Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T00:30:51.090Z Has data issue: false hasContentIssue false
  • English
  • Français

LARGE CARDINALS BEYOND CHOICE

Published online by Cambridge University Press:  20 August 2019

JOAN BAGARIA ,
PETER KOELLNER  and
W. HUGH WOODIN
JOAN BAGARIA
Affiliation:
ICREA (INSTITUCIÓ CATALANA DE RECERCA I ESTUDIS AVANÇATS) PASSEIG LLUÍS COMPANYS 23 BARCELONA 08010, CATALONIA, SPAIN E-mail:[email protected]
PETER KOELLNER
Affiliation:
DEPARTMENT OF PHILOSOPHY HARVARD UNIVERSITY CAMBRIDGE, MA02138, USA E-mail:[email protected]
W. HUGH WOODIN
Affiliation:
DEPARTMENT OF MATHEMATICS, DEPARTMENT OF PHILOSOPHY HARVARD UNIVERSITY CAMBRIDGE, MA02138, USA E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

03E2503E3503E4503E4703E5003E5500A30large cardinalsinner model theoryUltimate-LHODaxiom of choice
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 25 , Issue 3 , September 2019 , pp. 283 - 318
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory, vol. 2 (Kanamori, A. and Foreman, M., editors), Springer, Dordrecht, 2010, pp. 775–884.CrossRefGoogle Scholar
Devlin, K. J. and Jensen, R., Marginalia to a theorem of Silver, Proceedings of the ISILC logic conference (Kiel 1974) (Müller, G. H., Oberschelp, A., and Potthoff, K., editors), Lecture Notes in Mathematics, vol. 499, Springer, Berlin, 1975, pp. 115–142.Google Scholar
Goldberg, G., On the consistency strength of Reinhardt cardinals, unpublished, 2017.Google Scholar
Hayut, Y. and Karagila, A., Restrictions on forcings that change cofinalities. Archive for Mathematical Logic, vol. 55 (2016), pp. 373–384.CrossRefGoogle Scholar
Hugh Woodin, W., Suitable extender models I. Journal of Mathematical Logic, vol. 10 (2010), no. 1–2, pp. 101–339.CrossRefGoogle Scholar
Hugh Woodin, W., The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture, Set Theory, Arithmetic and the Foundations of Mathematics: Theorems, Philosophies (Kennedy, J. and Kossak, R., editors), Lecture Notes in Logic, vol. 36, Cambridge University Press, 2011, pp. 13–42.CrossRefGoogle Scholar
Hugh Woodin, W., In search of Ultimate-L: The 19th Midrasha Mathematicae lectures, this Bulletin, vol. 23 (2017), no. 1, pp. 1–106.Google Scholar
Kanamori, A., The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, second ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.Google Scholar
Kunen, K., Elementary embeddings and infinitary combinatorics. Journal of Symbolic Logic, vol. 36 (1971), pp. 407–413.CrossRefGoogle Scholar
Reinhardt, W., Topics in the metamathematics of set theory, Ph.D. thesis, University of California, Berkeley, 1967.Google Scholar
Silver, J. H., Some applications of model theory to set theory, Ph.D. thesis, University of California, Berkeley, 1966.Google Scholar
Woodin, H., The weak Ultimate-L Conjecture, Infinity, Computability, and Metamathematics (Geschke, S., Loewe, B., and Schlicht, P., editors), Tributes, vol. 23, College Publications, London, 2014, pp. 309–329.Google Scholar