An official website of the United States government
Abstract The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ , then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V , or “far” from V ? There is a program aimed at establishing the first alternative—the “close” 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 model—Ultimate- L —and he has isolated a natural conjecture associated with the model—the Ultimate- L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” 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 alternative—the “far” 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
Model theoretic characterizations of large cardinals revisited
In Boney [Israel J. Math. 236 (2020), pp. 133–181], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.
more »
« less
- Award ID(s):
- 2339018
- PAR ID:
- 10610263
- Publisher / Repository:
- American Mathematical Society
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society
- ISSN:
- 0002-9947
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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
-
Abstract We strengthen a result of Bagaria and Magidor (Trans. Amer. Math. Soc.366(2014), no. 4, 1857–1877) about the relationship between large cardinals and torsion classes of abelian groups, and prove thattheMaximum Deconstructibilityprinciple introduced in Cox (J. Pure Appl. Algebra226(2022), no. 5) requires large cardinals; it sits, implication‐wise, between Vopěnka's Principle and the existence of an ‐strongly compact cardinal.While deconstructibility of a class of modules always implies the precovering property by Saorín and Šťovíček (Adv. Math.228(2011), no. 2, 968–1007), the concepts are (consistently) nonequivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.more » « less
-
Abstract We present an alternative proof that from large cardinals, we can force the tree property at $$\kappa ^+$$ and $$\kappa ^{++}$$ simultaneously for a singular strong limit cardinal $$\kappa $$ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $$\kappa =\aleph _{\omega ^2}$$ .more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/tran/9067
