On extensions of elementary submodels by forcing

LÚCIA R. JUNQUEIRA, Universidade de São Paulo, São Paulo, Brazil. E-mail: lucia@ime.usp.br

PAUL LARSON, Miami University, Oxford, Ohio, USA. E-mail: larsonpb@muohio.edu

MARCELO D. PASSOS, Universidade de São Paulo, São Paulo, Brazil. E-mail: md passos@yahoo.com

Abstract

We study when a partial order ℙ preserves the properties of an elementary submodel M (of a structure of the form H_{θ}) be countably closed and ω-covering. In particular, we discuss the existence of ωcovering elementary submodels of different sizes.

Keywords: forcing,elementary submodels,ω-covering

1 Introduction

Throughout this article whenever we say “M is an elementary submodel,” we mean “M is an elementary submodel H_{θ} for some regular cardinal θ large enough.”^{1} If θ is regular, H_{θ} ⊧ ZF C − P. By “large enough θ” we mean so that H_{θ} contains all sets of interest in the context under discussion. For purposes of intuition, one may sloppily think of H_{θ} as V, the class of all sets.

For basic results on forcing and for fixing notation, we refer to [Kun80], [Jec86] or [Jec03]. For “self-containedness,” we will mention some. All partial orders will have a unique maximum element and we will always name it 1. Any element of a partial order will be called a condition. If ℙ is a partial order, a ℙ-name τ is the empty set or a relation such that, for all ⟨σ, p⟩ ∈ τ, σ is a ℙ-name and p ∈ ℙ. For a set x, x̌ = {⟨y̌, 1⟩: y ∈ x}.

The purpose of this article is to study how forcing affects two important properties of elementary submodels: countable closure and ω-covering. These properties play an important role when we use elementary submodels to help us prove results in general topology or set theory (see e.g. [Dow88]).

If M is an elementary submodel of H_{θ}, for a regular cardinal θ, ℙ ∈ M is a partial order, and G is a ℙ-generic filter (over H_{θ}), we have, by [She82], that

M[G] ≺ H_{θ}[G] = H_{θ}^{V[G]} ,

1Recall that, for a set x, we can define ⋃^{0} x = x and, for a natural number n, ⋃^{n+1} x = ⋃ (⋃^{n} x). Furthermore, its transitive closure will be trcl(x) = ⋃_{n<ω} (⋃^{n} x). For a cardinal θ, H_{θ} = {x: |trcl(x)| < θ} is a set (see e.g. [Kun80]).

1 L. J. of the IGPL, Vol. 0 No. 0, pp. 1–15 0000 \begin{matrix}c \\ ◯ \end{matrix} Oxford University Press

2 On extensions of elementary submodels by forcing

where M[G] = {τ_{G}: τ ∈ M is a ℙ-name}.

As ℙ ∈ M, Γ = {⟨p̌, p⟩: p ∈ ℙ} ∈ M. So, G = Γ_{G} ∈ M[G]. Furthermore, M ⊆ M[G].

In [She82], we have the definition of proper forcing and, for that, we have the definition of (ℙ, M)-generic condition. Some results concerning the equality M[G] ∩ V = M are presented there. For instance:

Theorem 1.1 ([She82]) Under the conditions above, the following are equivalent:

1. G ∩ M is ℙ-generic under M (i.e. , G ∩ M ∩ D ≠ ∅, whenever D ∈ M is a dense subset of ℙ);

2. M[G] ∩ V = M; and

3. M[G] ∩ ON = M ∩ ON.

Let us remember the definition of (ℙ, M)-generic (see e.g. [She82, Jec86]):

Definition 1.2 Let M be an elementary submodel and ℙ ∈ M be a partial order. We say that p ∈ ℙ is (ℙ, M)-generic if, for every D ∈ M, dense in ℙ, D ∩ M is predense below p.

In the following proposition we have the main equivalences of p being (ℙ, M)- generic:

Proposition 1.3 ([She82]) Let M be an elementary submodel, ℙ ∈ M be a partial order, and p ∈ ℙ be a condition. The following are equivalent:

1. p is (ℙ, M)-generic;

2. M[G] ∩ V = M, whenever G is a generic such that p ∈ G; and

3. M[G] ∩ ON = M ∩ ON, whenever G is a generic such that p ∈ G.

If ℙ ⊆ M, all conditions are (ℙ, M)-generic and this seems to be the most trivial condition under which everything is valid. Looking carefully at the proof of these theorems, we see that the important point is to get some antichains to be subsets of M. It is well known that all countable elements of M are subsets of M, so we easily have:

Proposition 1.4 If ℙ ∈ M has c.c.c.,^{2} then all its conditions are (ℙ, M)-generic. Furthermore, M[G] ∩ V = M, for all generic G.

Proof. Let D ∈ M be a dense subset of M. By elementarity, we can find A ∈ M such that A is a maximal antichain and A ⊆ D. Since A is countable, A ⊆ M. Then A ⊆ D ∩ M and D ∩ M is predense below 1.

More generally we have:

Theorem 1.5 Let M be an elementary submodel and ℙ ∈ M be a partial order. The following are equivalent:

2A partial order has c.c.c. (countable chain condition) if all its antichains are countable.

1. M[G] ∩ V = M, whenever G is generic, and

2. A ⊆ M, whenever A ∈ M is an antichain.

Proof. (If) We follow the same ideas from the proof of the previous result.

(Only if) Let A ∈ M be an antichain in ℙ. By elementarity, we can get ⟨p_{β}: β < γ⟩ ∈ M, an enumeration of A. We define

τ = {⟨α, p_{β}⟩: α < β < γ},

that is a ℙ-name and is in M, by definibility. If G is a generic, such that p_{β} ∈ G, we get τ_{G} = β ∈ M[G] ∩ ON ⊆ M. Since ⟨p_{β}: β < γ⟩ ∈ M, p_{β} ∈ M.

The paper is organized as follows: in Section 2 we give some basic results about preservation of properties of M by different classes of partial orders; in Section 3 we look more carefully at the property of M being ω-covering and prove several results about preservation and existence of such models of different cardinalities; in Section 4 we look at the question of getting M[G] to be ω-covering.

2 Preserving properties of M

We first recall:

Definition 2.1 We say that an elementary submodel M is countably closed, if [M]^{ω} ⊆ M that is, if all countable subsets of M are elements of M.

A straightforward density argument shows that for ω_{1}-Baire^{3} partial orders we have:

Lemma 2.2 Let ℙ be an ω_{1}-Baire partial order. Let σ and τ be ℙ-names and p ∈ ℙ, such that

p ⊩ “τ ⊆ σ ∧ τ is countable”.

Then there exists q ≤ p and a countable ℙ-name ρ ⊆ σ, such that q ⊩ “τ = ρ”.

We thus have:

Theorem 2.3 Let M be a countably closed elementary submodel and let ℙ ∈ M be an ω_{1}-Baire partial order. If G is ℙ-generic, then M[G] is countably closed.

Proof. Let us define

𝕄 = {⟨τ, 1⟩: τ ∈ M is ℙ-name}.

If G is a ℙ-generic, we have 𝕄_{G} = M[G]. This can be useful since it allows us to work just with ℙ-names.

Let p ∈ ℙ and τ be a ℙ-name such that

p ⊩ “τ ⊆ 𝕄 ∧ τ is countable”.

3The ω_{1}-Baire partial orders are those that do not add any countable subset for a set of the ground model.

By the previous lemma, there are a countable ℙ-name ρ ⊆ 𝕄 and an extension q of p such that

q ⊩ “τ = ρ”.

Since M is countably closed, ρ is an element of M.

Countably closed elementary submodels can be very large (they have at least cardinality greater or equal to the continuum). We can weaken such condition on the countable subsets and still have some “covering property” (see e.g. [Dow88]):

Definition 2.4 An elementary submodel M is ω-covering if for every countable set A ⊆ M there is a countable set B ∈ M such that A ⊆ B.

To get that M[G] still is ω-covering if M is, we can use orders in a much larger class:

Definition 2.5 We say that a partial order ℙ satisfies the countable covering property if, in any extension by ℙ, any countable subset of the ground model can be covered by a countable element of the ground model.

It is well-known that proper orders and ω_{1}-Baire orders satisfy the countable covering property. Actually, we have:

Lemma 2.6 Let ℙ be a partial order. The following are equivalent:

1. ℙ satisfies the countable covering property and

2. for p ∈ ℙ, and ℙ-names σ and τ, such that

p ⊩ “τ ⊆ σ ∧ τ is countable”,

there exists q ≤ p and a countable ℙ-name ρ ⊆ σ, such that q ⊩ “τ ⊆ ρ”.

This lemma is all we need to prove:

Theorem 2.7 Let M be an ω-covering elementary submodel and let ℙ ∈ M be a partial order. If ℙ satisfies the countable covering property and G is ℙ-generic, then M[G] is ω-covering.

Proof. Let p ∈ ℙ and τ be a ℙ-name such that

p ⊩ “τ ⊆ 𝕄 ∧ τ is countable”.

By the previous lemma, there exists a countable ℙ-name ρ ⊆ 𝕄 and an extension q of p such that q ⊩ “τ ⊆ ρ”. Since M is ω-covering, there is a countable X ∈ M such that ρ ⊆ X. Let

ψ = {⟨π, r⟩ ∈ X: r ∈ ℙ ∧ π is ℙ-name }.

By elementary, we have ψ ∈ M as {X, ℙ} ∈ M. Moreover, ψ is a countable ℙ-name and ρ ⊆ ψ. Thus q ⊩ “τ ⊆ ψ”.

It seems natural to define:

Definition 2.8 We say that a partial order ℙ preserves ω-covering if, for any regular cardinal θ and any ω-covering elementary submodel M ≺ H_{θ} with ℙ ∈ M, M[G] is ω-covering, whenever G is a ℙ-generic filter.

We will need the following result about ω-covering elementary submodels (see e.g. [Jun00]):

Lemma 2.9 If M is an ω-covering elementary submodel, then ω_{1} ⊆ M.

In the next section we will see how to have ω-covering elementary submodels with cardinality ℵ_{1} (this is also done in [Dow88]). We then get:

Theorem 2.10 If ℙ preserves ω-covering, then ℙ does not collapse ω_{1}.

Proof. Let M be an ω-covering elementary submodel of cardinality ℵ_{1} such that ℙ ∈ M. Since ℙ preserves ω-covering, by the previous lemma, if G is ℙ-generic, we have, in V [G], that ω_{1}^{V [G]} ⊆ M[G]. Thus, M[G] is uncountable in V [G], and since |M[G]| = |M| = |ℵ_{1}|^{V [G]}, we have ℵ_{1}^{V [G]} = ℵ_{1}^{V}.

The next diagram will represent the implications between the presented classes of partial orders:

\xymatrix{c.c.c. \ar[d] & countably closed \ar[dl] \ar[d] \\ proper \ar[dr] & ω_{1}-Baire \ar[d] \ar[r] & preserves countable closure \\  & countable covering property \ar[d] \\ preserves ω_{1} & preserves ω-covering \ar[l]}

Of course every countably closed elementary submodel is ω-covering. L. Junqueira showed in [Jun00] the following:

Theorem 2.11 ([Jun00]) CH is equivalent to “every ω-covering elementary submodel is countably closed.”

We finish this section providing two interesting examples about preservation of countable closure:

Example 2.12 If ℙ = F n(κ, 2) and we start with CH in V, we can take a countably closed elementary submodel M of cardinality 𝔠 = ℵ_{1}. In V[G], M[G] is still ω-covering but is not countably closed (as the continuum becomes greater than κ).

Example 2.13 Let ℙ = F n(I, 2, ω_{1})^{4} for an uncountable set I and assume ¬CH. Then ℙ is countably closed^{5} and thus ω_{1}-Baire. In V[G], we have CH. So, if M is ω-covering but not countably closed, M[G] becomes countably closed.

3 ω-covering elementary submodels

In [Dow88], we find a construction of ω-covering elementary submodels of cardinality ℵ_{1}. We can characterize such submodels by the following:

Proposition 3.1 Let M be an elementary submodel of cardinality ℵ_{1}. Then M is ω-covering if, and only if, there is a sequence ⟨(M_{α}, A_{α}): α < ω_{1}⟩ such that M = ⋃_{α<ω_{1}} M_{α} and:

1. ∀α < β < ω_{1} ( M_{α} ⊆ A_{α} ∈ M_{β} ) and

2. ∀α < ω_{1} ( M_{α} ≺ M ∧ A_{α} ∈ M ∧ |M_{α}| = |A_{α}| = ℵ_{0} ).

Proof. (Only if) Since M has cardinality ℵ_{1}, we can write M = ⋃_{α<ω_{1}} X_{α}, where each X_{α} is countable. Suppose the sequence is already constructed up to β < ω_{1}. Since {A_{α}: α < β} ∪ X_{β} ∈ [M]^{ω}, there exists countable M_{β} ≺ M such that {A_{α}: α < β} ∪ X_{β} ⊆ M_{β}. There exists A_{β} ∈ M, countable, such that M_{β} ⊆ A_{β}, as M is ω-covering.

(If) Let X ∈ [M]^{ω}. Since X is countable there is α < ω_{1} such that X ∈ [M_{α}]^{ω}. Then X ⊆ A_{α} and A_{α} ∈ M.

It is worthy to observe that this proof teaches us how to construct ω-covering elementary submodels of cardinality ℵ_{1}. Note that we thus have the following corollary (also showed in [Dow88]):

Corollary 3.2 For any set X of cardinality ℵ_{1}, there exists an ω-covering elementary submodel M such that X ⊆ M and |M| = ℵ_{1}.

With this characterization we can prove:

Theorem 3.3 Let ℙ be a partial order. Then ℙ preserves ω_{1} if, and only if, ℙ preserves ω-covering for elementary submodels of cardinality ℵ_{1}.

Proof. (Only if) Let M be an ω-covering elementary submodel of cardinality ℵ_{1} and let ⟨(M_{α}, A_{α}): α < ω_{1}⟩ be a sequence as in the previous proposition. Without loss of generality, we can suppose ℙ ∈ ⋂_{α<ω_{1}} M_{α}.

4For the sets I and J, and a cardinal λ, F n(I, J, λ) = {p ⊆ I × J: p is function ∧ |p| < λ}. Moreover, F n(I, J) = F n(I, J, ω). Both are ordered by p ≤ q if q ⊆ q.

5For a cardinal κ, a partial order is said κ-closed if, for all decreasing sequence of conditions ⟨p_{α}: α < λ⟩, with λ < κ, we have a commom extension for all conditions. For the case κ = ω, we say that it is countably closed.

For each α, let

τ_{α} = {⟨σ, 1⟩: σ ∈ A_{α} is a ℙ-name }.

Whenever α < β, by elementarity, τ_{α} ∈ M_{β}, since {A_{α}, ℙ} ∈ M_{β}.

Thus, if G is a ℙ-generic, we have M[G] = ⋃_{α<ω_{1}} M_{α}[G] and

1. ∀α < β < ω_{1} ( M_{α}[G] ⊆ (τ_{α})_{G} ∈ M_{β}[G] ) and

2. ∀α < ω_{1} ( M_{α}[G] ≺ M[G] ∧ (τ_{α})_{G} ∈ M[G] ∧ |M_{α}[G]| = |(τ_{α})_{G}| = ℵ_{0} ).

Since ω_{1} is preserved, M[G] is ω-covering.

As a consequence we have:

Corollary 3.4 ℙ preserves ω-covering for all elementary submodels of cardinality ℵ_{1} if, and only if, ℙ preserves ω-covering of any elementary submodel of cardinality ℵ_{1}.

Inspired by the construction for cardinality ℵ_{1}, we got a way to construct ω-covering elementary submodels of some greater cardinalities:

Lemma 3.5 Let n ≥ 1 be a natural number. For any set X of cardinality ℵ_{n}, there exists an ω-covering elementary submodel M such that X ⊆ M and |M| = ℵ_{n}.

Proof. Let us use induction on n. Note that we are already done for n = 1.

Let X be a set of cardinality ℵ_{n+1}. We can write X = ⋃_{α<ω_{n+1}} X_{α}, where all X_{α}’s have cardinality ℵ_{n}. We will construct an increasing sequence of ω-covering elementary submodels ⟨M_{α}: α < ω_{n+1}⟩ such that

∀α < ω_{n+1} ( X_{α} ⊆ M_{α} ∧ |M_{α}| = ℵ_{n} ).

Suppose we already have the sequence up to β < ω_{n+1}. Then, using the induction hypothesis, there exists M_{β}, ω-covering elementary submodel of cardinality ℵ_{n}, such that (⋃_{α<β} M_{α}) ∪ X_{β} ⊆ M_{β}.

Since ω_{n+1} is regular, M = ⋃_{α<ω_{n+1}} M_{α} will be ω-covering.

Theorem 3.6 Let M be an elementary submodel of cardinality ℵ_{n}, for some n such that 1 < n < ω. Then M is ω-covering if, and only if, there exists an increasing sequence of ω-covering elementary submodels, ⟨M_{α}: α < ω_{n}⟩, such that

M = ⋃_{α<ω_{n}} M_{α} ∧ ∀α < ω_{n} ( |M_{α}| = ℵ_{n−1} ).

Proof. It is similar to the proof of the previous lemma. We just have to ask the M_{α}’s to be elementary submodels of a fixed ω-covering elementary submodel.

Now let us present a way to construct elementary submodels that are not ωcovering:

Proposition 3.7 Let M = ⋃_{α<λ} M_{α}, where ⟨M_{α}: α < λ⟩ is a strictly increasing sequence of elementary submodels and λ is a limit ordinal of countable cofinality. Then M cannot be ωcovering.

Proof. Take a strictly increasing sequence ⟨α_{n}: n < ω⟩ which is cofinal in λ. For each n < ω, fix x_{n} ∈ M_{α_{n+1}} \ M_{α_{n}}. Let X = {x_{n}: n < ω}. Then X witnesses that M is not ω-covering.

Corollary 3.8 Let M be an ω-covering elementary submodel. Then |M| has uncountable cofinality.

Proof. Let μ = |M| and λ = cf(μ). We can write M = ⋃_{α<λ} X_{α}, where ℵ_{0} ≤ |X_{α}| < μ, for all α < λ. Using induction we can construct an increasing sequence of elementary submodels such that

∀α < λ ( X_{α} ⊆ M_{α} ≺ M ∧ |M_{α}| = |X_{α}| < μ ).

Then M = ⋃_{α<λ} M_{α} and, by its cardinality, we can suppose the sequence to be strictly increasing. We conclude that λ > ω, by the previous lemma.

Corollary 3.9 If α is a limit ordinal of countable cofinality, then there are no ω-covering elementary submodels of cardinality ℵ_{α}.

Using the characterization given by Theorem 3.6 we can prove:

Theorem 3.10 If ℙ is a partial order that does not collapse the cofinality of ω_{n} to ω, for any natural n ≥ 1, then ℙ preserves ω-covering for elementary submodels of cardinality less than ℵ_{ω}.

Proof. Since ω_{1} is preserved, we already have that ω-covering is preserved for elementary submodels of cardinality ℵ_{1}. The proof follows by induction on n.

Let M be an ω-covering elementary submodel of cardinality ℵ_{n+1}. By Theorem 3.6, M = ⋃_{α<ω_{n}+1} M_{α}, where ⟨M_{α}: α < ω_{n+1}⟩ is an increasing sequence of ω-covering elementary submodels of cardinality ℵ_{n}. If G is ℙ-generic, by induction hypothesis, each M_{α}[G] is ω-covering and, since (cf(ω_{n+1}))^{V [G]} > ω, M[G] is ω-covering.

In the proof of Proposition 3.7, it was quite important to have the sequence to be strictly increasing. Then, for the next theorem, we guarantee that submodels will be different in any extension by the axiom of regularity.

Theorem 3.11 Let ℙ be a partial order that preserves ω-covering. Then, for any cardinal κ of uncountable cofinality, κ still will have uncountable cofinality in any extension by ℙ.

Proof. Let κ be a cardinal such that λ = cf(κ) > ω and let θ ≥ (max{κ^{ω}, |trcl(ℙ)|})^{+}. So κ^{ω}, ℙ ∈ H_{θ}.

We will construct by induction a sequence ⟨M_{α}: α < λ⟩, such that:

1. ∀α < λ ( M_{α} ≺ H_{θ} ∧ |M_{α}| = κ^{ω} ) and

2. ∀α < β < λ ( {M_{α}} ∪ [M_{α}]^{ω} ⊆ M_{β} ).

Take M_{0} ≺ H_{θ} such that |M_{0}| = κ^{ω} and ℙ ∈ M_{0}. Suppose the sequence built up to β < λ. Since each M_{α} ∈ H_{θ}, there exists M_{β} ≺ H_{θ} such that |M_{β}| = κ^{ω} and {M_{α}: α < β} ∪ [⋃_{α<β} M_{α}]^{ω} ⊆ M_{β}.

Since λ > ω and the sequence is increasing, M = ⋃_{α<λ} M_{α} is an ω-covering elementary submodel.

For each α < λ, let τ_{α} = {⟨σ, 1⟩: σ ∈ M_{α} is a ℙ-name }. Whenever α < β, by elementarity, τ_{α} ∈ M_{β}, since {M_{α}, ℙ} ∈ M_{β}.

Let G be a ℙ-generic. We have M[G] = ⋃_{α<λ} M_{α}[G] and

1. ∀α < λ ( M_{α}[G] ≺ H_{θ}^{V [G]} ) and

2. ∀α < β < λ ( M_{α}[G] = (τ_{α})_{G} ∈ M_{β}[G] ).

Now, by Proposition 3.7, we get (cf(κ))^{V [G]} > ω.

The following example shows us that preservation of ω-covering elementary submodels of size ℵ_{1} does not imply preservation for models of bigger size:

Example 3.12 Let ℕ be the Namba forcing (see e.g. [Nam71, Zap98]). It is known that ℕ preserves ω_{1} but collapses the cofinality of ω_{2} to ω. Then ℕ is an example of partial order such that preserves ω-covering for elementary submodels of cardinality ℵ_{1} but not for those of cardinality ℵ_{2}.

Note that we could prove the existence of ω-covering elementary submodels of cardinality less than ℵ_{ω}. Since we do not have ω-covering elementary submodels of cardinality ℵ_{ω}, it is natural to ask: “What about of cardinality ℵ_{ω+1}?” The answer is “It depends on which additional axioms we assume together with ZFC!”

For this discussion we will need the following definitions, which can be found for instance in [AM05]:

Definition 3.13 Let X be a set and μ ≤ |X| be a cardinal. We say that C ⊆ [X]^{μ} is cofinal in [X]^{μ} if, for every x ∈ [X]^{μ}, there is y ∈ C, such that x ⊆ y (i.e. if C is cofinal with respect to ⊆). For cardinals μ ≤ κ, we define cf([κ]^{μ}, ⊆) as the least cardinality of a cofinal family in [κ]^{μ}.

Definition 3.14 For an infinite cardinal κ, we define Cov_{ω}(κ) = cf([κ]^{ω}, ⊆).

In [AM05], we can also find the following result:

{Lemma 3.15 \atop 1.} If ω ≤ μ ≤ κ are cardinals, then

|[κ]^{μ}| = 2^{μ} · cf([κ]^{μ}).

2. The function κ ↦ Cov_{ω}(κ) is increasing.

3. If κ is uncountable, Cov_{ω}(κ) ≥ κ.

4. If α is an ordinal Cov_{ω}(ℵ_{α+1}) = Cov_{ω}(ℵ_{α}) · ℵ_{α+1}.

Corollary 3.16 Cov_{ω}(ℵ_{0}) = 1, and, for any natural n ≥ 1, Cov_{ω}(ℵ_{n}) = ℵ_{n}.

Now, M is ω-covering if, and only if, M ∩ [M]^{ω} is cofinal in [M]^{ω}. Thus, if M is an ω-covering elementary submodel of cardinality κ, we get Cov_{ω}(κ) = κ. We have that this condition is also sufficient:

Theorem 3.17 Let κ be an infinite cardinal. The following are equivalent:

1. There is an ω-covering elementary submodel of cardinality κ, and

2. Cov_{ω}(κ) = κ.

Proof. Suppose Cov_{ω}(κ) = κ. Then κ is uncountable. Let X ⊆ H_{θ} be a set of cardinality κ. We will construct an increasing sequence ⟨M_{α}: α < ω_{1}⟩ such that:

1. ∀α ( X ⊆ M_{α} ≺ H_{θ}) ∧ |M_{α}| = κ ) and

2. ∀α∀Y ∈ [M_{α}]^{ω} ∃Z ∈ M_{α+1} ( |Z| = ℵ_{0} ∧ Y ⊆ Z ).

Let M_{0} be any elementary submodel of cardinality κ such that X ⊆ M_{0}. For any limit ordinal β < ω_{1}, let M_{β} = ⋃_{α<β} M_{α}. For the successor case, fix a cofinal C ⊆ [M_{α}]^{ω} of cardinality κ. Since C ⊆ H_{θ}, we can find M_{α+1} ≺ H_{θ} of cardinality κ such that M_{α} ∪ C ⊆ M_{α+1}.

Then M = ⋃_{α<ω_{1}} M_{α} is an ω-covering elementary submodel of cardinality κ.

We conclude:

Corollary 3.18 There is an ω-covering elementary submodel of cardinality ℵ_{ω+1} if, and only if, Cov_{ω}(ℵ_{ω+1}) = ℵ_{ω+1}.

Since there are no ω-covering elementary submodels of cardinality ℵ_{ω}, we get:

Proposition 3.19 Cov_{ω}(ℵ_{ω}) = Cov_{ω}(ℵ_{ω+1}). More generally, if α is a limit ordinal of countable cofinality, Cov_{ω}(ℵ_{α}) = Cov_{ω}(ℵ_{α+1}).

Proof. Let α be an ordinal of countable cofinality. By Corollary 3.8 and last theorem, Cov_{ω}(ℵ_{α}) ≥ ℵ_{α+1}. Now, by Lemma 3.15, Cov_{ω}(ℵ_{α+1}) = Cov_{ω}(ℵ_{α}) · ℵ_{α+1} = Cov_{ω}(ℵ_{α}).

It is interesting then to discuss which cardinals can be the cardinality of some ω-covering elementary submodel. We have the following results:

Lemma 3.20 If κ^{ω} = κ, then there are ω-covering elementary submodels of cardinality κ.

Proof. By Lemma 3.15, we get κ ≤ Cov_{ω}(κ) ≤ |[κ]^{ω}| = κ^{ω} = κ.

Proposition 3.21 (SCH) If κ is a cardinal such that cf(κ) > ω and κ ≥ 2^{ω}, then there are ω-covering elementary submodels of cardinality κ.

Proof. κ will satisfy κ^{ω} = κ (see e.g. [Jec03]).

Proposition 3.22 (GCH) If κ is a cardinal with uncountable cofinality, there are ω-covering elementary submodels of cardinality κ.

On extensions of elementary submodels by forcing 11

Proof. κ will satisfy κ^{ω} = κ.

Using pcf theory (created by Saharon Shelah; see e.g. [Jec95, JS96, KS98, She00, AM05]) one can prove:

Theorem 3.23 Cov_{ω}(ℵ_{ω}) < ℵ_{(2^{ω})^{+}}.

Theorem 3.24 ℵ_{ω}^{ω} < ℵ_{(2^{ω})^{+}}.

Shelah also proved:

Theorem 3.25 (Shelah) Cov_{ω}(ℵ_{ω}) < ℵ_{ω_{4}}.

We will get the consistency of “Cov_{ω}(ℵ_{ω+1}) = ℵ_{ω+1}” using the following proposition:

Proposition 3.26 If ℵ_{ω} is a strong limit, then Cov_{ω}(ℵ_{ω}) = 2^{ℵ_{ω}}. Proof. Since ℵ_{ω} is strong limit, we have 2^{ℵ_{ω}} = ℵ_{ω}^{ω} (see e.g. [Jec03]) and 2^{ω} < ℵ_{ω}. By Lemma 3.15, Cov_{ω}(ℵ_{ω}) = ℵ_{ω}^{ω} = 2^{ℵ_{ω}}.

We conclude:

Corollary 3.27 If ℵ_{ω} is a strong limit, “Cov_{ω}(ℵ_{ω+1}) = ℵ_{ω+1}” is equivalent to 2^{ℵ_{ω}} = ℵ_{ω+1}.

The following results give us the independence of “Cov_{ω}(ℵ_{ω+1}) = ℵ_{ω+1},” assuming the existence of large cardinals.

Theorem 3.28 ([Mag77a, Mag77b]) If there is a supercompact cardinal then there is a generic extension in which ℵ_{ω} is strong limit and 2^{ℵ_{ω}} = ℵ_{ω+2}.

Theorem 3.29 ([Git89]) If there exists a measurable cardinal κ of Mitchell order κ^{++}, then there exists a generic extension in which GCH holds below ℵ_{ω} and 2^{ℵ_{ω}} = ℵ_{ω+2}.

Therefore, we can get “Cov_{ω}(ℵ_{ω+1}) > ℵ_{ω+1}” assuming the existence of large cardinals. Moreover it does not depend on the value of the continuum:

Theorem 3.30 It is consistent that 2^{ω} = ℵ_{ω+2} and “Cov_{ω}(ℵ_{ω+1}) = ℵ_{ω+1}” holds. Proof. Let us start with a model for GCH and add ℵ_{ω+2} Cohen reals. Since a partial order with c.c.c. preserves cofinalities and cardinals and it also preserves ωcovering, we have what we want.

M.Kojman, in [Koj01], says that we cannot drop the assumption of the existence of large cardinals to violate “Cov_{ω}(ℵ_{ω+1}) = ℵ_{ω+1}.”

4 Getting M[G] ω-covering

We now look at the other direction of the “preservation problem.” If M[G] is ωcovering, what can be said about M? Was M already ω-covering? Can any partial order make M[G] ω-covering, even if M was not?

We can then prove:

Proposition 4.1 Let ℙ ∈ M be a partial order that satisfies the countable covering property, where M is an elementary submodel. Suppose that p ⊩ “𝕄 is ω-covering” for some (ℙ, M)- generic condition p. Then M is ω-covering.

Proof. Let X ⊆ M be a countable set. Then, there exist q ≤ p and τ ∈ M, ℙ-name, such that

q ⊩ “X̌ ⊆ τ ∧ τ is countable”,

because p ⊩ “𝕄 is ω-covering”.

Since ℙ satisfies the countable covering property,

D = {r ∈ ℙ: (∃Y, countable, (r ⊩ “τ ∩ V ⊆ Y̌ ”)) ∨ r ⊩ “τ ∩ V is not countable”}

is a dense subset of ℙ. Given that {τ, ℙ} ∈ M, by elementarity, D ∈ M. Then, D ∩M is predense below p, since p is (ℙ, M)-generic.

Let r ∈ D ∩ M be compatible to q and take s as a common extension of r and q. Given that q ⊩ “τ is countable”, fix, by elementarity, Y ∈ M, countable, such that r ⊩ “τ ∩ V ⊆ Y̌ ”. Then

s ⊩ “X̌ ⊆ τ ∩ V ⊆ Y̌ ”

and we are done.

Corollary 4.2 Let M be an elementary submodel and ℙ ∈ M be a partial order with c.c.c. Whenever G is ℙ-generic, M[G] is ω-covering if, and only if, M is ω-covering.

Proof. Since ℙ has c.c.c., all its conditions are (ℙ, M)-generic (Proposition 1.4).

Very similarly we can prove:

Proposition 4.3 Let ℙ ∈ M be an ω_{1}-Baire partial order, where M is an elementary submodel. Suppose that p ⊩ “𝕄 is countably closed” for some (ℙ, M)-generic condition p. Then M is countably closed.

Proof. Let X ⊆ M be a countable set. Then, there exist q ≤ p and τ ∈ M, ℙ-name, such that q ⊩ “X̌ = τ”. Since ℙ is ω_{1}-Baire,

D = {r ∈ ℙ: (∃Y, countable, (r ⊩ “τ ∩ V = Y̌ ”)) ∨ r ⊩ “τ ∩ V is not countable”}

is a dense subset of ℙ. Given that {τ, ℙ} ∈ M, by elementarity, D ∈ M. Then, D ∩M is predense below p, since p is (ℙ, M)-generic.

Let r ∈ D ∩ M be compatible to q and take s as a common extension of r and q. Given that q ⊩ “τ is countable”, fix, by elementarity, Y ∈ M, countable, such that r ⊩ “τ ∩ V = Y̌ ”. Then s ⊩ “X̌ = τ ∩ V = Y̌ ” and we are done.

If we drop the requirement that p be (ℙ, M)-generic (and consider elementary submodels of H_{|ℙ|^{+}}) then the previous proposition is false, as shown by the following example.

Example 4.4 Let θ = ℵ_{5} and suppose that 2^{ℵ_{n}} = ℵ_{n+1} for all n ∈ {0, 1, 2, 3}. Since we have CH, all ω-covering elementary submodels will be countably closed (Theorem 2.11).

We will construct L ≺ H_{θ}, countably closed, such that

ω_{1} ⊆ L, cf(L ∩ ω_{3}) = cf(L ∩ ω_{4}) = ω_{1}, and cf(L ∩ ω_{2}) = ω.

Our partial order ℙ will be F n(ω_{3}, ω_{3}, ℵ_{3}). It is known that this partial order preserves all cofinalities and all cardinalities. It is also straightforward to check that ℙ preserves “∀n < 4 ( 2^{ℵ_{n}} = ℵ_{n+1} ).”

After we construct L we will construct an f ∈ ℙ such that

cf(L[G] ∩ ω_{2}) = cf(L[G] ∩ ω_{3}) = cf(L[G] ∩ ω_{4}) = ω_{1},

whenever G is ℙ-generic and f ∈ G.

To complete the example, we will let K = L∩H_{ℵ_{4}}. Then we will have that K ≺ H_{ℵ_{4}} and that K is not ω-covering, as cf(K ∩ ω_{2}) = ω. If G is a ℙ-generic filter, then

K[G] = (L ∩ H_{ℵ_{4}})[G] = L[G] ∩ H_{ℵ_{4}}^{V[G]}.

Then K[G] will be countably closed, even though K is not.

Let us get Q ≺ H_{θ} that will “control” cf(L ∩ ω_{4}). By induction, we construct a sequence of ω-covering elementary submodels ⟨Q_{α}: α < ω_{1}⟩ such that

1. ∀α < ω_{1} ( ω_{3} ⊂ Q_{α} ≺ H_{θ} ∧ |Q_{α}| = ℵ_{3} ) and

2. ∀α < β < ω_{1} ( Q_{α} ⊆ Q_{β} ∧ Q_{α} ∩ ω_{4} ∈ Q_{β} ).

Note that item 1 will always provide Q_{α} ∩ ω_{4} ∈ ω_{4}. Take Q_{0} ≺ H_{θ}, ω-covering, such that ω_{3} ⊆ Q_{0} and |Q_{0}| = ℵ_{3}. For α < ω_{1}, let

Q_{α+1} = {f(Q_{α} ∩ ω_{4}): f ∈ Q_{α} is function of domain ω_{4}}.

We get that Q_{α+1} ≺ H_{θ} and Q_{α} ∩ ω_{4} ∈ Q_{α+1} (see e.g. [Lar04]). Moreover Q_{α+1} is ω-covering of size ℵ_{3}. For β < ω_{1}, a limit ordinal, take Q_{β} ≺ H_{θ}, ω-covering of size ℵ_{3}, such that ⋃_{α<β} Q_{α} ⊆ Q_{β} (Lemma 3.5). Let Q be ⋃_{α<ω_{1}} Q_{α}. Note that ⟨Q_{α} ∩ ω_{4}: α < ω_{1}⟩ is a strictly increasing cofinal sequence in the limit ordinal Q ∩ ω_{4}.

For “controlling” cf(L ∩ ω_{3}), similarly to Q, we construct a sequence of ω-covering elementary submodels ⟨N_{α}: α < ω_{1}⟩ such that

1. ∀α < ω_{1} ( ω_{2} ∪ {Q_{γ} ∩ ω_{4}: γ < ω_{1}} ⊂ N_{α} ≺ Q ∧ |N_{α}| = ℵ_{2} ) and

2. ∀α < β < ω_{1} ( N_{α} ⊆ N_{β} ∧ N_{α} ∩ ω_{3} ∈ N_{β} ).

Let N = ⋃_{α<ω_{1}} N_{α}. Thus N ≺ Q is ω-covering of size ℵ_{2}, and ⟨N_{α} ∩ ω_{3}: α < ω_{1}⟩ is a strictly increasing cofinal sequence in the limit ordinal N ∩ ω_{3}.

Finally we get L!

Let L_{0} ≺ N be ω-covering of size ℵ_{1} such that

{Q_{α} ∩ ω_{4}: α < ω_{1}} ∪ {N_{α} ∩ ω_{3}: α < ω_{1}} ⊆ L_{0}.

For each natural number n, let

L_{n+1} = {f(L_{n} ∩ ω_{2}): f ∈ L_{n} is function of domain ω_{2}}.

We define

L = ⋃_{n<ω} L_{n}.

Then L ≺ N has size ℵ_{1} and ⟨L_{n} ∩ ω_{2}: n < ω⟩ is strictly increasing and cofinal in L ∩ ω_{2}. By construction, ⟨Q_{α} ∩ ω_{4}: α < ω_{1}⟩ is cofinal in L ∩ ω_{4} and ⟨N_{α} ∩ ω_{3}: α < ω_{1}⟩ is cofinal in L ∩ ω_{3}.

For the construction of the condition f, we will need the help of another elementary submodel. Let M ≺ N be ω-covering, such that |M| = ℵ_{1} and L ⊆ M. Since |M| = ℵ_{1}, we can fix a strictly increasing sequence ⟨y_{α}: α < ω_{1}⟩ which is cofinal in M ∩ ω_{2}. Let ⟨D_{α}: α < ω_{1}⟩ be an enumeration of all the dense subsets of ℙ which are elements of M. Using induction, we can build two sequences, ⟨f_{α}: α < ω_{1}⟩ and ⟨x_{α}: α < ω_{1}⟩, satisfying:

1. ∀α < ω_{1} ( f_{α} ∈ D_{α} ∩ M ∧ x_{α} ∈ (dom f_{α}) ∩ L ∧ f_{α}(x_{α}) = y_{α} ) and

2. ∀α < β ( dom f_{α} ⊂ x_{β} ∧ f_{α} ⊂ f_{β} ).

Recall that M is countably closed and L ∩ ω_{3} is cofinal in M ∩ ω_{3}. Let x_{0} be any element of L_{0} ∩ ω_{3}. There exists f_{0} ∈ D_{0} ∩ M such that f_{0} ≤ {⟨x_{0}, y_{0}⟩}, as D_{0} ∈ M is dense, and by elementarity. If we suppose the sequences have been built up to β < ω_{1}, we have that (⋃_{α<β} f_{α}) ∈ M. Let x_{β} be any element of L ∩ ω_{3} greater than sup{sup dom f_{α}: α < β}. Since

( ⋃_{α<β} f_{α}) ∪ {⟨x_{β}, y_{β}⟩} ∈ M ∩ ℙ

and D_{β} ∈ M is dense, there is f_{β} ∈ D_{β} ∩ M such that f_{β} ≤ (⋃_{α<β} f_{α}) ∪ {⟨x_{β}, y_{β}⟩}.

Let f = ⋃_{α<ω_{1}} f_{α}. Then f is (ℙ, M)-generic. Fix G, a ℙ-generic such that f ∈ G. Then F = ⋃ G will be a function from ω_{3} to ω_{3} such that f ⊆ F and F ∈ L[G]. Moreover, for all α < ω_{1}, F (x_{α}) = y_{α} ∈ L[G]. So ⟨y_{α}: α < ω_{1}⟩ will be a strictly increasing cofinal sequence in L[G]∩ω_{2}. Since f is (ℙ, M)-generic, we get that M[G]∩ ON = M ∩ ON. So ⟨Q_{α} ∩ ω_{4}: α < ω_{1}⟩ is cofinal in L[G] ∩ ω_{4} and ⟨N_{α} ∩ ω_{3}: α < ω_{1}⟩ is cofinal in L[G] ∩ ω_{3}. Then

cf(L[G] ∩ ω_{i}) = ω_{1}, for all i ∈ {1, 2, 3, 4}.

Note that ω_{1} ⊆ L[G], thus L[G] is ω-covering for subsets of ω_{1}. Then L[G] is ωcovering for subsets of ω_{2}, as L[G] ∩ ω_{2} does not have countable cofinality. By similar arguments, we get that L[G] is ω-covering for subsets of ω_{3} and for subsets of ω_{4}. Since |H_{ℵ_{4}}^{V[G]}| = ℵ_{4}, we can prove that K[G] = L[G] ∩ H_{ℵ_{4}}^{V[G]} is ω-covering.

We thus get K, non ω-covering, and f ∈ ℙ, ℙ countably closed, such that K[G] is ω-covering, whenever f ∈ G and G is ℙ-generic.

Acknowledgements

The work in this paper was supported in part by NSF Grant DMS-0401603 (second author) and FAPESP Grant 2005/60183-5 (first and second authors).

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