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Notes toThe Continuum Hypothesis

1.See Hallett (1984) for further historical information on the role ofCH in the early foundations of set theory.

2.We have of necessity presupposed muchin the way of set theory. The reader seeking additionaldetail—for example, the definitions of regular and singularcardinals and other fundamental notions—is directed to one ofthe many excellent texts in set theory, for example Jech (2003).

3.To say that GCH holds below δ isjust to say that 2^ℵ_α =ℵ_α+1 for all ω ≤ α <δ and to say that GCH holds at δ is just to say that2^ℵ_δ = ℵ_δ+1).

4.To see this argue asfollows: Assume large cardinal axioms at the level involved in (A) and(B) and assume that there is a proper class of Woodincardinals. Suppose for contradiction that there is a prewellorderinginL(ℝ) of length ℵ₂. Now, using (A) force toobtain a saturated ideal on ℵ₂ without collapsingℵ₂. In this forcing extension, the originalprewellordering is still a prewellordering inL(ℝ) of lengthℵ₂, which contradicts (B). Thus, the original largecardinal axioms imply that Θ^L(ℝ)≤ ℵ₂. The same argument applies in the moregeneral case where the prewellordering is universally Baire.

5.For more on the topicof invariance under set forcing and the extent to which this has beenestablished in the presence of large cardinal axioms, see §4.4and §4.6 of the entry “Large Cardinals and Determinacy”.

6.The non-stationaryidealI_NS is a proper class from the point of viewofH(ω₂) and it manifests (through Solovay’stheorem on splitting stationary sets) a non-trivial application ofAC. For further details concerningA^G see§4.6 of the entry “Large Cardinals and Determinacy”.

7.Here are the details: LetA ∈Γ^∞ andM be a countable transitive model ofZFC. We say thatM isA-closed if for all setgeneric extensionsM[G] ofM,A ∩M[G] ∈M[G]. LetT be a set of sentences and φ be a sentence. Wesay thatT ⊢_Ω φ if there is a setA ⊆ℝ such that

1. L(A, ℝ) ⊧ AD⁺,
2. 𝒫 (ℝ) ∩L(A, ℝ) ⊆ Γ^∞, and
3. for all countable transitiveA-closedM,

   M ⊧ “T ⊧_Ω φ”,

   where here AD⁺ is a strengthening of AD.

8.Here are the details: First weneed another conjecture: (The AD⁺ Conjecture) SupposethatA andB are sets of reals such thatL(A, ℝ)andL(B, ℝ) satisfy AD⁺. Suppose every set

X ∈ 𝒫 (ℝ) ∩ (L(A, ℝ) ∪L(B,ℝ))

is ω₁-universally Baire. Then either

(Δ̰21)^L(A,ℝ)⊆(Δ̰21)^L(B,ℝ)

or

(Δ̰21)^L(B,ℝ)⊆(Δ̰21)}^L(A,ℝ).

(Strong Ω conjecture) Assume there is a proper class ofWoodin cardinals. Then the Ω Conjecture holds and theAD⁺ Conjecture is Ω-valid.

9.Asmentioned at the end ofSection 2.2 it could be the case (given ourpresent knowledge) that large cardinal axioms imply thatΘ^L(ℝ) < ℵ₃ and, moregenerally, rule out the definable failure of 2^ℵ₀= ℵ₂. This would arguably further buttress the casefor 2^ℵ₀ = ℵ₂.