The DDG and Very Large Cardinals Daniel Marini Universit` a degli - - PowerPoint PPT Presentation

The DDG and Very Large Cardinals

Daniel Marini Universit` a degli Studi di Torino

26/03/2019

Daniel Marini Universit ` a degli Studi di Torino

Set-Theoretic Geology

It concerns a switch in perspective of the forcing

• method. One asks himself if the universe 푉 might have

arisen by generic extension 푊[퐺] for some class 푊 and some 푊-generic filter 퐺 ⊆ 퐏 ∈ 푊. We shall assume some knowledge of set theory and basic notions of forcing. Some notations and preliminary results We will work in ZFC set theory, consisting of the axioms

• f Extensionality (Ext), Foundation (Fnd), Pairing (Prn),

Union (Unn), Power set (Pwr), Infinity (Inf), Separation (Spr), Collection (Clt), and Choice (AC). Sometimes, we need to refer to some weakened form

• f this theory. Mostly, ZFC − Pwr, ZFC훿, and ZF.

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Theorem A transitive class 푊 is an inner model of ZF if and only if it contains all ordinals, is almost universal, and closed under the G¨

• del operations. In particular,

being an inner model of ZFC is first-order expressible. Theorem Let ̇ 퐐 be a 퐏-name for a forcing in 푉. Let 퐺 ⊆ 퐏 ∈ 푉 be 푉-generic and 퐻 ⊆ val( ̇ 퐐, 퐺) be 푉[퐺]-

• generic. There exists a filter 퐺 ∗ 퐻 contained in the

forcing 퐏 ∗ ̇ 퐐 ∈ 푉 such that 푉[퐺][퐻] = 푉[퐺 ∗ 퐻].

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Sometimes it is useful to think about forcing with complete Boolean algebras. In such cases, we consider the Boolean completion of our forcing. Theorem Let 퐺 ⊆ 퐁 be generic. If 푊 is an inner model

• f ZFC and 푉 ⊆ 푊 ⊆ 푉[퐺], then there is a complete

sub-algebra 퐃 of 퐁 such that 푊 = 푉[퐃 ∩ 퐺]. Moreover, 푉[퐺] is a generic extension of 푊, since 퐁 is forcing equivalent to the iteration 퐃 ∗ ̌ 퐁∕ ̇ 퐺0, where 퐺0 = 퐃 ∩ 퐺 and ̌ 퐁∕ ̇ 퐺0 = { ̌ 푏 ∈ ̌ 퐵 ∶ ∀ ̌ 푝 ∈ ̇ 퐺0 (̌ 횤( ̌ 푝) ∥ ̌ 푞)}.

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Definition Let 푊 ⊆ 푉 be a transitive model of ZFC and 훿 ∈ 푉 be a cardinal. We say that 푊 exhibits the

• 훿-cover property for 푉 if for every 퐴 ∈ 푉 with 퐴 ⊆ 푊

and |퐴|푉 < 훿, there is 퐵 ∈ 푊 such that 퐴 ⊆ 퐵 and |퐵|푊 < 훿;

• 훿-approximation property for 푉 if for all 퐴 ∈ 푉 such

that – 퐴 ⊆ 푊; – for all 퐵 ∈ 푊 with |퐵|푊 < 훿, 퐴 ∩ 퐵 ∈ 푊; then 퐴 ∈ 푊. Lemma Let 푊[퐺] ⊇ 푊 be a generic extension by a non-trivial forcing notion 퐏 ∈ 푊 with 퐺 a ⟨푊, 퐏⟩-generic

• filter. If 훿 = |퐏|+, then 푊 satisfies the 훿-approximation

and 훿-cover properties for 푊[퐺].

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Definability of Grounds

A ground 푊 of 푉 is an inner model of ZFC such that 푉 can be obtained by set-forcing extension over 푊. We will also write GRD(푊, 푉). Theorem [The Ground-Model Definability Theorem] There exists a specific first-order formula ΦGRD(푦, 푥) such that 푉 = 푊[퐺] is a generic extension of a ground 푊 by ⟨푊, 퐏⟩-generic filter 퐺 with 퐏 ∈ 푊 if and only if there is 푟 ∈ 푊 such that 푊 = {푥 ∈ 푉 ∶ Φ푉

GRD(푟, 푥)}.

We will write ΦGRD(푟, 푉) in place of 푊.

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The formula ΦGRD Let Φ′

GRD(훿, 푧, 퐏, 퐺) be the formula:

(I) 훿 is a regular cardinal; (II) 퐏 ∈ 푧 is a forcing of size less than 훿; (III) 퐺 is a ⟨푧, 퐏⟩-generic; (IV) for every ℶ-fixed point 훾 > 훿 of cofinality greater than 훿, there exists a transitive structure 푀 of height 훾 such that (i) 푀 is a model of ZFC훿; (ii) 푧 = (2<훿)푀; (iii) 푀[퐺] = 푉훾; (iv) 푀 has the 훿-cover and 훿-approximation properties for 푉. ΦGRD(푟, 푥) is ∃훾 (Φ′

GRD(훿, 푟, 퐏, 퐺) ∧ 푥 ∈ 푊훾

), where 훾 and 푊훾 are as in (IV), 훿 = (|퐏|+)푉, and 푟 = (2<훿)푊.

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Hence, we are able to define the class of all grounds

• f 푉 as {ΦGRD(푟, 푉) ∶ 푟 ∈ 푉}.

How are grounds seen in different universes?

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Hence, we are able to define the class of all grounds

• f 푉 as {ΦGRD(푟, 푉) ∶ 푟 ∈ 푉}.

How are grounds seen in different universes? Proposition If 푈 is an inner model of ZFC satisfying ΦGRD(푟, 푉) ⊆ 푈 ⊆ 푉, then ΦGRD(푟, 푈) = ΦGRD(푟, 푉). On the other hand, for every generic extension 푉[퐺] and every 푟 ∈ 푉 there exists 푠 ∈ 푉 such that ΦGRD(푟, 푉) = ΦGRD(푠, 푉) = ΦGRD(푠, 푉[퐺]).

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Hence, we are able to define the class of all grounds

• f 푉 as {ΦGRD(푟, 푉) ∶ 푟 ∈ 푉}.

How are grounds seen in different universes? Proposition If 푈 is an inner model of ZFC satisfying ΦGRD(푟, 푉) ⊆ 푈 ⊆ 푉, then ΦGRD(푟, 푈) = ΦGRD(푟, 푉). On the other hand, for every generic extension 푉[퐺] and every 푟 ∈ 푉 there exists 푠 ∈ 푉 such that ΦGRD(푟, 푉) = ΦGRD(푠, 푉) = ΦGRD(푠, 푉[퐺]). Proof: The parameter 푟 in ΦGRD(푟, 푉) can be 2<훿 relativized in the ground itself, with any regular cardinal 훿 ≥ (|퐏|+)푉. Let us assume 훿 > (| 퐑퐎(퐏)|)푉. The class 푈 is a generic extension by a complete subalgebra of 퐑퐎(퐏), so the same parameter 푟 suffices.

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Hence, we are able to define the class of all grounds

• f 푉 as {ΦGRD(푟, 푉) ∶ 푟 ∈ 푉}.

How are grounds seen in different universes? Proposition If 푈 is an inner model of ZFC satisfying ΦGRD(푟, 푉) ⊆ 푈 ⊆ 푉, then ΦGRD(푟, 푈) = ΦGRD(푟, 푉). On the other hand, for every generic extension 푉[퐺] and every 푟 ∈ 푉 there exists 푠 ∈ 푉 such that ΦGRD(푟, 푉) = ΦGRD(푠, 푉) = ΦGRD(푠, 푉[퐺]). Proof: For the remaining implication, ΦGRD(푟, 푉) is a ground of 푉[퐺], hence there is a parameter 푠 ∈ 푉[퐺] which represents it. We can conclude applying the previous result to ΦGRD(푠, 푉[퐺]) ⊆ 푉 ⊆ 푉[퐺]. □

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The DDG

Definition

• The Downward Directed Ground hypothesis, DDG, is

the formula: ∀푟, 푠 ∈ 푉 ∃푡 ∈ 푉 (ΦGRD(푡, 푉) ⊆ ΦGRD(푟, 푉) ∩ ΦGRD(푠, 푉)).

• The strong DDG is the formula:

∀푋 ∈ 푉 ∃푡 ∈ 푉 (ΦGRD(푡, 푉) ⊆ ⋂

푟∈푋

ΦGRD(푟, 푉)). Definition The mantle ℳ푊 of a model of set theory 푊 is the intersection of all of its grounds. Formally, ℳ푊 = ⋂

푟∈푊

ΦGRD(푟, 푊). We will simply write ℳ instead of ℳ푉.

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Remark: The mantle is a parameter-free first-order definable class that is transitive and contains all

• rdinals.

Consequences of the DDG

Theorem The following holds: (i) If DDG holds, then for any ground 푊 of 푉 we have ℳ푊 = ℳ푉. (ii) If for every ground 푊 of 푉 one has ℳ푊 = ℳ푉, then the mantle is an inner model of ZF. (iii) Furthermore, if the strong DDG holds, then the mantle is an inner model of ZFC.

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Theorem The following holds: (i) If DDG holds, then for any ground 푊 of 푉 we have ℳ푊 = ℳ푉. Proof: Any ground of 푊 is a ground of 푉, hence ℳ푉 ⊆ ℳ푊. Vice versa, if 푎 ∉ ℳ푉, there is a ground 푊′ such that 푎 ∉ 푊′ and so 푎 ∉ 푊 ∩ 푊′. By directedness, there exists a ground 푊 ⊆ 푊 ∩푊′, which is a ground of 푊. As before, this means that ℳ푊 ⊆ ℳ푊. But 푎 ∉ 푊 implies 푎 ∉ ℳ푊, as requested.

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Theorem The following holds: (ii) If for every ground 푊 of 푉 one has ℳ푊 = ℳ푉, then the mantle is an inner model of ZF. Proof: Since every ground is a transitive class which contains all ordinals and is closed under the G¨

• del
• perations, then by definition so is ℳ (remember that

G¨

• del operations are absolute for transitive models).

It suffices to show that ℳ is almost universal. Let 퐴 ⊆ ℳ and 훼 = rank(퐴). In 푉, for any ground 푊 we have 푉훼 ∩ ℳ = 푉훼 ∩ (푊 ∩ ℳ) = (푉훼)푊 ∩ ℳ푊 ∈ 푊. By the arbitrary choice of 푊, we deduce that 퐴 ⊆ 푉훼 ∩ ℳ ∈ 푊, that is, ℳ is almost universal.

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Theorem The following holds: (iii) Furthermore, if the strong DDG holds, then the mantle is an inner model of ZFC. Proof: By (i) and (ii), we know that the mantle is an inner model of ZF definable in each ground. Take 푦 ∈ ℳ and consider an arbitrary ground ΦGRD(푠, 푉). Being a ground, it satisfies the Axiom of Choice, so it admits a well-ordering 푧 of 푦. For each well-ordering 푧 of 푦 with 푧 ∉ ℳ, there is 푠 such that 푧 ∉ ΦGRD(푠, 푉). Define (using Clt) the set (since 푧 ⊆ 푦 × 푦 which is a set) 퐴 = {푠 ∶ ∃푧 (푧 well-orders 푦 ∧ 푧 ∉ ΦGRD(푠, 푉))}. By strong DDG, ∃푟 (ΦGRD(푟, 푉) ⊆ ⋂

푠∈퐴 ΦGRD(푠, 푉)).

But ΦGRD(푟, 푉) is a ground so there is 푧 that well-orders 푦 in ΦGRD(푟, 푉). Then 푧 ∈ ℳ, for if not, 푧 ∉ ℳ means ∃푠 ∈ 퐴 (푧 ∉ ΦGRD(푠, 푉)), but ΦGRD(푟, 푉) ⊆ ΦGRD(푠, 푉), a contradiction. □

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Theorem The following are equivalent: (i) the DDG holds in 푉; (ii) the DDG holds in some generic extension of 푉; (iii) the DDG hold in every ground of 푉. Proof: (i)→(ii) and (iii)→(i) are immediate by means of the trivial forcing. Suppose that 푈 is a ground of 푉. Let 푊, 푊′ be grounds

• f 푈 and consider the generic extension 푉[퐺] such

that 푉[퐺] ⊨ DDG. Using iterated forcing, 푊 and 푊′ are also grounds of 푉[퐺]. Then, there is a ground 푊 ⊂ 푊 ∩ 푊′ in 푉[퐺]. But then 푊 ⊆ 푊 ∩ 푊′ ⊆ 푈, so 푊 is a ground of 푈. Hence, 푈 ⊨ DDG. □

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The generic multiverse

Woodin was among the first ones who investigate the concept of the generic multiverse of a model 푈

• f set theory.

It is the smallest collection of models containing 푈 and closed under set-forcing extensions and grounds. Definition A generic ground of 푉 is a ground of some generic extension of 푉, that is, an inner model 푊 for which there is a ⟨푊, 퐐⟩-generic filter 퐻 with 퐐 ∈ 푊 such that 푊[퐻] = 푉[퐺], where 퐺 ⊆ 퐏 ∈ 푉 is a 푉-generic filter. Definition The generic DDG for 푉 is the assertion: the DDG holds in all forcing extensions of 푉.

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Definition The generic mantle, denoted by 푔ℳ, is the class 푔ℳ = {푥 ∶ ∀퐏 (ퟏ ⊩퐏 ∀푧 ( ̌ 푥 ∈ ΦGRD(푧, 푉[퐺])))}, where 푉[퐺] is a generic extension by some 퐺 ⊆ 퐏 and in ΦGRD(푧, 푉[퐺]) the parameters are 퐏-names. Equivalently, 푔ℳ = {푥 ∶ ∀퐏 (ퟏ ⊩퐏 ̌ 푥 ∈ ̇ ℳ)}, where ̇ ℳ is an abuse of notation.

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Proposition The generic mantle of 푉 is a definable transitive class in 푉, containing ORD, and invariant under set forcing, that is for a generic extension 푉[퐺]

• f 푉, 푔ℳ푉 = 푔ℳ푉[퐺]. Moreover, it is the largest forcing-

invariant definable class. Proof: As for the mantle, ORD ⊆ 푔ℳ and it is a first-

• rder definable transitive class.

Moreover, for every generic extension 푉[퐺], 푔ℳ ⊆ 푔ℳ푉[퐺]. Now we assume 푥 ∉ 푔ℳ푉 and prove that 푥 ∉ 푔ℳ푉[퐺]. By

• ur assumption, there is a 푉[퐻] with 퐻 ⊆ 퐐 such

that ∃푟 (푥 ∉ ΦGRD(푟, 푉[퐻])). Take 푞 ∈ 퐻 such that 푞 ⊩ ∃ ̇ 푟 (¬ΦGRD( ̇ 푟, ̌ 푥)), where val( ̇ 푟, 퐻) = 푟. By forcing with ↓푞 over 푉[퐺], we may choose a ⟨푉[퐺], ↓푞⟩-generic filter that for simplicity we call anyway 퐻. Then, 푉[퐻] ⊨ 푥 ∉ ΦGRD( ̇ 푟, 푉[퐻]). Applying the product forcing, 퐺 × 퐻 is 푉-generic. Thus, ΦGRD(푟, 푉[퐻]) is a ground of 푉[퐻][퐺] = 푉[퐺][퐻]. Therefore 푥 ∉ 푔ℳ푉[퐺]. □

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Theorem The following are equivalent: (i) the generic DDG holds for 푉; (ii) the DDG holds for 푉 and the grounds of 푉 are dense below the generic grounds; (iii) the DDG holds for 푉 and the grounds of 푉 are dense below the grounds of every ground extension.

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Theorem The following are equivalent: (i) the generic DDG holds for 푉; (ii) the DDG holds for 푉 and the grounds of 푉 are dense below the generic grounds; (iii) the DDG holds for 푉 and the grounds of 푉 are dense below the grounds of every ground extension. Proof: Clearly the generic DDG implies the DDG since 푉 is a generic extension of itself. (i)→(iii) Suppose that 푊푟 = ΦGRD(푟, 푉) is a ground of 푉. Let 푊푟[퐺푟] be a generic extension of 푊푟 with 퐺푟 ⊆ 퐏푟 and 푊 a ground of 푊푟[퐺푟]. Thus, 푊 = ΦGRD(푡, 푊푟[퐺푟]) for some 푡 ∈ 푊푟[퐺푟]. Let ̇ 푡 be any 퐏푟-name in 푊푟 such that 푡 = val( ̇ 푡, 퐺푟). We first make the assumption 퐺푟 is ⟨푉, 퐏푟⟩-generic.

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By iterated forcing, 푊푟 is a ground of 푉[퐺푟]. Then, 푊푟 ⊆ 푊푟[퐺푟] ⊆ 푉[퐺푟], so 푊푟[퐺푟] is also a ground of 푉[퐺푟] and therefore, 푊 is a ground of 푉[퐺푟]. Using the DDG in 푉[퐺푟], there is a ground 푊푠 of 푉[퐺푟] contained in 푊푟 and 푊. We can choose 푟, 푠 and 푡 such that ΦGRD(⋅, 푉[퐺푟]) = ΦGRD(⋅, 푊푟[퐺푟]) for any of them. Hence, 푊푟[퐺푟] ⊨ 푊푠 ⊆ 푊 ∩ 푊푟. Pick 푝 ∈ 퐏푟 such that 푝 ⊩ Ψ, where Ψ is ∃̌ 푠 ∀푥 (ΦGRD(̌ 푠, 푥) → ΦGRD( ̇ 푡, 푥) ∧ ΦGRD(̌ 푟, 푥)). We want to prove that 퐴 = {푝 ∈ 퐏푟 ∶ 푝 ⊩ Ψ} is dense in 퐏푟. Take an arbitrary 푞 ∈ 퐏푟. Consider the poset ↓푞. For an arbitrary 퐺 ⟨푊푟, ↓푞⟩-generic, assume genericity also in 푉 and proceed as above. We obtain a 푝 ≤ 푞 in 퐴. Hence, 퐺푟 ∩ 퐴 ≠ ∅ and so 푊푟[퐺푟] ⊨ Ψ as requested.

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Theorem The following are equivalent: (ii) the DDG holds for 푉 and the grounds of 푉 are dense below the generic grounds; (iii) the DDG holds for 푉 and the grounds of 푉 are dense below the grounds of every ground extension. Proof: (iii)→(ii) Since 푉 is a ground of itself, a generic ground can be seen as a ground of a ground extension 푉[퐺] as shown below. 푊 푉 푉 푉[퐺] Thus (ii) is a special case of (iii).

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Theorem The following are equivalent: (i) the generic DDG holds for 푉; (ii) the DDG holds for 푉 and the grounds of 푉 are dense below the generic grounds; Proof: (ii)→(i) Let ΦGRD(푟, 푉[퐺]) and ΦGRD(푠, 푉[퐺]) be grounds

• f 푉[퐺]. By (ii), we may find grounds ΦGRD(푟′, 푉) and

ΦGRD(푠′, 푉) such that ΦGRD(푟′, 푉) ⊆ ΦGRD(푟, 푉[퐺]) and ΦGRD(푠′, 푉) ⊆ ΦGRD(푠, 푉[퐺]). Using the DDG, we know that there is ΦGRD(푡, 푉) ⊆ ΦGRD(푟′, 푉) ∩ ΦGRD(푠′, 푉), hence ΦGRD(푡, 푉) ⊆ ΦGRD(푟, 푉[퐺]) ∩ ΦGRD(푠, 푉[퐺]), and ΦGRD(푡, 푉) ⊆ 푉 ⊆ 푉[퐺], that is, ΦGRD(푡′, 푉[퐺]) = ΦGRD(푡′, 푉) = ΦGRD(푡, 푉) witnesses the DDG holds in 푉[퐺] for some parameter 푡′. □

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Corollary The generic DDG implies that ℳ = 푔ℳ. Proof: Clearly 푔ℳ ⊆ ℳ. On the other hand, by the previous theorem, the grounds are dense below the generic grounds. Thus, also the converse hold. □ Theorem [Usuba] ZFC ⊢ strong DDG. Remark: Since ZFC proves the strong DDG, then the generic DDG holds.

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We now summarize the above results. Corollary The following hold. (I) ℳ is a transitive model of ZFC and 푔ℳ = ℳ; (II) ℳ is a forcing invariant class; (III) The following are equivalent: (i) every generic extension of 푉 has only set many grounds; (ii) 푉 has only set many grounds; (iii) ℳ is the minimum ground

• f

all generic extensions of 푉; (iv) ℳ is the minimum ground of 푉; (v) ℳ is a ground of 푉; (vi) 푉 admits a minimal ground.

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We now summarize the above results. Corollary The following hold. (I) ℳ is a transitive model of ZFC and 푔ℳ = ℳ; (II) ℳ is a forcing invariant class; Proof: (I) Already proved. (II) Since 푔ℳ = ℳ.

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We now summarize the above results. Corollary The following hold. (III) The following are equivalent: (i) every generic extension of 푉 has only set many grounds; (ii) 푉 has only set many grounds; (iii) ℳ is the minimum ground

• f

all generic extensions of 푉; Proof: (i)→(ii) It is trivial, because 푉 is a generic extension of itself.

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(ii) 푉 has only set many grounds; (iii) ℳ is the minimum ground of all generic extensions

• f 푉;

(ii)→(iii) Let 푋 be a set such that {ΦGRD(푟, 푉) ∶ 푟 ∈ 푋} is the collection of all grounds of 푉. By the strong DDG, there is 푡 ∈ 푋 such that ΦGRD(푡, 푉) ⊆ ⋂

푟∈푋

ΦGRD(푟, 푉) = ℳ ⊆ ΦGRD(푡, 푉). Then the mantle is the minimum ground of 푉. For all generic extensions 푉[퐺], one has ℳ ⊆ 푉 ⊆ 푉[퐺], so ℳ is a ground of a ground of 푉[퐺] and then also a ground

• f 푉[퐺]. Since ℳ = ℳ푉[퐺], we deduce that ℳ is also

the minimum ground of 푉[퐺], for an arbitrary generic extension of 푉.

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(iii) ℳ is the minimum ground of all generic extensions

• f 푉;

(iv) ℳ is the minimum ground of 푉; (v) ℳ is a ground of 푉; (vi) 푉 admits a minimal ground. Proof: (iii)→(iv) As for (i)→(ii). (iv)→(v) Clear. (v)→(vi) If ℳ is a ground of 푉 then it is a ground which is the minimum under inclusion, that is, a minimum ground. (vi)→(v) Let ΦGRD(푟, 푉) be a minimal ground. Towards contradiction, let us assume that ℳ ⊊ ΦGRD(푟, 푉). This means that there is 푠 such that ∃푥 ∈ ΦGRD(푟, 푉) (푥 ∉ ΦGRD(푠, 푉)). By the DDG, ∃푡 (ΦGRD(푡, 푉) ⊆ ΦGRD(푟, 푉) ∩ ΦGRD(푠, 푉)). By minimality, ΦGRD(푟, 푉) = ΦGRD(푡, 푉) ⊆ ΦGRD(푠, 푉), so 푥 ∈ ΦGRD(푠, 푉), a contradiction.

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(iii) ℳ is the minimum ground of all generic extensions

• f 푉;

(v) ℳ is a ground of 푉; (v)→(iii) Let 푉[퐺] be a generic extension of 푉 and suppose ℳ is a ground of 푉. By iterated forcing, ℳ is a ground of 푉[퐺]. Since 푉[퐺] ⊨ ZFC, DDG holds in 푉[퐺] and so for all grounds ΦGRD(푟, 푉[퐺]) there is ΦGRD(푠, 푉[퐺]) ⊆ ℳ ∩ ΦGRD(푟, 푉[퐺]). Now, from ΦGRD(푠, 푉[퐺]) ⊆ ℳ = ℳ푉[퐺] we deduce ΦGRD(푠, 푉[퐺]) = ℳ and so ℳ ⊆ ΦGRD(푟, 푉[퐺]). Hence, ℳ is the minimum ground in an arbitrary generic extension of 푉.

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(i) every generic extension of 푉 has only set many grounds; (iii) ℳ is the minimum ground of all generic extensions

• f 푉;

Proof: (iii)→(i) Let 푉[퐺] be a generic extension of 푉. Being the minimum ground of every generic extension of 푉, there is a forcing 퐐 ∈ ℳ and a ⟨ℳ, 퐐⟩-generic filter 퐻 such that 푉[퐺] = ℳ[퐻]. Furthermore, for every ground ΦGRD(푟, 푉[퐺]), ℳ ⊆ ΦGRD(푟, 푉[퐺]) ⊆ ℳ[퐻]. Hence, ΦGRD(푟, 푉[퐺]) is a generic extension of ℳ by a complete subalgebra of 퐑퐎(퐐)ℳ. Therefore, 푉[퐺] has less than (2|(퐑퐎(퐐))ℳ|)푉[퐺] many grounds. □

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The DDG and Very Large Cardinals

Part Two

2/04/2019

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Definition Let 휅 < 휒 be regular uncountable cardinals. Let 푀 ⊆ H휒 be a transitive model of ZFC − Pwr with 휒 ⊆ 푀. We say that 푀 satisfies the: (a) 휅-covering property for H휒, if for every set of ordinals 퐴 with |퐴| < 휅 and 퐴 ⊆ 휒 (so that 퐴 ∈ H휒), there is a set of ordinals 퐵 ∈ 푀 such that 퐴 ⊆ 퐵 and |퐵|푀 < 휅; (b) 휅-approximation property for H휒, if for all bounded subsets of ordinals 퐴 ⊆ 휒 (so that 퐴 ∈ H휒) such that for every 퐵 ∈ 푀 ∩ [휒]<휅 one has 퐴 ∩ 퐵 ∈ 푀, then 퐴 ∈ 푀; (c) 휅-global covering property for H휒, if for every 훼 < 휒 and every function 푓∶ 훼 → 휒 (so that 푓 ∈ H휒), there is a function 퐹 ∈ 푀 such that dom(퐹) = 훼 and ∀훽 < 훼 (푓(훽) ∈ 퐹(훽) ∧ |퐹(훽)|푀 < 휅).

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Lemma Fix a regular cardinal 훿 ∈ 푉. Let 푊, 푊′, and 푉 be transitive models of ZFC훿 and suppose that

• 푊 ⊆ 푉 and 푊′ ⊆ 푉;
• 푊 and 푊′ have the 훿-approximation and 훿-cover

properties for 푉;

• (<훿2)푊 = (<훿2)푊′;
• (훿+)푊 = (훿+)푊′ = (훿+)푉.

Then 푊 = 푊′.

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Daniel Marini Universit ` a degli Studi di Torino

Lemma Fix a regular cardinal 훿 ∈ 푉. Let 푊, 푊′, and 푉 be transitive models of ZFC훿 and suppose that

• 푊 ⊆ 푉 and 푊′ ⊆ 푉;
• 푊 and 푊′ have the 훿-approximation and 훿-cover

properties for 푉;

• (<훿2)푊 = (<훿2)푊′;
• (훿+)푊 = (훿+)푊′ = (훿+)푉.

Then 푊 = 푊′. Lemma Let 휅 < 휒 be regular uncountable cardinals with 휅+ < 휒. Let 푀, 푁 be transitive models of ZFC−Pwr. Suppose that

• 푀 ⊆ H휒, 푁 ⊆ H휒, and 휒 ⊆ 푀 ∩ 푁;
• 푀 and 푁 have the 휅-approximation and 휅-global

covering properties for H휒;

• 풫(휅) ∩ 푀 = 풫(휅) ∩ 푁.

Then 푀 = 푁.

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Theorem [Bukovsk` y] Suppose 푀 ⊆ 푉 is an inner model of ZFC and 휅 is a regular uncountable cardinal. The following are equivalent: (i) 푀 satisfies the 휅-global covering property for 푉; (ii) there is a notion of forcing 퐏 ∈ 푀 and an ⟨푀, 퐏⟩- generic filter 퐺 such that 퐏 is 휅-cc in 푀 and 푉 = 푀[퐺].

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Daniel Marini Universit ` a degli Studi di Torino

Theorem [Bukovsk` y] Suppose 푀 ⊆ 푉 is an inner model of ZFC and 휅 is a regular uncountable cardinal. The following are equivalent: (i) 푀 satisfies the 휅-global covering property for 푉; (ii) there is a notion of forcing 퐏 ∈ 푀 and an ⟨푀, 퐏⟩- generic filter 퐺 such that 퐏 is 휅-cc in 푀 and 푉 = 푀[퐺]. Proof: (Sketches) We use the following results.

• ∀퐴 ⊆ 푀 ∃퐵 ⊆ ORD (푀[퐴] = 푀[퐵]).
• Any generic extension made with a 휅-cc separative

poset 퐏, adds a new subset of 휅.

• If 푀 has the 휅-global covering property for 푉 and

퐵 ⊆ 휆 is in 푉, then 푀[퐵] is a 휅-cc generic extension

• f 푀 and has the 휅-global covering for 푉.

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(ii) → (i) Since 푓 ∈ 푉, we can pick the 퐏-name ̇ 푓 such that val( ̇ 푓, 퐺) = 푓 and 푝 ∈ 퐺 which verify 푝 ⊩ Fun( ̇ 푓) ∧ dom( ̇ 푓) = ̌ 퐴 ∧ ran( ̇ 푓) ⊆ ̌ 퐵. Define 퐹(푎) = {푏 ∈ 퐵 ∶ ∃푞 ≤ 푝 (푞 ⊩ ̇ 푓( ̌ 푎) = ̌ 푏)} in 푀 by means of Separation axiom. Then, define 퐹 in 푀 as the sets {⟨푎, 퐹(푎)⟩ ∶ 푎 ∈ 퐴} using Replacement and Separation axioms. If for 푎 ∈ 퐴, 푓(푎) = 푏, then there is 푟 ∈ 퐺 such that 푟 ⊩ ̇ 푓( ̌ 푎) = ̌ 푏 and so we can find 푞 ≤ 푝, 푟 which forces the same. Hence 푓(푎) = 푏 ∈ 퐹(푎). Using the Axiom of Choice, we can choose such a 푞푏 ∈ 퐏. The set {푞푏 ∶ 푏 ∈ 퐹(푎)} is an antichain, since if 푠 ≤ 푞푏, 푞푏′ for 푏, 푏′ ∈ 퐹(푎), then 푠 ⊩ ̇ 푓( ̌ 푎) = ̌ 푏 ∧ ̇ 푓( ̌ 푎) = ̌ 푏′, against 푠 ≤ 푝 forcing that ̇ 푓 is a function. By assumption, antichains have size less than 휅, that implies |퐹(푎)| < 휅 as required.

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(i) → (ii) Let 퐴 = 풫(휅)푉. Since 푀[퐴] is transitive, then any subset of 휅 in 푉 is in 푀[퐴]. 푀[퐵] turns out to be a 휅-cc generic extension of 푀 for every 퐵 ⊆ 푀. Then, also 푀[퐴] is 휅-cc generic. Moreover, it has the 휅- global covering property for 푉. For each 퐵 ⊆ 푀[퐴], 푀[퐴][퐵] is a 휅-cc generic extension, but such an extension cannot add other subsets of 휅, since they are already in 푀[퐴]. Therefore, every 푀[퐴][퐵] must be necessarely trivial. Hence, 푉 = 푀[퐴], for they share the same sets of ordinals. □

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Lemma Suppose 푊, an inner model of ZFC, satisfies the 훿-global covering property for 푉. (i) For every cardinal 휅 > 훿, 푊 satisfies the 휅-global covering property for 푉. (ii) If 훿 is regular, then 푊 satisfies the 훿-covering property for 푉.

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Daniel Marini Universit ` a degli Studi di Torino

Lemma Suppose 푊, an inner model of ZFC, satisfies the 훿-global covering property for 푉. (i) For every cardinal 휅 > 훿, 푊 satisfies the 휅-global covering property for 푉. Proof: (i) Immediately follows by definition.

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Lemma Suppose 푊, an inner model of ZFC, satisfies the 훿-global covering property for 푉. (ii) If 훿 is regular, then 푊 satisfies the 훿-covering property for 푉. Proof: (ii) Assume 훿 regular. Let 퐴 ⊆ 푊 be a set of ordinals in 푉 such that |퐴| < 훿. Consider the inclusion map 푖∶ 퐴 → ORD, and the isomorphism 푗∶ 훼 → 퐴 given by 훼 = ot(퐴) < 훿. Let 푓∶ 훼 → ORD be 푖◦푗. By assumption, there is 퐹 ∈ 푊 with dom(퐹) = 훼 and ∀훽 < 훼 (푓(훽) ∈ 퐹(훽) ∧ |퐹(훽)|푊 < 훿). Since 푓(훽) ∈ 퐹(훽) ∈ ran(퐹), ran(푓) ⊆ ⋃ ran(퐹). Therefore, defining 퐵 = ⋃ ran(퐹) in 푊, it suffices to see that by construction 퐴 ⊆ 퐵, and |퐵| < 훿 since 훿 is regular.

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Lemma Let 휃 be a strong limit cardinal and 휅 < 휃 be a regular uncountable cardinal. Let us call 휒 = 휃+. Suppose that 푊 ⊆ 푉 is an inner model of ZFC and 푀 = H푊

휒 = H휒 ∩푊. If 푀 satisfies the 휅-global covering

property for H휒, then 푀 satisfies the 휅+-approximation property for H휒.

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Theorem [Usuba] ZFC ⊢ strong DDG.

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Theorem [Usuba] ZFC ⊢ strong DDG. Proof: Fix a set 푋 and a family of grounds of 푉 {푊푟 ∶ 푟 ∈ 푋}. We will construct a ground 푊 of 푉 contained in every 푊푟 for 푟 ∈ 푋. Let 퐏푟 and 퐺푟 ⊆ 퐏푟 be such that 푉 = 푊푟[퐺푟]. Let us take 휅 a regular cardinal such that for every 푟 ∈ 푋, |푋| < 휅 and |퐏푟| < 휅.

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Daniel Marini Universit ` a degli Studi di Torino

Theorem [Usuba] ZFC ⊢ strong DDG. Proof: Fix a set 푋 and a family of grounds of 푉 {푊푟 ∶ 푟 ∈ 푋}. We will construct a ground 푊 of 푉 contained in every 푊푟 for 푟 ∈ 푋. Let 퐏푟 and 퐺푟 ⊆ 퐏푟 be such that 푉 = 푊푟[퐺푟]. Let us take 휅 a regular cardinal such that for every 푟 ∈ 푋, |푋| < 휅 and |퐏푟| < 휅. Each 푊푟 satisfies the 휅-global covering property (Bukovsk` y)and 휅-approximation property for 푉. Now

• fix a strong limit cardinal 휃 > 휅;
• let 휒 = 휃+;
• let 훾 = 휒<휒;
• take an enumeration ⟨푓휉 ∶ 휉 < 훾⟩ of <휒휒;
• define ℎ∶ 휒 × 훾 → 휒 as

ℎ(훼, 휉) = {푓휉(훼), if 훼 ∈ dom(푓휉) 0,

• therwise.

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The first step is to construct a transitive model 푀 of ZFC − Pwr such that 휒 ⊆ 푀 ⊆ H휒, satisfying the 휅+- global covering property. Then, 푀 satisfies the 휅++- global covering and 휅++-approximation properties.

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Daniel Marini Universit ` a degli Studi di Torino

The first step is to construct a transitive model 푀 of ZFC − Pwr such that 휒 ⊆ 푀 ⊆ H휒, satisfying the 휅+- global covering property. Then, 푀 satisfies the 휅++- global covering and 휅++-approximation properties. CLAIM: There is a function 퐻 such that 퐻 ∈ ⋂

푟∈푋 푊푟,

dom(퐻) = 휒 × 훾, and for all ⟨훼, 휉⟩ ∈ 휒 × 훾, ℎ(훼, 휉) ∈ 퐻(훼, 휉) ⊆ 휒 and |퐻(훼, 휉)| < 휅+.

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Daniel Marini Universit ` a degli Studi di Torino

The first step is to construct a transitive model 푀 of ZFC − Pwr such that 휒 ⊆ 푀 ⊆ H휒, satisfying the 휅+- global covering property. Then, 푀 satisfies the 휅++- global covering and 휅++-approximation properties. CLAIM: There is a function 퐻 such that 퐻 ∈ ⋂

푟∈푋 푊푟,

dom(퐻) = 휒 × 훾, and for all ⟨훼, 휉⟩ ∈ 휒 × 훾, ℎ(훼, 휉) ∈ 퐻(훼, 휉) ⊆ 휒 and |퐻(훼, 휉)| < 휅+. We define 퐻푖,푟 by induction on 푖 < 휅 and 푟 ∈ 푋:

• 1. 퐻푖,푟 is a function in 푊푟;
• 2. dom(퐻푖,푟) = 휒 × 훾, and for all ⟨훼, 휉⟩ ∈ 휒 × 훾, ℎ(훼, 휉) ∈

퐻푖,푟(훼, 휉) ⊆ 휒 and |퐻푖,푟(훼, 휉)| < 휅;

• 3. for all ⟨훼, 휉⟩ ∈ 휒 × 훾,

⋃

푗<푖 푠∈푋

퐻푗,푠(훼, 휉) ⊆ 퐻푖,푟(훼, 휉).

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Suppose 퐻푗,푠 defined for all 푠 ∈ 푋 and 푗 < 푖 < 휅. Fix 푟 ∈ 푋 and let 퐻′ be such that for all ⟨훼, 휉⟩ ∈ 휒 × 훾, 퐻′(훼, 휉) = ⋃

푗<푖 푠∈푋

퐻푗,푠(훼, 휉). In such a way, for all ⟨훼, 휉⟩ ∈ 휒 × 훾 we have |퐻′(훼, 휉)| ≤ |푖| ⋅ |푋| ⋅ sup

푗<푖, 푠∈푋

|퐻푗,푠(훼, 휉)| < 휅 by construction (since 푘 is regular).

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Suppose 퐻푗,푠 defined for all 푠 ∈ 푋 and 푗 < 푖 < 휅. Fix 푟 ∈ 푋 and let 퐻′ be such that for all ⟨훼, 휉⟩ ∈ 휒 × 훾, 퐻′(훼, 휉) = ⋃

푗<푖 푠∈푋

퐻푗,푠(훼, 휉). In such a way, for all ⟨훼, 휉⟩ ∈ 휒 × 훾 we have |퐻′(훼, 휉)| ≤ 푖 ⋅ |푋| ⋅ sup

푗<푖, 푠∈푋

|퐻푗,푠(훼, 휉)| < 휅 by construction (since 푘 is regular). Satisfying the 휅- global covering property for 푉, 푊푟 admits a function 퐻푖,푟 such that 퐻′(훼, 휉) ⊆ 퐻푖,푟(훼, 휉) ⊆ 휒 and |퐻푖,푟(훼, 휉)| < 휅 for all ⟨훼, 휉⟩ ∈ 휒 × 훾. Notice that ℎ(훼, 휉) ∈ 퐻푖,푟(훼, 휉). So we can define 퐻(훼, 휉) = 퐻푟(훼, 휉) ∶= ⋃

푖<휅

퐻푖,푟(훼, 휉) for some (all) 푟 ∈ 푋.

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Daniel Marini Universit ` a degli Studi di Torino

This is well defined, indeed by construction ∀푟 ∈ 푋 (⋃

푗<푖 푠∈푋

퐻푗,푠(훼, 휉) ⊆ 퐻푖,푟(훼, 휉)), hence 퐻푟(훼, 휉) = ⋃

푖<휅

퐻푖,푟(훼, 휉) = ⋃

푖<휅

⋃

푗<푖

퐻푗,푟(훼, 휉) ⊆ ⋃

푖<휅

⋃

푗<푖 푠∈푋

퐻푗,푠(훼, 휉) ⊆ 퐻푟′(훼, 휉), for an arbitrary 푟′ ∈ 푋, therefore 퐻푟 = 퐻푟′ for all 푟, 푟′ ∈ 푋 and we can simply denote 퐻푟 with 퐻. Clearly, ℎ(훼, 휉) ∈ 퐻(훼, 휉) and |퐻(훼, 휉)| < 휅+.

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This is well defined, indeed by construction ∀푟 ∈ 푋 (⋃

푗<푖 푠∈푋

퐻푗,푠(훼, 휉) ⊆ 퐻푖,푟(훼, 휉)), hence 퐻푟(훼, 휉) = ⋃

푖<휅

퐻푖,푟(훼, 휉) = ⋃

푖<휅

⋃

푗<푖

퐻푗,푟(훼, 휉) ⊆ ⋃

푖<휅

⋃

푗<푖 푠∈푋

퐻푗,푠(훼, 휉) ⊆ 퐻푟′(훼, 휉), for an arbitrary 푟′ ∈ 푋, therefore 퐻푟 = 퐻푟′ for all 푟, 푟′ ∈ 푋 and we can simply denote 퐻푟 with 퐻. Clearly, ℎ(훼, 휉) ∈ 퐻(훼, 휉) and |퐻(훼, 휉)| < 휅+. It is left to show that for all 푟 ∈ 푋, 퐻 ∈ 푊푟. It suffices to show that the set 퐸 = {⟨훼, 휉, 휂⟩ ∈ 휒 × 훾 × 휒 ∶ 휂 ∈ 퐻(훼, 휉)} lies in 푊푟 for an arbitrary 푟 ∈ 푋.

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Fix such an 푟. We shall apply the 휅-approximation property of 푊푟. Taking any 푎 ∈ 푊푟 ∩ [휒 × 훾 × 휒]<휅, we will show that 퐸 ∩ 푎 ∈ 푊푟. Consider 푑 = {⟨훼, 휉⟩ ∈ 휒 × 훾 ∶ ∃휂 (⟨훼, 휉, 휂⟩ ∈ 퐸 ∩ 푎)}. Since |푎| < 휅, then |푑| < 휅. For each ⟨훼, 휉⟩ ∈ 푑, the set {휂 < 휒 ∶ ⟨훼, 휉, 휂⟩ ∈ 퐸 ∩ 푎} is contained in 퐻(훼, 휉), thus it has cardinality less than 휅.

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Fix such an 푟. We shall apply the 휅-approximation property of 푊푟. Taking any 푎 ∈ 푊푟 ∩ [휒 × 훾 × 휒]<휅, we will show that 퐸 ∩ 푎 ∈ 푊푟. Consider 푑 = {⟨훼, 휉⟩ ∈ 휒 × 훾 ∶ ∃휂 (⟨훼, 휉, 휂⟩ ∈ 퐸 ∩ 푎)}. Since |푎| < 휅, then |푑| < 휅. For each ⟨훼, 휉⟩ ∈ 푑, the set {휂 < 휒 ∶ ⟨훼, 휉, 휂⟩ ∈ 퐸 ∩ 푎} is contained in 퐻(훼, 휉), thus it has cardinality less than or equal to 휅. Moreover, by construction of 퐻, for every such ⟨훼, 휉⟩ we can find 푖(훼, 휉) such that {휂 < 휒 ∶ ⟨훼, 휉, 휂⟩ ∈ 퐸 ∩ 푎} ⊆ 퐻푖(훼,휉),푟(훼, 휉). Put 푖∗ = sup{푖(훼, 휉) ∶ ⟨훼, 휉⟩ ∈ 푑}, less than 휅 by construction. Then, for all ⟨훼, 휉⟩ ∈ 푑, {휂 < 휒 ∶ ⟨훼, 휉, 휂⟩ ∈ 퐸 ∩ 푎} ⊆ 퐻푖∗,푟(훼, 휉), and so 퐸 ∩ 푎 = {⟨훼, 휉, 휂⟩ ∈ 푎 ∶ 휂 ∈ 퐻푖∗,푟(훼, 휉)} ∈ 푊푟 by means of Separation axiom in 푊푟, as required. ⊲

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Since L is an inner model of ZFC, we know that there is a bijection 휋∶ 휒 × 훾 × 휒 → 훾, with 휋 ∈ L. Let 퐴 = 휋 ″ 퐻. Moreover, L is contained in every inner model of ZFC and 휋 ∈ L, we have 퐴 ∈ 푊푟 for every 푟 ∈ 푋. But 퐴 ⊆ ORD, so L[퐴] ⊨ ZFC and 퐻 ∈ 퐿[퐴] by Replacement axiom. Now let 푀 = HL[퐴]

휒

= H휒 ∩ L[퐴]. Since 휒 ∈ L and 휒 ⊆ H휒, we have 휒 ⊆ 푀 ⊆ H휒 and 푀 ⊆ ⋂

푟∈푋 푊푟.

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CLAIM: 푀 satisfies the 휅+-global covering property for H휒. Since ⟨푓휉 ∶ 휉 < 훾⟩ is an enumeration of <휒휒, for every 푓∶ 훼 → 휒 with 훼 < 휒 there is some 휉훼 < 훾 such that 푓 = 푓휉훼. Define the function 퐹 with dom(퐹) = 훼 and 퐹(훽) = 퐻(훽, 휉훼). By the choice of 퐻, for every 훽 < 훼, we have 푓(훽) = 푓휉훼(훽) = ℎ(훽, 휉훼) ∈ 퐻(훽, 휉훼) = 퐹(훽). Furthermore, |퐹(훽)| < 휅+ for every such 훽. Now, 퐻 ∈ L[퐴] implies 퐹 ∈ L[퐴], hence 퐹 ∈ HL[퐴]

휒

= 푀, since by construction ran(퐻) ⊆ 휒. ⊲

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We have just obtained the model of ZFC−Pwr we were looking for. Summarizing, for each strong limit cardinal 휃 > 휅, we are able to find a transitive model 푀 of ZFC− Pwr such that (1) 푀 ⊆ ⋂

푟∈푋 푊푟;

(2) 휃+ ⊆ 푀 ⊆ H휃+; (3) 푀 satisfies the 휅++-approximation and 휅++-global covering properties for H휃+; where the third follows by remark at the beginning of the proof. Letting 푟 = 풫(휅++) ∩ 푀, we know the model 푀 is the unique transitive model 푁 of ZFC − Pwr that satisfies (2) and (3) above and such that 풫(휅++)∩푁 = 푟.

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For each 휌 ⊆ 풫(휅++), we define the class 퐼휌 = {휃 ∶ 휃 > 휅 is a strong limit cardinal ∧ ∃푀 (푀 ⊨ ZFC−Pwr∧휌 = 풫(휅++)∩푀∧(1), (2), and (3) hold)}.

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For each 휌 ⊆ 풫(휅++), we define the class 퐼휌 = {휃 ∶ 휃 > 휅 is a strong limit cardinal ∧ ∃푀 (푀 ⊨ ZFC−Pwr∧휌 = 풫(휅++)∩푀∧(1), (2), and (3) hold)}. By the pigeonhole arguments, there must be one set 휌 such that 퐼휌 is a proper class. Fix such 휌. For each 휃 ∈ 퐼휌, fix the unique transitive model 푀휃 given by the definition of 퐼휌 for that particular 휃.

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Daniel Marini Universit ` a degli Studi di Torino

For each 휌 ⊆ 풫(휅++), we define the class 퐼휌 = {휃 ∶ 휃 > 휅 is a strong limit cardinal ∧ ∃푀 (푀 ⊨ ZFC−Pwr∧휌 = 풫(휅++)∩푀∧(1), (2), and (3) hold)}. By the pigeonhole arguments, there must be one set 휌 such that 퐼휌 is a proper class. Fix such 휌. For each 휃 ∈ 퐼휌, fix the unique transitive model 푀휃 given by the definition of 퐼휌 for that particular 휃. The sequence ⟨푀휃 ∶ 휃 ∈ 퐼휌⟩ is coherent: for 휃 < 휃′ both in 퐼휌, we have 푀휃 = 푀휃′ ∩ H휃+. This is so because 푀휃′ ∩ H휃+ = {푥 ∈ 푀휃′ ∶ 푀휃′ ⊨ | TC(푥)| ≤ 휃} ∈ 푀휃′ is the unique transitive model of ZFC − Pwr (it is transitive and closed under G¨

• del operations, so a model of

ZF − Pwr), hence 푀휃 = 푀휃′ ∩ H휃+.

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Daniel Marini Universit ` a degli Studi di Torino

Finally, we define 푊 = ⋃{푀휃 ∶ 휃 ∈ 퐼휌} which is definable in 푉, and prove the following.

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Daniel Marini Universit ` a degli Studi di Torino

Finally, we define 푊 = ⋃{푀휃 ∶ 휃 ∈ 퐼휌} which is definable in 푉, and prove the following. CLAIM: 푊 is an inner model of ZFC. Transitivity and 푊 ⊇ ORD immediately follow by definition of 푊 and the fact that all such 푀휃 are transitive models and 휃+ ⊆ 푀휃. Let us verify that 푊 satisfies the Axiom of Choice, that is almost universal, and closed under G¨

• del operations.

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Daniel Marini Universit ` a degli Studi di Torino

Finally, we define 푊 = ⋃{푀휃 ∶ 휃 ∈ 퐼휌} which is definable in 푉, and prove the following. CLAIM: 푊 is an inner model of ZFC. Transitivity and 푊 ⊇ ORD immediately follow by definition of 푊 and the fact that all such 푀휃 are transitive models and 휃+ ⊆ 푀휃. Let us verify that 푊 satisfies the Axiom of Choice, that is almost universal, and closed under G¨

• del operations.

Since the latter are absolute between transitive models, then closurness of 푊 follows as above from coherency of the sequence ⟨푀휃 ∶ 휃 ∈ 퐼휌⟩ and the fact that 푀휃 are models.

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Finally, we define 푊 = ⋃{푀휃 ∶ 휃 ∈ 퐼휌} which is definable in 푉, and prove the following. CLAIM: 푊 is an inner model of ZFC. Transitivity and 푊 ⊇ ORD immediately follow by definition of 푊 and the fact that all such 푀휃 are transitive models and 휃+ ⊆ 푀휃. Let us verify that 푊 satisfies the Axiom of Choice, that is almost universal, and closed under G¨

• del operations.

Since the latter are absolute between transitive models, then closurness of 푊 follows as above from coherency of the sequence ⟨푀휃 ∶ 휃 ∈ 퐼휌⟩ and the fact that 푀휃 are models. If 퐴 ⊆ 푊, then there exists 휃 ∈ 퐼휌 such that, for a 휃′ > 휃, 퐴 ⊆ 푀휃 = 푀휃′ ∩ H휃+ ∈ 푀휃′ ⊆ 푊. Hence 푊 is almost universal.

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Finally, we prove that the AC holds in 푊. For each 푥 ∈ 푊, there is 휃 ∈ 퐼휌 such that 푥 ∈ 푀휃, with 푀휃 a transitive model of ZFC − Pwr. Therefore, 푀휃 admits a well-ordering of 푥 which belongs to 푊. ⊲

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Finally, we prove that the AC holds in 푊. For each 푥 ∈ 푊, there is 휃 ∈ 퐼휌 such that 푥 ∈ 푀휃, with 푀휃 a transitive model of ZFC − Pwr. Therefore, 푀휃 admits a well-ordering of 푥 which belongs to 푊. ⊲ Since 푀휃 ⊆ ⋂

푟∈푋 푊푟, we deduce that 푊 is contained

in every 푊푟 for 푟 ∈ 푋. Furthermore, if 푓∶ 훼 → ORD, then there is 휃 > 훼 strong limit such that 푓∶ 훼 → 휃+, 푓 ∈ H휃+ and, using the notation above, there exists 휉 < 훾 such that 푓 = 푓휉 with 푓휉(훽) = ℎ(훽, 휉) ∈ 퐻(훽, 휉) ∈ L[퐴] ∩ H휃+ = 푀휃. We can thus apply the 푘++-global covering property of 푀휃 for H휃+, obtaining the resulting function 퐹 ∶ 훼 → 휃+ in 푀휃 ⊆ 푊.

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Daniel Marini Universit ` a degli Studi di Torino

Finally, we prove that the AC holds in 푊. For each 푥 ∈ 푊, there is 휃 ∈ 퐼휌 such that 푥 ∈ 푀휃, with 푀휃 a transitive model of ZFC − Pwr. Therefore, 푀휃 admits a well-ordering of 푥 which belongs to 푊. ⊲ Since 푀휃 ⊆ ⋂

푟∈푋 푊푟, we deduce that 푊 is contained

in every 푊푟 for 푟 ∈ 푋. Furthermore, if 푓∶ 훼 → ORD, then there is 휃 > 훼 strong limit such that 푓∶ 훼 → 휃+, 푓 ∈ H휃+ and, using the notation above, there exists 휉 < 훾 such that 푓 = 푓휉 with 푓휉(훽) = ℎ(훽, 휉) ∈ 퐻(훽, 휉) ∈ L[퐴] ∩ H휃+ = 푀휃. We can thus apply the 푘++-global covering property of 푀휃 for H휃+, obtaining the resulting function 퐹 ∶ 훼 → 휃+ in 푀휃 ⊆ 푊. The arbitrary choice of 푓 allows us to conclude that 푊 satisfies the 푘++-global covering property. Bukovsk` y’s theorem leads to the conclusion, namely 푊 is a ground model contained in every 푊푟, for 푟 ∈ 푋. □

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Definition For a cardinal 휅, we call a ground 푊 of 푉 a 휅-ground if there is a forcing 퐏 ∈ 푊 and a ⟨푊, 퐏⟩- generic filter 퐺 such that |퐏| < 휅 and 푉 = 푊[퐺]. Analogously, the 휅-mantle 휅-ℳ is the intersection of all 휅-grounds. By the Ground Model Definability theorem, the 휅- mantle is a definable transitive class. Usuba sketched also the proof about the downward directedness of 휅- grounds if 휅 is strong limit. This implies that the 휅-mantle is absolute between all 휅-grounds which in turn implies that the 휅-mantle models ZF.

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Lemma Let 휅 be a regular uncountable cardinal. There is a ground 푊 of 푉 such that 푊 ⊆ 휅-ℳ. Proof: Any 휅-ground has the 휅-cover and 휅- approximation properties. Hence, every 휅-ground 푊′ is uniquely determined by 푃 = 풫(휅) ∩ 푊′ ∈ 풫(풫(휅)), thus there are at most 22휅 many 휅-grounds of 푉. Let 푋 be the set of size less than 22휅 such that {푊푟 ∶ 푟 ∈ 푋} consists of all the 휅-grounds of 푉. By the strong DDG, there exists a ground 푊 of 푉 such that 푊 ⊆ ⋂

푟∈푋 푊푟,

as required. □

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Daniel Marini Universit ` a degli Studi di Torino

Lemma Let 휅 be a regular uncountable cardinal and 휃 > 휅 be an inaccessible cardinal. Let 휅-ℳ푉휃 be the 휅-mantle of 푉휃, that is, the intersection of all 휅-grounds

• f 푉휃. The following holds true.

(i) If 푊 is a 휅-ground of 푉, then 푊휃 = 푊 ∩ 푉휃 is a 휅- ground of 푉휃. (ii) 휅-ℳ푉휃 ⊆ 휅-ℳ휃. Moreover, if there are proper class many inaccessible cardinals, then there is 훼 > 휅 such that for every inaccessible cardinal 훾 > 훼, 휅-ℳ푉훾 = 휅-ℳ훾.

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Definition An uncountable cardinal 휅 is extendible if for every ordinal 훼 ≥ 휅, there is 훽 > 훼 and an elementary embedding 푗∶ 푉훼 → 푉훽 such that the critical point of 푗 is 휅 and 훼 < 푗(휅) .

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Daniel Marini Universit ` a degli Studi di Torino

Definition An uncountable cardinal 휅 is extendible if for every ordinal 훼 ≥ 휅, there is 훽 > 훼 and an elementary embedding 푗∶ 푉훼 → 푉훽 such that the critical point of 푗 is 휅 and 훼 < 푗(휅) . Theorem [ Kunen ] For any 훼 ∈ ORD there is no 푗∶ 푉훼+2 ⪯ 푉훼+2.

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Daniel Marini Universit ` a degli Studi di Torino

Definition An uncountable cardinal 휅 is extendible if for every ordinal 훼 ≥ 휅, there is 훽 > 훼 and an elementary embedding 푗∶ 푉훼 → 푉훽 such that the critical point of 푗 is 휅 and 훼 < 푗(휅) . Theorem [ Kunen ] For any 훼 ∈ ORD there is no 푗∶ 푉훼+2 ⪯ 푉훼+2. Lemma If 휅 is an extendible cardinal, then there is a proper class of inaccessible cardinals. Proof: Let 푗∶ 푉훼 → 푉훽 be the elementary map given by the definition of extendible cardinal. Towards contradiction, let us assume that there exists 훾 < 훼 large enough such that 푉훾 contains all inaccessible cardinals. Define inductively the sequence 휅0 = 휅, 휅푛+1 = 푗(휅푛), and 휅휔 = sup푛 휅푛. We can therefore restrict

• urselves to only two cases.

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Daniel Marini Universit ` a degli Studi di Torino

(i) ∃푛 (휅푛+1 > 훾 ∧ 휅푛 ≤ 훾): This would mean that 휅푛+1 = 푗(휅푛) ≤ 푗(훾) < 푗(훼) = 훽. Then, 푉훽 ⊨ 휅푛+1 is measurable. Indeed, 휅 is measurable because is the critical point of 푗, so also 푗(휅) = 휅1 is measurable, and so on up to 휅푛+1. We could conclude that 휅푛+1 is measurable in 푉 (Σ2-formula) and so inaccessible, against our assumption. (ii) ∀푛 (휅푛 < 훾 < 훼): We have sup푛(휅푛) = 휅휔 ≤ 훾. Call 휆 = 휅휔 for simplicity. Since 푗(sup푛(휅푛)) = sup푛(푗(휅푛)) = sup푛(휅푛+1) = 휆, we have 푗∶ 푉휆 → 푉휆 elementary map. Let us assume 휆 + 2 < 훼. We know that 푗(휆 + 2) = 푗(휆) + 푗(2) = 휆 + 2, hence 푗↾푉휆+2∶ 푉휆+2 → 푉휆+2 against Kunen’s theorem. □

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Theorem If there is an extendible cardinal, then the mantle is a ground of 푉. In fact, if 휅 is extendible, then the 휅-mantle of 푉 is the solid bedrock.

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Daniel Marini Universit ` a degli Studi di Torino

Theorem If there is an extendible cardinal, then the mantle is a ground of 푉. In fact, if 휅 is extendible, then the 휅-mantle of 푉 is the solid bedrock. Proof: Assume 휅 an extendible cardinal. Take a ground 푊 contained in 휅-ℳ and so ℳ ⊆ 푊 ⊆ 휅-ℳ. If we prove that 휅-ℳ is the mantle of 푉, then ℳ = 휅-ℳ = 푊 is a ground of 푉 and we would have finished the proof.

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Daniel Marini Universit ` a degli Studi di Torino

Theorem If there is an extendible cardinal, then the mantle is a ground of 푉. In fact, if 휅 is extendible, then the 휅-mantle of 푉 is the solid bedrock. Proof: Assume 휅 an extendible cardinal. Take a ground 푊 contained in 휅-ℳ and so ℳ ⊆ 푊 ⊆ 휅-ℳ. If we prove that 휅-ℳ is the mantle of 푉, then ℳ = 휅-ℳ = 푊 is a ground of 푉 and we would have finished the proof. Procede by contradiction and assume that there is a ground 푊 of 푉 such that 푊 ⊊ 휅-ℳ. Fix an inaccessible cardinal 휆 > 휅 such that 푊 is a 휆-ground of 푉 and 푊휆 ⊊ 휅-ℳ휆.

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Daniel Marini Universit ` a degli Studi di Torino

Theorem If there is an extendible cardinal, then the mantle is a ground of 푉. In fact, if 휅 is extendible, then the 휅-mantle of 푉 is the solid bedrock. Proof: Assume 휅 an extendible cardinal. Take a ground 푊 contained in 휅-ℳ and so ℳ ⊆ 푊 ⊆ 휅-ℳ. If we prove that 휅-ℳ is the mantle of 푉, then ℳ = 휅-ℳ = 푊 is a ground of 푉 and we would have finished the proof. Procede by contradiction and assume that there is a ground 푊 of 푉 such that 푊 ⊊ 휅-ℳ. Fix an inaccessible cardinal 휆 > 휅 such that 푊 is a 휆-ground of 푉 and 푊휆 ⊊ 휅-ℳ휆. The sets 푉휆 and 푊휆 are transitive models of ZFC. For every sufficiently large inaccessible cardinal 휃 > 휆, 휅-ℳ푉휃 = 휅-ℳ휃, with 휅-ℳ푉휃 the 휅-mantle of 푉휃.

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By definition of 휅, we can find an elementary embedding 푗∶ 푉휃+1 → 푉푗(휃)+1 with critical point 휅 and such that 휃 < 푗(휅). Being 푗 elementary, 푗(휃) is inaccessible, so also 푉푗(휃) and 푊푗(휃) are transitive models of ZFC.

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Daniel Marini Universit ` a degli Studi di Torino

By definition of 휅, we can find an elementary embedding 푗∶ 푉휃+1 → 푉푗(휃)+1 with critical point 휅 and such that 휃 < 푗(휅). Being 푗 elementary, 푗(휃) is inaccessible, so also 푉푗(휃) and 푊푗(휃) are transitive models of ZFC. Similarly, 푗(휅-ℳ푉휃) is the 푗(휅)-mantle

• f

푗(푉휃) = 푉푗(휃), thanks to the definability of 휅-mantle and absoluteness of 푉훼’s. Since 푊 is a 휆-ground of 푉, 푊푗(휃) is a 휆-ground of 푉푗(휃) and so 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), since 휆 < 휃 < 푗(휅).

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Daniel Marini Universit ` a degli Studi di Torino

By definition of 휅, we can find an elementary embedding 푗∶ 푉휃+1 → 푉푗(휃)+1 with critical point 휅 and such that 휃 < 푗(휅). Being 푗 elementary, 푗(휃) is inaccessible, so also 푉푗(휃) and 푊푗(휃) are transitive models of ZFC. Similarly, 푗(휅-ℳ푉휃) is the 푗(휅)-mantle

• f

푗(푉휃) = 푉푗(휃), thanks to the definability of 휅-mantle and absoluteness of 푉훼’s. Since 푊 is a 휆-ground of 푉, 푊푗(휃) is a 휆-ground of 푉푗(휃) and so 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), since 휆 < 휃 < 푗(휅). We know that 퐸휆

휔 is stationary in 휆, so there is a

sequence 푆 = ⟨푆훼 ∶ 훼 < 휆⟩ ∈ 푊 of pairwise disjoint stationary (in 푊) subsets of 휆. Being a 휆-cc forcing extension of 푊, 푉 sees each 푆훼 as a stationary subset

• f (퐸휆

휔)푉 as well. Let ⟨푆∗ 훼 ∶ 훼 < 푗(휆)⟩ = 푗(푆) ∈ 푉푗(휃).

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CLAIM: The following holds. 푗 ″ 휆 ={훼 < sup(푗 ″ 휆) ∶ 푆∗

훼 ∩ sup(푗 ″ 휆) is stationary in

sup(푗 ″ 휆)}. (⊆) Let 훼 ∈ 푗 ″ 휆. Clearly 훼 ≤ sup(푗 ″ 휆). In order to prove 푆∗

훼 to be stationary, we consider 퐶 ⊆ sup(푗 ″ 휆)

a club. Let 퐷 = 푗−1 ″ 퐶. The set 퐷 is a 휔-club in 휆, that is, is unbounded and 휔-closed (every sequence of length 휔 of elements in 퐷 has limit in 퐷). It is clear that 퐷 is unbounded because 푗 is an elementary map. In order to see that it is 휔-closed, consider ⟨푐푖 ∶ 푖 ∈ 휔⟩ a sequence of elements in 퐶. Then ⟨푗−1(푐푖) ∶ 푗−1(푖) ∈ 푗−1(휔)⟩ is a sequence of elements in 퐷. Since 푘 > 휔 is the critical point of 푗 we know that 푗−1(푖) = 푖, 푗−1(휔) = 휔, and taking 푐 = sup푖∈휔 푐푖 ∈ 퐶 (because from 퐶 closed we derive in particular 퐶 휔-closed) we have that 푗−1(푐) = sup푖∈휔 푗−1(푐푖) ∈ 퐷.

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Now, since 푆훼 ⊆ 퐸휆

휔 is stationary, 푆훼 ∩ 퐷 ≠ ∅ and then

푗(푆훼)∩퐶 = 푆∗

푗(훼)∩퐶 ≠ ∅. By arbitrarity of 퐶, we deduce

that 푆∗

훼 ∩ sup(푗 ″ 휆) is stationary in sup(푗 ″ 휆).

(⊇) Let us assume that 훼 < sup(푗 ″ 휆) with 푆∗

훼 ∩ sup(푗 ″ 휆)

stationary in sup(푗 ″ 휆). From 푗 ″ 휆 휔-club (as above) and 푆∗

훼 ⊆ 푗 ″ 퐸휆 휔, we know that there is 휂 < 휆 such that

푗(휂) ∈ 푆∗

훼. Since 푗 ″ 휆 is partitioned by 푗 ″ 푆훼′ for 훼′ < 휆,

there must be an 훼′ < 휆 such that 푗(휂) ∈ 푗 ″ 푆훼′. By elementarity, 푗 ″ 푆훼′ ⊆ 푗(푆훼′) = 푆∗

푗(훼′), so 푆∗ 훼 ∩ 푆∗ 푗(훼′) ≠ ∅.

But the 푆훼’s are disjoint, hence 훼 = 푗(훼′) and we are done. ⊲

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CLAIM: 푗 ″ 휆 ∈ 푊푗(휃). Since 휃 > 휆, 푆 ∈ 푊휃 but 휅-ℳ푉휃 = 휅-ℳ휃 and so 푆 ∈ 푊휃 ⊆ 휅-ℳ휃 = 휅-ℳ푉휃. Thus, 푗(휅-ℳ푉휃) ⊆ 푊푗(휃) implies that 푗(푆) ∈ 푗(휅-ℳ푉휃) ⊆ 푊푗(휃). Also 푉푗(휃) is a 휆-cc forcing extension

• f 푊푗(휃), hence for each 푆′ ⊆ sup(푗 ″ 휆), the stationarity
• f 푆′ is absolute between 푊푗(휃) and 푉푗(휃). This means

that the previous Claim applied also to 푊푗(휃), that is, 훼 ∈ 푗 ″ 휆 if and only if 훼 < sup(푗 ″ 휆) and 푆∗

훼 ∩ sup(푗 ″ 휆) is

stationary in sup(푗 ″ 휆) in 푊푗(휃). Therefore, 푗 ″ 휆 ∈ 푊푗(휃). ⊲

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Finally, we prove that 푊휆 = 휅-ℳ휆, which yields to the contradiction. Since 푊 ⊊ 휅-ℳ by our assumption, clearly 푊휆 ⊆ 휅-ℳ휆. Conversely, we shall prove by trasfinite induction on 훼 < 휆 that 휅-ℳ훼 ⊆ 푊훼.

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Finally, we prove that 푊휆 = 휅-ℳ휆, which yields to the contradiction. Since 푊 ⊊ 휅-ℳ by our assumption, clearly 푊휆 ⊆ 휅-ℳ휆. Conversely, we shall prove by trasfinite induction on 훼 < 휆 that 휅-ℳ훼 ⊆ 푊훼. Assume 훼 = 휅. The critical point of 푗 is 휅, so 푗(휅-ℳ푉휃) ∩ 푉휅 = 푗 ″ 휅-ℳ푉휃∩푉휅 = 푗 ″ 휅-ℳ휃∩푉휅 = 푗 ″ 휅-ℳ휅 = 휅-ℳ휅. Since 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), 푗(휅-ℳ푉휃) ∩ 푉휅 ⊆ 푊휅, that is 휅-ℳ휅 ⊆ 푊휅.

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Daniel Marini Universit ` a degli Studi di Torino

Finally, we prove that 푊휆 = 휅-ℳ휆, which yields to the contradiction. Since 푊 ⊊ 휅-ℳ by our assumption, clearly 푊휆 ⊆ 휅-ℳ휆. Conversely, we shall prove by trasfinite induction on 훼 < 휆 that 휅-ℳ훼 ⊆ 푊훼. Assume 훼 = 휅. The critical point of 푗 is 휅, so 푗(휅-ℳ푉휃) ∩ 푉휅 = 푗 ″ 휅-ℳ푉휃∩푉휅 = 푗 ″ 휅-ℳ휃∩푉휅 = 푗 ″ 휅-ℳ휅 = 휅-ℳ휅. Since 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), 푗(휅-ℳ푉휃) ∩ 푉휅 ⊆ 푊휅, that is 휅-ℳ휅 ⊆ 푊휅. Now, assume that for 휅 ≤ 훼 < 휆 we have 휅-ℳ훼 ⊆ 푊훼, so that they are equal. In order to show that 휅-ℳ훼+1 ⊆ 푊훼+1, we first prove that 휅-ℳ훼+1 ⊆ 푊.

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Daniel Marini Universit ` a degli Studi di Torino

Finally, we prove that 푊휆 = 휅-ℳ휆, which yields to the contradiction. Since 푊 ⊊ 휅-ℳ by our assumption, clearly 푊휆 ⊆ 휅-ℳ휆. Conversely, we shall prove by trasfinite induction on 훼 < 휆 that 휅-ℳ훼 ⊆ 푊훼. Assume 훼 = 휅. The critical point of 푗 is 휅, so 푗(휅-ℳ푉휃) ∩ 푉휅 = 푗 ″ 휅-ℳ푉휃∩푉휅 = 푗 ″ 휅-ℳ휃∩푉휅 = 푗 ″ 휅-ℳ휅 = 휅-ℳ휅. Since 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), 푗(휅-ℳ푉휃) ∩ 푉휅 ⊆ 푊휅, that is 휅-ℳ휅 ⊆ 푊휅. Now, assume that for 휅 ≤ 훼 < 휆 we have 휅-ℳ훼 ⊆ 푊훼, so that they are equal. In order to show that 휅-ℳ훼+1 ⊆ 푊훼+1, we first prove that 휅-ℳ훼+1 ⊆ 푊. Take 푋 ∈ 휅-ℳ훼+1. The 휅-mantle 휅-ℳ is transitive, so 푋 ∈ 휅-ℳ훼+1 = 휅-ℳ∩푉훼+1 = 휅-ℳ∩풫(푉훼) implies 푋 ⊆ 휅-ℳ∩푉훼 = 휅-ℳ훼.

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Daniel Marini Universit ` a degli Studi di Torino

Furthermore, 푋 ∈ 휅-ℳ훼+1 = 휅-ℳ훼+1 ∩ 푉휃 = 휅-ℳ푉휃

훼+1 ⊆

휅-ℳ푉휃, thus 푗(푋) ∈ 푗(휅-ℳ푉휃).

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Daniel Marini Universit ` a degli Studi di Torino

Furthermore, 푋 ∈ 휅-ℳ훼+1 = 휅-ℳ훼+1 ∩ 푉휃 = 휅-ℳ푉휃

훼+1 ⊆

휅-ℳ푉휃, thus 푗(푋) ∈ 푗(휅-ℳ푉휃). From 휅-ℳ훼 = 푊훼 ∈ 푊휆 and 푊휆 ⊨ ZFC, we can find a 훾 ∈ 푊휆 and a bijection 푓∶ 훾 → 휅-ℳ훼 in 푊휆. But 푊휆 ⊆ 휅-ℳ휆, so 푓 ∈ 휅-ℳ휆 = 휅-ℳ푉휃 ∩ 푉휆 ⊆ 휅-ℳ푉휃 and 푗(푓) ∈ 푗(휅-ℳ푉휃).

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Daniel Marini Universit ` a degli Studi di Torino

Furthermore, 푋 ∈ 휅-ℳ훼+1 = 휅-ℳ훼+1 ∩ 푉휃 = 휅-ℳ푉휃

훼+1 ⊆

휅-ℳ푉휃, thus 푗(푋) ∈ 푗(휅-ℳ푉휃). From 휅-ℳ훼 = 푊훼 ∈ 푊휆 and 푊휆 ⊨ ZFC, we can find a 훾 ∈ 푊휆 and a bijection 푓∶ 훾 → 휅-ℳ훼 in 푊휆. But 푊휆 ⊆ 휅-ℳ휆, so 푓 ∈ 휅-ℳ휆 = 휅-ℳ푉휃 ∩ 푉휆 ⊆ 휅-ℳ푉휃 and 푗(푓) ∈ 푗(휅-ℳ푉휃). By the latter Claim, 푗 ″ 휆 ∈ 푊푗(휃) ⊨ ZFC, so 푗 ″ 훾 ∈ 푊푗(휃) and by the above, 푗(푓) ∈ 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), hence 푗(푓) ″(푗 ″ 훾) = 푗 ″ 휅-ℳ훼 ∈ 푊푗(휃) by definition of 푓.

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Daniel Marini Universit ` a degli Studi di Torino

Furthermore, 푋 ∈ 휅-ℳ훼+1 = 휅-ℳ훼+1 ∩ 푉휃 = 휅-ℳ푉휃

훼+1 ⊆

휅-ℳ푉휃, thus 푗(푋) ∈ 푗(휅-ℳ푉휃). From 휅-ℳ훼 = 푊훼 ∈ 푊휆 and 푊휆 ⊨ ZFC, we can find a 훾 ∈ 푊휆 and a bijection 푓∶ 훾 → 휅-ℳ훼 in 푊휆. But 푊휆 ⊆ 휅-ℳ휆, so 푓 ∈ 휅-ℳ휆 = 휅-ℳ푉휃 ∩ 푉휆 ⊆ 휅-ℳ푉휃 and 푗(푓) ∈ 푗(휅-ℳ푉휃). By the latter Claim, 푗 ″ 휆 ∈ 푊푗(휃) ⊨ ZFC, so 푗 ″ 훾 ∈ 푊푗(휃) and by the above, 푗(푓) ∈ 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), hence 푗(푓) ″(푗 ″ 훾) = 푗 ″ 휅-ℳ훼 ∈ 푊푗(휃) by definition of 푓. Now 푗(푋) ∈ 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), thus 푗 ″ 푋 = 푗(푋)∩푗 ″ 휅-ℳ훼 ∈ 푊푗(휃). Let 휋 ∈ 푊푗(휃) be the collapsing map of 푗 ″ 휅-ℳ훼.

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Daniel Marini Universit ` a degli Studi di Torino

Furthermore, 푋 ∈ 휅-ℳ훼+1 = 휅-ℳ훼+1 ∩ 푉휃 = 휅-ℳ푉휃

훼+1 ⊆

휅-ℳ푉휃, thus 푗(푋) ∈ 푗(휅-ℳ푉휃). From 휅-ℳ훼 = 푊훼 ∈ 푊휆 and 푊휆 ⊨ ZFC, we can find a 훾 ∈ 푊휆 and a bijection 푓∶ 훾 → 휅-ℳ훼 in 푊휆. But 푊휆 ⊆ 휅-ℳ휆, so 푓 ∈ 휅-ℳ휆 = 휅-ℳ푉휃 ∩ 푉휆 ⊆ 휅-ℳ푉휃 and 푗(푓) ∈ 푗(휅-ℳ푉휃). By the latter Claim, 푗 ″ 휆 ∈ 푊푗(휃) ⊨ ZFC, so 푗 ″ 훾 ∈ 푊푗(휃) and by the above, 푗(푓) ∈ 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), hence 푗(푓) ″(푗 ″ 훾) = 푗 ″ 휅-ℳ훼 ∈ 푊푗(휃) by definition of 푓. Now 푗(푋) ∈ 푗(휅-ℳ푉휃) ⊆ 푊푗(휃), thus 푗 ″ 푋 = 푗(푋)∩푗 ″ 휅-ℳ훼 ∈ 푊푗(휃). Let 휋 ∈ 푊푗(휃) be the collapsing map of 푗 ″ 휅-ℳ훼. By uniqueness of the Mostowski collapse and the transitivity of 휅-ℳ훼, 푗 is the inverse of 휋. Hence, 휋 ″(푗 ″ 휅-ℳ훼) = 휅-ℳ훼, and 푋 = 휋 ″(푗 ″ 푋) ∈ 푊푗(휃). Therefore, 푋 ∈ 푊, which means that 휅-ℳ훼+1 ⊆ 푊. Then we can conclude, observing that 휅-ℳ훼+1 ⊆ 푉훼+1, that 휅-ℳ훼+1 ⊆ 푊 ∩ 푉훼+1 = 푊훼+1. □

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