终极L（数学论文）一 (14-5)

Definition 2.1. A cardinal κ is said to be an l3 cardinal if it is the critical point of an elementary embedding j：Vδ ≺ Vδ. I3 is the asser-tion that an l3 cardinal exists，and l3(κ，δ) is the assertion that the first statement holds for a particular pair of ordinals κ，δ such that κ＜δ.

Definition 2.2. A cardinal κ is said to be an I2 cardinal if it is the critical point of an elementary embedding j：V ≺ M such that Vδ ⊂ M where δ is the least ordinal greater than κ such that j(δ)＝δ. l2 is the assertion that an I2 cardinal exists，and I2(κ，δ) is the assertion that the first statement holds for a particular pair of ordinals κ，δ suchthat

κ＜δ.

In this section we wish to show that the α-tremendous cardinals and hyper-tremendous cardinals have consistency strength strictly between I3 and I2.

Theorem 2.3. Suppose that κ is ω-tremendous αs ωitnessed by 〈κᵢ： i＜ω〉.Then there is α normαl ultrαflter U on κ₀ such thαt the set of αll κ'＜κ₀ such thαt I3(κ'，δ) for some δ＜κ₀，is α member of U.

Proof.Suppose that κ is ω-tremendous and that 〈κᵢ：i ∈ ω〉to-gether with a certain family F of elementary embeddings witness the ω-tremendousness of κ .It can be assumed without loss of generality that all the embeddings in F with critical point κ₀ give rise to the same

NEWLARGE-CARDINALAXIOMSANDTHEULTIMATE-LPROGRAM 5

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