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Strongly uplifting cardinals Boldface Resurrection Axioms

Boldface resurrection & the strongly uplifting, the superstrongly unfoldable, and the almost-hugely unfoldable cardinals

Joel David Hamkins

The City University of New York College of Staten Island The CUNY Graduate Center

& MathOverflow ;-)

Mathematics, Philosophy, Computer Science

Boise Extravaganza in Set Theory (BEST)

American Association for the Advancement of Science - Pacific Division 95th annual meeting Riverside, CA, June 18-20, 2014 Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Acknowledgements

This is joint work with Thomas A. Johnstone, New York City Tech, CUNY.

• J. D. Hamkins, T. A. Johnstone, “Strongly uplifting cardinals and

the boldface resurrection axioms,” under review, arxiv.org/abs/1403.2788.

• J. D. Hamkins, T. A. Johnstone, “Resurrection axioms and

uplifting cardinals,” Archive for Mathematical Logic 53:3-4(2014), p. 463–485.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Recall the uplifting cardinals

Definition (Hamkins, Johnstone) An inaccessible cardinal κ is uplifting, if there are arbitrarily large inaccessible cardinals θ with Vκ ≺ Vθ. Vκ In consistency strength, “ORD is Mahlo” < uplifting < Mahlo Pseudo-uplifting: do not require the target θ to be inaccessible.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Recall the uplifting cardinals

Definition (Hamkins, Johnstone) An inaccessible cardinal κ is uplifting, if there are arbitrarily large inaccessible cardinals θ with Vκ ≺ Vθ. Vκ Vλ ≺ In consistency strength, “ORD is Mahlo” < uplifting < Mahlo Pseudo-uplifting: do not require the target θ to be inaccessible.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Recall the uplifting cardinals

Definition (Hamkins, Johnstone) An inaccessible cardinal κ is uplifting, if there are arbitrarily large inaccessible cardinals θ with Vκ ≺ Vθ. Vκ Vη ≺ In consistency strength, “ORD is Mahlo” < uplifting < Mahlo Pseudo-uplifting: do not require the target θ to be inaccessible.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Recall the uplifting cardinals

Definition (Hamkins, Johnstone) An inaccessible cardinal κ is uplifting, if there are arbitrarily large inaccessible cardinals θ with Vκ ≺ Vθ. Vκ Vθ ≺ In consistency strength, “ORD is Mahlo” < uplifting < Mahlo Pseudo-uplifting: do not require the target θ to be inaccessible.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Recall the uplifting cardinals

Definition (Hamkins, Johnstone) An inaccessible cardinal κ is uplifting, if there are arbitrarily large inaccessible cardinals θ with Vκ ≺ Vθ. Vκ Vθ ≺ In consistency strength, “ORD is Mahlo” < uplifting < Mahlo Pseudo-uplifting: do not require the target θ to be inaccessible.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Recall the uplifting cardinals

Definition (Hamkins, Johnstone) An inaccessible cardinal κ is uplifting, if there are arbitrarily large inaccessible cardinals θ with Vκ ≺ Vθ. Vκ Vθ ≺ In consistency strength, “ORD is Mahlo” < uplifting < Mahlo Pseudo-uplifting: do not require the target θ to be inaccessible.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Connection with forcing axioms

We introduced the uplifting cardinals precisely to prove: Theorem (Hamkins, Johnstone) The following are equiconsistent over ZFC:

1 There is an uplifting cardinal. 2 The resurrection axiom RA(all). 3 The resurrection axiom RA(ccc) for c.c.c. forcing 4 The resurrection axiom RA(proper) + ¬CH 5 The resurrection axiom RA(semi-proper) + ¬CH 6 and many other instances RA(Γ) + ¬CH.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Connection with forcing axioms

We introduced the uplifting cardinals precisely to prove: Theorem (Hamkins, Johnstone) The following are equiconsistent over ZFC:

1 There is an uplifting cardinal. 2 The resurrection axiom RA(all). 3 The resurrection axiom RA(ccc) for c.c.c. forcing 4 The resurrection axiom RA(proper) + ¬CH 5 The resurrection axiom RA(semi-proper) + ¬CH 6 and many other instances RA(Γ) + ¬CH.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Strongly uplifting cardinals

We now generalize this concept to a boldface notion. Definition (Hamkins, Johnstone) A cardinal κ is strongly uplifting, if for every A ⊆ κ there are unboundedly many inaccessible cardinals θ and A∗ with Vκ, ∈, A ≺ Vθ, ∈, A∗

Vκ, ∈, A A

• Vθ, ∈, A∗

≺

A∗

• Initial aim: equiconsistency with boldface resurrection.

But the cardinals themselves turned out to be very interesting.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Strongly uplifting cardinals

We now generalize this concept to a boldface notion. Definition (Hamkins, Johnstone) A cardinal κ is strongly uplifting, if for every A ⊆ κ there are unboundedly many inaccessible cardinals θ and A∗ with Vκ, ∈, A ≺ Vθ, ∈, A∗

Vκ, ∈, A A

• Vθ, ∈, A∗

≺

A∗

• Initial aim: equiconsistency with boldface resurrection.

But the cardinals themselves turned out to be very interesting.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Surprising strength on the target for free

Suppose every A ⊆ κ has arbitrarily large θ and A∗ with Vκ, ∈, A ≺ Vθ, ∈, A∗

Vκ, ∈, A A

• Vθ, ∈, A∗

≺

A∗

• Claim. Without loss, θ is inaccessible, w. compact and more.

strongly uplifting = pseudo strongly uplifting = strongly uplifting with indescribable targets

Proof. Fix A ⊆ κ. Let C = { δ < κ | Vδ, ∈, A ∩ δ ≺ Vκ, ∈, A }, club. Get Vκ, ∈, A, C ≺ Vθ, ∈, A∗, C∗. Since κ ∈ C∗ and is inaccessible, weakly compact, indescribable, etc. in Vθ, there are many such γ ∈ C∗. It follows that Vκ, ∈, A ≺ Vγ, ∈, A∗ ∩ γ for such γ.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Surprising strength on the target for free

Suppose every A ⊆ κ has arbitrarily large θ and A∗ with Vκ, ∈, A ≺ Vθ, ∈, A∗

Vκ, ∈, A A

• Vθ, ∈, A∗

≺

A∗

• Claim. Without loss, θ is inaccessible, w. compact and more.

strongly uplifting = pseudo strongly uplifting = strongly uplifting with indescribable targets

Proof. Fix A ⊆ κ. Let C = { δ < κ | Vδ, ∈, A ∩ δ ≺ Vκ, ∈, A }, club. Get Vκ, ∈, A, C ≺ Vθ, ∈, A∗, C∗. Since κ ∈ C∗ and is inaccessible, weakly compact, indescribable, etc. in Vθ, there are many such γ ∈ C∗. It follows that Vκ, ∈, A ≺ Vγ, ∈, A∗ ∩ γ for such γ.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Surprising strength on the target for free

Suppose every A ⊆ κ has arbitrarily large θ and A∗ with Vκ, ∈, A ≺ Vθ, ∈, A∗

Vκ, ∈, A A

• Vθ, ∈, A∗

≺

A∗

• Claim. Without loss, θ is inaccessible, w. compact and more.

strongly uplifting = pseudo strongly uplifting = strongly uplifting with indescribable targets

Proof. Fix A ⊆ κ. Let C = { δ < κ | Vδ, ∈, A ∩ δ ≺ Vκ, ∈, A }, club. Get Vκ, ∈, A, C ≺ Vθ, ∈, A∗, C∗. Since κ ∈ C∗ and is inaccessible, weakly compact, indescribable, etc. in Vθ, there are many such γ ∈ C∗. It follows that Vκ, ∈, A ≺ Vγ, ∈, A∗ ∩ γ for such γ.

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Strongly uplifting cardinals Boldface Resurrection Axioms

Absolute to L

Theorem Every strongly uplifting cardinal is strongly uplifting in L. Proof. Fix any A ⊆ κ with A ∈ L. Let C be club of δ < κ with Lδ, ∈, A ∩ δ ≺ Lκ, ∈, A. Extend Vκ, ∈, A, C ≺ Vθ, ∈, A∗, C∗ as before. It follows that Lκ, ∈, A ≺ Lγ, ∈, A∗ ∩ γ and A∗ ∩ γ ∈ L for γ ∈ C∗.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Embedding characterizations

We have many familiar large cardinal embedding characterizations, even very low in the large cardinal hierarchy. A cardinal κ is measurable if it is the critical point of an elementary embedding j : V → M. κ is weakly compact if for every A ⊆ κ there is M | = ZFC with A ∈ M and j : M → N with critical point κ. κ is θ-unfoldable if there is such j : M → N with j(κ) ≥ θ. κ is strongly θ-unfoldable if also Vθ ⊆ N. Strong unfoldability is in essence a transfinite continuation of total indescribability.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Embedding characterizations

We have many familiar large cardinal embedding characterizations, even very low in the large cardinal hierarchy. A cardinal κ is measurable if it is the critical point of an elementary embedding j : V → M. κ is weakly compact if for every A ⊆ κ there is M | = ZFC with A ∈ M and j : M → N with critical point κ. κ is θ-unfoldable if there is such j : M → N with j(κ) ≥ θ. κ is strongly θ-unfoldable if also Vθ ⊆ N. Strong unfoldability is in essence a transfinite continuation of total indescribability.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Embedding characterizations

We have many familiar large cardinal embedding characterizations, even very low in the large cardinal hierarchy. A cardinal κ is measurable if it is the critical point of an elementary embedding j : V → M. κ is weakly compact if for every A ⊆ κ there is M | = ZFC with A ∈ M and j : M → N with critical point κ. κ is θ-unfoldable if there is such j : M → N with j(κ) ≥ θ. κ is strongly θ-unfoldable if also Vθ ⊆ N. Strong unfoldability is in essence a transfinite continuation of total indescribability.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Embedding characterizations

We have many familiar large cardinal embedding characterizations, even very low in the large cardinal hierarchy. A cardinal κ is measurable if it is the critical point of an elementary embedding j : V → M. κ is weakly compact if for every A ⊆ κ there is M | = ZFC with A ∈ M and j : M → N with critical point κ. κ is θ-unfoldable if there is such j : M → N with j(κ) ≥ θ. κ is strongly θ-unfoldable if also Vθ ⊆ N. Strong unfoldability is in essence a transfinite continuation of total indescribability.

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Strongly uplifting cardinals Boldface Resurrection Axioms

A few new large cardinal concepts

In the same vein, consider the following weak analogues of superstrongness and almost hugeness. Definition (Hamkins, Johnstone) A cardinal κ is weakly superstrong if for every A ⊆ κ there is j : M → N, critical point κ with A ∈ M | = ZFC and Vj(κ) ⊆ N. The cardinal κ is weakly almost huge if for every A ⊆ κ there is j : M → N, critical point κ with A ∈ M | = ZFC and N<j(κ) ⊆ N.

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Extending to arbitrarily large targets

In the unfoldability style, it is natural to ask for arbitrarily large targets. Definition (Hamkins, Johnstone) A cardinal κ is superstrongly unfoldable (or weakly superstrong with arbitrarily large targets), if for every A ⊆ κ there is j : M → N with critical point κ, A ∈ M | = ZFC, j(κ) arbitrarily large and Vj(κ) ⊆ N. A cardinal κ is almost-hugely unfoldable (or weakly almost huge with arbitrarily large targets), if for every A ⊆ κ there is j : M → N with critical point κ, A ∈ M | = ZFC, j(κ) arbitrarily large and N<j(κ) ⊆ N.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

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A surprising equivalence

Nevertheless, these notions are actually equivalent, and both are equivalent to our previous notion of strongly uplifting! Theorem The following large cardinal concepts are equivalent.

1 κ is strongly uplifting. 2 κ is superstrongly unfoldable. 3 κ is almost-hugely unfoldable.

Let’s sketch the proof. 3 = ⇒ 2 = ⇒ 1 is easy.

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

A surprising equivalence

Nevertheless, these notions are actually equivalent, and both are equivalent to our previous notion of strongly uplifting! Theorem The following large cardinal concepts are equivalent.

1 κ is strongly uplifting. 2 κ is superstrongly unfoldable. 3 κ is almost-hugely unfoldable.

Let’s sketch the proof. 3 = ⇒ 2 = ⇒ 1 is easy.

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The nontrivial implication: κ strongly uplifting = ⇒ almost-hugely unfoldable.

Proof. Fix A ⊆ κ and M | = ZFC with A ∈ M ⊇ M <κ. Find π : M, ∈ ∼ = κ, E. By strong uplifting, Vκ, ∈, E ≺ Vγ, ∈, E∗ with γ inaccessible. Since E is well-founded and γ is regular, E∗ is well-founded. Mostowski collapse τ : γ, E∗ ∼ = N, ∈. Let j = τ ◦ π : M → N. Note j ↾ κ = id since if α is coded by ξ with respect to E, then it is also coded by ξ with respect to E∗, and so j(α) = α. Similarly, j(κ) = γ, and so cp(j) = κ. Since Vκ, ∈, E sees that every set is coded via E, it follows by elementarity that every x ∈ Vγ is coded via E∗, and so Vj(κ) = Vγ ⊆ N, giving superstrongness. Similarly, M <κ ⊆ M implies that Vκ, ∈, E believes that κ, E is closed under <κ-sequences, and so for Vγ, ∈, E∗, which implies N<j(κ) ⊆ N, giving almost hugeness.

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Weakly superstrong = weakly almost huge

Similarly, the local concepts are also equivalent. Theorem A cardinal κ is weakly superstrong if and only if κ is weakly almost huge. As a result, many techniques from the superstrongness and almost hugeness context have fruitful analogues with the corresponding weak notions. General phenomenon: weak versions of strongness and supercompactness coincide weak superstrongness and weak almost hugeness coincide

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Quick upper bound

Theorem 0♯ exists = ⇒ every Silver indiscernible is strongly uplifting in L. Proof. If κ < δ are Silver indiscernibles, let j : L → L have j(κ) = δ. It follows that Lκ, ∈, A ≺ Lδ, ∈, j(A) witnesses strong uplifting. Consequently, if there is a Ramsey cardinal, for example, then ℵ1 is weakly superstrong, weakly almost huge and more in L.

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Strongly uplifting cardinals Boldface Resurrection Axioms

A better upper bound

A cardinal δ is subtle, if for any Bη | η < δ with Bη ⊆ η and any club C ⊆ δ there are κ < η in C such that Bκ = Bη ∩ κ. Theorem If δ is a subtle cardinal, then the strongly uplifting cardinals in Vδ are stationary. Proof. If not, fix club C ⊆ δ with no κ ∈ C strongly uplifting in Vδ. Fix θκ, Aκ ⊆ κ so Vκ, ∈, Aκ has no extension Vγ, ∈, A∗ any γ > θκ in Vδ. Thin the club. Let Bκ ⊆ κ code diagram of Vκ, ∈, Aκ. Since δ subtle, ∃κ < η in C with Bκ = Bη ∩ κ. So Vκ, ∈, Aκ ≺ Vη, ∈, Aη, contradiction.

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Laver functions

Theorem Every superstrongly unfoldable cardinal has an

• rdinal-anticipating Laver function.

A Laver function is ℓ : κ → κ such that for any α and any A ⊆ κ there is j : M → N, critical point κ, with ℓ, A ∈ M | = ZFC and j(κ) as large as desired, Vj(κ) ⊆ N, and j(ℓ)(κ) = α. Strongly uplifting version: ∀α, A ⊆ κ there are arbitrarily large extensions Vκ, ∈, A, ℓ ≺ Vθ, ∈, A∗, ℓ∗ with ℓ∗(κ) = α.

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Strongly uplifting cardinals Boldface Resurrection Axioms

Laver functions

Theorem Every superstrongly unfoldable cardinal has an

• rdinal-anticipating Laver function.

A Laver function is ℓ : κ → κ such that for any α and any A ⊆ κ there is j : M → N, critical point κ, with ℓ, A ∈ M | = ZFC and j(κ) as large as desired, Vj(κ) ⊆ N, and j(ℓ)(κ) = α. Strongly uplifting version: ∀α, A ⊆ κ there are arbitrarily large extensions Vκ, ∈, A, ℓ ≺ Vθ, ∈, A∗, ℓ∗ with ℓ∗(κ) = α.

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Proof of Laver functions

Theorem Every superstrongly unfoldable cardinal has an

• rdinal-anticipating Laver function.

Proof. Define ℓ(δ) = α, if the number of θ for which Vθ | = δ is strongly uplifting is β, α + 1 some β. Fix α and A ⊆ κ, and let η be (θ, α + 2)th with Vη | = κ is strongly uplifting (by reflection there are many). So get Vκ, ∈, A, ℓ ≺ Vγ, ∈, A∗, ℓ∗ with ℓ∗(κ) = α. Can actually anticipate all OD sets. Under V = HOD, all sets. Open question: must there always be a full Laver function?

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Resurrection

Let us turn now to the topic of Resurrection axioms. These are forcing axioms, involving the idea that truths killed by forcing might be resurrected again.

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Existential closure

The resurrrection axioms are inspired by the concept of existential closure in model theory. Definition A model M is existentially closed if whenever M is a submodel

• f N, then existential witnesses in N exist already in M.

In other words, M ≺Σ1 N. Existential closure asserts that objects which could exist in a larger model, already exist in the ground model.

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Forcing Axioms as Existential Closure

Many classical forcing axioms can be viewed as expressing to a degree that the universe is existentially closed. The essence of these axioms is the assertion that certain filters, which could exist in a forcing extension, exist already in V. V ⊆ V[G] Martin’s Axiom, the Proper Forcing Axiom, Martin’s Maximum are all expressible as instances of existential closure. Meanwhile, the universe V is never actually existentially closed in all its forcing extensions. But the collection Hc = { sets of hereditary size < c } can be existentially closed in forcing extensions, and this is precisely what the forcing axioms express.

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Forcing Axioms as Existential Closure

Many classical forcing axioms can be viewed as expressing to a degree that the universe is existentially closed. The essence of these axioms is the assertion that certain filters, which could exist in a forcing extension, exist already in V. V ⊆ V[G] Martin’s Axiom, the Proper Forcing Axiom, Martin’s Maximum are all expressible as instances of existential closure. Meanwhile, the universe V is never actually existentially closed in all its forcing extensions. But the collection Hc = { sets of hereditary size < c } can be existentially closed in forcing extensions, and this is precisely what the forcing axioms express.

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Forcing Axioms as existential closure

Theorem (Stavi(80s), Bagaria) Martin’s Axiom MA is equivalent to the assertion that for any c.c.c. forcing extension V[g] Hc ≺Σ1 HV[g]

c

. Theorem (Bagaria) The Bounded Proper Forcing Axiom is equivalent to the assertion that for any proper forcing extension V[g] Hω2 ≺Σ1 HV[g]

ω2

. Thus, Hc is existentially closed in forcing extensions.

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Existential Closure ⇐ ⇒ Resurrection

• Theorem. The Following are equivalent

A model M is existentially closed. M has Resurrection. That is, whenever M ⊆ N0, then there is M ⊆ N0 ⊆ N1 with M ≺ N1. (⇐) Resurrection implies existential closure, since witnesses in N0 still exist in N1, which is fully elementary over M. (⇒) If M is existentially closed and M ⊆ N0, then the full elementary diagram of M is consistent with the atomic diagram

• f N0. A model of this theory is the desired N1. QED

The Key Point. Equivalence can break down when the class of models is restricted. But resurrection remains stronger.

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Existential Closure ⇐ ⇒ Resurrection

• Theorem. The Following are equivalent

A model M is existentially closed. M has Resurrection. That is, whenever M ⊆ N0, then there is M ⊆ N0 ⊆ N1 with M ≺ N1. (⇐) Resurrection implies existential closure, since witnesses in N0 still exist in N1, which is fully elementary over M. (⇒) If M is existentially closed and M ⊆ N0, then the full elementary diagram of M is consistent with the atomic diagram

• f N0. A model of this theory is the desired N1. QED

The Key Point. Equivalence can break down when the class of models is restricted. But resurrection remains stronger.

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The main idea

This suggests using resurrection to formulate forcing axioms, in place of Σ1 elementarity. That is, we can formulate forcing axioms by means of the resurrection concept, considering not just Σ1 elementarity in the relevant forcing extensions ∀Q M ≺Σ1 MV Q but instead asking for full elementarity in a further extension ∀Q ∃ ˙ R M ≺ MV Q∗ ˙

R Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Resurrection Axiom

This led naturally to the Resurrection Axioms. Definition (Hamkins, Johnstone) Suppose Γ is a definable class of forcing notions. The resurrection axiom RA(Γ) is the assertion that for every Q ∈ Γ there is ˙ R ∈ ΓV Q such that Hc ≺ HV[g∗h]

c

whenever g ∗ h ⊆ Q ∗ ˙ R is V-generic.

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Equiconsistency Strength of RA(Γ)

Theorem (Hamkins, Johnstone) The following are equiconsistent over ZFC:

1 There is an uplifting cardinal. 2 The resurrection axiom RA(all). 3 The resurrection axiom RA(ccc) for c.c.c. forcing 4 The resurrection axiom RA(proper) + ¬CH 5 The resurrection axiom RA(semi-proper) + ¬CH 6 and many other instances RA(Γ) + ¬CH.

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Boldface resurrection

Now, we extend the analysis to the boldface context. Specifically: By allowing predicates, we generalize the uplifting cardinals to the strongly uplifting cardinals. Vκ, ∈, A ≺ Vθ, ∈, A∗ By allowing predicates, we generalize the resurrection axiom RA to the boldface resurrection axiom RA ∼ . Hc, ∈, A ≺ HV[g][h]

c

, ∈, A∗

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Boldface resurrection

Definition The boldface resurrection axiom RA ∼ (Γ) asserts that for every Q ∈ Γ and A ⊆ c there is ˙ R ∈ ΓV Q and A∗ with Hc, ∈, A ≺ HV[g∗h]

c

, ∈, A∗ in the corresponding extension V[g ∗ h].

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Embedding characterization

Theorem If |Hc| = c, then the following are equivalent for any class Γ.

1 The boldface resurrection axiom RA

∼ (Γ).

2 For every Q ∈ Γ and every transitive set M |

= ZFC− with |M| = c ∈ M and Hc ⊆ M, there is ˙ R ∈ ΓV Q, such that in the corresponding extension V[g ∗ h], there is elementary j : M → N with j ↾ Hc = id and j(c) = cV[g∗h] and HV[g∗h]

c

⊆ N. Similarly, the weak boldface axiom wRA ∼ (Γ) is equivalent to the embedding characterization obtained by omitting the requirement that ˙ R ∈ ΓV Q.

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Strength of Boldface Resurrection

Main Theorem The following theories are equiconsistent over ZFC.

1 There is a strongly uplifting cardinal. 2 The boldface Resurrection Axiom RA

∼ (all).

3 The boldface Reserrection Axiom RA

∼ (ccc) for c.c.c. forcing.

4 The boldface Resurrection Axiom for proper forcing. 5 The boldface Resurrection Axiom for semi-proper forcing.

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strongly uplifting = ⇒ RA ∼

Assume κ is strongly uplifting. We produce a forcing extension with RA ∼ (proper). Let f : κ → κ be a strongly uplifting Menas

• function. Let P be the PFA lottery preparation, which forces at

stage γ < κ with the lottery sum Qγ = ⊕{ Q ∈ HV[Gγ]

f(γ)+ | Q is proper }.

Let G ⊆ P be V-generic. Since the generic will often opt to add a Cohen real, κ = cV[G]. If Q is proper in V[G] and A ⊆ κ, find extension Vκ, ∈, A, f, P ≺ Vθ, ∈, A∗, f ∗, P∗ with f ∗(κ) > |Q|. Opt for Q in the stage κ lottery of P∗, which becomes P ∗ Q ∗ ˙ R, where ˙ R is the rest of the forcing. So Vκ[G], ∈, A ≺ Vθ[G][g][h], ∈, A∗. In other words, HV[G]

c

, ∈, A ≺ HV[G][g][h]

c

, ∈, A∗, which witnesses RA ∼ in V[G].

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RA ∼ = ⇒ strongly uplifting

For this direction, the basic fact is that if RA ∼ (Γ), then κ = c is strongly uplifting in L. In the case of RA ∼ (proper), for example, fix any A ⊆ κ with A ∈ L. Let Q = Coll(ω1, θ). By resurrection, there is ˙ R, such that Hc, ∈, A ≺ HV[g][h]

c

, ∈, A∗ in the corresponding extension V[g][h]. It follows that Lκ, ∈, A ≺ LcV[g][h], ∈, A∗, and all initial segments of A∗ are in L, which is enough to conclude κ is strongly uplifting in L. For RA ∼ (ccc), use Add(ω, θ).

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York

Strongly uplifting cardinals Boldface Resurrection Axioms

Thank you.

Slides and articles available on http://jdh.hamkins.org. Joel David Hamkins The City University of New York

Boldface resurrection and superstrongly uplifting cardinals Joel David Hamkins, New York