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## Fundamenta Mathematicae

2003 | 179 | 1 | 43-60
Tytuł artykułu

### The gap between I₃ and the wholeness axiom

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∃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.
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Tom
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43-60
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2003
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• 310 East Washington Ave., Fairfield, IA 52556, U.S.A.
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