The spectrum of elementary embeddings j: V→ V

Annals of Pure and Applied Logic 139 (1):327-399 (2006)
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Abstract

In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j:V→V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V→V relative to models of ZFC. We do this by working in the extended language , using as axioms all the usual axioms of ZFC , along with an axiom schema that asserts that j is a nontrivial elementary embedding. Without additional axiomatic assumptions on j, we show that that the resulting theory is weaker than an ω-Erdös cardinal, but stronger than n-ineffables. We show that natural models of ZFC+BTEE give rise to Schindler’s remarkable cardinals. The approach to inconsistency from ZFC+BTEE forks into two paths: extensions of ZFC+BTEE+Cofinal Axiom and ZFC+BTEE+¬Cofinal Axiom, where Cofinal Axiom asserts that the critical sequence is cofinal in the ordinals. We describe near-minimal inconsistent extensions of each of these theories. The path toward inconsistency from ZFC+BTEE+¬Cofinal Axiom is paved with a sequence of theories of increasing large cardinal strength. Indeed, the extensions of the theory ZFC +“j is a nontrivial elementary embedding” form a hierarchy of axioms, ranging in strength from Con to the existence of a cardinal that is super-n-huge for every n, to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way. We also isolate several intermediate-strength axioms which, when added to ZFC+BTEE, produce theories having strengths in the vicinity of a measurable cardinal of high Mitchell order, a strong cardinal, ω Woodin cardinals, and n-huge cardinals. We also determine precisely which combinations of axioms, of the form result in inconsistency

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Citations of this work

The Axiom of Infinity and Transformations j: V → V.Paul Corazza - 2010 - Bulletin of Symbolic Logic 16 (1):37-84.
Lifting elementary embeddings j: V λ → V λ. [REVIEW]Paul Corazza - 2007 - Archive for Mathematical Logic 46 (2):61-72.
Large transitive models in local ZFC.Athanassios Tzouvaras - 2014 - Archive for Mathematical Logic 53 (3-4):233-260.
Lifting elementary embeddings j: Vλ → Vλ.Paul Corazza - 2007 - Archive for Mathematical Logic 46 (2):61-72.

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References found in this work

Elementary embeddings and infinitary combinatorics.Kenneth Kunen - 1971 - Journal of Symbolic Logic 36 (3):407-413.
Proper forcing and remarkable cardinals.Ralf-Dieter Schindler - 2000 - Bulletin of Symbolic Logic 6 (2):176-184.
The wholeness axiom and Laver sequences.Paul Corazza - 2000 - Annals of Pure and Applied Logic 105 (1-3):157-260.
The Wholeness Axioms and V=HOD.Joel David Hamkins - 2001 - Archive for Mathematical Logic 40 (1):1-8.
Consistency of V = HOD with the wholeness axiom.Paul Corazza - 2000 - Archive for Mathematical Logic 39 (3):219-226.

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