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The gap between I₃ and the wholeness axiom

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Corazza P.

Fundamenta Mathematicae

2003

tom 179

nr 1

43-60

∃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.

A new large cardinal and Laver sequences for extendibles

100%

Corazza P.

Fundamenta Mathematicae

1997

tom 152

nr 2

183-188

We define a new large cardinal axiom that fits between $A_3$ and $A_4$ in the hierarchy of axioms described in [SRK]. We use this new axiom to obtain a Laver sequence for extendible cardinals, improving the known large cardinal upper bound for the existence of such sequences.

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2 Fundamenta Mathematicae

2 Corazza P.

1 2003

1 1997

Wyniki wyszukiwania

The gap between I₃ and the wholeness axiom

A new large cardinal and Laver sequences for extendibles