终极L（数学论文）一 (14-12)

Definition5.1.Suppose that α is a limit ordinal such that α＞0，and that〈κᵦ：β＜α〉together with a family F of elementary embeddings witness that κ is α-tremendous，with just one embedding in the family F witnessing α-tremendousness for each finite sequence of ordinals less than α.Suppose that，given any ω-sequence of ordinals〈βᵢ：i＜ω〉less than α. there is an elementary embedding j：Vλ₊₁ ≺ Vλ₊₁ with critical sequence 〈κᵦᵢ：i＜ω〉，obtained by gluing together the obvious ω-sequence of embeddings from F，where λ：＝supₙ∈ω κᵦₙ .Then the cardinal κ is said to be α-enormous*.

Definition 5.2.Suppose that a cardinal κ is κ-enormous*.Then κ is said to be hyper-enormous*.

In this section we wish to prove the following theorem.

Theorem 5. is not consistent ωith ZF thαt there exists αn ordinαl λ αnd α non-triυiαl elementαry embedding j：Vλ₊₂ ≺ Vλ₊₂.

Proof.The same reasoning that shows that every I2 cardinal κ has a normal ultrafilter U concentrating on the hyper-tremendous cardinals，also shows in ZF that if κ is a critical point of an elementary embedding

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 9

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