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 2 ℵ 0 > ℵ 2 so they will contradict PFA, yet we want them to retain and extend some of the important consequences of 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 R θ = H θ (or V θ ). M ≺ R θ and let M be the transitive collapse of M . Let j M ∶ M → M be the inverse of the collapsing map π M . Let κ = min { α ∈ M ∶ M ∩ α ≠ α } . Let κ M be the critical point of j M . So j M ( κ M ) = 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 ∈ P γ ( 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 = ω 2 and γ = ω 1 .

Write P ∗ κ ( R θ ) for the set of all M ≺ R θ such that M ∩ κ ∈ κ . For γ ≤ κ we let G κ,γ ( R θ ) = { M ∈ P ∗ κ ( R θ ) ∶ M is γ -guessing } . Definition (Viale) GM ( κ,γ,R θ ) is the statement that G κ,γ ( R θ ) is stationary. GM ( κ,γ ) is the principle: GM ( κ,γ,R θ ) holds, for all sufficiently large θ . Remark If M is γ -guessing and γ ≤ γ ′ ≤ κ then M is γ ′ -guessing. If M is ℵ 0 -guessing then it is 0 -guessing.

Lemma (Viale) If M is ℵ 0 -guessing then κ M and κ are inaccessible. 1 M ≺ V δ is ℵ 0 -guessing iff M = V δ , for some δ . 2 The following is a reformulation of Magidor’s characterization of supercompactness in terms of ℵ 0 -guessing models. Theorem (Magidor) κ is supercompact iff GM ( κ, ℵ 0 ) holds. Remark For this reason we use the term Magidor models for ℵ 0 -guessing models.

Theorem (Weiss) GM ( ω 2 ,ω 1 ) implies the failure of ◻( λ ) , for all regular λ ≥ ω 2 , and the tree property at ω 2 , in fact, the two cardinal tree property TP ( ω 2 ,λ ) , for λ ≥ ω 2 . Theorem (Viale, Weiss) PFA implies GM ( ω 2 ,ω 1 ) . It is not known if GM ( ω 2 ,ω 1 ) implies the Singular Cardinal Hypothesis, but a slight strengthening of it does imply SCH .

Definition Suppose M ≺ R θ is of size ω 1 . We say that M is: internally unbounded if P ω 1 ( M ) ∩ M is unbounded in P ω 1 ( M ) , 1 internally stationary if P ω 1 ( M ) ∩ M is stationary in P ω 1 ( M ) , 2 internally club (IC) if P ω 1 ( M ) ∩ M contains a club in P ω 1 ( M ) . 3 Theorem (Viale) Suppose that the set of internally unbounded guessing models is stationary in P ∗ ω 2 ( R θ ) , 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 ω 1 . Suppose M is an IC ω 1 -guessing model. For countable X ∈ M we let F ( X ) = {( Z,f ) ∶ Z ∈ X,f ∈ M ∩ 2 Z ∩ X } . For a sequence ⃗ X = ( X ξ ) ξ of elements of M , let F ( ⃗ X ) = ⋃ ξ F ( X ξ ) . Definition We say that M is special if there is an increasing continuous sequence ⃗ X = ( X ξ ) 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 ∈ 2 Z ∩ X ξ , g ∈ 2 Z ∩ X η , f ⊆ g , and s ( Z,f ) = s ( Z,g ) then f is guessed in X ξ .

It is easy to see that if M is a special IC-model of size ω 1 then M is a guessing model in any ω 1 -preserving extension of V , i.e. M is an indestructible guessing model . If M is an IC ω 1 -guessing model there is a natural proper poset that specializes it. Definition Elements of P M are triples p = (M p ,s p ,d p ) where: M p is a finite ∈ -chain of countable elementary submodels of M , 1 s p is a finite partial specializing map on F (M p ) , 2 d p ∶ M p → [ M ] < ω is such that if P ∈ Q then d p ( P ) ∈ Q . 3 We let q ≤ p if M p ⊆ M q , s p ⊆ s q and d p ( P ) ⊆ d q ( P ) , for all P ∈ M p . The role of d p is to make sure that the generic sequence of countable elementary submodels is continuous.

We let SGM ( ω 2 ,ω 1 ) denote the statement that for every large enough θ there are stationary many special IC-guessing models in P ∗ ω 2 ( R θ ) . Viale-Weiss proof actually shows that PFA implies SGM ( ω 2 ,ω 1 ) . Proposition SGM ( ω 2 ,ω 1 ) implies: there are no ω 1 -Souslin trees 1 there are no weak Kurepa trees on ω 1 . 2 Remark SGM ( ω 2 ,ω 1 ) 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 ( ω 3 ,ω 2 ) holds. However, in Trang’s model CH holds, so this is too weak for what we want. How about GM ( ω 3 ,ω 1 ) ? It implies the tree property at ω 3 , but it is not clear if it implies GM ( ω 2 ,ω 1 ) so we may lose some of the consequences we already had, such as the tree property at ω 2 . 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 = ( a ξ ∶ 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: otp ( c ) < δ , in particular δ is singular, 1 for all γ < δ , there is η < δ such that c ∩ γ = a η . 2 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 NS λ .

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 S κ λ = { α < λ ∶ cof ( α ) = κ } . Theorem (Shelah) Suppose λ is a regular cardinal. Then S < λ λ + ∈ I [ λ + ] . 1 Suppose κ is regular and κ + < λ . Then there is a stationary subset of 2 S κ λ which belongs to I [ λ ] . The approachability property AP κ + states that κ + ∈ I [ κ + ] . For a regular cardinal κ the issue is to understand I [ κ + ] ↾ S κ κ + .

Approachability ideal We concentrate on the case κ = ω 1 . Fact Suppose ¯ a = ( a ξ ∶ ξ < ω 2 ) is a sequence of bounded subsets of ω 2 . Let M ≺ H θ be an ω 1 -guessing model of size ω 1 such that ¯ a ∈ M . Then M ∩ ω 2 ∉ B ( ¯ a ) . Therefore, GM ( ω 2 ,ω 1 ) implies that S ω 1 ω 2 ∉ I [ ω 2 ] . However, one can ask a stronger question. Question (Shelah) Can I [ ω 2 ] ↾ S ω 1 ω 2 consistently be the nonstationary ideal on S ω 1 ω 2 ?

Approachability ideal Note that this cannot follow from GM ( ω 2 ,ω 1 ) since it requires the continuum to be at least ω 3 . Theorem (Mitchell) Suppose κ is κ + -Mahlo. Then there is a generic extension in which κ = ω 2 and I [ ω 2 ] ↾ S ω 1 ω 2 is the non stationary ideal on S ω 1 ω 2 . Remark In Mitchell’s model ω 3 ∈ I [ ω 3 ] . It is not known if one can have Mitchell’s result for two consecutive cardinals, say ω 2 and ω 3 .

Strong guessing models Definition Let θ > ω 2 be a regular cardinal. We say that M ∈ P ∗ ω 3 ( R θ ) is a strong ω 1 -guessing model if M can be written as the union of an increasing ω 1 -continuous chain ( M ξ ∶ ξ < ω 2 ) of special ω 1 -guessing models of size ω 1 . G + ω 3 ,ω 1 ( R θ ) = { M ∈ P ∗ ω 3 ( R θ ) ∶ M is a strong ω 1 -guessing model } . Definition GM + ( ω 3 ,ω 1 ) states that G + ω 3 ,ω 1 ( R θ ) is stationary, for all large enough θ . Remark GM + ( ω 3 ,ω 1 ) obviously implies Mitchell’s result.

Strong guessing models Theorem GM + ( ω 3 ,ω 1 ) implies the following: all ω 1 -Aronszajn trees are special 1 there are no weak ω 1 - Kurepa trees 2 the tree property at ω 2 and ω 3 3 Singular Cardinal Hypothesis 4 I [ ω 2 ] ↾ S ω 1 ω 2 is the non stationary ideal on S ω 1 ω 2 . 5 Moreover, GM + ω 3 ,ω 1 has the right structural form, i.e. if a particular instance can be forced by a poset in the appropriate class then by meeting ω 2 dense sets we can ’pull’ this back to V . Question Is GM + ( ω 3 ,ω 1 ) consistent?