 Award ID(s):
 1945592
 NSFPAR ID:
 10320417
 Date Published:
 Journal Name:
 The bulletin of symbolic logic
 ISSN:
 19435894
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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 LSAoveruB be the statement that in all (set) generic extensions there is a model of LSA whose Suslin, coSuslin 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 LSAoverUB fails; this implies that LSAoverUB 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 Steelβs wellknown theorem that βAD in L(R) holds in all generic extensionsβ is equivalent to βthe theory of L(R) is sealedβ in the presence of a proper class of measurable cardinals.more » « less

Abstract A set of reals is
universally Baire if all of its continuous preimages in topological spaces have the Baire property. 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.$\mathsf {Sealing}$ The
($\mathsf {Largest\ Suslin\ Axiom}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let$\mathsf {LSA}$ be the statement that in all (set) generic extensions there is a model of$\mathsf {LSAoveruB}$ whose Suslin, coSuslin sets are the universally Baire sets.$\mathsf {LSA}$ We show that over some mild large cardinal theory,
is equiconsistent with$\mathsf {Sealing}$ . In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that$\mathsf {LSAoveruB}$ is weaker than the theory β$\mathsf {Sealing}$ there is a Woodin cardinal which is a limit of Woodin cardinalsβ.$\mathsf {ZFC} +$ A variation of
, called$\mathsf {Sealing}$ , is also shown to be equiconsistent with$\mathsf {Tower\ Sealing}$ over the same large cardinal theory.$\mathsf {Sealing}$ The result is proven via Woodinβs
technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of$\mathsf {Core\ Model\ Induction}$ as explained in the paper.$\mathsf {CMI}$ 
The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper.more » « less

This paper studies structural consequences of supercompact ness of Ο1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from βΟ1 is supercompactβ. βΟ1 is supercompactβ also implies that AD+, a strengthening of the Axiom of Determinacy (AD), is equiv alent to ADR. It is shown that βΟ1 is supercompactβ does not imply AD. The most one can hope for is Suslin coSuslin determinacy. We show that this follows from βΟ1 is supercompactβ and Hod Pair Capturing (HPC), an innermodel theoretic hypothesis that imposes certain smallness con ditions on the universe of sets. βΟ1 is supercompactβ on its own implies that every Suslin coSuslin set is the projection of a determined (in fact, homogenously Suslin) set. βΟ1 is supercompactβ also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.more » « less

Abstract The HOD Dichotomy Theorem states that if there is an extendible cardinal, Ξ΄ , then either HOD is βcloseβ to V (in the sense that it correctly computes successors of singular cardinals greater than Ξ΄ ) or HOD is βfarβ from V (in the sense that all regular cardinals greater than or equal to Ξ΄ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD βcloseβ to V , or βfarβ from V ? There is a program aimed at establishing the first alternativeβthe βcloseβ side of the HOD Dichotomy. This is the program of inner model theory . In recent years the third author has provided evidence that there is an ultimate inner modelβUltimate L βand he has isolated a natural conjecture associated with the modelβthe Ultimate L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holdsβHOD is βcloseβ to V . This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternativeβthe βfarβ side of the HOD Dichotomy. This is the program of large cardinals beyond choice . Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate L Conjecture must fail. This is the future where chaos prevails.more » « less