is closed under H.

The posets QH can be iterated using the machinery of virtual models provided we have sufficiently many IC ω1-guessing models to begin with and each of these models remains guessing throughout the iteration. More precisely, if Gα is the generic for the iteration up to α then N[Gα] is ω1-guessing, for each such N with α ∈ N. This requires overcoming a number of technical difficulties. Finally, we get the following. Theorem Suppose there are two supercompact cardinals κ < λ. Then there is a generic extension of V in which GM+(ω3,ω1) holds.

The above example provides a blueprint for many similar results. Definition Let Q be a poset and θ sufficiently large. We say that a model M ≺ Rθ is Q-IC provided M is an IC model of size ω1, Q ∈ M, and there is an M-generic filter over Q ∩ M. PFA implies that for every proper Q there are stationary many Q-IC models in P∗

ω2(Rθ). Let us say that a model M ≺ Rθ is a strong Q-model if there is

an ∈-increasing ω1-continuous chain (Mξ ∶ ξ < ω2) of Q-IC models whose union in M. Definition FA∗

ω2(Q) states that for all θ large enough the set of Q-strong models of size

ω2 is stationary in P∗

ω3(Rθ).

Question For which Q is FA∗

ω2(Q) consistent?

Note that we are not requiring the existence of generic filters meeting ω2 dense sets, only the existence of an ω1-continuous chain (Mξ ∶ ξ < ω2) and an Mξ-generic filter Gξ over Q ∩ Mξ, for each ξ. In each case we would start with a large supply of 2nd order elementary submodels of appropriate Rθ. There are two issues:

1

prove that the class of posets adding a witness for FA∗

ω2(Q) can be

iterated without collapsing ω1 and ω2,

2

prove that some 2nd order property of these models is preserved at stage α of the iteration in order to define the α-th poset. One possible example is a higher version of the Mapping Reflection

• Principles. It implies 2ω ≤ ω3, but we don’t know if it is consistent.