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

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C(n) -cardinals with Forcing

Alejandro Poveda

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

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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

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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

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

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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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).

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).

some specific property.

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).

some specific property.

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).

some specific property.

plus Replacement, modulo de remaining ZF axioms.

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The classes C(n)

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

How C(n) looks like?

absoluteness).

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)

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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

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

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

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

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

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

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

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

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

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

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

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What about C(n) -extendibility?

Some properties of C(n) -extendibility

inaccessible).

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What about C(n) -extendibility?

Some properties of C(n) -extendibility

inaccessible).

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What about C(n) -extendibility?

Some properties of C(n) -extendibility

inaccessible).

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What about C(n) -extendibility?

Some properties of C(n) -extendibility

inaccessible).

C(m)-extendibles.

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What about C(n) -extendibility?

Some properties of C(n) -extendibility

inaccessible).

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)

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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)

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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)

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)

hierarchy in terms of consistency strength?

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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A historical interlude

first measurable and the first supercompact.

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A historical interlude

first measurable and the first supercompact.

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A historical interlude

first measurable and the first supercompact.

Theorem (Magidor)

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

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

Magidor).

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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

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

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:

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Theorem (Bagaria-Casacuberta-Mathias-Rosický)

TFAE:

Theorem (Bagaria)

For n ≥ 1, TFAE:

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Theorem (Bagaria)

For n ≥ 1, TFAE:

Corollary (Bagaria)

TFAE:

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Corollary (Bagaria)

TFAE:

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

notions?

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A Magidor-like analysis

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

Question

notions?

consistency strength?

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A Magidor-like analysis

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

Question

notions?

consistency strength?

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

n ≥ 1, “k is C(n)-extendible

n ≥ 1, “k is C(n)-supercompact

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

for any λ ≥ κ

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Proof of Main Theorem 1

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

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

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

there is S ⊆ Sκ

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Proof of Main Theorem 1

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

there is S ⊆ Sκ

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

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)

θ ∈ (Sj(κ)

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

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)

θ ∈ (Sj(κ)

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

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

Definition

Let Pℓ

α < κ and Pℓ

Qα = Pα” and Pℓ

Qα trivial”, otherwise.

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Proof of Main Theorem 1

Definition

Let Pℓ

α < κ and Pℓ

Qα = Pα” and Pℓ

Qα trivial”, otherwise.

Proposition

The iteration Pℓ

pattern.

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Definition

Let Pℓ

α < κ and Pℓ

Qα = Pα” and Pℓ

Qα trivial”, otherwise.

standard arguments

Proposition

The iteration Pℓ

pattern.

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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.

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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.

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:

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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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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

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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

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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

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

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

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

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

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

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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

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

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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

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

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

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

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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

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

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

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

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, ≤∗

Magidor iteration of Prikry forcings if the following conditions holds:

Qα, ≤ ˙

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

Qα = {1}, otherwise.

α < κ, Mα ˙ Qα = P ˇ

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∗

Definition (Magidor Product)

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

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∗

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) ≤⋆

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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

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

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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Steps of the proof

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.

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

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Steps of the proof

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.

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

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

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Steps of the proof

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.

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

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

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

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Proof of Main Theorem 2

Steps of the proof

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

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Proof of Main Theorem 2

Steps of the proof

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

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Proof of Main Theorem 2

Steps of the proof

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

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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)”

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

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

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)”

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

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

P Indϕ(x)(κ, {Add(κ, 1)}) (Hamkins)

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

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?

(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?

(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¯

elementary embedding j : V¯

→ Vλ such that:

δ and j(¯ δ) = δ.

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:

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Theorem (Magidor)

For a cardinal δ, the following statements are equivalent:

Theorem (Bagaria-P)

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

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Theorem (Magidor)

For a cardinal δ, the following statements are equivalent:

Theorem (Bagaria-P)

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

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,

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

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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¯

• 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¯

• f κ.We want to lift j to j∗ : V [G] ¯

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¯

• f κ.We want to lift j to j∗ : V [G] ¯

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¯

• f κ.We want to lift j to j∗ : V [G] ¯

three issues:

G¯

G] ¯

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¯

• f κ.We want to lift j to j∗ : V [G] ¯

three issues:

G¯

G] ¯

Gλ] = V [ ˙ G]λ”.

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

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

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

Definition (C(k)

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

a ∈ Vα, ΣL

We shall denote by C(k)

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)

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)

(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¯

j : V¯

→ Vλ such that:

δ and j(¯ δ) = δ.

a) = a.

λ is P-Σn-reflecting.

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Proposition

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

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Proposition

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

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Proposition

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

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.

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.

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:

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

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

Let’s illustrate with an example

P is the Jensen’s iteration

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

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

Let’s illustrate with an example

P is the Jensen’s iteration

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

is Πn+1-definable.

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

Let’s illustrate with an example

P is the Jensen’s iteration

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

is Πn+1-definable.

Πn+1-Mahlo.

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

Let’s illustrate with an example

P is the Jensen’s iteration

P V [ ˙ G]λ ⊆ Vλ[ ˙ Gλ].

is Πn+1-definable.

Πn+1-Mahlo.

∩ 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,

V P.

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 ♦+

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 ♦+

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

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

If δ is supercompact

∗

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!