Draft C ( n ) -cardinals with Forcing Alejandro Poveda Departament - - PowerPoint PPT Presentation

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Draft C ( n ) -cardinals with Forcing Alejandro Poveda Departament - - PowerPoint PPT Presentation

Draft C ( n ) -cardinals with Forcing Alejandro Poveda Departament de Matemtiques i Informtica Reflections on Set Theoretic Reflection A Conference in honor to Joan Bagaria 18/10/2018 - Catalonia Partially supported by MECD (Ministerio de


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C(n) -cardinals with Forcing

Alejandro Poveda

Departament de Matemàtiques i Informàtica

Reflections on Set Theoretic Reflection A Conference in honor to Joan Bagaria 18/10/2018 - Catalonia

Partially supported by MECD (Ministerio de Educación, cultura y Deporte) Grant no FPU15/00026.
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This talk is based upon joint projects with J. Bagaria and with Y. Hayut and M. Magidor

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Our aim today is to explore some connections between

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Our aim today is to explore some connections between

1 Reflection.
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Our aim today is to explore some connections between

1 Reflection. 2 Large Cardinals between the first supercompact and VP.
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Our aim today is to explore some connections between

1 Reflection. 2 Large Cardinals between the first supercompact and VP.

We’ll divide the talk in two parts: I Magidor-like analysis of C(n) -supercompact cardinals (with Hayut and Magidor).

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Our aim today is to explore some connections between

1 Reflection. 2 Large Cardinals between the first supercompact and VP.

We’ll divide the talk in two parts: I Magidor-like analysis of C(n) -supercompact cardinals (with Hayut and Magidor). II: Robusteness of C(n) -extendible cardinals under class forcing iterations (joint with Bagaria)

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1 Reflection.
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Motivation and basic definitions

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Reflection

It is well-known that one of the fundamental properties of V is reflection. For any metatheoretic n ≥ 0, the following is a theorem of ZFC

Reflection Theorem (Lévy-Montague)

There is a club proper class of ordinals C(n) such that for any α ∈ C(n), Vα ≺Σn V . Namely, for any Σn-formula ϕ( x) and any set of parameters

  • a ∈ Vα,

Vα ϕ( a) ⇐ ⇒ n ϕ( a).

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Reflection Theorem (Lévy-Montague)

There is a club proper class of ordinals C(n) such that for any α ∈ C(n), Vα ≺Σn V . Namely, for any Σn-formula ϕ( x) and any set of parameters

  • a ∈ Vα,

Vα ϕ( a) ⇐ ⇒ n ϕ( a).

◮ Morally, this means that the universe is too large to be definable by

some specific property.

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Reflection Theorem (Lévy-Montague)

There is a club proper class of ordinals C(n) such that for any α ∈ C(n), Vα ≺Σn V . Namely, for any Σn-formula ϕ( x) and any set of parameters

  • a ∈ Vα,

Vα ϕ( a) ⇐ ⇒ n ϕ( a).

◮ Morally, this means that the universe is too large to be definable by

some specific property.

◮ Lévy observed that the Reflection Theorem is equivalent to Infinity

plus Replacement, modulo de remaining ZF axioms.

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Reflection Theorem (Lévy-Montague)

There is a club proper class of ordinals C(n) such that for any α ∈ C(n), Vα ≺Σn V . Namely, for any Σn-formula ϕ( x) and any set of parameters

  • a ∈ Vα,

Vα ϕ( a) ⇐ ⇒ n ϕ( a).

◮ Morally, this means that the universe is too large to be definable by

some specific property.

◮ Lévy observed that the Reflection Theorem is equivalent to Infinity

plus Replacement, modulo de remaining ZF axioms.

◮ These are exactly the two axioms that makes Set Theory what it is: namely, the theory of the infinity.
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Reflection Theorem (Lévy-Montague)

There is a club proper class of ordinals C(n) such that for any α ∈ C(n), Vα ≺Σn V . Namely, for any Σn-formula ϕ( x) and any set of parameters

  • a ∈ Vα,

Vα ϕ( a) ⇐ ⇒ n ϕ( a).

◮ Morally, this means that the universe is too large to be definable by

some specific property.

◮ Lévy observed that the Reflection Theorem is equivalent to Infinity

plus Replacement, modulo de remaining ZF axioms.

◮ These are exactly the two axioms that makes Set Theory what it is: namely, the theory of the infinity. ◮ The Reflection phenomenon is an essential feature of V .
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The classes C(n)

Let us give some basic properties of the C(n) classes:

How C(n) looks like?

1 If n = 0, C(0) = On since any Σ0-formula is absolute. 2 If n = 1, C(1) = {κ ∈ CARD : κ is strong limit} (by Lévy

absoluteness).

3 If n ≥ 2, then there is no local combinatorial characterization of

C(n) . Namely, there is no property ϕ(x) such that κ ∈ C(n) iff ∃α (Vα ϕ(κ)).

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The classes C(n)

Some easy properties of C(n)

1 For each n ≥ 0, C(n+1) ⊆ C(n). 2 For each n ≥ 1, C(n) is Πn-definable (but not Σn). 3 For each n ≥ 0, C(n+1) ⊆ Lim(C(n)).
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When Large Cardinals meet the C(n) classes

We can define the C(n) -version of classical large cardinals.

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When Large Cardinals meet the C(n) classes

We can define the C(n) -version of classical large cardinals.

C(n) -Large Cardinals (Bagaria)

Let n ≥ 1

1 A cardinal κ is C(n) -measurable if there is an elementary embedding

j : V → M with crit(j) = κ, j(κ) ∈ C(n).

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When Large Cardinals meet the C(n) classes

We can define the C(n) -version of classical large cardinals.

C(n) -Large Cardinals (Bagaria)

Let n ≥ 1

1 A cardinal κ is C(n) -measurable if there is an elementary embedding

j : V → M with crit(j) = κ, j(κ) ∈ C(n).

2 A cardinal κ is C(n) -strong if for every λ > κ there is j : V → M

such that crit(j) = κ, j(κ) > λ, Vλ ⊆ M and j(κ) ∈ C(n).

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When Large Cardinals meet the C(n) classes

We can define the C(n) -version of classical large cardinals.

C(n) -Large Cardinals (Bagaria)

Let n ≥ 1

1 A cardinal κ is C(n) -measurable if there is an elementary embedding

j : V → M with crit(j) = κ, j(κ) ∈ C(n).

2 A cardinal κ is C(n) -strong if for every λ > κ there is j : V → M

such that crit(j) = κ, j(κ) > λ, Vλ ⊆ M and j(κ) ∈ C(n).

3 A cardinal κ is C(n) -supercompact if for every λ > κ there is

j : V → M such that crit(j) = κ, j(κ) > λ, Mλ ⊆ M and j(κ) ∈ C(n).

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When Large Cardinals meet the C(n) classes

We can define the C(n) -version of classical large cardinals.

C(n) -Large Cardinals (Bagaria)

Let n ≥ 1

1 A cardinal κ is C(n) -measurable if there is an elementary embedding

j : V → M with crit(j) = κ, j(κ) ∈ C(n).

2 A cardinal κ is C(n) -strong if for every λ > κ there is j : V → M

such that crit(j) = κ, j(κ) > λ, Vλ ⊆ M and j(κ) ∈ C(n).

3 A cardinal κ is C(n) -supercompact if for every λ > κ there is

j : V → M such that crit(j) = κ, j(κ) > λ, Mλ ⊆ M and j(κ) ∈ C(n).

4 A cardinal κ is C(n) -extendible if for every λ > κ there is θ ∈ ON

and j : Vλ → Vθ such that crit(j) = κ, j(κ) > λ and j(κ) ∈ C(n).

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Notice that if κ is C(n) -LC then κ is LC, were LC stands for some classical notion of Large Cardinal.

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Notice that if κ is C(n) -LC then κ is LC, were LC stands for some classical notion of Large Cardinal. C(n) -measurability and C(n) -strongness do not lead to stronger large cardinal notions.

LC which are also C(n) -LC

For instance, let κ be measurable and let µ ∈ C(n), µ > κ. Let U be a measure witnessing the measurability of κ and define a µ-length iteration

  • f ultrapowers. Since µ is strong limit, jµ : V → Mµ is an elementary

embedding with crit(jµ) = κ and jµ(κ) = µ ∈ C(n). (Similar arguments leads to the same conclusion for C(n) -strong cardinals

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Notice that if κ is C(n) -LC then κ is LC, were LC stands for some classical notion of Large Cardinal. C(n) -measurability and C(n) -strongness do not lead to stronger large cardinal notions.

LC which are also C(n) -LC

For instance, let κ be measurable and let µ ∈ C(n), µ > κ. Let U be a measure witnessing the measurability of κ and define a µ-length iteration

  • f ultrapowers. Since µ is strong limit, jµ : V → Mµ is an elementary

embedding with crit(jµ) = κ and jµ(κ) = µ ∈ C(n). (Similar arguments leads to the same conclusion for C(n) -strong cardinals

Not too interesing C(n) -LC

1 C(n) -measurability. 2 C(n) -strongness.
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What about C(n) -extendibility?

Some properties of C(n) -extendibility

1 Extendibility is equivalent to C(1)-extendibility (since j(κ) is (a real)

inaccessible).

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What about C(n) -extendibility?

Some properties of C(n) -extendibility

1 Extendibility is equivalent to C(1)-extendibility (since j(κ) is (a real)

inaccessible).

2 If κ is a C(n) -extendible cardinal then κ ∈ C(n+2).
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What about C(n) -extendibility?

Some properties of C(n) -extendibility

1 Extendibility is equivalent to C(1)-extendibility (since j(κ) is (a real)

inaccessible).

2 If κ is a C(n) -extendible cardinal then κ ∈ C(n+2). 3 C(n) -extendibility is a Πn+2-definable property.
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What about C(n) -extendibility?

Some properties of C(n) -extendibility

1 Extendibility is equivalent to C(1)-extendibility (since j(κ) is (a real)

inaccessible).

2 If κ is a C(n) -extendible cardinal then κ ∈ C(n+2). 3 C(n) -extendibility is a Πn+2-definable property. 4 Let m < n. From (2) and (3), any C(n)-extendible cardinal is limit of

C(m)-extendibles.

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What about C(n) -extendibility?

Some properties of C(n) -extendibility

1 Extendibility is equivalent to C(1)-extendibility (since j(κ) is (a real)

inaccessible).

2 If κ is a C(n) -extendible cardinal then κ ∈ C(n+2). 3 C(n) -extendibility is a Πn+2-definable property. 4 Let m < n. From (2) and (3), any C(n)-extendible cardinal is limit of

C(m)-extendibles.

C(n) -extendibility forms a hierarchy

Hence, C(n) -extendibility induces an increasing hierarchy in terms of consistency strength.

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C(n) -extendibility forms a hierarchy

Hence, C(n) -extendibility induces an increasing hierarchy in terms of consistency strength.

C(n) -extendibility and reflection

As we will discuss later, this phenomenon is deeply connected with strong forms of (structural) reflection

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The difficult case: C(n) -supercompactnes

On the contrary, in this case we mainly have questions:

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The difficult case: C(n) -supercompactnes

On the contrary, in this case we mainly have questions:

Questions about C(n) -supercompactness (Bagaria-Tsaprounis)

1 Does supercompactness imply C(1)-supercompactness?
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The difficult case: C(n) -supercompactnes

On the contrary, in this case we mainly have questions:

Questions about C(n) -supercompactness (Bagaria-Tsaprounis)

1 Does supercompactness imply C(1)-supercompactness? 2 Let n ≥ 2. Does any C(n) -supercompact cardinal κ lie in C(n+1)? ◮ Notice that for n = 1, C(1)-supercompact cardinals are supercompact and thus are C(2)-cardinals.
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The difficult case: C(n) -supercompactnes

On the contrary, in this case we mainly have questions:

Questions about C(n) -supercompactness (Bagaria-Tsaprounis)

1 Does supercompactness imply C(1)-supercompactness? 2 Let n ≥ 2. Does any C(n) -supercompact cardinal κ lie in C(n+1)? ◮ Notice that for n = 1, C(1)-supercompact cardinals are supercompact and thus are C(2)-cardinals. 3 Does the family of C(n) -supercompact cardinals form an increasing

hierarchy in terms of consistency strength?

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The difficult case: C(n) -supercompactnes

On the contrary, in this case we mainly have questions:

Questions about C(n) -supercompactness (Bagaria-Tsaprounis)

1 Does supercompactness imply C(1)-supercompactness? 2 Let n ≥ 2. Does any C(n) -supercompact cardinal κ lie in C(n+1)? ◮ Notice that for n = 1, C(1)-supercompact cardinals are supercompact and thus are C(2)-cardinals. 3 Does the family of C(n) -supercompact cardinals form an increasing

hierarchy in terms of consistency strength?

4 Which is the relation between C(n) -supercompactness and

C(n) -extendibility?

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But, where did these cardinals appear? Or, in other words, why are they worth to be studied?

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1 Reflection. 2 Large Cardinals between the first supercompact and VP.
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A historical interlude

◮ One of the prominent regions of V is that encompassed between the

first measurable and the first supercompact.

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A historical interlude

◮ One of the prominent regions of V is that encompassed between the

first measurable and the first supercompact.

◮ Magidor discovered two of the main configurations of this stratum
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A historical interlude

◮ One of the prominent regions of V is that encompassed between the

first measurable and the first supercompact.

◮ Magidor discovered two of the main configurations of this stratum

Theorem (Magidor)

1 Assume that κ is a strongly compact cardinal. Then there is a generic

extension of the universe where κ is strongly compact and the first measurable cardinal.

2 Assume that κ is a supercompact cardinal. Then there is a generic

extension of the universe where κ is supercompact and the first strongly compact cardinal.

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A historical interlude

What it can be said about the immediate upper region? Namely, between the first supercompact cardinal and Vopěnka’s principle?

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A historical interlude

What it can be said about the immediate upper region? Namely, between the first supercompact cardinal and Vopěnka’s principle?

Definition (Vopěnka Principle)

Vopěnka’s principle (VP) holds if for any proper class C of structures in the same vocaculary there are two different A, B ∈ C and j : A → B elementary.

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A historical interlude

What it can be said about the immediate upper region? Namely, between the first supercompact cardinal and Vopěnka’s principle?

Definition (Vopěnka Principle)

Vopěnka’s principle (VP) holds if for any proper class C of structures in the same vocaculary there are two different A, B ∈ C and j : A → B elementary. This has many consequences:

Some consequences of VP

◮ Implies that extendible cardinals form a stationary proper class (≈

Magidor).

◮ Any strong logic L has a compactness number, LST number...
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A historical interlude

Bagaria gave a level-by-level equivalence of VP.

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A historical interlude

Bagaria gave a level-by-level equivalence of VP. Firstly, let us recall Magidor characterization of supercompact cardinals

Theorem (Magidor)

TFAE

1 κ is supercompact. 2 For all λ ∈ C(1) there are ¯

κ < ¯ λ < κ and an elementary embedding j : V¯

λ, ∈ → Vλ, ∈ such that crit(j) = ¯

κ, ¯ λ ∈ C(1) and j(¯ κ) = κ (i.e. the Π1-definable class {Vλ, ∈ λ : λ ∈ C(1)} reflects below κ)

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Theorem (Bagaria-Casacuberta-Mathias-Rosický)

TFAE:

1 VP(Π1) holds. 2 VP(κ, Σ2) holds, for some κ. 3 There is a supercompact cardinal.
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Theorem (Bagaria-Casacuberta-Mathias-Rosický)

TFAE:

1 VP(Π1) holds. 2 VP(κ, Σ2) holds, for some κ. 3 There is a supercompact cardinal.

Theorem (Bagaria)

For n ≥ 1, TFAE:

1 VP(Πn+1) holds. 2 VP(κ, Σn+2) holds, for some κ. 3 There is a C(n) -extendible cardinal.
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Theorem (Bagaria)

For n ≥ 1, TFAE:

1 VP(Πn+1) holds. 2 VP(κ, Σn+2) holds, for some κ. 3 There is a C(n) -extendible cardinal.

Corollary (Bagaria)

TFAE:

1 VP holds. 2 For every n ≥ 1, VP(Πn) holds. 3 VP(κ, Σn+2) holds, for a proper class of κ and for every n ≥ 1. 4 For every n ≥ 1, there is a C(n) -extendible cardinal.
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Corollary (Bagaria)

TFAE:

1 VP holds. 2 For every n ≥ 1, VP(Πn) holds. 3 VP(κ, Σn+2) holds, for a proper class of κ and for every n ≥ 1. 4 For every n ≥ 1, there is a C(n) -extendible cardinal.

Conclusion

C(n) -extendible cardinals are the canonical representatives in the Large Cardinal hierarchy in that region

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Part I: Magidor-like analysis of the class of C(n) -supercompact cardinals

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A Magidor-like analysis

A standard analysis will have to based on the following questions:

Question

1 Are supercompactness and C(1)-supercompactness equivalent

notions?

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A Magidor-like analysis

A standard analysis will have to based on the following questions:

Question

1 Are supercompactness and C(1)-supercompactness equivalent

notions?

2 Does C(n) -supecompactness entail a strict hierarchy in terms of

consistency strength?

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A Magidor-like analysis

A standard analysis will have to based on the following questions:

Question

1 Are supercompactness and C(1)-supercompactness equivalent

notions?

2 Does C(n) -supecompactness entail a strict hierarchy in terms of

consistency strength?

3 How are related the notions of C(n) -supercompactness and

C(n) -extendibility?

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C(1)-supercompactness is not equivalent to supercompactness

Main Theorem 1 (Hayut-Magidor-P.)

Assume GCH holds and let κ be a supercompact cardinal. Then there is a generic extension V P where κ remains supercompact, GCH holds and there are no elementary embeddings j : V P → M such that crit(j) = κ, Mω ⊆ M and j(κ) is a limit cardinal. In particular, in V P the cardinal κ is supercompact but not C(1)-supercompact cardinal.

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C(1)-supercompactness is not equivalent to supercompactness

Main Theorem 1 (Hayut-Magidor-P.)

Assume GCH holds and let κ be a supercompact cardinal. Then there is a generic extension V P where κ remains supercompact, GCH holds and there are no elementary embeddings j : V P → M such that crit(j) = κ, Mω ⊆ M and j(κ) is a limit cardinal. In particular, in V P the cardinal κ is supercompact but not C(1)-supercompact cardinal.

Answer to our first question

Are supercompactness and C(1)-supercompactness equivalent? No.

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C(1)-supercompactness is not equivalent to supercompactness

Main Theorem 1 (Hayut-Magidor-P.)

Assume GCH holds and let κ be a supercompact cardinal. Then there is a generic extension V P where κ remains supercompact, GCH holds and there are no elementary embeddings j : V P → M such that crit(j) = κ, Mω ⊆ M and j(κ) is a limit cardinal. In particular, in V P the cardinal κ is supercompact but not C(1)-supercompact cardinal. Working a bit more we can get the following

Corollary

Assume that the theory “ZFC + GCH + ∃λ, κ ∈ S, ∃µ ∈ S(1)(λ < κ < µ)” is consistent. Then it is also consistent the theory “ZFC + m´ ın M < m´ ın K < m´ ın S < m´ ın S(1)”.

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The first C(n) -supercompact may be the first strongly compact

Main Theorem 2 (Hayut-Magidor-P.)

Let n ≥ 1 and κ be a C(n) -supercompact cardinal. Assume that κ carries a S(n)-fast function (i.e. a function ℓ : κ → κ such that for each λ > κ there is j : V → M a λ-C(n) -supercompact embedding such that j(ℓ)(κ) > λ). Then, there is a generic extension V M where m´ ın K = m´ ın S = m´ ın S(n) < m´ ın E.

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The first C(n) -supercompact may be the first strongly compact

Main Theorem 2 (Hayut-Magidor-P.)

Let n ≥ 1 and κ be a C(n) -supercompact cardinal. Assume that κ carries a S(n)-fast function (i.e. a function ℓ : κ → κ such that for each λ > κ there is j : V → M a λ-C(n) -supercompact embedding such that j(ℓ)(κ) > λ). Then, there is a generic extension V M where m´ ın K = m´ ın S = m´ ın S(n) < m´ ın E.

Answer to questions 2 and 3

2 Does C(n) -supercompactness entails a strict hierarchy in terms of consistency strength? No.

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The first C(n) -supercompact may be the first strongly compact

Main Theorem 2 (Hayut-Magidor-P.)

Let n ≥ 1 and κ be a C(n) -supercompact cardinal. Assume that κ carries a S(n)-fast function (i.e. a function ℓ : κ → κ such that for each λ > κ there is j : V → M a λ-C(n) -supercompact embedding such that j(ℓ)(κ) > λ). Then, there is a generic extension V M where m´ ın K = m´ ın S = m´ ın S(n) < m´ ın E.

Answer to questions 2 and 3

2 Does C(n) -supercompactness entails a strict hierarchy in terms of consistency strength? No. 3 How are related C(n) -supercompactness and C(n) -extendibility? Consistently, first extendible greater than first C(n) -supercompact

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Working a little bit we can get more:

Corollary

Let V , ∈, κ be a transitive model of ZFC∗ plus C(ω)-EXT, then there is a generic extension V M, ∈, κ witnessing ZFC⋆ plus C(ω)-SUP and m´ ın K = m´ ın S = m´ ın S(ω) < m´ ın E. Here we are working with and extended language L = {∈, k} and

◮ C(ω)-EXT is the schema asserting that for every (metatheoretic)

n ≥ 1, “k is C(n)-extendible

◮ C(ω)-SUP is the schema asserting that for every (metatheoretic)

n ≥ 1, “k is C(n)-supercompact

◮ ZFC⋆ is the version of ZFC where we allow the constant symbol k to

be used in any instance of replacement and separation.

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A sketch of the proofs

In the following slides we are giving a sketch of the two main results:

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Proof of Main Theorem 1

◮ By a classical result of Solovay, if κ is strongly compact then λ fails

for any λ ≥ κ

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Proof of Main Theorem 1

◮ By a classical result of Solovay, if κ is strongly compact then λ fails

for any λ ≥ κ → Forcing unboundely many λ-sequences below a cardinal κ kills any supercompact below κ

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Proof of Main Theorem 1

◮ By a classical result of Solovay, if κ is strongly compact then λ fails

for any λ ≥ κ → Forcing unboundely many λ-sequences below a cardinal κ kills any supercompact below κ

Question

How many λ-sequences are permitted to be below a supercompact κ?

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Proof of Main Theorem 1

◮ By a classical result of Solovay, if κ is strongly compact then λ fails

for any λ ≥ κ → Forcing unboundely many λ-sequences below a cardinal κ kills any supercompact below κ

Question

How many λ-sequences are permitted to be below a supercompact κ?

Towards an answer

◮ Let λ < κ. There is a generic extension where κ is supercompact and

there is S ⊆ Sκ

λ stationary such that θ holds, each θ ∈ S.
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Proof of Main Theorem 1

◮ By a classical result of Solovay, if κ is strongly compact then λ fails

for any λ ≥ κ → Forcing unboundely many λ-sequences below a cardinal κ kills any supercompact below κ

Question

How many λ-sequences are permitted to be below a supercompact κ?

Towards an answer

◮ Let λ < κ. There is a generic extension where κ is supercompact and

there is S ⊆ Sκ

λ stationary such that θ holds, each θ ∈ S.

This is close to be optimal as if κ is supercompact there is no club C ⊆ κ where λ holds, for each λ ∈ S.

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Proof of Main Theorem 1

Nonetheless, the situation is quite different with C(1)-supercompact cardinals:

Proposition

Assume GCH holds. Let κ be a supercompact cardinal, λ < κ and assume that for each θ ∈ Sκ

≤λ, θ-holds. Then there is no elementary embedding

j : V → M such that crit(j) = κ, Mλ ⊆ M and j(κ) being a limit cardinal.

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Proof of Main Theorem 1

Proof

Suppose such embedding exists. Notice that cof(j(κ)) > λ and thus Sj(κ)

≤λ = (Sj(κ) ≤λ )M is a (real) stationary set. By elementarity, for every

θ ∈ (Sj(κ)

≤λ )M, there is a θ-sequence in M. Since j(κ) is a limit cardinal

with cof(j(κ)) > λ, we can pick θ ∈ Sj(κ)

≤λ , θ > κ a θ is a cardinal. Let us

prove there is a θ-sequence in V which will yield to the desired

  • contradiction. For this it will be enough to show that θ+ = (θ+)M.
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Proof of Main Theorem 1

Proof

Suppose such embedding exists. Notice that cof(j(κ)) > λ and thus Sj(κ)

≤λ = (Sj(κ) ≤λ )M is a (real) stationary set. By elementarity, for every

θ ∈ (Sj(κ)

≤λ )M, there is a θ-sequence in M. Since j(κ) is a limit cardinal

with cof(j(κ)) > λ, we can pick θ ∈ Sj(κ)

≤λ , θ > κ a θ is a cardinal. Let us

prove there is a θ-sequence in V which will yield to the desired

  • contradiction. For this it will be enough to show that θ+ = (θ+)M.Let

µ = |(θ+)M| and notice that µ is a cardinal with cof(µ) > λ. Since GCH holds we have the following inequalities: θ+ = θλ ≤ µλ = µ. Therefore, there is a θ sequence with θ ≥ κ hence κ is not longer

  • supercompact. Contradiction.
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Proof of Main Theorem 1

1 Let ℓ : κ → κ be a Laver function and define the iteration Pℓ κ

Definition

Let Pℓ

κ denote the κ-Easton support iteration defined in such a way that if

α < κ and Pℓ

α was defined, if α ∈ cl(ℓ) ∩ Sκ ω then Pℓ α “ ˙

Qα = Pα” and Pℓ

α “ ˙

Qα trivial”, otherwise.

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Proof of Main Theorem 1

1 Let ℓ : κ → κ be a Laver function and define the iteration Pℓ κ

Definition

Let Pℓ

κ denote the κ-Easton support iteration defined in such a way that if

α < κ and Pℓ

α was defined, if α ∈ cl(ℓ) ∩ Sκ ω then Pℓ α “ ˙

Qα = Pα” and Pℓ

α “ ˙

Qα trivial”, otherwise.

2 Using the fast behaviour of ℓ we can show the following:

Proposition

The iteration Pℓ

κ preserves the supercompactness of κ and the GCH

pattern.

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1 Let ℓ : κ → κ be a Laver function and define the iteration Pℓ κ

Definition

Let Pℓ

κ denote the κ-Easton support iteration defined in such a way that if

α < κ and Pℓ

α was defined, if α ∈ dom(ℓ) ∩ Eκ ω then Pℓ α “ ˙

Qα = Pα” and Pℓ

α “ ˙

Qα trivial”, otherwise.

2 Using the fast behaviour of ℓ we can show the following using

standard arguments

Proposition

The iteration Pℓ

κ preserves the supercompactness of κ and the GCH

pattern.

3 Finally, using the previous proposition the theorem follows.
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Proof of Main Theorem 2

Let us now sketch the proof of

Main Theorem 2

Let n ≥ 1 and κ be a C(n) -supercompact cardinal. Assume κ carries a S(n)-fast function. Then, there is a generic extension V M where m´ ın K = m´ ın S = m´ ın S(n) < m´ ın E.

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Proof of Main Theorem 2

First of all, it is worth to emphasize that the preservation by forcing of C(n) -supercompact (C(n) -extendible cardinals) is pretty much harder than with supercompact ones.

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Proof of Main Theorem 2

First of all, it is worth to emphasize that the preservation by forcing of C(n) -supercompact (C(n) -extendible cardinals) is pretty much harder than with supercompact ones.

◮ They are not derivable by measures but by (long) extenders.
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Proof of Main Theorem 2

First of all, it is worth to emphasize that the preservation by forcing of C(n) -supercompact (C(n) -extendible cardinals) is pretty much harder than with supercompact ones.

◮ They are not derivable by measures but by (long) extenders. ◮ There is no standard (i.e. combinatorial) characterization for the class

C(n) (Main difficulty).

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Proof of Main Theorem 2

Let κ be C(n) -supercompact and P be some κ-length iteration. Typically we face up with two possible strategies to show that κ remains C(n) -supercompact in V P:

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Proof of Main Theorem 2

Let κ be C(n) -supercompact and P be some κ-length iteration. Typically we face up with two possible strategies to show that κ remains C(n) -supercompact in V P:

1 Lifting the embeddings.
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Proof of Main Theorem 2

Let κ be C(n) -supercompact and P be some κ-length iteration. Typically we face up with two possible strategies to show that κ remains C(n) -supercompact in V P:

1 Lifting the embeddings. 2 Define suitable extenders in the generic extension.
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First strategy: lifting the embeddings

Let λ > κ and j : V → M be a λ-C(1)-supercompact embedding. It is not a big deal to show that

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First strategy: lifting the embeddings

Let λ > κ and j : V → M be a λ-C(1)-supercompact embedding. It is not a big deal to show that

◮ j⋆ lifts to j : V P → Mj(P) (for instance, if P has Easton support).
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First strategy: lifting the embeddings

Let λ > κ and j : V → M be a λ-C(1)-supercompact embedding. It is not a big deal to show that

◮ j⋆ lifts to j : V P → Mj(P) (for instance, if P has Easton support). ◮ Mλ ⊆ M (doable using some fast function guiding P).
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First strategy: lifting the embeddings

Let λ > κ and j : V → M be a λ-C(1)-supercompact embedding. It is not a big deal to show that

◮ j⋆ lifts to j : V P → Mj(P) (for instance, if P has Easton support). ◮ Mλ ⊆ M (doable using some fast function guiding P). ◮ V P “j⋆(κ) ∈ C(n)” (As the forcing is mild).
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First strategy: lifting the embeddings

Let λ > κ and j : V → M be a λ-C(1)-supercompact embedding. It is not a big deal to show that

◮ j⋆ lifts to j : V P → Mj(P) (for instance, if P has Easton support). ◮ Mλ ⊆ M (doable using some fast function guiding P). ◮ V P “j⋆(κ) ∈ C(n)” (As the forcing is mild).

Issue

There is no guarantee that j⋆ is definable within V P.

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First strategy: lifting the embeddings

Let λ > κ and j : V → M be a λ-C(1)-supercompact embedding. It is not a big deal to show that

◮ j⋆ lifts to j : V P → Mj(P) (for instance, if P has Easton support). ◮ Mλ ⊆ M (doable using some fast function guiding P). ◮ V P “j⋆(κ) ∈ C(n)” (As the forcing is mild).

Issue

There is no guarantee that j⋆ is definable within V P. If j(κ) was a small cardinal (in V ), we could find a generics for j(P)/G definable in V [G] via Diagonalization/Distributiviness arguments. Notice this is not our case.

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First strategy: lifting the embeddings

Let λ > κ and j : V → M be a λ-C(1)-supercompact embedding. It is not a big deal to show that

◮ j⋆ lifts to j : V P → Mj(P) (for instance, if P has Easton support). ◮ Mλ ⊆ M (doable using some fast function guiding P). ◮ V P “j⋆(κ) ∈ C(n)” (As the forcing is mild).

Issue

There is no guarantee that j⋆ is definable within V P. If j(κ) was a small cardinal (in V ), we could find a generics for j(P)/G definable in V [G] via Diagonalization/Distributiviness arguments. Notice this is not our case.

Conclusion

The previous comment suggest that one has to somehow build by hand the generic for j(P)/G.

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Second strategy: Defining extenders in a generic extension

Let λ > κ and j : V → M be a λ-C(n)-supercompact embedding. We want to build an extender E = Ea : a ∈ [η]<ω witnessing that κ is λ-C(1)-supercompact in the generic extension.

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Second strategy: Defining extenders in a generic extension

Let λ > κ and j : V → M be a λ-C(n)-supercompact embedding. We want to build an extender E = Ea : a ∈ [η]<ω witnessing that κ is λ-C(1)-supercompact in the generic extension. A natural candidate is the extender derived by the potential lifted

  • embedding. Namely,

X ∈ Ea ← → ∃p ∈ G ∃q≤j(p) \ κ (p ⌢ q M

j(P) ˙

a ∈ τ( ˙ X)).

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Second strategy: Defining extenders in a generic extension

Let λ > κ and j : V → M be a λ-C(n)-supercompact embedding. We want to build an extender E = Ea : a ∈ [η]<ω witnessing that κ is λ-C(1)-supercompact in the generic extension. A natural candidate is the extender derived by the potential lifted

  • embedding. Namely,

X ∈ Ea ← → ∃p ∈ G ∃q≤j(p) \ κ (p ⌢ q M

j(P) ˙

a ∈ τ( ˙ X)). If (Mκ, ≤, ≤⋆) is a Magidor iteration such that (j(Mκ)/Mκ, ≤⋆) is λ+-closed and ≤=≤⋆, then

◮ The Ea are κ-complete normal measures.
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Second strategy: Defining extenders in a generic extension

Let λ > κ and j : V → M be a λ-C(n)-supercompact embedding. We want to build an extender E = Ea : a ∈ [η]<ω witnessing that κ is λ-C(1)-supercompact in the generic extension. A natural candidate is the extender derived by the potential lifted

  • embedding. Namely,

X ∈ Ea ← → ∃p ∈ G ∃q≤j(p) \ κ (p ⌢ q M

j(P) ˙

a ∈ τ( ˙ X)). If (Mκ, ≤, ≤⋆) is a Magidor iteration such that (j(Mκ)/Mκ, ≤⋆) is λ+-closed and ≤=≤⋆, then

◮ The Ea are κ-complete normal measures. ◮ We can manage to get Mλ E ⊆ ME.
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Second strategy: Defining extenders in a generic extension

Let λ > κ and j : V → M be a λ-C(n)-supercompact embedding. We want to build an extender E = Ea : a ∈ [η]<ω witnessing that κ is λ-C(1)-supercompact in the generic extension. A natural candidate is the extender derived by the potential lifted

  • embedding. Namely,

X ∈ Ea ← → ∃p ∈ G ∃q≤j(p) \ κ (p ⌢ q M

j(P) ˙

a ∈ τ( ˙ X)). If (Mκ, ≤, ≤⋆) is a Magidor iteration such that (j(Mκ)/Mκ, ≤⋆) is λ+-closed and ≤=≤⋆, then

◮ The Ea are κ-complete normal measures. ◮ We can manage to get Mλ E ⊆ ME.

Issue

How can we make sure that jE(κ) ∈ C(n)?

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Second strategy: Defining extenders in a generic extension

A natural candidate is the extender defined in the following way: X ∈ Ea ← → ∃p ∈ G ∃q ≤j(p) \ κ (p ⌢ q M

j(P) ˙

a ∈ τ( ˙ X)). If (Mκ, ≤, ≤⋆) is a Magidor iteration such that (j(Mκ)/Mκ, ≤⋆) is λ+-closed and ≤=≤⋆, then

◮ The Ea are κ-complete normal measures. ◮ It is possible to get Mλ E ⊆ ME. ◮ j(κ) ∈ C(n).
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Second strategy: Defining extenders in a generic extension

A natural candidate is the extender defined in the following way: X ∈ Ea ← → ∃p ∈ G ∃q ≤j(p) \ κ (p ⌢ q M

j(P) ˙

a ∈ τ( ˙ X)). If (Mκ, ≤, ≤⋆) is a Magidor iteration such that (j(Mκ)/Mκ, ≤⋆) is λ+-closed and ≤=≤⋆, then

◮ The Ea are κ-complete normal measures. ◮ It is possible to get Mλ E ⊆ ME. ◮ j(κ) ∈ C(n).

Issue

How can we make sure jE(κ) ∈ C(n)? As there is no combinatorial characterization for the class C(n), a natural strategy is to make sure that jE(κ) = j(κ).

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Second strategy: Defining extenders in a generic extension

Issue

How can we make sure jE(κ) ∈ C(n)? Notice that this is difficult since there is no combinatorial description for the class C(n).

Conclusion

The previous suggest that we have somehow manage to get jE(κ) = j(κ) as j(κ) is still a C(n)-cardinal in the generic extension.

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Proof of Main Theorem 2

For the proof of Main Theorem 2 we followed the first strategy and thus we have to handmade the j(P)/G-generic.

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Proof of Main Theorem 2

1 κ be a C(n) -supercompact cardinal. 2 ℓ : κ → κ be a S(n)-fast function (i.e. For all λ > κ there is

j; V → M witnessing λ-C(n) -supercompactness of κ and j(ℓ)(κ) > λ).

3 ran(ℓ) = κα : α < κ which are measurable not limit of the previous

measurables (κα > supβ<α κβ).

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We will need the concept of Magidor iteration of Prikry-type forcings:

Magidor iteration of Prikry-type forcings (Gitik)

Let κ be a cardinal and Mκ = Mα, ˙ Qβ : β < α ≤ κ be a κ-stage iteration of forcings. We will say that Mκ, ≤Mκ, ≤∗

Mκ is a κ-stage

Magidor iteration of Prikry forcings if the following conditions holds:

1 For all α < κ, Mα ˙

Qα, ≤ ˙

Qα, ≤∗ ˙ Qα has the Prikry property 2 For all p, q ∈ Mκ, p ≤Mκ q iff 1 For all α < κ, p ↾ α ≤Mα q ↾ α, 2 There is b ∈ [κ]<ω such that for every α ∈ κ \ b, p ↾ α Mα ˙ p(α) ≤∗ Mα ˙ q(α). 3 For all p, q ∈ Mκ, p ≤∗ Mκ q iff p ≤Mκ q and the witness b for the

condition 2.1 is the empty set.

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For each α < κ let Uα be a normal measure over κα. Denote by PUα the corresponding Prikry forcing.

Magidor iteration of Prikry forcings with respect to ran(ℓ)

Let Mκ be the Magidor iteration where M0 is the trivial forcing and for every ordinal α < κ if Mα “ ˇ Uα is a normal measure over κα” then Mα ˙ Qα = P ˇ

Uα, and Mα ˙

Qα = {1}, otherwise.

◮ Since our measurables are not limit of the previous ones, for all

α < κ, Mα ˙ Qα = P ˇ

Uα. ◮ Mκ, ≤, ≤∗ satisfies the Prikry property. ◮ Our forcing will be Mκ/1α : α < κ. For the ease of clarity, let us

also denote it by Mκ.

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Proof of Main Theorem 2

By the previous comments, Mκ is essentially a product. Formally, Mκ is isomorphic to M∗

ran(ℓ),κ, where

Definition (Magidor Product)

The κ-Magidor product with respect to A = κα : α < κ, M∗

A,κ, is the

set of all sequences p = s(α), Aα : α < κ such that (a) For every α < κ, (s(α), Aα) ∈ PUα, where PUα stands for the Prikry forcing with respect some normal measure Uα over κα ∈ A. (b) {α < κ : s(α) = ∅} ∈ [κ]<ℵ0. Given two conditions p, q ∈ M∗

A,κ, p ≤ q (p is stronger than q) if for every

α < κ, p(α) ≤PUα q(α). We will also say that p is a direct extension of q, p ≤⋆ q if for every α < κ, p(α) ≤⋆

PUα q(α)
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On the sequel we will denote by Mκ the κ-Magidor product with respect to ran(ℓ). A typical condition p of this forcing is of the form (∅, A0), · · · (s(α0), Aα0), · · · , (s(αn), Aαn), (∅, Aαn+1), · · · )

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On the sequel we will denote by Mκ the κ-Magidor product with respect to ran(ℓ). A typical condition p of this forcing is of the form (∅, A0), · · · (s(α0), Aα0), · · · , (s(αn), Aαn), (∅, Aαn+1), · · · )

Stem

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On the sequel we will denote by Mκ the κ-Magidor product with respect to ran(ℓ). A typical condition p of this forcing is of the form (∅, A0), · · · (s(α0), Aα0), · · · , (s(αn), Aαn), (∅, Aαn+1), · · · )

Stem Large sets

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Proof of Main Theorem 2

Steps of the proof

1 Mκ has the Mathias-Prikry property:

For every dense open set D and for every p ∈ Mκ there is a p⋆ ≤⋆ p and a length sequence γ for a stem such that for all q ≤ p⋆ with

  • γ ≤p len(sq) then q ∈ D.
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Proof of Main Theorem 2

Steps of the proof

1 Mκ has the Mathias-Prikry property:

For every dense open set D and for every p ∈ Mκ there is a p⋆ ≤⋆ p and a length sequence γ for a stem such that for all q ≤ p⋆ with

  • γ ≤p len(sq) then q ∈ D.
◮ This implies that any Magidor generic C is of the form Cα : α < κ with Cα being Prikry generic. Namely, C = Cα : α < κ is generic iff ∀α ∈ κ ∀A ∈ Uα |A \ Cα| < ℵ0 (Mathias criterion)
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Steps of the proof

1 Mκ has the Mathias-Prikry property:

For every dense open set D and for every p ∈ Mκ there is a p⋆ ≤⋆ p and a length sequence γ for a stem such that for all q ≤ p⋆ with

  • γ ≤p len(sq) then q ∈ D.
2 Let j : V → M be a λ-C(n)-supercompact embedding such that

j(ℓ)(κ) > λ (S(n)-fast behaviour of ℓ). Set µ = j(κ) and M⋆ = MM

j(κ),j(A)\A.
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Steps of the proof

1 Mκ has the Mathias-Prikry property:

For every dense open set D and for every p ∈ Mκ there is a p⋆ ≤⋆ p and a length sequence γ for a stem such that for all q ≤ p⋆ with

  • γ ≤p len(sq) then q ∈ D.
2 Let j : V → M be a λ-C(n)-supercompact embedding such that

j(ℓ)(κ) > λ (S(n)-fast behaviour of ℓ). Set µ = j(κ) and M⋆ = MM

j(κ),j(A)\A.We iterate ω-many times each M-measurable

cardinal of j(A) \ A leading to a µ-length iteration of ultrapowers within M.

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Steps of the proof

1 Mκ has the Mathias-Prikry property:

For every dense open set D and for every p ∈ Mκ there is a p⋆ ≤⋆ p and a length sequence γ for a stem such that for all q ≤ p⋆ with

  • γ ≤p len(sq) then q ∈ D.
2 Let j : V → M be a λ-C(n)-supercompact embedding such that

j(ℓ)(κ) > λ (S(n)-fast behaviour of ℓ). Set µ = j(κ) and M⋆ = MM

j(κ),j(A)\A.We iterate ω-many times each M-measurable

cardinal of j(A) \ A leading to a µ-length iteration of ultrapowers within M.Collecting all the critical sequences Cα we get a generic filter for jµ(M⋆) over Mµ, the µ-iterate of M. (Works by discreteness

  • f the measurables).
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Proof of Main Theorem 2

Steps of the proof

1 MA,κ has the Strong Prikry property. 2 Define a j⋆ : V → Mµ and build H a jµ(M⋆)-generic over Mµ using

(long) iterated ultrapowers. (Of course, this is definable in V )

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Proof of Main Theorem 2

Steps of the proof

1 MA,κ has the Strong Prikry property. 2 Define a j⋆ : V → Mµ and build H a jµ(M⋆)-generic over Mµ using

(long) iterated ultrapowers. (Of course, this is definable in V )

3 Lift the embedding to j⋆ : V [G] → Mµ[G × H] ◮ This can be done as the stems are finite.
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Proof of Main Theorem 2

Steps of the proof

1 MA,κ has the Strong Prikry property. 2 Define a j⋆ : V → Mµ and build H a jµ(M⋆)-generic over Mµ using

(long) iterated ultrapowers. (Of course, this is definable in V )

3 Lift the embedding to j⋆ : V [G] → Mµ[G × H] ◮ This can be done as the stems are finite. 4 Mµ[G × H]λ ⊆ Mµ[G × H]. ◮ Let jµ(fα)( ρβα 0 , . . . , ρβα nα ) : α < λ such that ρβα i ⊳ Cβα i (i < nα). Since j(ℓ)(κ) > λ then crit(jµ) > λ and thus jµ(fα) : α < λ ∈ Mµ. Now the result follows as the sequence ( ρβα 0 , . . . , ρβα nα ) : α < λ is definable in M[H]. Thus, Mµ[H]λ ⊆ Mµ[H]. The results follows since |Mκ| < λ.
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Proof of Main Theorem 2

Steps of the proof

1 MA,κ has the Strong Prikry property. 2 Define a j⋆ : V → Mµ and build H a jµ(M⋆)-generic over Mµ using

(long) iterated ultrapowers. (Of course, this is definable in V )

3 Lift the embedding to j⋆ : V [G] → Mµ[G × H] ◮ This can be done as the stems are finite. 4 Mµ[G × H]λ ⊆ Mµ[G × H]. ◮ Let jµ(fα)( ρβα 0 , . . . , ρβα nα ) : α < λ such that ρβα i ⊳ Cβα i (i < nα). Since j(ℓ)(κ) > λ then crit(jµ) > λ and thus jµ(fα) : α < λ ∈ Mµ. Now the result follows as the sequence ( ρβα 0 , . . . , ρβα nα ) : α < λ is definable in M[H]. Thus, Mµ[H]λ ⊆ Mµ[H]. The results follows since |Mκ| < λ. 5 Obviously j⋆(κ) = j(κ) ∈ C(n) as the forcing is mild.
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A comment on the Main Theorem 2

Main Theorem 2 (Hayut-Magidor-P.)

Let n ≥ 1 and κ be a C(n) -supercompact cardinal. Moreover, let us assume that κ carries a S(n)-fast function. Then, there is a generic extension V M where m´ ın K = m´ ın S = m´ ın S(n) < m´ ın E.

Remark

The same result holds if instead of considering C(n) -supercompact one consider κ being LC-supercompact with LC a Large cardinal notion indestructible by mild forcing. For instance, κ is E-supercompact if for every λ > κ then there is j : V → M such that crit(j) = κ, j(κ) > λ, Mλ ⊆ M and j(κ) ∈ E

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Open questions

Question

Exhibit any (non trivial) forcing notion different to the Magidor product which preserves C(n) -supercompactness.

Question

Is the theory “m´ ın S < m´ ın S(1) < m´ ın E” consistent?

Question

Let n ≥ 1. Are the notions of C(n) -supercompactness and C(n+1)-supercompactness different?

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Part II: Robusteness of C(n) -extendible cardinals under class forcing iterations

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Our aim

Study the robustness of C(n) -extendible cardinals under forcing extensions.

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Our aim

Study the robustness of C(n) -extendible cardinals under forcing extensions. This is part of a long-standing project in Set Theory:

The indestructibility phenomenon

Let κ be a cardinal, ϕ(x) some LC property of κ and Γ be some definable class of forcings closed by iterations/products. We denote by Indϕ(x)(κ, Γ) the formula “∀Q ∈ Γ Q ϕ(κ)”

◮ ϕ(x) is supercompactness: If ϕ(κ), there is P such that

P Indϕ(x)(κ, κ-directed closed) (Laver).

◮ ϕ(x) is strongness: If ϕ(κ), there is P such that

P Indϕ(x)(κ, κ+-wc with Prikry Property) (Gitik-Shelah)

◮ ϕ(x) is strong compactness:If ϕ(κ), there is P such that

P Indϕ(x)(κ, {Add(κ, 1)}) (Hamkins)

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The indestructibility phenomenon

Let κ be a cardinal, ϕ(x) some LC property of κ and Γ be some definable class of forcings closed by iterations/products. We denote by Indϕ(x)(κ, Γ) the formula “∀Q ∈ Γ Q ϕ(κ)”

◮ ϕ(x) is supercompactness: If ϕ(κ), there is P such that

P Indϕ(x)(κ, κ-directed closed) (Laver).

◮ ϕ(x) is strongness: If ϕ(κ), there is P such that

P Indϕ(x)(κ, κ+-wc with Prikry Property) (Gitik-Shelah)

◮ ϕ(x) is strong compactness:If ϕ(κ), there is P such that

P Indϕ(x)(κ, {Add(κ, 1)}) (Hamkins)

◮ ϕ(x) is to be a Vopěnka’s cardinal:If ϕ(κ),

Indϕ(x)(κ, Suitable κ-iterations) (Brooke-Taylor)

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Question:

What can we say about cardinals from upper regions?

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Question:

What can we say about cardinals from upper regions? Notice that the first candidates for studying are extendible cardinals.

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Question:

What can we say about cardinals from upper regions? Notice that the first candidates for studying are extendible cardinals.

Extendible cardinals are never Laver indestructible

This is a fancy argument due to Tsaprounis. Assume κ is extendible and force with P, the Jensen iteration for GCH. Now P forces GCH and preserves the extendibility of κ (Tsaprounis). Call this model V . Assume there is some P ⊆ Vκ making κ Laver indestructible, then in V P

1 There are unboundedly many places below κ where GCH fails. 2 The formula formula asserting that is Π3 and thus, since κ is C(3) in

V P, there are class many ordinals where GCH fails. This yields to a contradiction.

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Theorem (Tsaprounis)

Extendible cardinals are never Laver indestructible.

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Theorem (Tsaprounis)

Extendible cardinals are never Laver indestructible. This is a particular case of the next general theorem:

Theorem (Bagaria-Hamkins-Tsaprounis-Usuba)

Suppose that Vκ ≺Σ2 Vλ and G ⊆ P is a V -generic filter for nontrivial strategically <κ-closed forcing P ∈ Vη, where η ≤ λ . Then for every θ ≥ η, Vκ = V [G]κ ⊀Σ3 V [G]θ.

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Theorem (Tsaprounis)

Extendible cardinals are never Laver indestructible. This is not completely surprising as the preservation of (very) large cardinals by nice forcing imposes strong forms of agreement between V P and V .

Observation

Let κ be a C(n) -cardinal and P a κ-distributive forcing. Assume that P “κ ∈ ˙ C(n)”. Then V ≡Σn(Vκ) V P.

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Theorem (Tsaprounis)

Extendible cardinals are never Laver indestructible. This is not completely surprising as the preservation of (very) large cardinals by nice forcing imposes strong forms of agreement between V P and V .

Observation

Let κ be a C(n) -cardinal and P a κ-distributive forcing. Assume that P “κ ∈ ˙ C(n)”. Then V ≡Σn(Vκ) V P.

The moral

The more correct a cardinal is, the harder is to preserve its correctness, and therefore the more fragile it becomes.

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Theorem (Tsaprounis)

Extendible cardinals are never Laver indestructible. Despite Tsaprounis’ non-indestructibility result, one may ask the following:

Problem

Do nice forcing notions preserve extendible cardinals?

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Theorem (Tsaprounis)

Extendible cardinals are never Laver indestructible. Despite Tsaprounis’ non-indestructibility result, one may ask the following:

Problem

Do nice forcing notions preserve extendible cardinals?

◮ By the theorem from [BHTU] we presented so far, the answer is

(more than likely) negative.

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Theorem (Tsaprounis)

Extendible cardinals are never Laver indestructible. Despite Tsaprounis’ non-indestructibility result, one may ask the following:

Problem

Do nice forcing notions preserve extendible cardinals?

◮ By the theorem from [BHTU] we presented so far, the answer is

(more than likely) negative.

Natural approach

Focus on nice class forcing iterations and discuss whether a general theory

  • f preservation of extendibles is plausible.
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Our project

We seek for a general theory for preservation of C(n) -extendible cardinals by nice class forcing iterations.

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Our project

We seek for a general theory for preservation of C(n) -extendible cardinals by nice class forcing iterations. The first notion we introduce is the following:

Definition (Σn-supercompact cardinal)

Let n ≥ 1. If λ > δ is in C(n), then we say that δ is λ-Σn-supercompact if for every a ∈ Vλ, there exist ¯ δ < ¯ λ < δ and ¯ a ∈ V¯

λ, and there exists

elementary embedding j : V¯

λ −

→ Vλ such that:

◮ cp(j) = ¯

δ and j(¯ δ) = δ.

◮ j(¯

a) = a.

◮ ¯

λ ∈ C(n). We say that δ is a Σn−supercompact cardinal if it is λ-Σn-supercompact for every λ > δ in C(n).

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Theorem (Magidor)

For a cardinal δ, the following statements are equivalent:

1 δ is a supercompact cardinal. 2 δ is a Σ1-supercompact cardinal.
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Theorem (Magidor)

For a cardinal δ, the following statements are equivalent:

1 δ is a supercompact cardinal. 2 δ is a Σ1-supercompact cardinal.

Theorem (Bagaria-P)

Let n ≥ 1. For a cardinal δ, the following statements are equivalent:

1 δ is a C(n) -extendible cardinal. 2 δ is a Σn+1-supercompact cardinal. 3 δ is a C(n)+-extendible cardinal. ((1)≡(3) also due to Tsaprounis)
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Theorem (Magidor)

For a cardinal δ, the following statements are equivalent:

1 δ is a supercompact cardinal. 2 δ is a Σ1-supercompact cardinal.

Theorem (Bagaria-P)

Let n ≥ 1. For a cardinal δ, the following statements are equivalent:

1 δ is a C(n) -extendible cardinal. 2 δ is a Σn+1-supercompact cardinal. 3 δ is a C(n)+-extendible cardinal. ((1)≡(3) also due to Tsaprounis)

Morally, C(n) -extendibility is the natural model-theoretic strengthening of supercompactness.

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It turns out that Σn+1-supercompactness is a useful reformulation of C(n) -extendibility in the context of forcing iterations. Of course, not every forcing iteration is allowed (for instance, a Magidor iteration of Prikry/Magidor/Radin forcings).

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It turns out that Σn+1-supercompactness is a useful reformulation of C(n) -extendibility in the context of forcing iterations. Of course, not every forcing iteration is allowed (for instance, a Magidor iteration of Prikry/Magidor/Radin forcings).Even though, there is a broad family of class iterations which are harmless with respect to the preservation of C(n) -extendibles:

Definition (Suitable iteration)

A forcing iteration Pα; ˙ Qα : α ∈ ORD is suitable if it is the direct limit

  • f an Easton support iteration such that for each λ,
1 There is some θ greater than λ such that

Pν “ ˙ Qν is λ-directed closed ” for all ν ≥ θ.

2 If λ is an inaccessible cardinal, then Pλ ⊆ Vλ.
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Global idea for the preservation of C(n) -extendibles

Let n ≥ 1 and κ be a Σn+1-supercompact cardinal.

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Global idea for the preservation of C(n) -extendibles

Let n ≥ 1 and κ be a Σn+1-supercompact cardinal. Let λ > κ be a C(n+1)-cardinal and j : V¯

λ → Vλ witnessing λ − Σn+1-supercompactness
  • f κ.
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Global idea for the preservation of C(n) -extendibles

Let n ≥ 1 and κ be a Σn+1-supercompact cardinal. Let λ > κ be a C(n+1)-cardinal and j : V¯

λ → Vλ witnessing λ − Σn+1-supercompactness
  • f κ.We want to lift j to j∗ : V [G] ¯
λ → V [G]λ. Thus we have to deal with

three issues:

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Global idea for the preservation of C(n) -extendibles

Let n ≥ 1 and κ be a Σn+1-supercompact cardinal. Let λ > κ be a C(n+1)-cardinal and j : V¯

λ → Vλ witnessing λ − Σn+1-supercompactness
  • f κ.We want to lift j to j∗ : V [G] ¯
λ → V [G]λ. Thus we have to deal with

three issues:

1 Of course, the lifting argument.
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Global idea for the preservation of C(n) -extendibles

Let n ≥ 1 and κ be a Σn+1-supercompact cardinal. Let λ > κ be a C(n+1)-cardinal and j : V¯

λ → Vλ witnessing λ − Σn+1-supercompactness
  • f κ.We want to lift j to j∗ : V [G] ¯
λ → V [G]λ. Thus we have to deal with

three issues:

1 Of course, the lifting argument. 2 P “V¯ λ[ ˙

λ] = V [ ˙

G] ¯

λ” and P “Vλ[ ˙

Gλ] = V [ ˙ G]λ”.

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Global idea for the preservation of C(n) -extendibles

Let n ≥ 1 and κ be a Σn+1-supercompact cardinal. Let λ > κ be a C(n+1)-cardinal and j : V¯

λ → Vλ witnessing λ − Σn+1-supercompactness
  • f κ.We want to lift j to j∗ : V [G] ¯
λ → V [G]λ. Thus we have to deal with

three issues:

1 Of course, the lifting argument. 2 P “V¯ λ[ ˙

λ] = V [ ˙

G] ¯

λ” and P “Vλ[ ˙

Gλ] = V [ ˙ G]λ”.

3 P “¯

λ ∈ ˙ C(n+1)” and P “λ ∈ ˙ C(n+1)”. We need two new notions to fulfil requirements (2) and (3).

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Definition (P-reflecting cardinal)

Let P be a suitable iteration. A cardinal λ is P-reflecting if

1 λ is inaccessible, and 2 P forces that V [ ˙

G]λ ⊆ Vλ[ ˙ Gλ]. (Hence, if G is P-generic over V , then V [G]λ = Vλ[Gλ].)

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Definition (P-reflecting cardinal)

Let P be a suitable iteration. A cardinal λ is P-reflecting if

1 λ is inaccessible, and 2 P forces that V [ ˙

G]λ ⊆ Vλ[ ˙ Gλ]. (Hence, if G is P-generic over V , then V [G]λ = Vλ[Gλ].)

Definition (C(k)

P -cardinal)

For k ≥ 0, an ordinal α, and a definable suitable iteration P, we shall write Vα, ∈, P ∩ Vα ≺Σk V , ∈, P if for every ϕ ∈ ΣL

k and every ¯

a ∈ Vα, ΣL

k ϕ(a) iff Vα, ∈, P ∩ Vα ϕ(a).

We shall denote by C(k)

P

the class of all ordinals α such that Vα, ∈, P ∩ Vα ≺Σk V , ∈, P.

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Combining both definitions we get:

Definition (P-Σk-reflecting cardinal)

If k ≥ 1 and P is a definable suitable iteration, then we say that a cardinal κ is P-Σk-reflecting if it is P-reflecting and, moreover, it belongs to C(k)

P .

It is not hard to show that P-Σk-reflecting cardinal fulfil requirement (2):

Proposition

Suppose P is a suitable iteration. If κ is a P-reflecting cardinal in V such that κ ∈ C(k)

P

(and therefore Vκ, ∈, Pκ ≺Σk V , ∈, P), then P forces V [ ˙ G]κ ≺Σk V [ ˙ G].

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Thus, we are not interested in Σn+1-supercompact embeddings but in the next ones:

Definition (P-Σn-supercompact cardinal)

If n ≥ 1 and P is a suitable iteration, then we say that a cardinal δ is P-Σn-supercompact if there exists a proper class of P-Σn-reflecting cardinals, and for every such cardinal λ > δ and every a ∈ Vλ there exist ¯ δ < ¯ λ < δ and ¯ a ∈ V¯

λ, and there exists an elementary embedding

j : V¯

λ −

→ Vλ such that:

◮ cp(j) = ¯

δ and j(¯ δ) = δ.

◮ j(¯

a) = a.

◮ ¯

λ is P-Σn-reflecting.

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Proposition

Assume there are class many P-Σn+1-reflecting cardinals.

1 Then every P-Σn+1-supercompact cardinals is Σn+1-supercompact.
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Proposition

Assume there are class many P-Σn+1-reflecting cardinals.

1 Then every P-Σn+1-supercompact cardinals is Σn+1-supercompact. 2 The converse is also true if P is Σ1 or Π1 definable.
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Proposition

Assume there are class many P-Σn+1-reflecting cardinals.

1 Then every P-Σn+1-supercompact cardinals is Σn+1-supercompact. 2 The converse is also true if P is Σ1 or Π1 definable.

The existence of class many P-Σn+1-reflecting cardinals follows if ORD is Mahlo enough.

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Proposition

Assume there are class many P-Σn+1-reflecting cardinals.

1 Then every P-Σn+1-supercompact cardinals is Σn+1-supercompact. 2 The converse is also true if P is Σ1 or Π1 definable.

The existence of class many P-Σn+1-reflecting cardinals follows if ORD is Mahlo enough.

Main Theorem

Suppose m, n ≥ 1 and m ≤ n + 1. Suppose P is a weakly homogeneous Γm-definable suitable iteration and there exists a proper class of P-Σn+1-reflecting cardinals. If δ is a P-Σn+1-supercompact cardinal, then P “ δ is C(n) -extendible”.

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Proposition

Assume there are class many P-Σn+1-reflecting cardinals.

1 Then every P-Σn+1-supercompact cardinals is Σn+1-supercompact. 2 The converse is also true if P is Σ1 or Π1 definable.

The existence of class many P-Σn+1-reflecting cardinals follows if ORD is Mahlo enough.

Main Corollary

Suppose n ≥ 1, P is a weakly homogeneous Γ1-definable suitable iteration, δ is a C(n) -extendible cardinal, and there is a proper class of P-Σn+1-reflecting cardinals. Then P “ δ is C(n) -extendible ”.

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The previous results are too general but rely into two (apparently) restrictions:

1 Weak homogeneity. 2 Proper class many P-Σn+1-reflecting cardinals.
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The previous results are too general but rely into two (apparently) restrictions:

1 Weak homogeneity. (Not avoidable) 2 Proper class many P-Σn+1-reflecting cardinals.
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The previous results are too general but rely into two (apparently) restrictions:

1 Weak homogeneity. (Not avoidable) 2 Proper class many P-Σn+1-reflecting cardinals.(Avoidable)

Let’s illustrate with an example

P is the Jensen’s iteration

1 It is not hard to show that any inaccessible λ is P-reflecting (i.e.

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

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The previous results are too general but rely into two (apparently) restrictions:

1 Weak homogeneity. (Not avoidable) 2 Proper class many P-Σn+1-reflecting cardinals.(Avoidable)

Let’s illustrate with an example

P is the Jensen’s iteration

1 It is not hard to show that any inaccessible λ is P-reflecting (i.e.

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

2 P is Π1 definable so C(n+1) P

is Πn+1-definable.

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The previous results are too general but rely into two (apparently) restrictions:

1 Weak homogeneity. (Not avoidable) 2 Proper class many P-Σn+1-reflecting cardinals.(Avoidable)

Let’s illustrate with an example

P is the Jensen’s iteration

1 It is not hard to show that any inaccessible λ is P-reflecting (i.e.

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

2 P is Π1 definable so C(n+1) P

is Πn+1-definable.

3 If there is a C(n) -extendible → VP(Πn+1) holds → ORD is

Πn+1-Mahlo.

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The previous results are too general but rely into two (apparently) restrictions:

1 Weak homogeneity. (Not avoidable) 2 Proper class many P-Σn+1-reflecting cardinals.(Avoidable)

Let’s illustrate with an example

P is the Jensen’s iteration

1 It is not hard to show that any inaccessible λ is P-reflecting (i.e.

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

2 P is Π1 definable so C(n+1) P

is Πn+1-definable.

3 If there is a C(n) -extendible → VP(Πn+1) holds → ORD is

Πn+1-Mahlo.

4 C(n+1) P

∩ Reg is a proper class contained in the class of P-Σn+1-reflecting cardinals.

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Applications

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Applications: C(n) -extendible cardinals and the power set function on regular cardinals

Theorem (Tsaprounis and Bagaria-P.)

Let n ≥ 1, P be the Jensen’s iteration for GCH and κ be a C(n) -extendible

  • cardinal. Forcing with P preserves the C(n) -extendibility of κ.

More generally,

Theorem (Bagaria-P.)

If E is a Π1-definable Easton function, then PE preserves C(n)-extendible cardinals, all n ≥ 1. More generally, if E is a Πm-definable Easton function (m > 1) and λ is C(m+n−1)-extendible, then PE forces that λ is C(n)-extendible, all n ≥ 1 such that m ≤ n + 1.

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Applications: Preservation of VP

Theorem (Brooke-Taylor and Bagaria-P.)

Let P be a weakly-homogeneous definable suitable iteration. If VP holds in V , then VP holds in V P.

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Applications: Preservation of VP

Theorem (Brooke-Taylor and Bagaria-P.)

Let P be a weakly-homogeneous definable suitable iteration. If VP holds in V , then VP holds in V P.

Theorem (Bagaria-P.)

Let n, m ≥ 1 be such that m ≤ n + 1, and let P be a weakly-homogeneous Γm-definable suitable iteration. Then,

1 If Γ = Σ or n > 1, and VP(Πm+n) holds, then VP(Πn+1) holds in

V P.

2 If Γ = Π and n = 1, VP(Πm+1) holds, and ORD is Πm+2-Mahlo,

then VP(Π2) holds in V P.

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Applications: C(n) -extendible cardinals and the HOD Conjecture

Regardless the HOD Conjecture was true, our arguments actually shows that there may be little agreement between V and HOD about the computation of successors of regular cardinals.

Definition

Let C = Pα; ˙ Qα : α ∈ ORD be the Easton support iteration where P0 is the trivial forcing and for each ordinal α, if Pα “α is regular” then Pα “ ˙ Qα = ˙ Coll(α, α+)”, and Pα “ ˙ Qα is trivial” otherwise.

Theorem (Bagaria-P.)

Forcing with C preserves C(n)-extendible cardinals and forces “ (λ+)HOD < λ+, for every regular cardinal λ”.

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Applications: C(n) -extendible cardinals and diamond principles

Theorem (Bagaria-P.)

Let n ≥ 1 and suppose δ is a C(n)-extendible cardinal. We can force that ♦S holds, for every κ and every stationary S ⊆ κ+ while preserving the C(n) -extendibility of the cardinal δ. In particular, if VP holds in V , we can force ♦S, for every κ and every stationary S ⊆ κ+ while preserving VP. The same is also true for ♦+

S

Theorem (Bagaria-P.)

Let n ≥ 1 and assume that the GCH holds. If δ is a C(n)-extendible cardinal, then in V D the cardinal δ is still C(n)-extendible and ♦+

κ+ holds

for every cardinal κ. Hence, if VP and the GCH hold in V , then VP also holds in V D, together with ♦+

κ+, for every cardinal κ.
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Applications: C(n) -extendible cardinals and square principles

If δ is supercompact

    

λ

fails if cof(λ) < δ, λ,<cof(λ) fails ∀λ ≥ δ, λ,cof(λ) may holds if cof(λ) ≥ δ.

Theorem (Bagaria-P.)

Let n ≥ 1. If δ is a C(n) -extendible cardinal, there is a generic extension where δ remains C(n) -extendible and there is a proper class of cardinals λ, δ ≤ cof(λ) < λ, for which λ,cof(λ) holds.

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Applications: C(n) -extendible cardinals and V = HOD

Here we run into troubles as the iteration for forcing V = HOD is not weakly homogeneous. Nonetheless, there is a strong analogue of our main theorem for non-homogeneous iterations:

Theorem (Bagaria-P.)

Let P be a (not necessarily definable) suitable iteration. If δ is a P- C(n)-extendible cardinal, and there is a proper class of P-reflecting cardinals, then P forces that δ is C(n)-extendible.

Theorem (Bagaria-P.)

Let n ≥ 1 and let P be the standard class forcing that forces V=HOD. If δ is P- C(n)-extendible, then P forces that δ is C(n)-extendible.

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Moltes felicitats i per molts anys més, Joan

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Enjoy your stay in Catalonia!