DIAMOND, GCH AND WEAK SQUARE

MARTIN ZEMAN

Abstract. Shelah proved recently that if κ > ω and S ⊆ κ^{+} is a stationary set of ordinals of cofinality different from cf(κ) then 2^{κ} = κ^{+} implies ♢_{κ}(S). We show that for singular κ, an elaboration on his argument allows to derive ♢_{κ}(T) from 2^{κ} = κ^{+} + □_{κ}^{∗} where T = {δ < κ^{+} | cf(δ) = cf(κ)}. This gives a strong restriction on the existence of saturated ideals on κ^{+}.

AMS Subject Classification: 03E04, 03E05 Keywords: Diamond, Weak Square, Generalized Continuum Hypothesis

It is a well-known fact that ♢_{λ} implies 2^{<λ} = λ. In many situations the converse is also true. Jensen [3] proved that CH does not imply ♢, so when looking for the converse one has to focus on λ > ω_{1}. Let

S_{μ}^{λ} = {δ < λ | cf(δ) = μ}.

and

T_{κ} = S_{cf(κ)}^{κ^{+}}.

Gregory observed that GCH below ω_{2} implies ♢_{ω_{2}}(S_{ω}^{ω_{2}}). A sequence of improvements on his result, mainly by Gregory [4], Jensen (unpublished) and Shelah [7], resulted in the following theorem whose proof can be found in [2].

Theorem 0.1 (Gregory, Jensen, Shelah). If 2^{<κ} = κ and 2^{κ} = κ^{+} then ♢_{κ^{+}}(S) holds whenever S ⊆ κ^{+} is a stationary set of ordinals of cofinality different from cf(κ). If κ is singular and additionally □_{κ} holds then ♢_{κ^{+}}(T_{κ}).

Shelah also proved that for regular κ, the condition 2^{κ} = κ^{+}+ □_{κ} is not sufficient to guarantee ♢_{κ^{+}}(T_{κ}), so the absolute ZFC result is possible only for singular κ.

The question remained whether the localized GCH, i.e. the equality 2^{κ} = κ^{+} alone implies ♢_{κ^{+}}. Shelah proved this to be true for sufficiently large κ, and recently [8] found an argument that proves it for every uncountable cardinal κ; see Komjáth’s paper [6] for a simplified proof and an elaboration on Shelah’s argument.

Theorem 0.2 (Shelah). Let κ > ω and 2^{κ} = κ^{+}. Then ♢_{κ^{+}}(S) holds for every stationary S ⊆ κ^{+} that is disjoint with T_{κ}.

This note combines arguments from the proof of Theorem 0.1 with Shelah’s argument for Theorem 0.2 to give a proof of ♢_{κ^{+}}(S) for S ⊆ T_{κ}.

Theorem 0.3 (Main Theorem). Assume κ is a singular cardinal and T ⊆ T_{κ} is stationary with stationarily many reflection points. Then

2^{κ} = κ^{+} + □_{κ}^{∗} ⟹ ♢_{κ^{+}}(T).

The author was supported in part by NSF grant DMS-0500799.

It was proved by Cummings, Foreman and Magidor [1] that for singular κ, the principle □_{κ}^{∗} is consistent with the requirement that every stationary T ⊆ T_{κ} has stationarily many reflection points. Consequently, in their model we have ♢_{κ^{+}}(S) for all stationary S ⊆ κ^{+}. Corollary 0.4. Assume κ is a singular cardinal. Then

2^{κ} = κ^{+} + □_{κ}^{∗} ⟹ ♢_{κ}(T_{κ}).

Both the above theorem and its corollary provides a very strong restriction on the existence of saturated ideals on κ^{+} and provide a close link between the study of such ideals and the PCF-theory.

Rinot recently extended the result of this paper to the situation where weak square is replaced by a variant of approachability property and also showed that, relatively to the existence to a supercompact cardinal, it is consistent that □_{κ}^{∗} fails but ♢_{κ^{+}}(S) holds for every stationary S ⊆ κ^{+}.

1. The argument

We begin with splitting Shelah’s argument into two steps. We first isolate a combinatorial statement that alone implies the existence of a ♢_{λ}(S)-sequence in ZFC; we denote this statement by ◯_{λ}(S). This statement is implicit the arguments in Shelah [8] and Komjáth [6]. It turns out that the implication ◯_{λ}(S) ⟹ ♢_{λ}(S) is true no matter whether λ is a successor cardinal or not. The second step is a proof that ◯_{λ}(S) holds, which relies on the localized GCH if λ = κ^{+} and S concentrates on points of cofinality distinct from cf(κ) which gives the original Shelah’s result, and on the weak square if κ is singular and S concentrates on points of cofinality cf(κ) which gives the result in Theorem 0.3. Our approach owes a lot to Komjáth’s exposition in [6]. Definition 1.1. Let λ be a regular cardinal and S ⊆ λ. We say that the pair ⟨x_{ξ} | ξ < λ⟩, ⟨A_{δ} | δ ∈ S⟩ witnesses ◯_{λ}(S) iff the following three conditions are met.

(a) ⟨x_{ξ} | ξ < λ⟩ is an enumeration of [λ]^{<λ}.

(b) A_{δ} ⊆ δ and card(A_{δ}) < card(δ) whenever δ ∈ S.

(c) For every Z ⊆ λ there is a stationary S′ ⊆ S such that for every δ ∈ S′ there are unboundedly many α < δ for which there is β < δ satisfying α, β ∈ A_{δ} and Z ∩ α = x_{β}.

We say that ◯_{λ}(S) holds iff there are ⟨x_{ξ}⟩_{ξ} and ⟨A_{δ}⟩_{δ} as above.

Notice that ◯_{λ}(S) postulates the existence of an enumeration of [λ]^{<λ} of length λ, so it imposes some constraints on the behaviour of the exponential function below λ. In particular, if λ = κ^{+} then ◯_{λ}(S) implies 2^{κ} = κ^{+}. Notice also that (b) in the above definition stipulates that the cardinality of A_{δ} is strictly smaller than that of δ, which together with (c) implies that without loss of generality S can be viewed as a set of singular ordinals. Of course, this has a non-trivial meaning only when λ is inaccessible. Finally observe that if there is a pair ⟨x_{ξ} | ξ < λ⟩, ⟨A_{δ} | δ ∈ S⟩ witnessing ◯_{λ}(S) then for every enumeration ⟨x′_{ξ} | ξ < λ⟩ there is a sequence ⟨A′_{δ} | δ ∈ S⟩ such that the pair ⟨x′_{ξ}⟩_{ξ}, ⟨A′_{δ}⟩_{δ} witnesses ◯_{λ}(S). To see this, pick any f : λ → λ such that x_{β} = x′_{f(β)} for all β < λ and let A′_{δ} = A_{δ} ∪ f[A_{δ}] for all δ ∈ S satisfying f[δ] ⊆ δ. Lemma 1.2. Let λ be a regular cardinal, S ⊆ λ and ◯_{λ}(S) hold. Then there is a pair ⟨x_{ξ} | ξ < λ⟩, ⟨A_{δ} | δ ∈ S⟩ satisfying the following.

(a) ⟨x_{ξ} | ξ < λ⟩ is an enumeration of [λ × λ]^{<λ}.

(b) A_{δ} ⊆ δ and card(A_{δ}) < card(δ) whenever δ ∈ S.

(c) For every Z ⊆ λ × λ there is a stationary S′ ⊆ S such that for every δ ∈ S′ there are unboundedly many α < δ for which there is β < δ satisfying α, β ∈ A_{δ} and Z ∩ (α × α) = x_{β} ∩ (α × α).

Proof. Pick a pair ⟨y_{ξ} | ξ < λ⟩, ⟨B_{δ} | δ ∈ S⟩ witnessing ◯_{λ}(S). Let f : λ × λ → λ be a bijection and C_{f} = {δ < λ | f[δ × δ] = δ}.

To each δ ∈ S pick C_{δ} to be a subset of lim(C_{f}) ∩ δ of size strictly smaller than card(δ) that is cofinal in δ if such a set exists; let C_{δ} = ∅ otherwise. Letting x_{β} = f^{−1}[y_{β}] and A_{δ} = B_{δ} ∪ C_{δ}, we obtain a pair ⟨x_{ξ} | ξ < λ⟩, ⟨A_{δ} | δ ∈ S⟩ as in the conclusion of the lemma. To see this, it suffices to verify clause (c) in the statement of the lemma.

Given any Z ⊆ λ × λ, let S′ ⊆ S be the stationary set obtained by applying ◯_{λ}(S) to f[Z]. Let δ ∈ S′ ∩ lim(C_{f}). If ᾱ < δ, pick α ∈ C_{δ} such that ᾱ ≤ α. Since S′ satisfies (c) in Definition 1.1 with f[Z], y_{β} and B_{δ} in place of Z, x_{β} and A_{δ}, there are α′, β ∈ B_{δ} such that α ≤ α′ and f[Z] ∩ α′ = y_{β}. Then f[Z] ∩ α = y_{β} ∩ α and the conclusion follows immediately from the fact that α ∈ C_{f}. ⊓ ⊔

With the statement ◯_{λ}(S) in hand, one can reformulate the first step in Shelah’s argument into the following proposition. It reduces the proof of ♢_{λ}(S) to the proof of ◯_{λ}(S) and works even for cardinals λ that are not successors, which is slightly more than Shelah has originally proved. The second setp in Shelah’s argument can be then viewed as a proof of ◯_{κ^{+}}(S) from the localized GCH. We will show how to obtain ◯_{κ^{+}}(S) from the additional assumption that □_{κ}^{∗} holds in situations where localized GCH does not seem to suffice. Proposition 1.3. Let λ be regular and S ⊆ λ be stationary. Then

◯_{λ}(S) ⟹ ♢_{λ}(S)

Proof. Let ⟨x_{ξ} | ξ < λ⟩, ⟨A_{δ} | δ ∈ S⟩ be a pair satisfying the conclusion of Lemma 1.2. For x ⊆ λ × λ we write (x)_{ξ} to denote {ζ < λ |⟨ξ, ζ⟩ ∈ x}. Consider sequences ⟨X_{ξ}, C_{ξ} | ξ < θ⟩ of length θ ≤ λ such that X_{ξ} ⊆ λ, C_{ξ} is closed unbounded in λ and, letting

V_{ξ}^{δ} = {⟨α, β⟩ ∈ A_{δ} × A_{δ} |(∀η < ξ)(X_{η} ∩ α = (x_{β})_{η} ∩ α)},

for every ξ < θ and δ ∈ S ∩ C_{ξ} either dom(V_{ξ+1}^{δ}) is bounded in δ or else V_{ξ}^{δ} ⫌ V_{ξ+1}^{δ}. Notice that the non-strict inclusion V_{ξ}^{δ} ⊇ V_{ξ′}^{δ} holds anyway whenever ξ ≤ ξ′.

The crucial observation is that any sequence ⟨X_{ξ}, C_{ξ} | ξ < θ⟩ as above has length strictly below λ. Assume for a contradiction that this fails, that is, there is such a sequence with θ = λ. Let S′ come from the application of Lemma 1.2 to the pair ⟨x_{ξ}⟩_{ξ}, ⟨A_{δ}⟩_{δ} and to set

Z = {⟨ξ, ζ⟩ | ζ ∈ X_{ξ}},

and let δ ∈ S′ ∩ △{C_{ξ} | ξ < λ} be such that δ > κ if λ = κ^{+} and δ is a cardinal if λ is inaccessible. We have arbitrarily large α < δ for which there exists β < δ such that α, β ∈ A_{δ} and Z ∩ (α × α) = x_{β} ∩ (α × α), so for each ξ < δ the set dom(V_{ξ}^{δ}) is unbounded in δ. Since δ ∈ S ∩ C_{ξ} whenever ξ < δ, from the properties of the sequence ⟨X_{ξ}, C_{ξ} | ξ < θ⟩ we obtain V_{ξ}^{δ} ⫌ V_{ξ′}^{δ} whenever ξ < ξ′ < δ. This is a contradiction, as V_{ξ}^{δ} ⊆ A_{δ} × A_{δ} and card(A_{δ}) < card(δ).

Pick a sequence ⟨X_{ξ}, C_{ξ} | ξ < θ⟩ as in the previous paragraph which has no proper extension. Then θ < λ. Letting

D_{δ} = ⋃{(x_{β})_{θ} ∩ α | ⟨α, β⟩ ∈ V_{θ}^{δ}},

the sequence ⟨D_{δ} | δ ∈ S⟩ is a ♢_{λ}(S)-sequence. To see this, pick an arbitrary X ⊆ λ and a closed unbounded C ⊆ λ. There exists some δ ∈ S ∩ C such that dom(V_{θ}^{δ}) is unbounded in δ and X ∩ α = (x_{β})_{θ} ∩ α for all ⟨α, β⟩ ∈ V_{θ}^{δ}, as otherwise we could extend the sequence ⟨X_{ξ}, C_{ξ} | ξ < θ⟩ by letting X_{θ} = X and C_{θ} = C, in contradiction with its maximality. But then X ∩ δ = D_{δ}. ⊓ ⊔

We now focus on proofs of ◯_{λ}(S). The point of introducing ◯_{λ}(S) is that it is often easier to give a direct proof of ◯_{λ}(S) than a direct proof of ♢_{λ}(S). This is clear from Shelah’s argument in [8] which in our notation is a proof of ◯_{κ^{+}}(S). As Proposition 1.3 also holds for inaccessible λ, our hope was that Shelah’s argument may be used for proofs of ♢_{λ}(S) for inaccessible λ. It seems, however, that for inaccessible λ the proofs of ♢_{λ}(S) may require more new ideas. For instance, the proofs of ♢_{λ}(S_{ε}^{λ}) for a Mahlo cardinal λ in [5] and [9] can be easily modified to give proofs of ◯_{λ}(S_{ε}^{λ}), but introducing a ◯_{λ}(S_{ε}^{λ})-sequence into the construction does not seem to enable any strengthening of the results or a simplification of the construction in [9]. For inaccessibles λ that are not Mahlo it is not clear either whether an argument using ◯_{λ}(S_{ε}^{λ}) may work. It is certainly clear that constructions of a ◯_{λ}(S_{ε}^{λ})-sequence from “below” as in Propositions 1.4 and 1.5 will not work, essentially for the same reason why constructions of ♢_{λ}(S_{ε}^{λ}) from “below” cannot work, as described in [9]. Analogously as in [9], given any fixed ◯_{λ}(S_{ε}^{λ})-witness ⟨x_{β} | β < λ⟩, ⟨A_{δ} | δ < λ⟩, there is a < λ-distributive forcing that “kills” such a witness. On the other hand, any construction of a ◯_{λ}(S_{ε}^{λ})-witness from “below” would give rise to the same witness in the ground model and in the generic extension.

Let us turn to the proof of ◯_{κ^{+}}(S). As already mentioned above, the next proposition can be viewed as the first step in Shelah’s argument. We include it, as it is a starting point for our variation with weak square. Proposition 1.4. Assume S ⊆ κ^{+} is stationary and disjoint from T_{κ}. Then

2^{κ} = κ^{+} ⟹ ◯_{κ^{+}}(S).

Proof. Pick an arbitrary enumeration ⟨y_{ξ} | ξ < κ^{+}⟩ of [κ^{+}]^{≤κ}. The existence of such an enumeration is guaranteed by the localized GCH. Let g : ε × κ^{+} → κ^{+} be a bijection where ε = cf(κ). For each δ ∈ S pick an increasing (with respect to the inclusion) sequence of sets ⟨A_{ι}^{δ} | ι < ε⟩ such that | A_{ι}^{δ} | < κ for all ι < ε and ⋃_{ι<ε} A_{ι}^{δ} = δ.

We show that there is an ι < ε such that for every Z ⊆ κ^{+} there are stationarily many ordinals δ ∈ S satisfying:

(1)

For unboundedly many α < δ there are β < δ such that α, β ∈ A_{ι}^{δ} and Z ∩ α = (g^{−1}[y_{β}])_{ι}.

It follows that letting A_{δ} = A_{ι}^{δ} and x_{β} = (g^{−1}[y_{β}])_{ι}, the pair ⟨x_{β} | β < κ^{+}⟩, ⟨A_{ι}^{δ} | δ ∈ S⟩, witness ◯_{κ^{+}}(S). ^{1}

Assume for a contradiction there is no ι as in the previous paragraph. Then for every ι < ε there is a set Z_{ι} ⊆ κ^{+} such that (1) holds only on a non-stationary subset of S. Let Z = {⟨ι, ξ⟩ | ξ ∈ Z_{ι}} and and Z′ = g[Z]. The set S′ consisting of all δ ∈ S such that

1See proof of Proposition 1.3 for the notation (u)_{η}.

• g[ε × α] = α for cofinally many α < δ and

• (∀α < δ)(∃β < δ)(Z′ ∩ α = y_{β}) is stationary in κ^{+}. To each δ ∈ S′ pick a cofinal strictly increasing sequence ⟨α_{η}^{δ} | η < cf(δ)⟩ such that g[ε × α_{η}] = α_{η} for each η < cf(δ), and to each η < cf(δ) pick β_{η} < δ such that Z′ ∩ α_{η} = y_{β_{η}}. This is possible by the above arrangements for elements of S′.

If δ ∈ S′ then there is an ι(δ) < ε such that α_{η}, β_{η} ∈ A_{ι(δ)}^{δ} for cofinally many η < cf(δ). This follows immediately if cf(δ) < ε, as the assignment

η ↦ the least ι such that α_{η}, β_{η} ∈ A_{ι}^{δ}

cannot be cofinal in ε, so in fact α_{η}, β_{η} ∈ A_{ι(δ)}^{δ} for all η < cf(δ). If cf(δ) > ε this follows by the pigeonhole principle, namely the inverse image of some A_{ι}^{δ} under this assignment must have size cf(δ). Applying the pigeonhole principle to the assignment δ ↦ ι(δ), we obtain a stationary S′′ ⊆ S′ and a ι < ε such that ι(δ) = ι for all δ ∈ S′′.

Pick δ ∈ S′′. By the above arrangements, there are cofinally many α < δ for which there are β < δ such that α, β ∈ A_{ι}^{δ} and Z′ ∩ α = y_{β}. Moreover, the ordinals α can be chosen so that g[ε × α] = α. It follows that

Z ∩ (ε × α) = g^{−1}[Z′ ∩ α] = g^{−1}[y_{β}],

so Z_{ι} ∩ α = (g^{−1}[y_{β}])_{ι} for all α, β as above. Since this is true of any δ ∈ S′′ we obtained a contradiction to the fact that Z_{ι} is a coutnerexample to (1). ⊓ ⊔

The following proposition shows how to apply a standard construction that uses □_{κ}^{∗} to prove ◯_{κ^{+}}(T). Proposition 1.5. Assume κ is singular and T ⊆ T_{κ} is a stationary subset of κ^{+} with stationarily many reflection points. Then

2^{κ} = κ^{+} + □_{κ}^{∗} ⟹ ◯_{κ^{+}}(T).

Proof. We elaborate on the argument from the proof of Proposition 1.4. Let ε = cf(κ). Fix the following objects:

• Sequences ⟨y_{ξ} | ξ < κ^{+}⟩, ⟨A_{ι}^{δ} | ι < ε⟩ and a bijection g : ε × κ^{+} → κ^{+} as in the proof of Proposition 1.4.

• For each ordinal δ < κ^{+} an injection h_{δ} : δ → κ.

• A □_{κ}^{∗}-sequence ⟨𝒞_{δ} | δ ∈ lim ∩(κ, κ^{+})⟩. For each δ fix an enumeration ⟨c_{ζ}^{δ} | ζ < κ⟩ of the set 𝒞_{δ}.

• An increasing (with respect to the inclusion) sequence of sets ⟨B_{ι} | ι < ε⟩

such that | B_{ι} | < κ for each ι and ⋃_{ι<ε} B_{ι} = κ × κ. For each δ ∈ lim ∩(κ, κ^{+}) and ζ < κ define a function f_{ζ}^{δ} : δ → κ × κ and a

sequence of sets ⟨A_{ζ,ι}^{δ} | ι < ε⟩ as follows.

f_{ζ}^{δ}(ξ) = ⟨η, h_{γ}(ξ)⟩

where γ is the least element of c_{ζ}^{δ} strictly above ξ and η = otp(c_{ζ}^{δ} ∩ γ) − 1 ^{2} and

A_{ζ,ι}^{δ} = (f_{ζ}^{δ})^{−1}[B_{ι}].

2Notice that otp(c_{ζ}^{δ} ∩ γ) is a successor ordinal if γ is as above.

Notice that each f_{ζ}^{δ} is an injection. By our choice of the sets B_{ι} we then have | A_{ζ,ι}^{δ} | < κ and ⋃_{ι<ε} A_{ζ,ι}^{δ} = δ. We also have the following coherency property for the sets A_{ζ,ι}^{δ}: If δ̄ is a limit point of c_{ζ}^{δ} then there is an ordinal ζ̄ < κ such that

(2)  A_{ζ,ι}^{δ} ∩ δ̄ = Aδ̄ ; ζ̄,ι.

To see this notice first that if δ̄ is a limit point of c_{ζ}^{δ} then there is ζ̄ < κ such that c_{ζ}^{δ} ∩δ̄ = cδ̄ ; ζ̄, and from the definition of f_{ζ}^{δ} we immediately conclude that f δ̄ ; ζ̄ = f_{ζ}^{δ} ↾ δ̄. The rest follows immediately from the definition of A_{ζ,ι}^{δ}.

Fix an increasing sequence ⟨κ_{ι} | ι < ε⟩ cofinal in κ. For each δ ∈ T and ι < ε set A_{ι}^{δ} = ⋃_{ζ<κ_{ι}} A_{ζ,ι}^{δ}. Notice that | A_{ι}^{δ} | < κ, as | A_{ζ,ι}^{δ} | < | B_{ι} | for all ζ < κ_{ι}. Following the ideas from the proof of Proposition 1.4 we prove: There is an ι < ε such that for every Z ⊆ κ^{+} there are stationarily many δ ∈ T satisfying:

(3)

For unboundedly many α < δ there are β < δ such that α, β ∈ A_{ι}^{δ} and Z ∩ α = (g^{−1}[y_{β}])_{ι}.

It follows that letting x_{β} = (g^{−1}[y_{β}])_{ι} and A_{δ} = A_{ι}^{δ} where ι is as above, the pair ⟨x_{β} | β < κ^{+}⟩, ⟨A_{δ} | δ ∈ T ⟩ witnesses ◯_{κ^{+}}(T).

Assume for a contradiction that no ι as above exists. As in the proof of Proposition 1.4 pick a counterexample Z_{ι} for each ι < ε, let Z = {⟨ι, ξ⟩ ∈ ε × κ^{+} | ξ ∈ Z_{ι}} and Z′ = g[Z]. Let C be a closed unbounded subset of κ^{+}. By our assumption on T, there is a reflection point δ′ of T such that:

• δ′ is a limit point of C.

• g[ε × α] = α for cofinally many α < δ′.

• (∀α < δ′)(∃β < δ′)(Z′ ∩ α = y_{β}). As δ′ is a reflection point of T, necessarily cf(δ′) > ε. Pick an increasing sequence ⟨α_{η} | η < cf(δ′)⟩ cofinal in δ′ such that g[ε × α_{η}] = α_{η} for each η < cf(δ′). To each η < cf(δ′) assign some β_{η} < δ′ satisfying Z′ ∩ α_{η} = x_{β_{η}}. It is convenient to pick β_{η} to be least posssible. Since cf(δ′) > ε, using the pigeonhole principle we conclude that there is some ι′ < ε such that α_{η}, β_{η} ∈ A_{0,ι′}^{δ′} for cofinally many η < cf(δ′). Let δ ∈ T ∩ C ∩ lim(c_{0}^{δ′}) be a limit point of {α_{η} | α_{η}, β_{η} ∈ A_{0,ι′}^{δ′}}. Such a δ exists by our choice of δ′ and ι′, and by the fact that T ∩ δ′ is stationary in δ′. Let ξ < κ be such that A_{ξ,ι′}^{δ} = A_{0,ι′}^{δ′} ∩ δ and let ι(δ) > ι′ be such that κ_{ι(δ)} > ξ. The existence of such a ξ follows from (2). Then A_{0,ι′}^{δ′} ∩ δ ⊆ A_{ι(δ)}^{δ}, as B_{ι′} ⊆ B_{ι(δ)}.

The previous paragraph proves that there is a stationary T′ ⊆ T such that for every δ ∈ T′ there is an ι(δ) < ε such that for cofinally many α < δ there are β < δ such that α, β ∈ A_{ι(δ)}^{δ}, Z′ ∩ α = y_{β} and g[ε × α] = α. The rest of the proof literally follows the proof of Proposition 1.4. We first find a stationary T′′ ⊆ T′ on which ι(δ) stabilize; let ι be the stabilized value. Then we unfold Z′ and y_{β} using g and conclude that for α, β as above we have Z_{ι} ∩ α = (g^{−1}[y_{β}])_{ι}. This yields a contradiction with the fact that Z_{ι} is a counterexample to (3). ⊓ ⊔

References

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[3] Devlin, K. J. and Johnsbraten, H., The Souslin Problem, Lecture Notes in Mathematics 405, Springer 1974 [4] Gregory, J., Higher Souslin Trees and the Generalized Continuum Hypothesis, Journal of

Symbolic Logic 14 (1976), 663-671

[5] Jensen, R., Diamond at Mahlo cardinals, Handwritten notes, Oberwolfach 1991

[6] Komjáth, P., Shelah’s proof of diamond, unpublished

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[9] Zeman, M., ♢ at Mahlo cardinals, Journal of Symbolic Logic 65(4) (2000), 1813-1822

Department of Mathematics, University of California at Irvine, Irvine, CA 92697

E-mail address: mzeman@math.uci.edu