终极L（数学论文）一 (14-11)

property is defined in the same way except by means of elementarv em- beddings j：(Vα)ⱽ ≺ (Vᵦ)ⱽ where j ∈ V [G] for a set generic extension of V. The notion of a virtually α-enormous or hyper-enormous cardinal is clear.We state a result about virtually hyper-enormous cardinals in this section and shall state a result about virtually ω-enormous cardi-nals later in Section 6.

Theorem 4.1. If κ is α meαsurαble cαrdinαl，αnd V＝HOD，then there is α sequence cofinαl in κ ωitnessing the υirtuαl huper-enormousness of κ.

Proof.Suppose that j：V ≺ M with critical point κ witnesses the measurability of κ.Then there is an elementary embedding j'：Vκ₊₁ ≺ (M∩Vⱼ₍κ₎₊₁)which appears in a generic extension of M(here using the hypothesis V＝HOD). Iterating reflection yields the desired result. ▢

5.NCONSISTENCY OF THE CHOICELESS CARDINALS

It is quite likely that the critical point of a non-trivial elementary embedding j：Vλ₊₂ ≺ Vλ₊₂ can be shown to be hyper-enormous assum-ing the Axiom of Depending Choice (but not full Choice，obviously). However，in what follows we shall only need to use a weaker statement，which can be proved without any form of Choice.

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