The Chang-type uB-powerset sealing conjecture

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Suppose κ\kappa is a supercompact cardinal and there is a proper class of inaccessible limits of Woodin cardinals. Suppose g⊆Col⁡(ω,22κ)g\subseteq {\operatorname{Col}}(\omega,2^{2^\kappa}) is VV-generic. Let

ηg=(ηuBg)L(uBg,Rg).\eta_g=(\eta_{{{\sf{uB}}}_g})^{L({{\sf{uB}}}_g,\mathbb{R}_g)}.

Then cf⁡(ηg)=ω\operatorname{cf}(\eta_g)=\omega.

Chang-type uB-powerset sealing conjecture. Moreover, the Sealing\sf{Sealing} Theorem holds for C−uBp\mathsf{C-uBp} in V[g]V[g].

Here C−uBp=L(ηω,uBp)\mathsf{C-uBp}=L(\eta^\omega,{{\sf{uBp}}}) is the Chang-type uB{{\sf{uB}}}-powerset model. The statement is presented under strong large-cardinal hypotheses, and the source gives no explicit resolution status.

References

Primary source

Sandra Müller and Grigor Sargsyan, “Gödel's Program in Set Theory”, arXiv:2412.07325 (2024).

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