终极L（数学论文）一 (14-3)

Definition 1.1. Suppose that α is a limit ordinal such that α＞0.We say that an uncountable regular cardinal κ is α-tremendous if there exists an increasing sequence of cardinals 〈κᵦ：β＜α〉such that Vκᵦ ≺Vκ for all β＜α ，and if n＞1 and〈βᵢ：i＜n〉is an increasing sequence of ordinals less than α，then if β₀ ≠ 0 then for all β'＜β₀ there is an elementary embedding j：Vκᵦₙ₋₂ ≺ Vκᵦₙ₋₁，with critical point κᵦ' and

j(κᵦ')＝κᵦ₀ and j(κᵦ₁)＝κᵦᵢ₊₁ for all i such that

0 ≤ i＜n – 2，and if β₀＝0 then there is an elementary embedding j：Vκᵦₙ₋₂ ≺ Vκᵦₙ₋₁ withcritical point κ'＜κ₀ and j(κ')＝κ₀ and j(κᵦᵢ) ＝κᵦᵢ₊₁ for all i such that 0 ≤ i＜n – 2.

Definition 1.2.A cardinal κ such that κ is κ-tremendous is said to be hyper-tremendous.

Definition 1.3.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 . Suppose further that there is an elementary embedding k：V ≺ M，fixing all regular cardinals greater than λ，with Vλ ⊆ M and (Vλ₊₁)ᴹ ≺ Vλ₊₁，and k│Vλ＝j│Vλ. If β：=supₙ∈ω βₙ＜α then let ρ：＝κᵦ，otherwise let ρ：＝κ . Suppose that.whenever we have Vλ₊₁ ⊆ S ⊆ Vᵨ and

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