∃κI₃(κ) is the assertion that there is an elementary embedding 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 induces a transitive model 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm179-1-4,
author = {Paul Corazza},
title = {The gap between I3 and the wholeness axiom},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {43-60},
zbl = {1044.03041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm179-1-4}
}
Paul Corazza. The gap between I₃ and the wholeness axiom. Fundamenta Mathematicae, Tome 177 (2003) pp. 43-60. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm179-1-4/