The gap between I₃ and the wholeness axiom

Paul Corazza. "The gap between I₃ and the wholeness axiom." Fundamenta Mathematicae 179.1 (2003): 43-60. <http://eudml.org/doc/283242>.

@article{PaulCorazza2003,
abstract = {∃κI₃(κ) is the assertion that there is an elementary embedding $i: V_\{λ\} → V_\{λ\}$ with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance of Replacement for j-formulas is included in WA, Kunen’s inconsistency argument is not applicable. It is known that an I₃ embedding $i: V_\{λ\} → V_\{λ\}$ induces a transitive model $⟨V_\{λ\},∈,i⟩$ of ZFC + WA. We study here the gap in consistency strength between I₃ and WA. We formulate a sequence of axioms ⟨Iⁿ₄: n ∈ ω⟩ each of which asserts the existence of a transitive model of ZFC + WA having strong closure properties. We show that I₃ represents the “limit” of the axioms Iⁿ₄ in a sense that is made precise.},
author = {Paul Corazza},
journal = {Fundamenta Mathematicae},
keywords = {large cardinal; ; wholeness axiom; elementary embedding; WA; huge cardinal; extender},
language = {eng},
number = {1},
pages = {43-60},
title = {The gap between I₃ and the wholeness axiom},
url = {http://eudml.org/doc/283242},
volume = {179},
year = {2003},
}

TY - JOUR
AU - Paul Corazza
TI - The gap between I₃ and the wholeness axiom
JO - Fundamenta Mathematicae
PY - 2003
VL - 179
IS - 1
SP - 43
EP - 60
AB - ∃κI₃(κ) is the assertion that there is an elementary embedding $i: V_{λ} → V_{λ}$ with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance of Replacement for j-formulas is included in WA, Kunen’s inconsistency argument is not applicable. It is known that an I₃ embedding $i: V_{λ} → V_{λ}$ induces a transitive model $⟨V_{λ},∈,i⟩$ of ZFC + WA. We study here the gap in consistency strength between I₃ and WA. We formulate a sequence of axioms ⟨Iⁿ₄: n ∈ ω⟩ each of which asserts the existence of a transitive model of ZFC + WA having strong closure properties. We show that I₃ represents the “limit” of the axioms Iⁿ₄ in a sense that is made precise.
LA - eng
KW - large cardinal; ; wholeness axiom; elementary embedding; WA; huge cardinal; extender
UR - http://eudml.org/doc/283242
ER -