终极L（数学论文）二 (7-5)

Σ₂-sentence is true in Ultimate-L，so we are required to find a univer-sallyBaire set of reals A in Ultimate-L such that the Σ₂-sentence in questions holds in (HOD)ᴸ⁽ᴬ，ℝ⁾∩V𝚹ᴸ₍ᴀ，R₎. From well-known generic absoluteness results which are known to hold assuming a proper class of Woodin cardinals，it is sufficient to prove that this does obtain in some set-generic extension of Ultimate-L.So choose an ordinal β such that (Vᵦ)Utimate-L is a Σ₂-elementary substructure of Ultimate-L，and choose a γ＜β such that (Vᵧ)Ultimate-L models the Σ₂-sentence. Now consider a generic extension of Ultimate-L where A is a universally Baire set chosen to contain enough data so that.in the generic exten-sion，𝚹ᴸ⁽ᴬ，ℝ⁾ ≤ β，and (HOD)ᴸ⁽ᴬ，ℝ⁾∩ Vᵧ in the generic extension is equal to the intersection of the Ultimate-L of the ground model and Vᵧ. This can be arranged by ensuring that each ordinal less than β is collapsed to be countable in the generic extension，and that all the data for sets of ordinals less than γ which are needed to generate(Ultimate L∩Vᵧ)ⱽ are coded into the universally Baire set A which appears as a set of reals in the generic extension. In this generic extension，the de-sired result obtains，so the aforementioned generic absoluteness results imply that it obtains in our ground model as well. This completes the proof of Theorem 6.4. ▢

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