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1. Sealing from iterability

   Sargsyan, Grigor; Trang, Nam (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   We show that if V has a proper class ofWoodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then Sealing holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that Sealing holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that Sealing is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of Sealing obtained by W.H. Woodin. The Largest Suslin Axiom (LSA) is a determinacy axiom isolated byWoodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let LSA-over-uB be the statement that in all (set) generic extensions there is a model of LSA whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming V has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe V [g], where g is V -generic for the collapse of the successor of the least strong cardinal to be countable, the theory LSA-over-UB fails; this implies that LSA-over-UB is not equivalent to Sealing (over the base theory of V [g]). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “AD in L(R) holds in all generic extensions” is equivalent to “the theory of L(R) is sealed” in the presence of a proper class of measurable cardinals. 

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2. SEALING OF THE UNIVERSALLY BAIRE SETS

   https://doi.org/10.1017/bsl.2021.29  

   Sargsyan, Grigor; Trang, Nam (October 2021, The bulletin of symbolic logic)

   A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The 𝖫𝖺𝗋𝗀𝖾𝗌𝗍 𝖲𝗎𝗌𝗅𝗂𝗇 𝖠𝗑𝗂𝗈𝗆 ( 𝖫𝖲𝖠 ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 be the statement that in all (set) generic extensions there is a model of 𝖫𝖲𝖠 whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is equiconsistent with 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is weaker than the theory “ 𝖹𝖥𝖢 + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 by Woodin. A variation of 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 , called 𝖳𝗈𝗐𝖾𝗋 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 , is also shown to be equiconsistent with 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 is not equivalent to 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 . 

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3. Strongly Compact Cardinals and Ordinal Definability

   https://doi.org/10.1142/S0219061322500106  

   Goldberg, Gabriel (May 2023, Journal of Mathematical Logic)

   This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. 

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4. The exact consistency strength of the generic absoluteness for the universally Baire sets

   https://doi.org/10.1017/fms.2023.127  

   Sargsyan, Grigor; Trang, Nam (January 2024, Forum of Mathematics, Sigma)

   Abstract A set of reals isuniversally Baireif all of its continuous preimages in topological spaces have the Baire property.$$\mathsf {Sealing}$$is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The$$\mathsf {Largest\ Suslin\ Axiom}$$($$\mathsf {LSA}$$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let$$\mathsf {LSA-over-uB}$$be the statement that in all (set) generic extensions there is a model of$$\mathsf {LSA}$$whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory,$$\mathsf {Sealing}$$is equiconsistent with$$\mathsf {LSA-over-uB}$$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that$$\mathsf {Sealing}$$is weaker than the theory ‘$$\mathsf {ZFC} +$$there is a Woodin cardinal which is a limit of Woodin cardinals’. A variation of$$\mathsf {Sealing}$$, called$$\mathsf {Tower\ Sealing}$$, is also shown to be equiconsistent with$$\mathsf {Sealing}$$over the same large cardinal theory. The result is proven via Woodin’s$$\mathsf {Core\ Model\ Induction}$$technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of$$\mathsf {CMI}$$as explained in the paper. 

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5. Do Subducted Seamounts Act as Weak Asperities?

   https://doi.org/10.1029/2023JB027551  

   Lee, Sungho; Choi, Eunseo; Scholz, Christopher H. (November 2023, Journal of Geophysical Research: Solid Earth)

   Abstract The additional work of ploughing makes seamounts more resistant to subduction and more strongly coupled than smoother areas. Nevertheless, the idea that subducted seamounts are weakly coupled and slip aseismically has become dominant in the last decade. This idea is primarily based on the claim that a seamount being subducted in the southern Japan Trench behaves this way. The key element in this assertion is that largeM ∼ 7 earthquakes that abut the leading edge of the seamount require that the seamount be aseismically sliding to initiate them. More recent observations show instead that the surrounding region is aseismically sliding while the seamount acts as a stationary buttress. Here we re‐examine this case and model it with both weak and strong asperity assumptions. Our modeling results show that only a strong asperity model can produce this type of earthquake. Strong asperities also rupture the seamount in great earthquakes with long recurrence times. This provides the previously unknown source for a series of great tsunami earthquakes that have occurred along the southern Japan Trench, the most recent being the 1677 M8.3–8.6 Enpo Boso‐oki tsunami earthquake. The “weak asperity” hypothesis is thus found to be false in this foundational example. 

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