终极L（数学论文）一 (14-4)

S ∈ L(Vλ₊₁) [X] where X：＝(eᵢ” δᵢ：i＜n) for some finite ordinal n where each eᵢ is an elementary embedding with critical point greater than λ with δᵢ the supremum of the critical sequence of eᵢ and the δᵢ are pairwise distinct，and k(S) ⊆ S，then we have k(S) ≺ S.If all these conditions are satisfied，then the cardinal κ is said to be α-enormous.

4 MᶜALLUM

Definition 1.4. A cardinal κ such that κ is κ-enormous is said to be hyper-enormous.

We will shortly establish that the α-tremendous cardinals and hyper-tremendous cardinals are consistent relative to I2.We shall also estab-lish that the α-enormous cardinals and hyper-enormous cardinals have greater consistency strength than I0，or any other previously consid-ered large-cardinal axiom not known to be inconsistent with ZFC.The finalsectionwill brieflydiscuss the source of inspirationforthe original formulation of the definitions，which may serve as some motivation for assuming that these large cardinals are consistent with ZFC，and the results proved in the sequel may provide some additional motivation for assuming consistency.

Let us begin by establishing that the α-tremendous cardinals for limit ordinals α＞0 and the hyper-tremendous cardinalshaveconsistency strength strictly between I3 and I2.

2.THE CONSISTENCYSTRENGTH OF THE-TREMENDOUS CARDINALS AND HYPER-TREMENDOUSCARDINALS

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