终极L（数学论文）一

注意：一共划分（1/2）篇章！

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM

RUPERT MCALLUM

ABSTRACT.We will consider a number of new large-cardinal prop- erties, the c-tremendous cardinals for each limit ordinal α＞0，the hyper-tremendous cardinals，the-enormous cardinals for each limit ordinal α＞0，and the hyper-enormous cardinals. For limit ordinals a α＞0，the α-tremendous cardinals and hyper-tremendous cardinals have consistency strength between I3 and I2. The α- enormous cardinals and hyper-enormous cardinalshaveconsistency strength greater than I0, and also all the large-cardinal axioms discussed in the second part of Hugh Woodin's paper on suitable extender models, not known to be inconsistent with ZFC and of greater consistency strength than I0.Ralf Schindler and Victoria Gitman have developed the notion of a virtual large-cardinal prop- erty, and a clear sense can be given to the notions of“virtuallv

x-enormous"and“virtually hyper-enormous”. On the assumption that V＝HOD，a measurable cardinal can be shown to be vir- tually hyper-enormous. Using a definition of Ultimate-L given in Section 6, claimed to be the correct definition on the assumption that there is a proper class of -enormous cardinals for each limit ordinal α＞0，it can be shown that，if V is equal to Ultimate-L in the sense of that definition,then it follows that a virtually w-enormous cardinal is alimitof Ramsey cardinals.

One can introduce the notion of a hyper-enormous* cardinal，of somewhat less strength than a hyper-enormous cardinal, and it can be shown that a cardinal κ which is a critical point of an clementary embedding j：Vλ＋₂ ≺ V＋₂，in a context not assuming choiceis necessarily a hyper-enormous* cardinal.(It is quitelikely that assuming the Axiom of Depending Choice it can be shown to be hyper-enormous，too，but the former proposition is all that is needed for what follows.) Building on this insight，we can obtain the result that the existence of such an elementary embedding is in fact outright inconsistent with ZF.

Finally,the assertion that there is a proper class of α-enormous cardinals for each limit ordinal α＞0 can be shown to imply a version of the Ultimate-L Conjecture.

Keywords：Ultimate-L program，large cardinals. MSC：03E45，03E55

To my beloved ωife Mαri Mnαtsαkαnyαn，ωithout ωhom this ωork

ωould not hαυe been possible.

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 3

ACKNOWLEDGEMENTS

Hugh Woodin provided very helpful feedback on a number of early drafts of this work in which a number of unsatisfactory definitions of the notion of an α-enormous cardinal were formulated，and I am very thankful for his assistance.

In what follows we will present a number of new large-cardinal ax- ioms,and applications of them.Let us begin by presenting the defini- tions of the newlarge-cardinal properties tobeconsidered.

1.DEFINITIONS OF THE NEW LARGE-CARDINAL PROPERTIES

Definition 1.1. Suppose that α is a limit ordinal such that α＞0.We say that an uncountable regular cardinal κ is α-tremendous if there exists an increasing sequence of cardinals 〈κᵦ：β＜α〉such that Vκᵦ ≺Vκ for all β＜α ，and if n＞1 and〈βᵢ：i＜n〉is an increasing sequence of ordinals less than α，then if β₀ ≠ 0 then for all β'＜β₀ there is an elementary embedding j：Vκᵦₙ₋₂ ≺ Vκᵦₙ₋₁，with critical point κᵦ' and

j(κᵦ')＝κᵦ₀ and j(κᵦ₁)＝κᵦᵢ₊₁ for all i such that

0 ≤ i＜n – 2，and if β₀＝0 then there is an elementary embedding j：Vκᵦₙ₋₂ ≺ Vκᵦₙ₋₁ withcritical point κ'＜κ₀ and j(κ')＝κ₀ and j(κᵦᵢ) ＝κᵦᵢ₊₁ for all i such that 0 ≤ i＜n – 2.

Definition 1.2.A cardinal κ such that κ is κ-tremendous is said to be hyper-tremendous.

Definition 1.3.Suppose that α is a limit ordinal such that α＞0，and that〈κᵦ：β＜α〉together with a family F of elementary embeddings witness that κ is α-tremendous，with just one embedding in the family F witnessing α-tremendousness for each finite sequence of ordinals less than α.Suppose that，given any ω-sequence of ordinals〈βᵢ：i＜ω〉less than α，there is an elementary embedding j：Vλ₊₁ ≺ Vλ₊₁ with critical sequence 〈κᵦᵢ：i＜ω〉，obtained by gluing together the obvious ω-sequence of embeddings from F，where λ：＝ supₙ∈ω κᵦₙ . Suppose further that there is an elementary embedding k：V ≺ M，fixing all regular cardinals greater than λ，with Vλ ⊆ M and (Vλ₊₁)ᴹ ≺ Vλ₊₁，and k│Vλ＝j│Vλ. If β：=supₙ∈ω βₙ＜α then let ρ：＝κᵦ，otherwise let ρ：＝κ . Suppose that.whenever we have Vλ₊₁ ⊆ S ⊆ Vᵨ and

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

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

normal ultrafilter on κ₀.denoted by U in what follows. We may use reflection to show the existence of a κ'₀＜κ₀ belonging to any fixed member of U，such that 〈κ'₀，κ₀，κ₁ . . .〉，together with a certain family F₀ of elementary embeddings，witness ω-tremendousness of κ.Then we can repeat this procedure to find a κ'₁ belonging to the same fixed member of U such that κ'₀＜κ'₁＜κ₀，such that 〈κ'₀，κ'₁，κ₀，κ₁，. . .〉，together with a certain family F₁ of elementarv embeddings，witness

ω-tremendousness of κ.We can continue in this way，and we can also ar- range things so that there is a sequence of embeddings，jₙ： Vκ'ₙ₋₁ ≺ Vκ'ₙ with critical point κ'₀ for all n＞1，which can be chosen by induc-tion，such that for each n＞1，jₙ coheres with jₘ for all m such that 1＜m＜n，and the embeddings from Fₙ that have critical sequence beginning with 〈κ'₀，κ'₁，. . .，κ'ₙ₋₂〉can be chosen so as to be coherent with jₙ. In this way we obtain a sequence 〈κ'ₙ：n＜ω〉and a sequence of embeddings jₙ with the previously stated properties. The existence of such a pair of sequences for any given element of U yields the claimed result. ▢

Theorem 2.4.Suppose thαt κ is αn l2 cαrdinαl. Then there is α nor-mαl ultrαfilter U on κ concentrαting on the hyper-tremendous cαrdinαls.

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

(1) N＝L(N) ∩Vλ₊₂ and crit(j)＜λ；

(2) Nλ ⊆ L(N)；

(3)for all F：Vλ₊₁ → N \ {∅} such that F ∈ L(N) there exists G：Vλ₊₁ → Vλ₊₁ such that G ∈ N and such that for all A ∈ Vλ₊₁，G(A) ∈ F(A).

We shall state one claim that makes reference to Laver’s axiom at the end of Section 6，but shall not refer to it further duringthis section.

Definition 3.2.We define the sequence〈E⁰α(Vλ₊₁)：α＜ΥVλ₊₁〉to be the maximum sequence such that the following hold.

(1)E⁰₀(V₊₁)＝L(V₊₁)∩Vλ₊₂ and E⁰₁(Vλ₊₁)＝L((Vλ₊₁)#)∩Vλ₊₂.

(2)Suppose α＜ΥVλ₊₁ and α is a limit ordinal. Then E⁰α(Vλ₊₁)＝L(U {E⁰ᵦ(Vλ₊₁)：β＜α})∩Vλ₊₂.

(3)Suppose α＋1＜ΥVλ₊₁. Then for some X ∈ E⁰α₊₁(Vλ₊₁)，E⁰α(Vλ₊₁)＜X，where by this we mean that there is a surjection π：Vλ₊₁ → E⁰α(Vλ₊₁) with π ∈ L(X，Vλ₊₁)，and B⁰α₊₁(Vλ₊₁)＝L(X，Vλ₊₁)∩Vλ₊₂，and if α＋2＜ΥVλ₊₁ then E⁰α₊₂(Vλ₊₁)＝L((X，Vλ₊₁)#)∩Vλ₊₂.

(4)Suppose α＜Υλ₊₁. Then there exists X ⊆ Vλ₊₁ such that E⁰α(Vλ₊₁) ⊆ L(X，Vλ₊₁) and such that there is a proper elementary em-bedding j：L(X，Vλ₊₁) ≺ L(X，Vλ₊₁)，where this means that j is non-trivial with critical point below λ，and for all X' ∈ L (X，Vλ₊₁)∩Vλ₊₂ there exists a Y ∈ L(X，Vλ₊₁)∩Vλ₊₂ such that 〈Xᵢ：i＜ω〉∈L(Y，Vλ₊₁)，where X₀＝X' and Xᵢ₊₁＝j(Xᵢ) for all i ≥ 0.

(5) Suppose α＜ΥVλ₊₁，α is a limit ordinal， and let N＝E⁰α(Vλ₊₁).Then either

(a) (cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＜λ，or

(b)(cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＞λ and for some Z ∈ N，L(N)＝(HODVλ₊₁∪{Z})ᴸ⁽ᴺ⁾.

Here 𝚹ᴺ＝sup{𝚹ᴸ⁽ˣ，ⱽλ⁺¹⁾：X ∈ N} where 𝚹ᴸ⁽ˣ，ⱽλ⁺¹⁾ is the supremum of the ordinals γ which can serve as the codomain of a suriection with domain Vλ₊₁ where the surjection is an element of L(X，Vλ₊₁).

(6)Suppose α＋1＜TVλ₊₁，α is a limit ordinal，and let N＝E⁰α(Vλ₊₁).Then either

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM7

(a) (cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＜λ，and E⁰α₊₁(Vλ₊₁)＝L(Nλ，N)∩Vλ₊₂，or

(b) (cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＞λ，and E⁰α₊₁(Vλ₊₁)＝L(ε(N)，N)∩Vλ₊₂，where ε(N)is the set of elementary embeddings k：N ≺ N.

Define N：＝L(∪{E⁰α(Vλ₊₁)│α＜ΥVλ₊₁}) ∩ Vλ₊₂. Suppose that

cof(𝚹ᴺ)＞λ and L(N) ≠ (HODVλ₊₁∪{Z})ᴸ⁽ᴺ⁾ for all Z ∈ N，and further there is an elementary embedding j：L(N) ≺ L(N) with

crit(j)＜λ. Then we say that λ satisfies Woodin’s axiom.

Theorem 3.3.Suppose thαt κ is ω-enormous αs ωitnessed by the se-guence〈κₙ：n＜ω〉.Then Vκ₀ is α model for the αssertion thαt there is α proper clαss of λ sαtisfying Woodin’s αxiom.

Proof.Assume the hypotheses and notation given in the statement of the theorem. If we let λ：＝sup{κₙ：n＜ω}，then there is an elementary embedding j：Vλ₊₁ ≺ Vλ₊₁ with critical sequence〈κₙ：n＜ω〉.It is clearly sufficient to prove that λ satisfies Woodin’s axiom and can also beshown tosatisty Laver’s axiom on the assumption that V＝HOD，via embeddings extending j，in Vκ.

Suppose that a sequence of sets〈E⁰α(Vλ₊₁)：α＜β〉 satisfies require- ments (1)-(6) of the definition of Woodin’s axiom, relativized to Vκ，for some β ≤ ΥVλ₊₁，and define N to be the unique possible candi-date for E⁰ᵦ if it exists. It can be shown by transinite induction that L(j(N)∪Vλ₊₁)∩Vκ＝L(N)∩Vκ. Then.considering that the action of j on the elements of such an N is determined by j│Vλ，and using the hypothesis of ω-enormousness，it can be shown by transfinite induc-tion that the restriction of j to L(N)∩Vκ is an elementary embedding L(N)∩Vκ ≺ L(N)∩Vκ，proper in the case where β＜ΥVλ₊₁. Since this is so for every N satisfying all the previously stated requirements we may now conclude by transfinite induction that λ satisfies Woodin’s axiom in Vκ via an embedding extending the restriction of j to Vλ₊₁.This completes the argument. ▢

This completes the demonstration that α-enormous and hyper-enormous cardinals have greater consistency strength than any previously con-sidered extension of ZFC not known to be inconsistent.

4. VIRTUALLY α-ENORMOUS AND HYPER-ENORMOUS CARDINALS

Ralf Schindler and Victoria Gitman in [4] have introduced the notion of virtual large-cardinal properties.Given any large-cardinal property defined with reference to a set-sized elementary embedding j：Vα ≺ Vᵦ or family of such embeddings，the corresponding virtual large-cardinal

8 MᶜALLUM

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.

Definition5.1.Suppose that α is a limit ordinal such that α＞0，and that〈κᵦ：β＜α〉together with a family F of elementary embeddings witness that κ is α-tremendous，with just one embedding in the family F witnessing α-tremendousness for each finite sequence of ordinals less than α.Suppose that，given any ω-sequence of ordinals〈βᵢ：i＜ω〉less than α. there is an elementary embedding j：Vλ₊₁ ≺ Vλ₊₁ with critical sequence 〈κᵦᵢ：i＜ω〉，obtained by gluing together the obvious ω-sequence of embeddings from F，where λ：＝supₙ∈ω κᵦₙ .Then the cardinal κ is said to be α-enormous*.

Definition 5.2.Suppose that a cardinal κ is κ-enormous*.Then κ is said to be hyper-enormous*.

In this section we wish to prove the following theorem.

Theorem 5.3.It is not consistent ωith ZF thαt there exists αn ordinαl λ αnd α non-triυiαl elementαry embedding j：Vλ₊₂ ≺ Vλ₊₂.

Proof.The same reasoning that shows that every I2 cardinal κ has a normal ultrafilter U concentrating on the hyper-tremendous cardinals，also shows in ZF that if κ is a critical point of an elementary embedding

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 9

Vλ₊₂ ≺ Vλ₊₂，then there is a normal ultrafilter U concentrating on a sequence 〈κα：α＜κ〉 which witnesses that κ is hyper-enormous*.In[6]，Gabriel Goldberg has also shown，using iterated collapse forcing，that if the existence of such an embedding is consistent with ZF then it is also consistent with Vλ being well-orderable (using a well-ordering where if m＜n and 〈κ'ᵢ：i ∈ ω〉is the critical sequence of j ，the map jⁿ⁻ᵐ maps the restriction of the well-ordering to Vκ'ₘ₊₁ \ Vκ'ₘ to the restriction of the well-ordering to Vκ'ₙ₊₁ \ Vκ'ₙ).

So suppose the conjunction of these two hypotheses，in ZF，and let κ be the critical point of the embedding and let S be the sequence 〈κα：α＜κ〉 mentioned before. For each α＜κ，let Eα be the equivalence relation on [κα]ω which holds of two sets of ordinals less than κα whose elements in order constitute two sequences of countably infinite length，if and only if the two seguences in question have the same tail. There is a sequence 〈Cα：α＜κ〉 such that for each α＜κ，Cα is a choice set for the equivalence classes of Eα，and for each pair (α，β) with

α＜β，when one is choosing an elementary embedding j' from a fixed family of embeddings witnessing the hyper-enormous*ness of κ，one can without loss of generality choose it so that j'(Cα)＝Cᵦ. Then using the embedding j one can extend this to a family of choice sets 〈Cα：α＜λ〉，such that if α ＜β ＜κ'ₙ then an elementary embedding j' can be chosen which is part of a fixed family of embeddings witnessing the hyper-enormous*ness of κ'ₙ，such that j'(Cα)＝Cᵦ.

This allows one to construct a choice set C for the corresponding equivalence relation E on [λ]ω. The method is as follows. Given an X ∈ [λ]ω， it follows from our stated assumptions that one may find an X' ∈ Vκ'ₙ for any given n＞0 such that X' ∈ [ρ]ω for a ρ of cofinality ω between κ'ₙ₋₁ and κ'ₙ and an embedding ex.ₙ：Vᵨ₊₁ ≺ Vλ₊₁ which carries a sequence of hyper-enormous* cardinals cofinal in ρ to the critical sequence of j or a tail thereof，such that ex，ₙ(X')＝X. This can be used together with the sequence of choice sets 〈Cα：α＜λ〉to choose a member of the equivalence class of X，depending on n. Using the relation mentioned earlier between the different choice sets Cα，one can argue that this data can be chosen in such a way that the function mapping n to the chosen member of the equivalence class of X is in fact eventually constant，and that a choice set for the equivalence relation E can be constructed in this way.

However，this gives rise to a contradiction using the method of proof of Kunen’s inconsistency theorem. And this contradiction was obtained

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