终极L（数学论文）二

注意：现状（2/2）篇章！

10 MᶜCALLUM

from a set of assumptions which are provably consistent byforcing relative only to ZF plus the existence of an elementary embedding Vλ₊₂ ≺ Vλ₊₂. Thus the existence of an elementary embedding Vλ₊₂ ≺ Vλ₊₂ is in fact inconsistent with ZF. ▢

6.A PROOF OF THE ULTIMATE-L CONJECTURE

In this section，we will seek to give a proof of Hugh Woodin's Ultimate-L Conjecture.The most important sources for Hugh Woodin’s Ultimate-L program are [1]，[2]，and [3]. We must begin by giving the statement of the axiom V＝Ultimate-L，following Definition 7.14 of [3].

Definition 6.1. The axiom V＝Ultimate-L is defined to be the asser-tion that

(1)There is a proper class of Woodin cardinals.

(2) Given any Σ₂-sentence ф which is true in V，there exists a univer-sally Baire set of reals A，such that，if 𝚹ᴸ⁽ᴬ，ℝ⁾ is defined to be the least ordinal 𝚹 such that there is no surjection from ℝ onto 𝚹 in L(A，ℝ)，then the sentence ф is true in HODᴸ⁽ᴬ，ℝ⁾∩V𝚹ᴸ₍ᴀ，R₎.

Now let us recall a set of definitions from [3].

Definition 6.2. Suppose that N is a transitive proper class model of ZFC and that δ is a supercompact cardinal in V. We say that N is a weak extender model for δ supercompact，if for all γ＞δ，there exists on Pδ(γ) a normal fine δ-complete measure μ，with μ (N∩Pδ(λ))＝1and μ∩N ∈ N.

Definition 6.3. A sequence N：＝ 〈Nα：α ∈ Ord〉is weakly Σ₂-definable if there is a formula ф(x)such that

(1)For all β＜η₁＜η₂＜η₃， if (Nф)ⱽη₁│β＝(Nф)ⱽη₃│β then (Nф)ⱽη₁│β＝(Nф)ⱽη₂│β＝(Nф)ⱽη₃│β；

(2)For all β ∈ Ord，N│β＝(Nф)ⱽη│β for sufficiently large η，where，for all γ， (Nф)ⱽγ＝{α ∈ Vᵧ：Vᵧ╞ф[α]}. Suppose N ⊂ V is an inner model such that N╞ ZFC. Then N is weakly Σ₂-definable if the sequence〈N∩Vα：α ∈ Ord〉is weakly Σ₂-deinable.

We can now state the result we plan to prove in this section.

Theorem 6.4.Suppose thαt there is α proper clαss of α-enormous cαrdinαls for eαch limit ordinαl α＞0.Then the folloωing υersion of the Ultimαte-L conjecture，giυen αs Conjecture 7.41 in [3]，holds.

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 11

Suppose thαt δ is αn extendible cαrdinαl (in fαct one cαn eυen suppose only thαt δ is α supercompαct cαrdinαl).Then there is α ωeαk extender model N for the supercompαctness of δ such thαt

(1) N is ωeαkly Σ₂-definαble αnd N ⊂ HOD；

(2) N＝“V＝Ultimαte-L”.

(3) N ╞ GCH.

Proof of Theorem 6.4.Let us give the long awaited definition of Ultimate-L. We claim that what follows is the correct definition of Ultimate-L，assuming that there are sufficiently many large cardinals in V as out-lined in the hypotheses for Theorem 6.4.The correct way to deline it when we are making weaker large-cardinal assumptions still remains to be discovered.

Suppose that κ is ω-enormous as witnessed by a sequence 〈κₙ：n＜ω〉，where clearly we may assume without loss of generality that the latter sequence is in HOD，and we will do so. Then we may consider all the sets ofordinals of the form j”λ where λ：＝sup{κₙ：n ∈ ω} for some sequence 〈κₙ：n ∈ ω〉 with the properties previously described，and j is an elementary embedding Vλ₊₁ ≺ Vλ₊₁ with critical sequence〈κₙ：n ∈ ω〉. Some of these sets of ordinals will be members of HOD. We define Ultimate-L to bethe smallest enlargement of L containing every member of a proper-class-length sequence of such set of ordinals in HOD， obtained in this way from ω-enormous cardinals κ，with exactly one such set of ordinals j” λ in the sequence for every possible value of λ. It will follow from the results of this section together with known results about the Ultimate-L Coniecture that Ultimate-L so defined does not in fact depend on the choice of the sequence. In this model，there will indeed exist at least one elementary embedding j：Vλ₊₁ ≺ Vλ₊₁ with critical sequence 〈κₙ：n ∈ ω〉，for every possible critical sequence in HOD arising from an embedding partially witnessing α-enormousness of some cardinal in V. The necessary elementary embedding within the model can be constructed using similar arguments to those of Section

3. Thus this model will still remain a model for the assertion that there is a proper class of α-enormous cardnals for each limit ordinal α＞0. Further clearly this inner model will satisfy GCH，and it is easily seen to be weakly Σ₂-definable and a subclass of HOD， and a weak extender model for the supercompactness of any cardinal δ which is supercompact in V，given that the stated large-cardinal hypothesis holds in V. In order to see the last point，it is necessary to observe that given the stated large-cardinal assumptions，any supercompact cardinal is necessarily hyper-enormous，and all necessary elementary

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embeddings for witnessing this do descend to the model Ultimate-L. We must now show that this model is indeed a model for the axiom V＝Ultimate-L as stated at the start of this section.

Clearly，our version of Ultimate-L is a model for the assertion that there is a proper class of Woodin cardinals. Suppose then，that some

Σ₂-sentence is true in Ultimate-L，so we are required to find a univer-sallyBaire set of reals A in Ultimate-L such that the Σ₂-sentence in questions holds in (HOD)ᴸ⁽ᴬ，ℝ⁾∩V𝚹ᴸ₍ᴀ，R₎. From well-known generic absoluteness results which are known to hold assuming a proper class of Woodin cardinals，it is sufficient to prove that this does obtain in some set-generic extension of Ultimate-L.So choose an ordinal β such that (Vᵦ)Utimate-L is a Σ₂-elementary substructure of Ultimate-L，and choose a γ＜β such that (Vᵧ)Ultimate-L models the Σ₂-sentence. Now consider a generic extension of Ultimate-L where A is a universally Baire set chosen to contain enough data so that.in the generic exten-sion，𝚹ᴸ⁽ᴬ，ℝ⁾ ≤ β，and (HOD)ᴸ⁽ᴬ，ℝ⁾∩ Vᵧ in the generic extension is equal to the intersection of the Ultimate-L of the ground model and Vᵧ. This can be arranged by ensuring that each ordinal less than β is collapsed to be countable in the generic extension，and that all the data for sets of ordinals less than γ which are needed to generate(Ultimate L∩Vᵧ)ⱽ are coded into the universally Baire set A which appears as a set of reals in the generic extension. In this generic extension，the de-sired result obtains，so the aforementioned generic absoluteness results imply that it obtains in our ground model as well. This completes the proof of Theorem 6.4. ▢

We should also note that if.V＝Ultimate-L，then if κ is ω-enormous as witnessed by〈κᵢ：i＜ω〉then Vκ₀ models the assertion that there is a proper class of λ satisfying Laver’s axiom，as defined in Section 3，and if κ is virtually ω-enormous as witnessed by〈κᵢ：i＜ω〉then Vκ₀ models the assertion that there is a proper class of Ramsey cardinals.

7. CONCLUDING REMARKS

The new large cardinals were inspired by Victoria Marshall’s work on reflection principles in[5] and are plausibly the correct generali-sation of the reflection principles which were demonstrated by her in that work to imply the existence of n-huge cardinals. The large cardi-nal axiom used to prove the Ultimate-L conjecture certainly has quite substantial consistency strength and some skepticism about its consis-tency would certainly be quite reasonable at this stage，but it may be

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 13

that the further study of the inner model theory of Ultimate-L and inner models which approximate it from within will provide new in- sights and increased confidence in consistency. In the mean time.it may very well be that the Ultimate-L conjecture is provable from just an extendible cardinal as originally envisaged by Hugh Woodin. so in that sense much work remains to be done.

If these new large cardinals are indeed consistent then the study of them appears to be quite fruitful，and it may be that the addition to ZFC of an axiom schema asserting for each n＜ω the existence of a hyper-enormous cardinal κ such that Vκ ≺ Σₙ V，together with the axiom V＝Ultimate-L，will eventually come to be accepted as the correct “effectively complete” theory of V，assuming that confidence develops over time that this theory is consistent.

REFERENCES

[1] Hugh Woodin. Suitable Extender Models I.Journαl of Mαthemαticαl Logic，Vol.10，Nos.1 & 2 (2010)，pp.101-339.

[2] Hugh Woodin. Suitable Extender Models II：Beyond ω-huge.Journαl of Mαth-emαticαl Logic，Vol.11，No. 2 (2011)，pp.151-436.

[3] Hugh W. Woodin.In Search Of Ultimate-L：The 19th Midrasha Mathematical Lectures. The Bulletin of Symbolic Loqic，23(1)，1-109.

[4] Victoria Gitman and Ralf Schindler. Virtual Large Cardinals，pre-print.

[5] M. Victoria Marshall R. Higher order reflection principles，Journαl of Symbolic Logic，vol. 54，no.2，1989，pp.474-489.

[6] Gabriel Goldberg. On the consistency strength of Reinhardt cardinals，pre-print.

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