终极L（数学论文）一 (14-7)

Proof. Suppose that κ is an I2 cardinal and let the elementary embed-ding j：V ≺ M with critical point κ witness that κ is an I2 cardinal， the supremum of the critical sequence being δ . If we let U be the ul-trafilter on κ arising from j we can easily show that the set of κ'＜κ such that there is an elementary embedding kκ'：Vδ ≺ Vδ，with critical sequence consisting of κ' followed by the critical sequence of j，is a member of U (denoted by X hereafter). Then the sequence of ordinals belonging to this set，together with a family of embeddings that can be derived from the sequence of embeddings 〈kκ'：κ' ∈ X〉witness that κ is hyper-tremendous. Since it also follows that is hyper-tremendous in M，the desired result follows. ▢

This completes the proof that the α-tremendous cardinalsand hyper-tremendous cardinals have consistency strength strictly between I3 and I2. In the next section we discuss the consistency strength of α-enormous and hyper-enormous cardinals.

3. CONSISTENCY STRENGTH OF α-ENORMOUS AND

HYPER-ENORMOUS CARDINALS

We wish to show that α-enormous cardinals and hyper-enormous cardinals have consistency strength greater than any previously con-sidered large-cardinal axiom not known to be inconsistent with ZFC.

6 MCALLUM

We shall begin by defining some large-cardinal axioms discussed in[2].

Definition 3.1. We say that an ordinal A satisies Laver’s axiom if the following holds.There is a set N such that Vλ₊₁ ⊆ N ⊊ Vλ₊₂ and an elementary embedding j：L (N) ≺ L(N)，such that

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