Increasing δ_{2}^{1} and Namba-style forcing^{∗}

Richard Ketchersid Miami University

Paul Larson ^{†} Miami University

Jindřich Zapletal ^{‡} University of Florida

April 17, 2007

Abstract We isolate a forcing which increases the value of δ_{2}^{1} while preserving ω_{1} under the assumption that there is a precipitous ideal on ω_{1} and a measurable cardinal.

1 Introduction

The problem of comparison between ordinals defined in descriptive set theory such as δ_{n}^{1}, n ∈ ω and cardinals such as ℵ_{n}, n ∈ ω has haunted set theorists for decades. In this paper, we want to make a humble comment on the comparison between δ_{2}^{1} and ω_{2}.

Hugh Woodin showed [6] that if the nonstationary ideal on ω_{1} is saturated and there is a measurable cardinal then δ_{2}^{1} = ℵ_{2}. Thus the iterations for making the nonstationary ideal saturated must add new reals, and they must increase δ_{2}^{1}. It is a little bit of a mystery how this happens, since the new reals must be born at limit stages of the iteration and no one has been able to construct a forcing increasing the ordinal δ_{2}^{1} explicitly. The paper [7] shed some light on this problem; it produced a single step Namba type forcing which can increase δ_{2}^{1} in the right circumstances. In this paper we clean up and optimize the construction and prove:

Theorem 1.1. Suppose that there is a normal precipitous ideal on ω_{1} and a measurable cardinal κ. For every ordinal λ ∈ κ there is an ℵ_{1} preserving poset forcing δ_{2}^{1} > λ.

An important disclaimer: this result cannot be immediately used to iterate and obtain a model where δ_{2}^{1} = ℵ_{2} from optimal large cardinal hypotheses.

∗2000 AMS subject classification.

†Partially supported by DMS NSF grant 0401603

‡Partially supported by GA ČR grant 201-03-0933 and DMS NSF grant 0300201.

The forcing obtained increases δ_{2}^{1} once, to a value less than ω_{2}. If the reader wishes to iterate the construction in order to obtain a model where δ_{2}^{1} = ω_{2}, he will encounter the difficult problem of forcing a precipitous ideal on ω_{1} by an ℵ_{1}-preserving poset. Forcing δ_{2}^{1} = ℵ_{2} may be possible with some other type of accumulation of partial orders obtained in this paper.

The notation in this paper is standard and follows [2]. After the paper was written we learned that a related construction was discovered by Jensen [4]: a Namba-type forcing in the model L[U] with one measurable cardinal introducing a mouse which iterates to any length given beforehand.

2 Generic ultrapowers, iterations, and δ_{2}^{1}

In order to prepare the ground for the forcing construction, we need to restate several basic definitions and claims regarding the generic ultrapowers and their iterations.

Definition 2.1. [3] Suppose that J is a σ-ideal on ω_{1}. If G ⊂ P(ω_{1}) \ J is a generic filter, then we consider the generic ultrapower j : V → N modulo the filter G, in which only the ground model functions are used. If the model N is wellfounded, it is identified with its transitive collapse, and the ideal J is called precipitous.

The following definitions and facts have been isolated in [6].

Definition 2.2. [6] Suppose that M is a countable transitive model, and M ⊧“J is a precipitous ideal”. An iteration of length β ≤ ω_{1} of the model M is a sequence M_{α} : α ≤ β of models together with commuting system of elementary embeddings; successor stages are obtained through a generic ultrapower, and limit stages through a direct limit. A model is iterable if all of its iterands are wellfounded.

Definition 2.3. [1] Suppose J is a precipitous ideal on ω_{1}. An elementary submodel M of a large structure with j ∈ M is selfgeneric if for every maximal antichain A ⊂ P(ω_{1}) \ J in the model M there is a set B ∈ A ∩ M such that M ∩ ω_{1} ∈ B. In other words, the filter {B ∈ M ∩ P(ω_{1}) \ J : M ∩ ω_{1} ∈ B} is an M-generic filter.

Note that if M is a selfgeneric submodel, N is the Skolem hull of M ∪ {M ∩ ω_{1}}, and j : M̄ → N̄ is the elementary embedding between the transitive collapses induced by id : M → N, then j is a generic ultrapower of the model M by the genric filter identified in the above definition. The key observation is that selfgeneric models are fairly frequent:

Proposition 2.4. Suppose that J is a precipitous ideal on ω_{1} and μ > 2^{ℵ_{1}} is a regular cardinal. The set of countable selfgeneric elementary submodels of H_{μ} is stationary in [H_{μ}]^{ℵ_{0}}.

Proof. Suppose that f : H_{μ}^{<ω} → H_{μ} is a function; we must find a selfgeneric submodel of H_{μ} closed under it. Let G ⊂ P(ω_{1}) \ J be a generic filter and j : V → N be the associated generic ultrapower embedding into a transitive model. Note that j′′H_{μ}^{V} is a selfgeneric submodel of j(H_{μ}) closed under the function j(f); it is not in general an element of the model N. Consider the tree T of all finite attempts to build a selfgeneric submodel of j(H_{μ}) closed under the function j(f). Then T ∈ N and the previous sentence shows that the tree T is illfounded in V [G]. Since the model N is transitive, it must be the case that the tree T is illfounded in N too, and so M ⊧there is a countable selfgeneric elementary submodel of j(H_{μ}) closed under the function j(f). An elementarity argument then yields a countable selfgeneric elementary submodel of the structure H_{μ} closed under the function f in the ground model as desired.

Our approach to increasing δ_{2}^{1} is in spirit the same as that of Woodin. We start with a ground model V with a precipitous ideal J on ω_{1}, a measurable cardinal κ, and an ordinal λ ∈ κ. Choose a regular cardinal μ between λ and κ. In the generic extension V [G], it will be the case that ω_{1}^{V} = ω_{1}^{V [G]} and κ is still measurable and moreover there is a countable elementary submodel M ≺ H_{μ}^{V} such that

• M is selfgeneric

• M̄ is iterable

• λ is a subset of one of the iterands of M̄.

In fact, it will be the case that writing M_{α}, α ∈ ω_{1} for the models obtained by transfinite inductive procedure M_{0} = M, M_{α+1} =Skolem hull of M_{α} ∪{M_{α} ∩ ω_{1}}, and M_{α} = ⋃_{β∈α} for limit ordinals α, and writing M̄_{α} for the respective transitive collapses, the models M_{α} are all selfgeneric, the models M̄_{α}, α ≤ ω_{1} constitute an iteration of the model M̄, and λ ⊂ ⋃_{α} M_{α}. By Lemma 4.7 of [6], δ_{2}^{1} must be larger than the cumulative hierarchy rank of the model M̄_{ω_{1}}, which by the third item is at least λ. Note that the model M cannot be an element of the ground model.

It may seem that adding a model M such that all the models M_{α}, α ∈ ω_{1} are selfgeneric is an overly ambitious project. The forcing will in fact add a countable set {f_{n} : n ∈ ω} ⊂ H_{μ}^{V} such that every countable elementary submodel containing it as a subset is necessarily selfgeneric. It will also add a countable set {g_{n} : n ∈ ω} ⊂ H_{μ}^{V} of functions from ω_{1}^{<ω} to ω such that λ = ⋃_{n} rng(g_{n}). This will be achieved by a variation of the classical Namba construction by an ℵ_{1}-preserving forcing of size < κ. In the generic extension, use the measurability of κ to find an elementary submodel N of a large structure containing J, μ, κ as well as the functions f_{n}, g_{n}, n ∈ ω such that the ordertype of N ∩ κ is ω_{1}, and consider the transitive collapse N̄ of the model N ∩ V. It is iterable by Lemma 4.5 of [6]. This means that even the transitive collapse M̄ of the model M = N ∩ H_{μ}^{V} is iterable, since it is a rank-initial segment of N̄ and every iteration of M̄ extends to an iteration of N̄. Thus the model M is as desired, and this will complete the proof.

3 A class of Namba-like forcings

Definition 3.1. Suppose that X is a set and I is a collection of subsets of X closed under subsets, X ∉ I. The forcing Q_{I} consists of all nonempty trees T ⊂ X^{<ω} such that every node t ∈ T has an extension s ∈ T such that {x ∈ X : s^{⌢}x ∈ T } ∉ I. The ordering is that of inclusion.

It is not difficult to see that the forcing Q_{I} adds a countable sequence of elements of the underlying set X. The only property of the generic sequence we will use is that it is not a subset of any ground model set in the collection I. The usual Namba forcing is subsumed in the above definition: just put X = ℵ_{2} and I =all subsets of ω_{2} of size ℵ_{1}. A small variation of the argument in [5] will show that whenever I is an < ℵ_{2}-complete ideal then the forcing Q_{I} preserves ℵ_{1} and if in addition CH holds then no new reals are added. We want to increase the ordinal δ_{2}^{1}, so we must add new reals, and so we must consider weaker closure properties of the collection I. The following definition is critical.

Definition 3.2. Suppose that J is an ideal on a set Y, X is a set, and I is a collection of subsets of X. We say that I is closed under J integration if for every J-positive set B ⊂ Y and every set D ⊂ B × X whose vertical sections are in I the set ∫_{B} D dJ = {x ∈ X : {y ∈ B : ⟨y, x⟩ ∉ D} ∈ J} ⊂ X is also in the collection I.

We will use this definition in the context of a precipitous ideal J on ω_{1}. In this case, the closure under J integration allows of an attractive reformulation:

Proposition 3.3. Suppose that J is a precipitous ideal on ω_{1} and I is a collection of subsets of some set X closed under inclusion. Then I is closed under J-integration if and only if P(ω_{1}) \ J forces that writing j : V → M for the generic ultrapower, the closure of I under J integration is equivalent to the statement that for every set A ⊂ X not in I, the set j′′A is not covered by any element of j(I).

Proof. For the left-to-right implication, assume that I is closed under J integration. Suppose that some condition forces that Ċ ∈ j(I) is a set; strengthening this condition of necessary we can find a set B ∈ P(ω_{1}) \ J and a function f : B → I such that B ⊩ Ċ = j(f)(ω̌_{1}). Let D ⊂ B × X be defined by ⟨α, x⟩ ∈ D ↔ x ∈ f(α) and observe that ∫_{B} D dJ ∈ I. Thus, if A ∉ I is a set, it contains an element x ∉ ∫_{B} D dJ, then the set B′ = {α ∈ B : x ∉ f(α)} ⊂ B is J-positive and as a P(ω_{1}) \ J condition it forces j(x) ∉ Ċ and j(Ǎ) ⊄ Ċ.

The opposite implication is similar.

The reader should note the similarity between the above definition and the Fubini properties of ideals on Polish spaces as defined in [8].

The basic property of the class of forcings we have just introduced is the following.

Proposition 3.4. Suppose that J is a precipitous ideal on ω_{1}, X is a set, and I is a collection of subsets of the set X closed under J integration. Then the forcing Q_{I} preserves ℵ_{1}.

Proof. Suppose that T ⊩ ḟ : ω̌ → ω̌_{1} is a function. A usual fusion argument provides for a tree S ⊂ T in the poset Q_{I} such that for every node t ∈ S on the n-th splitting level the condition S ↾ t decides the value of the ordinal ḟ(ň) to be some definite ordinal g(t) ∈ ω_{1}. Here, S ↾ t is the tree of all nodes of the tree S inclusion-compatible with t. To prove the theorem, it is necessary to find a tree U ⊂ S and an ordinal α ∈ ω_{1} such that the range g′′U is a subset of α.

For every ordinal α ∈ ω_{1} consider a game G_{α} between Players I and II in which the two players alternate for infinitely many rounds indexed by n ∈ ω, Player I playing nodes t_{n} ∈ T on the n-th splitting level of the tree T and Player II answering with a set A_{n} ∈ I. Player I is required to play so that t_{0} ⊂ t_{1} ⊂ … and the first element on the sequence t_{n+1} \ t_{n} is not in the set A_{n}. He wins if the ordinals g(t_{n}), n ∈ ω are all smaller than α.

It is clear that these games are closed for Player I and therefore determined. Note that if Player I has a winning strategy σ in the game G_{α} for some ordinal α ∈ ω_{1}, then the collection of all nodes which can arise as the answers of strategy σ to some play by Player II forms a tree U in Q_{I} and g′′U ⊂ α. Thus the following claim will complete the proof of the theorem.

Claim 3.5. There is an ordinal α ∈ ω_{1} such that Player I has a winning strategy in the game G_{α}.

Assume for contradiction that Player II has a winning strategy σ_{α} for every ordinal α ∈ ω_{1}. Let M ≺ H_{κ} be a selfgeneric countable elementary submodel of some large structure containing the sequence of these strategies as well as X, I, J. Let β = M ∩ ω_{1}. We will find a legal counterplay against the strategy σ_{β} in which Player I uses only moves from the model M. It is clear that in such a counterplay, the ordinals g(t_{n}), n ∈ ω stay below β. Therefore Player I will win this play, and that will be the desired contradiction.

The construction of the counterplay proceeds by induction. Build nodes t_{n}, n ∈ ω of the tree S as well as subsets B_{n}, n ∈ ω of ω_{1} so that

• B_{0} ⊃ B_{1} ⊃ … are all J-positive sets in the model M such that β ∈ B_{n} for every number n

• t_{0} ⊂ t_{1} ⊂ ⋯ ⊂ t_{n} are all in the model M and they form a legal finite counterplay against all strategies σ_{α}, α ∈ B_{n}, in particular, against the strategy σ_{β}.

Suppose that the node t_{n} ∈ S ∩M and the set B_{n} have been found. Consider the set D = {⟨α, x⟩ : α ∈ B_{n}, x ∈ σ_{α}(t_{n})} ⊂ B × X. Its vertical sections are sets in the collection I, and by the assumptions so are the integrals ∫_{C} D dJ for all J-positive sets C ⊂ B_{n}. Since the node t_{n} ∈ S has more than I many immediate successors, it follows that the set A = {C ⊂ B_{n} : C ∉ J and ∃x ∈ X ∀α ∈ C t_{n}^{⌢}x ∈ S ∧ x ∉ σ_{α}(t_{n})} is dense in P(ω_{1}) \ J below the set B_{n}. This set is also in the model M and by the selfgenericity there is a point x ∈ X ∩ M such that t_{n}^{⌢}x ∈ S and the set S_{n+1} = {α ∈ B_{n} : x ∉ σ_{α}(t_{n})} is in the set A ∩ M and contains the ordinal β. The node t_{n+1} ⊃ t_{n} is then just any node at n + 1-st splitting level extending t_{n}^{⌢}x. Clearly, t_{n+1} ∈ M by the elementarity of the model M. This concludes the inductive construction and the proof.

As the last remark in this section, the class of sets I closed under Jintegration is itself closed on various operations, and this leads to simple operations on the partial orders of the form Q_{I}. We will use the following operation. If X_{0}, X_{1} are disjoint sets and I_{0} ⊂ P(X_{0}) and I_{1} ⊂ P(X_{1}) are sets closed under subsets and J integration, then also the set K ⊂ P(X_{0} ∪ X_{1}) defined by A ∈ K if either A ∩ X_{0} ∈ I_{0} or A ∩ X_{1} ∈ I_{1} is closed under subsets and J-integration. It is easy to see that the forcing Q_{K} adds an ω sequence of elements of X_{0} ∪ X_{1} which cofinally often visits both sets and its intersection with X_{0} or X_{1} is not a subset of any ground model set in I_{0} or I_{1} respectively.

4 Wrapping up

Fix a normal precipitous ideal J on ω_{1}, a measurable cardinal κ, and an ordinal λ < κ. Theorem 1.1 is now proved through identification of several interesting collections of sets closed under J-integration. This does not refer to the precipitousness of the σ-ideal J anymore.

Definition 4.1. X_{0} is the set of all functions from ω_{1}^{<ω} to λ. I_{0} ⊂ P(X_{0}) is the closure of the set of its generators under subset and J-integration, where the generators of I_{0} are the sets A_{α} = {g ∈ X_{0} : α ∉ rng(g)} for α ∈ λ.

The obvious intention behind the definition is that if {g_{n} : n ∈ ω} ⊂ X_{0} is a set of functions which is not covered by any element of the set I_{0} then ⋃_{n} rng(g_{n}) = λ. With the previous section in mind, we must prove that X_{0} ∉ I_{0}. Unraveling the definitions, it is clear that it is just necessary to prove that whenever n is a natural number, S ⊂ ω_{1}^{n} is a J^{n}-positive set, and D ⊂ S × X_{0} is a set whose vertical sections are I_{0}-generators, then the integral ∫_{S} D dJ^{n} is not equal to X_{0}. Here J^{n} is the usual n-fold Fubini power of the ideal J. Let g : ω_{1}^{n} → λ be a function such that for every n-tuple β⃗ ∈ S, the vertical section Dβ⃗ is just the generator A_{g(β⃗)}. Then clearly g ∉ ⋃_{β⃗∈S} Dβ⃗, in particular g ∉ ∫_{S} D dJ^{n} and ∫_{S} D dJ^{n} ≠ X_{0}.

Definition 4.2. X_{1} is the set of all functions with domain ω_{1}^{<ω} ×𝔄 and range a subset of ω_{1} × P(ω_{1}). Here 𝔄 is the set of all maximal antichains in the forcing P(ω_{1})\J. The set I_{1} is the closure of the set of its generators under subset and Jintegration, where the generators of I_{1} are the sets of the form A_{α,Z} = {f ∈ X_{1} : for every finite sequence β⃗ ∈ α^{<ω}, f(β⃗, Z)(0) ∈ α and f(β⃗, Z)(1) is not a set in Z containing α}, where α ∈ ω_{1} and Z ∈ 𝔄 are arbitrary.

The obvious intention behind this definition is that whenever {f_{n} : n ∈ ω} is a countable subset of X_{1} which is not covered by any element of the set I_{1} then every countable elementary submodel M ≺ H_{μ} containing all these functions must be self-generic: whenever Z ∈ M is a maximal antichain in P(ω_{1}) \ J, writing α = M ∩ ω_{1}, there must be a number n such that f_{n} ∉ A_{α,Z}. Perusing the definition of the set A_{α,Z} and noting that M is closed under the function f_{n}, we conclude that it must be the case that for some finite sequence β⃗ ∈ α^{<ω} the value f_{n}(β⃗, Z) ∈ M must be a set in Z containing the ordinal α. Since the maximal antichain Z was arbitrary, this shows that M is self-generic as required.

We must prove that X_{1} ∉ I_{1}. This is a rather elementary matter, nevertheless it is somewhat more complicated than the 0 subscript case. Unraveling the definitions, it is clear that it is just necessary to prove that whenever n is a natural number, S ⊂ ω_{1}^{n} is a J^{n}-positive set, and D ⊂ S ×X_{0} is a set whose vertical sections are I_{1}-generators, then the integral ∫_{S} D dJ^{n} is not equal to X_{1}. Here J^{n} is the usual n-fold Fubini power of the ideal J. Fix then n ∈ ω, a J^{n}-positive set S ⊂ ω_{1}^{n}, and the set D ⊂ S × X_{1}; we must find a function f ∈ X_{1} and a J_{n}-positive set U ⊂ S such that ∀β⃗ ∈ U ⟨β⃗, f⟩ ∉ D. For every sequence β⃗ ∈ S choose a countable ordinal α(β⃗) and a maximal antichain Z(β⃗) ⊂ P(ω_{1}) \ J such that Dβ⃗ = A_{α(β⃗),Z(β⃗)}. Use standard normality arguments to find numbers m, k ≤ n and a J^{n}-positive set T ⊂ S consisting of increasing sequences such that

• for a sequence β⃗ ∈ T, the value of α(β⃗) depends only on β⃗ ↾ m and α(β⃗) ≥ β⃗(m − 1)

• the value of Z(β⃗) depends only on β⃗ ↾ k and the partial map π with domain ω_{1}^{k}, defined by Z(β⃗) = π(β⃗ ↾ k) whenever β⃗ ∈ T, is countable-to-one.

There are now several cases.

• There is a J^{n}-positive set U ⊂ T such that α(β⃗) > β⃗(m−1). Here, consider the function f ∈ X_{1} such that f(β⃗ ↾ m, Z) = α(β⃗) for every sequence β⃗ ∈ U and every maximal antichain Z. Clearly, f ∉ ⋃_{β⃗∈U} Dβ⃗ as required: for every sequence β⃗ ∈ U, it is the case that α(β⃗) = f(β⃗ ↾ m, Z(β⃗))(0) and so the ordinal α(β⃗) does not have the required closure property with respect to f.

• The first case fails and k ≥ m. Here, define the map f ∈ X_{1} by f(0, Z)(0) = sup{β⃗(k − 1) : β⃗ ∈ T and Z = Z(β⃗)} + 1 for every maximal antichain Z. The set U = {y ∈ T : α(β⃗) = β⃗(m − 1)} and the map f are as required: again, for every sequence β ∈ U the ordinal α(β⃗) ≤ β⃗(k − 1) < f(0, Z(β⃗))(0) does not have the required closure properties.

• The first case fails and k < m. Define the function f ∈ X_{1} in the following way. For every sequence γ⃗ ∈ ω_{1}^{m−1}, if the set Wγ⃗ = {α ∈ ω_{1} : ∃β⃗ ∈ T γ^{⌢}α ⊂ β⃗ and α = α(β⃗)} is J-positive, let f(γ⃗, π(γ⃗ ↾ k)) to be some element of the maximal antichain π(γ⃗ ↾ k) with J-positive intersection with Wγ⃗. The set U = {β⃗ ∈ T : α(β⃗) = β⃗(m) and β⃗(m) ∈ f(β⃗ ↾ (m − 1), π(β⃗ ↾ k)} is then J^{n} positive and f ∉ ⋃_{β⃗∈U} Dβ⃗ as required: the ordinal α(β⃗) belongs to the set f(β⃗ ↾ k, Z(β⃗)) ∈ Z(β⃗).

Thus X_{1} ∉ I_{1}.

To conclude the proof of Theorem 1.1, just form a collection K ⊂ P(X_{0}∪X_{1}) as in the end of the previous section and force with the poset Q_{K}. Since K is closed under J-integration, the forcing preserves ℵ_{1}. It also adds sets {f_{n} : n ∈ ω} ⊂ X_{1} and {g_{n} : n ∈ ω} ⊂ X_{0} with the required properties, showing that in the generic extension, δ_{2}^{1} > λ.

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