Splitting stationary sets from weak forms of Choice ^{∗}

Paul Larson and Saharon Shelah

February 16, 2008

Abstract Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below λ of cofinality θ into λ many stationary sets, where θ < λ are regular cardinals. This is a continuation of [5].

In this note we consider the issue of splitting stationary sets in the presence of weak forms of the Axiom of Choice plus the existence of certain types of ladder systems. Our primary interest is the theory ZF + DC plus the assertion that for some large enough cardinal λ, there is a ladder system for the members of λ of countable cofinality, that is, a function that assigns to every such α < λ a cofinal subset of ordertype ω. In this context, we show that for every γ < λ of uncountable cofinality the set of α < γ of countable cofinality can be uniformly split into cf(γ) many stationary sets. It follows from this and the results of [5] that there is no nontrivial elementary embedding from V into V, under the assumption of ZF + DC plus the assertion that the countable subsets of each ordinal can be wellordered. As a counterpoint to some of the results presented here, we give a symmetric forcing extension in which there are regressive functions on stationary sets not constant on stationary sets.

1 AC and DC

Given a nonempty set Z, the statement AC_{Z} says that whenever ⟨X_{a} : a ∈ Z⟩ is a collection of nonempty sets, there is a function f with domain Z such that f(a) ∈ X_{a} for each a ∈ Z. If γ is an ordinal, the statement AC_{<γ} says that AC_{η} holds for all ordinals η < γ.

A tree T is a set of functions such that the domain of each function is an ordinal, and such that, whenever f ∈ T and α ∈ dom(f), f↾α ∈ T. Two elements f, g of a tree T are compatible if f ⊂ g or g ⊂ f. A branch through a tree is a pairwise compatible collection of elements of T. A branch is maximal if it is not properly contained in any other branch.

∗The first author is supported in part by NSF grant DMS-0401603. The research of the second author is supported by the United States-Israel Binational Science Foundation. The research in this paper began during a visit by the first author to Rutgers University in October 2007, supported by NSF grant DMS-0600940. This is the second author’s publication number 925.

Given an ordinal γ, the statement DC_{γ} says that for every tree T ⊂ ^{<γ}X (for some set X) there is a b ⊂ T which is a maximal branch. The statement DC_{<γ} says that DC_{η} holds for all ordinals η < γ. It follows immediately from the definition of DC_{γ} that DC_{γ} implies DC_{η} for all η < γ. We write DC for DC_{ω} and AC for the statement that AC_{Z} holds for all sets Z.

Lemma 1.1 shows that DC_{γ} implies AC_{γ} for all ordinals γ.

Lemma 1.1. Suppose that γ is a limit ordinal such that DC_{γ} holds, and T is a tree such that

• every f ∈ T is a function with domain η, for some η < γ;

• for all limit ordinals η < γ, if f is a function with domain η such that f↾α ∈ T for all α ∈ η, then f ∈ T;

• for every f ∈ T there is a g ∈ T properly containing f.

Then there is a function f with domain γ such that f↾α ∈ T for all α < γ.

Proof. Let b be a maximal branch of T, and let f = ⋃ b. Then f is a function whose domain is an ordinal η ≤ γ. If η < γ, then f ∈ T and f has a proper extension in T, contradicting its supposed maximality.

2 Ladder systems

Notation. Given an ordinal δ, we let cf(δ) denote the cofinality of δ. Given an ordinal α and a set A, we let C_{A}^{α} denote the ordinals below α whose cofinality is in A. Given an ordinal λ and a function f, we let ϕ(λ, f) be the statement that there exists a sequence ⟨c_{δ} : δ ∈ C_{dom(f)}^{λ}⟩ such that each c_{δ} is a cofinal subset of δ of ordertype less than f(cf(δ)).

Note that ϕ(λ, f) implies that f(γ) ≥ γ + 1 for all regular cardinals γ ∈ dom(f).

Notation. We let ψ(λ, θ) be the statement ϕ(λ, {(θ, θ + 1)}). We say that a sequence ⟨c_{δ} : δ ∈ C_{dom(f)}^{λ}⟩ witnesses ϕ(λ, f) if each c_{δ} is a cofinal subset of δ of ordertype less than f(cf(δ)), and similarly for ψ(λ, θ).

The statement ψ(λ, ω) follows from the statement Ax_{λ}^{2} of [5] (in the case ∂ = ω), which says that there exists a well-orderable A ⊂ [λ]^{ℵ_{0}} such that every element of [λ]^{ℵ_{0}} has infinite intersection with a member of A. We will be primarily interested in statements ϕ(λ, f) where f is either the ordinal successor function or the cardinal successor function on some set of regular cardinals. The two following lemmas show that when the domain of f is a single regular cardinal, there is in some sense no statement strictly in between these two.

Lemma 2.1 (ZF). For each ordinal ordinal γ there exists a sequence

⟨e_{δ} : δ < γ⟩

such that each e_{δ} is a cofinal subset of δ of ordertype less than or equal to |γ|. Proof. Let π: |γ| → γ be a bijection. For each δ < γ, let e_{δ} be the set of ordinals of the form π(α), where α < |γ|, π(α) < δ and π(α) > π(β) for all β < α with π(β) < δ. Notation. Given a set x of ordinals, we let o.t.(x) denote the ordertype of x. Given an ordinal η < o.t.(x), we let x(η) be the η-th member of x, i.e., the unique α ∈ x such that o.t.(x ∩ α) = η. Lemma 2.2 (ZF). Let λ be an ordinal, let θ be a regular cardinal, and let η be an ordinal less than θ^{+}. Then ϕ(λ, {(θ, η)}) implies ψ(λ, θ). Proof. Let ⟨c_{δ} : δ ∈ C_{{θ}}^{λ}⟩ witness ϕ(λ, {(θ, η)}), and let ⟨e_{δ} : δ < η⟩ be such that each e_{δ} is a cofinal subset of δ of ordertype less than or equal to θ. For each δ ∈ C_{{θ}}^{λ}, letting α_{δ} be the ordertype of c_{δ}, let d_{δ} = {c_{δ}(β) | β ∈ e_{α_{δ}}}. Then each d_{δ} is a cofinal subset of δ of ordertype θ.

3 Splitting C_{θ}^{λ} from DC_{θ} and AC_{γ}

Notation. Given ordinals α, β, η and a sequence of sets of ordinals C̄ = ⟨c_{δ} : δ ∈ S⟩ (for some set S), we let S_{α,β}^{η}(C̄) be the set of δ ∈ S such that o.t.(c_{δ}) > η and c_{δ}(η) ∈ [α, β).

We are primarily interested in the following theorem in the case where θ and γ are both ω, in which case ψ(λ, ω) implies the existence of a sequence C̄ satisfying the stated hypotheses. Theorem 3.1 (ZF). Suppose that the following hold.

• θ ≥ ℵ_{0} is a regular cardinal such that DC_{θ} holds;

• γ ≥ θ is an ordinal such that AC_{γ} holds;

• λ is an ordinal of cofinality greater than γ;

• E is a club subset of λ;

• C̄ = ⟨c_{δ} : δ ∈ C_{{θ}}^{λ} ∩ E⟩ is a sequence such that each c_{δ} is a cofinal subset

of δ of ordertype less than or equal to γ. Then

1. there exists an η^{∗} < γ such that for each α < λ there exists a β ∈ (α, λ) such that S_{α,β}^{η^{∗}}(C̄) is a stationary subset of λ; 2. there exist functions g: C_{(γ,λ)}^{λ} → γ, h: C_{(γ,λ)}^{λ} → λ and a collection of

ordinals ⟨α_{β}^{ξ} : ξ ∈ C_{(γ,λ)}^{λ}, β < h(ξ)⟩ such that

• for each ξ ∈ C_{(γ,λ)}^{λ}, h(ξ) < cf(ξ)^{+};

• for each ξ ∈ C_{(γ,λ)}^{λ}, ⟨α_{β}^{ξ} : β < h(ξ)⟩ is a continuous increasing sequence cofinal in ξ,

• for each ξ ∈ C_{(γ,λ)}^{λ} and each β < h(ξ), S_{α_{β}^{ξ},α_{β+1}^{ξ}}^{g(ξ)}(C̄↾ξ) is stationary.

Proof. We prove the first part first. Supposing that there is no such η^{∗}, for each η < γ let α_{η}^{∗} < λ be the least α < λ such that S_{α,β}^{η}(C̄) is nonstationary for all β ∈ (α, λ). Using the fact that cf(λ) > γ, let α^{∗} be the least element of C_{{θ}}^{λ} ∩E greater than or equal to the supremum of {α_{η}^{∗} : η < γ}. Now, applying DC_{θ} and AC_{γ}, we choose a continuous increasing sequence of ordinals ⟨α_{ξ} : ξ < θ⟩ and sets D_{ξ,η} (ξ < θ, η < γ) by recursion on ξ < θ such that

1. α_{0} = α^{∗}

2. each D_{ξ,η} is a club subset of E disjoint from S_{α^{∗},α_{ξ}}^{η}(C̄)

3. if ξ < θ a limit ordinal then α_{ξ} = ⋃{α_{ζ} : ζ < ξ}

4. if ξ = ζ + 1, then α_{ξ} = min(⋂_{ρ≤ζ,η<γ} D_{ρ,η} \ (α_{ζ} + 1))

Let α_{θ} = ⋃{α_{ξ} : ξ < θ}. Then α_{θ} < λ as cf(λ) > θ, so α_{θ} ∈ C_{{θ}}^{λ} ∩ E. For some η < γ, c_{α_{θ}}(η) > α^{∗}, hence for some ξ < θ, c_{α_{θ}}(η) ∈ [α_{∗}, α_{ξ}). Then α_{θ} ∈ S_{α_{0},α_{ξ}}^{η}(C̄), contradicting the assumption that α_{θ} ∈ D_{ξ,η}.

To prove the second part of the lemma, fix ξ ∈ C_{(γ,λ)}^{λ}. Applying the first part of the lemma with ξ as λ, let g(ξ) be the least η ∈ γ such that for each α < ξ there exists a β ∈ (α, ξ) such that S_{α,β}^{η}(C̄↾ξ) is a stationary subset of ξ. Then by recursion on β < ξ we can choose an increasing continuous sequence of ordinals α_{β}^{ξ} < λ (β < ξ) such that α_{0}^{ξ} = 0, α_{β}^{ξ} = ∪{α_{ζ}^{ξ} : ζ < β} for limit β, and, if β = ζ + 1, then, if α_{ζ}^{ξ} = ξ then α_{β}^{ξ} = ξ, otherwise α_{β}^{ξ} is the minimal

ordinal δ ∈ (α_{ζ}^{ξ}, ξ) such that S_{α_{ζ}^{ξ},δ}^{g(ξ)}(C̄↾ξ) is stationary. Let h(ξ) be the least β

such that α_{β}^{ξ} = ξ if some such β exists, and let it be ξ otherwise. Since there is a

club subset of ξ of cardinality cf(ξ), and the sets S_{α_{β}^{ξ},α_{β+1}^{ξ}}^{g(ξ)}(C̄↾ξ) (β < h(ξ)) are

disjoint stationary subsets of ξ, h(ξ) < cf(ξ)^{+}. This completes the definitions of g, h and ⟨α_{β}^{ξ} : ξ ∈ C_{(γ,λ)}^{λ}, β < h(ξ)⟩.

Corollary 3.2. Suppose that the following hold.

• θ ≥ ℵ_{0} is a regular cardinal such that DC_{θ} holds;

• λ is an ordinal of cofinality greater than θ;

• A is the set of regular cardinals in the interval [θ, λ).

Then ψ(λ, θ) implies ϕ(λ, f), where f is the cardinal successor function on A.

The following is a consequence of the results of [5], Woodin’s proof of Kunen’s Theorem (see [2]) and the arguments in this section.

Corollary 3.3 (ZF + DC). Assume that for every ordinal λ there exists a wellorderable set A ⊂ [λ]^{ℵ_{0}} such that every element of [λ]^{ℵ_{0}} has infinite intersection with a member of A. Then there is no nontrivial elementary embedding from V into V.

Proof. Suppose towards a contradiction that j: V → V is an elementary embedding. Let κ_{0} be the critical point of j, and for each nonzero n < ω, let κ_{n+1} = j(κ_{n}). Let κ_{ω} = ∪{κ_{n} : n < ω}. Then j(κ_{ω}) = κ_{ω} and j(κ_{ω}^{+}) = κ_{ω}^{+}. For no α < κ_{0} is there is a surjection from V_{α} onto κ_{0} (to see this, consider j(π), where π is such a surjection, in light of the fact that j↾V_{κ_{0}} is the identity function). By elementarity, then, the same is true for each κ_{n}, and so the same is true for κ_{ω}. Then by the results of [5] (specifically, Lemma 2.13), κ_{ω}^{+} is regular.

Let C̄ = ⟨c_{δ} : δ ∈ S_{ω}^{κ_{ω}^{+}}⟩ witness ψ(κ_{ω}^{+}, ω). Applying Theorem 3.1, let n_{∗} ∈ ω and ᾱ = ⟨α_{ξ} : ξ < κ_{ω}^{+}⟩ be such that ᾱ is a continuous increasing sequence of elements of κ_{ω}^{+} and such that S_{α_{ξ},α_{ξ+1}}^{n_{∗}}(C̄) is a stationary subset of κ_{ω}^{+} for each ξ < κ_{ω}^{+}.

Let F be the set of limit ordinals δ < κ_{ω}^{+} such that j(α) < δ for every α < δ. Then F is a club. Let E be the set of members of F of cofinality less than κ_{0}. Then j↾E is the identity function, and no stationary set of ordinals of countable cofinality is disjoint from E.

Let ⟨S′_{ξ} : ξ < κ_{ω}^{+}⟩ = j(⟨S_{α_{ξ},α_{ξ+1}}^{n_{∗}} : ξ < κ_{ω}^{+}⟩). As j is an elementary embedding, V ⊧ “S′_{κ_{0}} is a stationary subset of λ disjoint from S′_{ξ} for ξ ∈ κ_{ω}^{+} \ {κ_{0}}.” Hence, S′_{κ_{0}} is disjoint from S′_{j(ξ)}, for all ξ < κ_{ω}^{+}. But

⋃_{ξ<κ_{ω}^{+}} S′_{j(ξ)} ⊃ ⋃_{ξ<κ_{ω}^{+}}(S′_{j(ξ)} ∩ E) = ⋃_{ξ<κ_{ω}^{+}}(S_{α_{ξ},α_{ξ+1}}^{n_{∗}} ∩ E) = E.

4 Club guessing

In this section we show that the standard club-guessing arguments go through under weak forms of Choice plus the existence of ladder systems. Theorem 4.1 uses forms of DC, and Theorem 4.3 uses AC.

Theorem 4.1 (ZF). Let θ < λ be regular cardinals, with θ^{+} < λ, and suppose that DC_{θ^{+}} holds. Suppose that ⟨c_{δ} : δ ∈ C_{{θ}}^{λ}⟩ is a sequence such that each c_{δ} is a closed cofinal subset of δ of ordertype less than θ^{+}. Then the following hold.

• There exists a sequence ⟨d_{δ} : δ ∈ C_{{ℵ_{0}}}^{λ}⟩ such that each d_{δ} is a cofinal subset of δ, and such that for every club subset D ⊂ λ there is a δ ∈ C_{{ℵ_{0}}}^{λ} with d_{δ} ⊂ D.

• If θ is uncountable, then there exists a sequence ⟨d_{δ} : δ ∈ C_{{θ}}^{λ}⟩ such that each d_{δ} is a closed cofinal subset of c_{δ}, and such that for every club subset D ⊂ λ there is a δ ∈ C_{{θ}}^{λ} with d_{δ} ⊂ D.

Proof. We argue as in [4], Chapter III.

For the first part, for any two sets A, B, let gl(A, B) denote the set

{sup(α ∩ B) | α ∈ A \ (min(B) + 1)}.

Note that if A and B ∩ γ are club subsets of an ordinal γ, then gl(A, B) is a club subset of B ∩ γ as well.

Supposing that the first conclusion of the theorem is false, choose for each ζ ≤ θ^{+} a club subset D_{ζ} ⊂ λ such that the following conditions are satisfied.

• D_{0} does not contain c_{δ} for any δ ∈ C_{{θ}}^{λ};

• for each ζ < θ^{+}, D_{ζ+1} is contained in the limit points of D_{ζ}, and D_{ζ+1} does not contain gl(c_{δ}, D_{ζ}) for any δ ∈ C_{{θ}}^{λ} which is a limit point of D_{ζ}.

• for each limit ordinal ζ ≤ θ^{+}, D_{ζ} = ⋂_{ξ<ζ} D_{ξ}.

Now fix a δ ∈ C_{{θ}}^{λ} which is a limit point of D_{θ^{+}}. For each α ∈ c_{δ}, either there is a ζ < θ^{+} such that α ≤ min(D_{ζ}), or ⟨sup(α ∩ D_{ζ}) : ζ < θ^{+}⟩ is a nonincreasing sequence which reaches an eventually constant value. Since |c_{δ}| < θ^{+}, there is a ζ < θ^{+} such that for each α ∈ c_{δ}, α > min(D_{ζ}) implies α > min(D_{ζ+1}), and, if α > min(D_{ζ}), then sup(α ∩ D_{ζ}) = sup(α ∩ D_{ζ+1}). Then gl(c_{δ}, D_{ζ}) = gl(c_{δ}, D_{ζ+1}). However, gl(c_{δ}, D_{ζ+1}) ⊂ D_{ζ+1} and D_{ζ+1} was chosen not to contain gl(c_{δ}, D_{ζ}), giving a contradiction.

For the second part, note that we can just take the intersection of c_{δ} and d_{δ} for each δ ∈ C_{{θ}}^{λ}, where d_{δ} is given by the first part.

4.2 Question. Does DC_{θ} suffice for Theorem 4.1?

Theorem 4.3 (ZF). Suppose that

• θ < λ are regular uncountable cardinals;

• there is no surjection from P(θ) to λ;

• AC_{X} holds, where X is the union of θ^{+} and the set of club subsets of θ;

• ⟨c_{δ} : δ ∈ C_{{θ}}^{λ}⟩ is a sequence such that each c_{δ} is a closed cofinal subset of δ of ordertype less than θ^{+}.

Then there exists a sequence ⟨e_{δ} : δ ∈ C_{{θ}}^{λ}⟩ such that each e_{δ} is a closed cofinal subset of c_{δ} of ordertype θ, and such that for every club subset D ⊂ λ there is a δ ∈ C_{{θ}}^{λ} with e_{δ} ⊂ D.

Proof. Applying AC_{X}, let D̄ = ⟨d_{δ} : δ ∈ C_{{θ}}^{θ^{+}}⟩ be such that each d_{δ} is a club subset of δ of ordertype θ. For each δ ∈ C_{{θ}}^{λ}, let c′_{δ} = c_{δ} ∩ d_{δ}. Then each c′_{δ} is a closed, cofinal subset of δ of ordertype θ.

For each δ ∈ C_{{θ}}^{λ} and for each club C ⊂ θ, let c(C)_{δ} = {c′_{δ}(β) | β ∈ C}. Supposing that the conclusion fails, choose ⟨E_{C} : C ⊂ θ club⟩ such that each E_{C} is a club subset of λ not containing c(C)_{δ} for any δ ∈ C_{{θ}}^{λ}. As there is no surjection from P(θ) to λ,

E = ⋂{E_{C} : C ⊂ θ club}

is a club subset of λ. Let δ be any limit member of E in C_{{θ}}^{λ}, and let

C = {α < γ | c′_{δ}(α) ∈ E}.

Then c(C)_{δ} = c′_{δ} ∩ E ⊂ E_{C}, contradicting the choice of E_{C}.

5 Splitting at higher cofinalities

In this section we consider the problem of using a ladder system to split C_{{θ}}^{λ} into stationary sets without the help of AC and DC. So the difference is that we try to split at cofinality θ without DC_{θ}.

Theorem 5.1 (ZF). Suppose that the following hold.

• θ < λ are regular uncountable cardinals;

• γ ∈ [θ, λ) is an ordinal;

• C̄ = ⟨c_{δ} : δ ∈ C_{{θ}}^{λ}⟩ is a sequence such that each c_{δ} is a cofinal subset of δ of ordertype less than or equal to γ.

Then either

1. there exist η < γ and a continuous increasing sequence ⟨α_{ξ} : ξ < λ⟩ such that each α_{ξ} ∈ λ and each S_{α_{ξ},α_{ξ+1}}^{η}(C̄) is stationary, or

2. the following two statements hold:

(a) for some club E ⊂ λ there exists a regressive function F on E ∩ C_{{θ}}^{λ} such that F^{−1}{β} not stationary for any β < λ, and

(b) if AC_{γ} holds, then for some α_{∗} < λ there is a regressive function G on C_{{θ}}^{λ} \ (α_{∗} + 1) such that for each β < λ the set of γ ∈ C_{{θ}}^{λ} such that G(γ) < β is not stationary.

Proof. Suppose first that there exists a η < γ such that for each α < λ there exists a β ∈ (α, λ) such that S_{α,β}^{η}(C̄) is a stationary subset of λ. Then we can choose α_{ξ} < λ (ξ < λ) by induction on ξ, increasing continuously with ξ, such that α_{0} = 0 and

α_{ξ+1} = min{α : α_{ξ} < α < λ ∧ S_{α_{ξ},α}^{η}(C̄) is stationary}.

Then the first conclusion of the lemma holds.

Suppose instead that there is no such η. Then for each η < γ, let α_{η}^{∗} < λ be minimal such that for all β ∈ (α_{η}^{∗}, λ), S_{α_{η}^{∗},β}^{η}(C̄) not a stationary subset of λ. Let α_{∗} = sup{α_{η}^{∗} : η < γ}. Then α_{∗} < λ, as λ = cf(λ) > γ.

Define F: C_{{θ}}^{λ} \ (α_{∗} + 1) → λ × γ by letting F (δ) = (α, η) if α is the least element of c_{δ} greater than α_{∗} and α = c_{δ}(η). Then for no (α, η) ∈ λ × γ is F^{−1}{(α, η)} stationary.

Let H: λ × γ → λ be the function H(α, η) = γ · α + η, and let E be the set of α ∈ (α_{∗}, λ) such that H(β, η) < α for all β < α and η < γ. Then E is a club set. Furthermore, the function H ∘ F is regressive on E ∩ C_{{θ}}^{λ} and not constant on a stationary set, as desired.

Finally, suppose that AC_{γ} holds. For each β ∈ (α_{∗}, λ) and each η < γ, S_{α_{∗},β}^{η}(C̄) is nonstationary. It follows (from AC_{γ}) that for each β ∈ (α_{0}, λ), S_{β} = ⋃_{η<γ} S_{α_{∗},β}^{η}(C̄) is nonstationary. Now define G: C_{{θ}}^{λ} \ (α_{∗} + 1) → λ by letting G(δ) be the least element of c_{δ} greater than α_{∗}. Then for every β ∈ λ, the set of δ ∈ C_{θ}^{λ} \ (α_{∗} + 1) with G(δ) < β is nonstationary.

6 A model of ZF and a regressive function

In this section we give a proof of the following theorem, which is complementary to Theorem 5.1.

Theorem 6.1 (ZFC). Let θ < λ be regular cardinals. There is a partial order P such that in the P -extension of V there is an inner model M with the following properties.

• M and V have the same ordinals of cofinality θ;

• λ is a regular cardinal in M;

• M satisfies ZF + DC_{<θ} + ϕ(λ, f), where f is the ordinal successor function on the regular cardinals below θ;

• there exists in M a regressive function on (C_{[θ,λ)}^{λ})^{M} which is not constant on a stationary set.

The strategy for the proof is a direct modification of Cohen’s original proof of the independence of AC (see [1]).

Assume that ZFC holds and that θ < λ are uncountable regular cardinals. Given a set X ⊂ λ × λ, let P_{X} be the partial order whose conditions consist of pairs (f, d) such that • f is a regressive function on C_{[θ,λ)}^{λ} whose domain is α ∩ C_{[θ,λ)}^{λ} for some

successor ordinal α < λ;

• d is a partial function whose domain is a subset of X of cardinality less than λ such that for each (α, β) in the domain of d, d(α, β) is a closed, bounded subset of max(dom(f)) + 1 disjoint from f^{−1}{α}.

The order on P_{X} is given by: (f, d) ≤ (g, e) if g ⊂ f, dom(e) ⊂ dom(d) and d(α, β) ∩ (max(dom(g)) + 1) = e(α, β) for all (α, β) ∈ dom(e).

The partial order P_{X} is closed under decreasing sequences of length less than θ and therefore does not add sets of ordinals of cardinality less than θ. Furthermore, if |X|^{+} < λ, then below densely many conditions (conditions (f, d) with |dom(f)| > |X|) every descending sequence in P_{X} of length less than λ has a lower bound, so P_{X} does not add sequences from V of length less than λ. We will see below that P_{X} is in some sense homogeneous.

Given X ⊂ λ × λ and a regressive function F on C_{[θ,λ)}^{λ}, let Q_{F,X} denote the partial order whose conditions are partial functions d with domain a subset of X of cardinality less than λ, such that for each (α, β) in the domain of d, d(α, β) is a closed, bounded subset of λ disjoint from F^{−1}{α}. If X is a subset of λ × λ such that |X|^{+} < λ, and Y ⊂ λ × λ is disjoint from X, then, since P_{X} does not add bounded subsets of λ, P_{X∪Y} is forcing-isomorphic to P_{X} ∗ Q_{Ḟ ,Y}, where Ḟ represents the generic regressive function added by P_{X}.

Let D̄ = ⟨d_{δ} : δ ∈ C_{{θ}}^{λ}⟩ be a sequence in V such that each d_{δ} is a cofinal subset of δ of ordertype cf(δ). Let^{<θ}Ord be the class of functions whose domain is an ordinal less than θ and whose range is contained in the ordinals.

A V -generic filter for P_{X} is naturally represented by a pair (F, C̄), where F is a regressive function on (C_{[θ,λ)}^{λ})^{V}, C̄ has the form ⟨C_{α,β} : (α, β) ∈ X⟩, and each C_{α,β} is a club subset of λ disjoint from F^{−1}{α}. Fixing such a pair, let M be the smallest transitive inner model of ZF containing D̄, F, ^{<θ}Ord and every function from γ to {C_{α,β} : (α, β) ∈ X}, for any γ < θ. Every set in M is definable in M from D̄, F, a member of ^{<θ}Ord and a function from a γ < θ to {C_{αβ} : (α, β) ∈ X}. It follows that M is closed under sequences of length less than θ in V [F, C̄], and therefore that M satisfies DC_{<θ}. Since D̄ is in M, and since V and V [F, C̄] have the same ordinals of cofinality less than θ, M satisfies ϕ(λ, f), where f is the ordinal successor function on the regular cardinals below θ. Since V [F, C̄] and V have the same sequences of ordinals of length less than θ, M is definable in V [F, C̄] from D̄, F and the (unordered) set {C_{α,β} : (α, β) ∈ X}.

Given Y ⊂ X, let N_{Y} denote V [F, ⟨C_{α,β} : (α, β) ∈ Y ⟩].

Lemma 6.2. Suppose that X = Z × Z, for some uncountable Z ⊂ λ, and that (F, C̄) is V -generic for P_{X}. Then every subset of V in M exists in N_{Y} for some Y ⊂ X of cardinality less than θ.

Proof. Given such a set A, we can fix Y ⊂ X of cardinality less than θ such that Y is of the form W ×W for some W ⊂ λ and such that A is definable in M from D̄, F, an element x of ^{<θ}Ord and ⟨C_{α,β} : (α, β) ∈ Y ⟩. Let ϕ be a formula such that

A = {a | M ⊧ ϕ(a, D̄, F, x, ⟨C_{α,β} : (α, β) ∈ Y ⟩)}.

We have that P_{X} is forcing-equivalent to P_{Y} ∗ Q_{Ḟ ,X\Y}. Suppose that there are two conditions d and e in Q_{F,X\Y} (in N_{Y}) and some a ∈ V such that

d⊩M ⊧ ϕ(a, D̄, F, x, ⟨C_{α,β} : (α, β) ∈ Y ⟩)

and

e⊩M ⊧ ¬ϕ(a, D̄, F, x, ⟨C_{α,β} : (α, β) ∈ Y ⟩).

There are conditions d′ ≤ d and e′ ≤ e in Q_{F,X\Y} such that

• for every (α, β) ∈ dom(d′) there is a β′ such that (α, β′) ∈ dom(e′) and e′(α, β′) = d′(α, β), and

• for every (α, β) ∈ dom(e′) there is a β′ such that (α, β′) ∈ dom(d′) and d′(α, β′) = e′(α, β).

There is then a natural isomorphism π between Q_{F,X\Y} below d′ and Q_{F,X\Y} below e′. This isomorphism π has the property that, given two generic filters G_{d′} and G_{e′} for Q_{F,X\Y} with π[G_{d′}] = G_{e′}, the (unordered) generic set {C_{α,β} : (α, β) ∈ X \ Y } is the same in the two extensions. Then M is the same model in the two extensions, contradicting the claim that

d⊩M ⊧ ϕ(a, D̄, F, x, ⟨C_{α,β} : (α, β) ∈ Y ⟩)

and

e⊩M ⊧ ¬ϕ(a, D̄, F, x, ⟨C_{α,β} : (α, β) ∈ Y ⟩).

It follows from Lemma 6.2 that M and V have the same sequences of ordinals of length less than λ, so λ is a regular cardinal in M. In the case that X = λ×λ, then, M satisfies ZF + DC_{<θ} + ϕ(λ, f), where f is the ordinal successor function on the regular cardinals below θ, and there exists in M a regressive function on C_{[θ,λ)}^{λ} which is not constant on a stationary set.

References

[1] T. Jech, Set Theory, Springer-Verlag, Berlin, 2003

[2] A. Kanamori, The Higher Infinite. Large cardinals in set theory from their beginnings, 2nd edition, Springer Monographs in Mathematics, 2003

[3] S. Shelah, Proper and Improper Forcing, Springer-Verlag, Berlin, 1998

[4] S. Shelah, Cardinal Arithmetic, Oxford University Press, 1994

[5] S. Shelah, PCF without choice, publication number 835, preprint