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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:joan.bagaria@icrea.cat
PETER KOELLNER
Affiliation:
DEPARTMENT OF PHILOSOPHY HARVARD UNIVERSITY CAMBRIDGE, MA02138, USA E-mail:koellner@fas.harvard.edu
W. HUGH WOODIN
Affiliation:
DEPARTMENT OF MATHEMATICS, DEPARTMENT OF PHILOSOPHY HARVARD UNIVERSITY CAMBRIDGE, MA02138, USA E-mail:woodin@math.harvard.edu
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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 

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