Strong guessing models

Boban Velickovic

IMJ-PRG Universite de Paris

CUNY Set Theory SeminarNew York, June 19 2020

Outline

Background and motivation

This is joint work with my PhD student R. Mohammadpour.

General form of forcing axioms. Let K be a class of forcing notions and κan uncountable cardinal.

FAκ(K)For every P ∈ K and a family D of κ dense subsets of P there is a filter G inP such that G ∩D ≠ ∅, for all D ∈ D.

MAκ ≡ FAκ(ccc)PFA ≡ FAℵ1(proper)MM ≡ FAℵ1(stationary preserving)

RemarkK cannot be the class of all posets or even all posets preserving ℵ1.

PFA implies2ℵ0 = 2ℵ1 = ℵ2

Singular Cardinal HypothesisThe tree property at ℵ2

the failure of ◻(κ), for regular κ > ℵ1.

MM impliesNSω1 is ℵ2-saturatedChang’s conjecture (ℵ2,ℵ1)→ (ℵ1,ℵ0)

We are looking for higher forcing axioms that have similar structuralconsequences. In particular we want to have 2ℵ0 > ℵ2.

Guessing models

We first search for some principles that follow from PFA, imply most of itsstructural properties, but are consistent with 2ℵ0 being bigger than ω2. Thekey notion is that of a guessing model.

Definition (Viale)Let R be a model of a fragment of set theory and M ≺ R. Let γ be acardinal. Let Z ∈M and f ∶ Z → 2 be a function.

1 f is γ-approximated in M if f ↾ C ∈M , for all C ∈ Pγ(Z) ∩M .2 f is guessed in M if there is f ∈M such that f ↾M = f ↾M .

We say that M is a γ-guessing model if every f ∈ R which isγ-approximated in M is guessed in M .

RemarkM ≺Hθ is a γ-guessing model iff the transitive collapse M of M has theγ-approximation property in the sense of Hamkins.

Write P∗κ(R) for the set of all M ≺ R such that M ∩ κ ∈ κ. For γ ≤ κ we let

Gκ,γ(R) = {M ∈ P∗κ(R) ∶M is γ-guessing}.

Definition (Viale)GM(κ, γ) is the statement that Gκ,γ(Hθ) is stationary, for all sufficientlylarge θ.

We are primarily interested in γ = ω1 and κ = ω2, i.e. ω1-guessing models ofsize ω1.

Lemma (Viale)1 If M is ℵ0-guessing then κM =M ∩ κ and κ are inaccessible.2 M ≺ Vδ is ℵ0-guessing iff M = Vδ , for some δ, where M is the

transitive collapse of M .

The following is a reformulation of Magidor’s characterization ofsupercompactness in terms of ℵ0-guessing models.

Theorem (Magidor)κ is supercompact iff GM(κ,ℵ0) holds.

RemarkFor this reason we use the term Magidor models for ℵ0-guessing models.

Theorem (Viale, Weiss)PFA implies GM(ω2, ω1).

Theorem (Weiss)GM(ω2, ω1) implies

1 the failure of ◻(λ), for all regular λ ≥ ω2.2 TP(ω2), in fact, TP(ω2, λ), for λ ≥ ω2.

Theorem (Viale, Krueger)GM(ω2, ω1) implies SCH.

Theorem (Cox, Krueger)GM(ω2, ω1) is consistent with 2ℵ0 arbitrarily large.

DefinitionLet θ > ω1 be a regular cardinal. Let N ≺Hθ be of size ℵ1.

We say that N is internally unbounded (I.U.) if there is an ∈-chain ofcountable models (Nξ ∶ ξ < ω1) such that N = ⋃ξNξ.We say that N is internally club (I.C.) if the above sequence can betaken to be continuous.

DefinitionLet M ≺Hθ be of size ℵ1. We say that M is locally internally unbounded ifPω1(X) ∩M is cofinal in Pω1(X ∩M), for every X ∈M .

FactSuppose θ0 < θ1 are regular and M ≺Hθ1 is locally internally unboundedwith θ0 ∈M . Then M ∩Hθ0 is internally unbounded.

Theorem (Krueger)If M ≺Hθ is an ω1-guessing model of size ℵ1, then M is locally internallyunbounded.

Proof.Let X ∈M and x ∈ Pω1(X ∩M). We need to find a countable y ∈M withx ⊆ y.Let f ∶ ω → x be a bijection, and set xn = f”n, hence xn ⊆ xn+1.Let A = {xn ∶ n ∈ ω}. Then A ⊆ [X]<ω ∈M .

If A is countably approximated in M , since M is an ω1-guessingmodel, A ∈M , and hence x = ⋃A ∈M . Set y = x.Otherwise there is a countable Y ⊆ [X]<ω in M such that A ∩ Y ∉M ,but then A ∩ Y is infinite, and x = ⋃(A ∩ Y ) ⊆ ⋃Y ∈M . Set y = ⋃Y .

Definition (Viale)Let λ be singular of cofinality ω. A = (A(n,α) ∶ n < ω,α < λ+) is a strongcovering matrix for λ+ if:

1 A(0, α) ⊆ A(1, α) ⊆ A(2, α) . . ., for all α,2 ⋃nA(n,α) = α, for all α,3 ∣A(n,α)∣ < λ, for all n and α,4 for all α < β there is n such that A(m,α) ⊆ A(m,β), for all m ≥ n,5 for all x ∈ Pω1(λ+) there is γx < λ+ such that for all α ≥ γx there is n

such that A(m,α) ∩ x = A(m,γx) ∩ x, for all m ≥ n.

PropositionAssume λ > 2ℵ0 is of countable cofinality. Then there is a strong coveringmatrix A for λ+.

Proposition (Viale)Assume for all λ > 2ℵ0 of countable cofinality and a strong covering matrixA for λ+, there is an unbounded set B ⊆ λ+ such that Pω1(B) is covered byA . Then SCH holds.

RemarkPω1(B) is covered by A if, for every x ∈ Pω1(B), there are n,α such thatx ⊆ A(n,α).

LemmaSuppose cof(λ) = ω and A is a strong covering matrix for λ+. Let θ besufficiently large regular cardinal. Let M ≺Hθ be an ω1-guessing internallyunbounded model of size ℵ1. Let δM = sup(M ∩ λ+). Then there is n suchthat A(m,δM) ∩ x ∈M , for all x ∈ Pω1(λ+) ∩M and m ≥ n.

Proof.Otherwise, for each n, pick xn ∈ Pω1(λ+) ∩M with A(n, δM) ∩ xn ∉M .By internal unboundedness of M find countable x ∈M such that ⋃n xn ⊆ x.By elementarity of M , γx ∈M .By definition of γx there is n0 such that for all n ≥ n0

A(n, δM) ∩ x = A(n, γx) ∩ x ∈M.

Given n ≥ n0 we have A(n, δM) ∩ x = A(n, γx) ∩ x ∈M , and hence:

A(n, δM) ∩ xn = A(n, δM) ∩ x ∩ xn ∈M.

This is a contradiction.

Theorem (Viale, Krueger)Assume GM(ω2, ω1). Then SCH holds.

Proof.Let λ > 2ℵ0 be of countable cofinality and let A be a strong covering matrixfor λ+. We find an unbounded B ⊆ λ+ such that Pω1(B) is covered by A .

Fix θ large enough and an I.U. ω1-guessing model M ≺Hθ of size ℵ1 withA ∈M . We may assume cof(δM) = ω1. By previous lemma there is n0such that A(m,δM) ∩ x ∈M , for all x ∈ Pω1(λ+) ∩M , and all m ≥ n0.

Since M is an ω1-guessing model we can find, for each m ≥ n0, Am ∈Msuch that A(m,δM) ∩M = Am ∩M . Since cof(δM) = ω1 we can findm ≥ n0 such that A(m,δM) ∩M is unbounded in δM , but since Am ∈Mand A(m,δM) ∩M = Am ∩M , it follows that Am is unbounded in λ+.

If x ∈ Pω1(Am) ∩M then x is covered by A(m,δM). By elementarity of Mit follows that every x ∈ Pω1(Am) is covered by some member of A .Hence, we can set B = Am.

Approachability ideal

Guessing models are closely related to the approachability ideal I[λ].

DefinitionLet λ be a regular cardinal and a = (aξ ∶ ξ < λ) a sequence of boundedsubsets of λ. We let B(a) denote the set of all δ < λ such that there is acofinal c ⊆ δ such that:

1 otp(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), forsequences 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 variationshave been extensively studied over the past 40 years.

For regular κ < λ we let Sκλ = {α < λ ∶ cof(α) = κ}.

Theorem (Shelah)Suppose λ is a regular cardinal.

1 Then S<λλ+ ∈ I[λ+].2 Suppose κ is regular and κ+ < λ. Then there is a stationary subset ofSκλ which belongs to I[λ].

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

Approachability ideal

PropositionAssume GM(κ+, κ). Then κ+ ∉ I[κ+].

Proof.Let a = (aξ ∶ ξ < κ+) be a sequence of bounded subsets of κ+.Fix M ≺Hκ++ a κ-guessing model of size κ with a ∈M .Let δ =M ∩ κ+. We claim that δ ∉ B(a).Suppose δ ∈ B(a), and let c ⊆ δ witness this. Thus µ = o.t.(c) < δ.For γ < δ there is η < δ such that c ∩ γ = aη ∈M .So c is κ-approximated in M .Since M is κ-guessing model, there is c∗ ∈M with c = c∗ ∩M .Then c is an initial segment of c∗, and c = c∗(µ) ∩ κ+, where c∗(µ) is theµ-th element of c∗.It follows that c ∈M , and hence also δ = sup(c) ∈M , a contradiction!

Question (Shelah)Can I[ω2] ↾ Sω1

ω2consistently be the nonstationary ideal on Sω1

ω2?

Theorem (Mitchell)Suppose κ is κ+-Mahlo. Then there is a generic extension in which κ = ω2

and I[ω2] ↾ Sω1ω2

is the non stationary ideal on Sω1ω2

.

Definition (Mitchell Property)

For λ regular, MP(λ+) denotes the statement that I[λ+] ↾ Sλλ+ is thenonstationary ideal on Sλλ+ .

RemarkMP(ω2) implies 2ℵ0 ≥ ℵ3.

Some questions

Some questions:1 Does GM(ω2, ω1) imply MP(ω2)?2 What about GM(ω3, ω2)?3 Does GM(ω2, ω1) bound the continuum?

Some answers:1 No! GM(ω2, ω1) is consistent with c = ℵ2. (Viale–Weiss)2 GM(ω3, ω2) is consistent with CH. (Trang)3 GM(ω2, ω1) is consistent with continuum large.(Cox– Krueger)

Strong guessing models

Idea: combine GM(ω2, ω1), GM(ω3, ω2) and MP(ω2).

DefinitionLet R be a model of a fragment of ZFC. We say that M ≺ R is a strongω1-guessing model if M can be written as the union of an increasingω1-continuous ∈-chain (Mξ ∶ ξ < ω2) of ω1-guessing models of size ω1.

RemarkEvery strongly ω1-guessing model is also an ω1-guessing model.

G+

ω3,ω1(R) = {M ∈ [R]ω2 ∶M is a strong ω1-guessing model}.

DefinitionGM+(ω3, ω1) states that G+

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

Strong guessing models

TheoremGM+(ω3, ω1) implies the following:

1 GM(ω3, ω2) and GM(ω2, ω1).2 MP(ω2) and hence 2ℵ0 ≥ ℵ3.3 there are no weak ω1-Kurepa trees nor weak ω2-Kurepa trees.4 the tree property at ω2 and ω3.5 the failure of ◻(λ), for all λ ≥ ω2.6 Singular Cardinal Hypothesis.

Theorem (Mohammadpour, V.)Assume there are two supercompact cardinals. There there is a genericextension in which GM+(ω3, ω1) holds.

Special guessing models

DefinitionSuppose (T,<) a tree of size and height ℵ1. T is weakly special if there is afunction σ ∶ T → ω such that if σ(r) = σ(s) = σ(t) with r < s, t, then s and tare comparable.

PropositionIf T is a tree of height and size ω1 and is weakly special then T has at mostℵ1 many cofinal branches.

Proof.Let f be a weak specializing map of T . If b is a cofinal branch there is aninteger nb such that ∣f−1(nb) ∩ b∣ = ℵ1. Let tb be the least element off−1(nb) ∩ b. Then the map b↦ tb is injective from the set of cofinalbranches to T .

Let X be a set.

TX = {(Z, f) ∶ Z ∈X is uncountable and f ∶ Z ∩X → 2}.

DefinitionSuppose that M is an ω1-guessing model. Let (Mξ ∶ ξ < ω1) be anIU-sequence. Let

T (M) = ⋃ξ<ω1

(TMξ∩M).

Define the order ≤ on T (M) be letting (Z, f) ≤ (W,g) if and only if Z =Wand f ⊆ g.

RemarkSuppose that M is an ω1-guessing model of size ℵ1. Then (T (M),≤) is atree of size and height ω1 with at most ℵ1 cofinal branches.

DefinitionWe say that M is a special guessing model if T (M) is weakly special.

PropositionIf M is a special ω1-guessing model of size ω1 then M remains ω1-guessingin any outer universe W of V with ωW1 = ωV1 .

Proof.

Suppose W is an outer universe with ωW1 = ωV1 . Let X ∈M and supposef ∶X → 2 with f ∈W is ω1-approximated in M . Then f gives a branchthrough T (M). But all the branches of T (M) are in V , hence f ∈ V . SinceM is ω1-guessing model in V , it follows that f ∈M .

Definition (SGM(ω2, ω1))

SGM(ω2, ω1) denotes the statement that the set of special ω1-guessingmodels of size ℵ1 is stationary in [Hθ]ℵ1 , for all sufficiently large regular θ.

Theorem (Cox-Krueger)SGM(ω2, ω1) is consistent with continuum arbitrary large, modulo theexistence of a supercompact cardinal.

Assume SGM(ω2, ω1). Then Souslin’s Hypothesis holds.

Assume SGM(ω2, ω1) and 2ℵ0 < ℵω1 . Then the principle AMP(ω1)holds: every forcing that adds a new subset of ω1 either adds a real orcollapsing some cardinal below 2ℵ0 .

DefinitionA model M of cardinality ω2 is special strongly ω1-guessing if it is theunion of an ∈-increasing chain (Mξ ∶ ξ < ω2) which is continuous atcofinality ω1 of special ω1-guessing models of cardinality ω1.

Definition (SGM+(ω3, ω1))

SGM+(ω3, ω1) states that the set of special strongly ω1-guessing models isstationary in [Hθ]ω2 , for all large enough regular θ.

Theorem (Mohammadpour, V.)Assume there are two supercompact cardinals. There there is a genericextension in which SGM+(ω3, ω1) holds.

Theorem (Mohammadpour, V.)

Assume SGM+(ω3, ω1), 2ℵ0 < ℵω1 and 2ℵ1 < ℵω2 . Then AMP(ω2) holds:every poset that adss a new subset of ω2 either adds a real or collapses somecardinal.

2ℵ0 ≥ ℵ3

¬ ◻ (ω2, λ) TP(ℵ2) SCH MP(ω2)

AMP(ℵ1) SH GM(ω2, ω1) TP(ℵ3) FS(ω2, ω1)

AMP(ℵ2) SGM(ω2, ω1) GM+(ω3, ω1) SFS(ω2, ω1)

SGM+(ω3, ω1)