Draft 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 Π n + 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.
Draft 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
Draft The difficult case: C ( n ) -supercompactnes On the contrary, in this case we mainly have questions:
Draft 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?
Draft 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.
Draft 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?
Draft 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?
Draft But, where did these cardinals appear? Or, in other words, why are they worth to be studied?
Draft 1 Reflection. 2 Large Cardinals between the first supercompact and VP .
Draft A historical interlude ◮ One of the prominent regions of V is that encompassed between the first measurable and the first supercompact.
Draft 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
Draft 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.
Draft A historical interlude What it can be said about the immediate upper region? Namely, between the first supercompact cardinal and Vopěnka’s principle?
Draft 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.
Draft 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...
Draft A historical interlude Bagaria gave a level-by-level equivalence of VP .
Draft 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 λ ∈ C ( 1 ) and j ( ¯ κ , ¯ j : � V ¯ λ , ∈� → � V λ , ∈� such that crit ( j ) = ¯ κ ) = κ (i.e. the Π 1 -definable class {� V λ , ∈ λ � : λ ∈ C ( 1 ) } reflects below κ )
Draft Theorem (Bagaria-Casacuberta-Mathias-Rosický) TFAE: 1 VP ( Π 1 ) holds. 2 VP ( κ , Σ 2 ) holds, for some κ . 3 There is a supercompact cardinal.
Draft Theorem (Bagaria-Casacuberta-Mathias-Rosický) TFAE: 1 VP ( Π 1 ) holds. 2 VP ( κ , Σ 2 ) holds, for some κ . 3 There is a supercompact cardinal. Theorem (Bagaria) For n ≥ 1 , TFAE: 1 VP ( Π n + 1 ) holds. 2 VP ( κ , Σ n + 2 ) holds, for some κ . 3 There is a C ( n ) -extendible cardinal.
Draft Theorem (Bagaria) For n ≥ 1 , TFAE: 1 VP ( Π n + 1 ) holds. 2 VP ( κ , Σ n + 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 ( Π n ) holds. 3 VP ( κ , Σ n + 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.
Draft Corollary (Bagaria) TFAE: 1 VP holds. 2 For every n ≥ 1 , VP ( Π n ) holds. 3 VP ( κ , Σ n + 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
Draft Part I: Magidor-like analysis of the class of C ( n ) -supercompact cardinals
Draft 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?
Draft 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?
Draft 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?
Draft 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 , M ω ⊆ M and j ( κ ) is a limit cardinal. In particular, in V P the cardinal κ is supercompact but not C ( 1 ) -supercompact cardinal.
Draft 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 , M ω ⊆ 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.
Draft 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 , M ω ⊆ 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 ın S ( 1 ) ” . “ ZFC + m´ ın M < m´ ın K < m´ ın S < m´
Draft 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 ın S ( n ) < m´ ın K = m´ ın S = m´ m´ ın E .
Draft 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 ın S ( n ) < m´ ın K = m´ ın S = m´ m´ ın E . Answer to questions 2 and 3 2 Does C ( n ) -supercompactness entails a strict hierarchy in terms of consistency strength? No.
Draft 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 ın S ( n ) < m´ ın K = m´ ın S = m´ 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
Draft 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 ın S ( ω ) < m´ ın K = m´ ın S = m´ 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.
Draft A sketch of the proofs In the following slides we are giving a sketch of the two main results:
Draft Proof of Main Theorem 1 ◮ By a classical result of Solovay, if κ is strongly compact then � λ fails for any λ ≥ κ
Draft 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
Draft 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 ?
Draft 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 ⊆ S κ λ stationary such that � θ holds, each θ ∈ S .
Draft 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 ⊆ S κ 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 .
Draft 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 ∈ S κ ≤ λ , � θ -holds. Then there is no elementary embedding j : V → M such that crit ( j ) = κ , M λ ⊆ M and j ( κ ) being a limit cardinal.
Draft Proof of Main Theorem 1 Proof Suppose such embedding exists. Notice that cof ( j ( κ )) > λ and thus S j ( κ ) ≤ λ = ( S j ( κ ) ≤ λ ) M is a (real) stationary set. By elementarity, for every θ ∈ ( S j ( 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 ∈ S j ( 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 .
Draft Proof of Main Theorem 1 Proof Suppose such embedding exists. Notice that cof ( j ( κ )) > λ and thus S j ( κ ) ≤ λ = ( S j ( κ ) ≤ λ ) M is a (real) stationary set. By elementarity, for every θ ∈ ( S j ( 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 ∈ S j ( 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 + = θ λ ≤ µ λ = µ . Therefore, there is a � θ sequence with θ ≥ κ hence κ is not longer supercompact. Contradiction.
Draft Proof of Main Theorem 1 1 Let ℓ : κ → κ be a Laver function and define the iteration P ℓ κ Definition Let P ℓ κ denote the κ -Easton support iteration defined in such a way that if α “ ˙ α < κ and P ℓ α was defined, if α ∈ cl ( ℓ ) ∩ S κ Q α = P � α ” and ω then � P ℓ α “ ˙ � P ℓ Q α trivial”, otherwise.
Draft Proof of Main Theorem 1 1 Let ℓ : κ → κ be a Laver function and define the iteration P ℓ κ Definition Let P ℓ κ denote the κ -Easton support iteration defined in such a way that if α “ ˙ α < κ and P ℓ α was defined, if α ∈ cl ( ℓ ) ∩ S κ Q α = P � α ” and ω then � P ℓ α “ ˙ � P ℓ Q α trivial”, otherwise. 2 Using the fast behaviour of ℓ we can show the following: Proposition The iteration P ℓ κ preserves the supercompactness of κ and the GCH pattern.
Draft 1 Let ℓ : κ → κ be a Laver function and define the iteration P ℓ κ Definition Let P ℓ κ denote the κ -Easton support iteration defined in such a way that if α “ ˙ α < κ and P ℓ α was defined, if α ∈ dom ( ℓ ) ∩ E κ ω then � P ℓ Q α = P � α ” α “ ˙ and � P ℓ Q α trivial”, otherwise. 2 Using the fast behaviour of ℓ we can show the following using standard arguments Proposition The iteration P ℓ κ preserves the supercompactness of κ and the GCH pattern. 3 Finally, using the previous proposition the theorem follows.
Draft 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 ın S ( n ) < m´ ın K = m´ ın S = m´ m´ ın E .
Draft 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.
Draft 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.
Draft 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).
Draft 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 :
Draft 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.
Draft 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.
Draft 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
Draft 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 → M j ( P ) (for instance, if P has Easton support).
Draft 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 → M j ( P ) (for instance, if P has Easton support). ◮ M λ ⊆ M (doable using some fast function guiding P ).
Draft 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 → M j ( P ) (for instance, if P has Easton support). ◮ M λ ⊆ M (doable using some fast function guiding P ). ◮ V P � “ j ⋆ ( κ ) ∈ C ( n ) ” (As the forcing is mild).
Draft 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 → M j ( P ) (for instance, if P has Easton support). ◮ M λ ⊆ 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 .
Draft 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 → M j ( P ) (for instance, if P has Easton support). ◮ M λ ⊆ 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.
Draft 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 → M j ( P ) (for instance, if P has Easton support). ◮ M λ ⊆ 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 .
Draft 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 = � E a : a ∈ [ η ] <ω � witnessing that κ is λ - C ( 1 ) -supercompact in the generic extension.
Draft 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 = � E a : 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, → ∃ p ∈ G ∃ q ≤ j ( p ) \ κ ( p ⌢ q � M a ∈ τ ( ˙ X ∈ E a ← X )) . j ( P ) ˙
Draft 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 = � E a : 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, → ∃ p ∈ G ∃ q ≤ j ( p ) \ κ ( p ⌢ q � M a ∈ τ ( ˙ X ∈ E a ← X )) . j ( P ) ˙ If ( M κ , ≤ , ≤ ⋆ ) is a Magidor iteration such that ( j ( M κ ) / M κ , ≤ ⋆ ) is λ + -closed and ≤ = ≤ ⋆ , then ◮ The E a are κ -complete normal measures.
Draft 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 = � E a : 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, → ∃ p ∈ G ∃ q ≤ j ( p ) \ κ ( p ⌢ q � M a ∈ τ ( ˙ X ∈ E a ← X )) . j ( P ) ˙ If ( M κ , ≤ , ≤ ⋆ ) is a Magidor iteration such that ( j ( M κ ) / M κ , ≤ ⋆ ) is λ + -closed and ≤ = ≤ ⋆ , then ◮ The E a are κ -complete normal measures. ◮ We can manage to get M λ E ⊆ M E .
Draft 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 = � E a : 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, → ∃ p ∈ G ∃ q ≤ j ( p ) \ κ ( p ⌢ q � M a ∈ τ ( ˙ X ∈ E a ← X )) . j ( P ) ˙ If ( M κ , ≤ , ≤ ⋆ ) is a Magidor iteration such that ( j ( M κ ) / M κ , ≤ ⋆ ) is λ + -closed and ≤ = ≤ ⋆ , then ◮ The E a are κ -complete normal measures. ◮ We can manage to get M λ E ⊆ M E . Issue How can we make sure that j E ( κ ) ∈ C ( n ) ?
Draft Second strategy: Defining extenders in a generic extension A natural candidate is the extender defined in the following way: → ∃ p ∈ G ∃ q ≤ j ( p ) \ κ ( p ⌢ q � M a ∈ τ ( ˙ X ∈ E a ← j ( P ) ˙ X )) . If ( M κ , ≤ , ≤ ⋆ ) is a Magidor iteration such that ( j ( M κ ) / M κ , ≤ ⋆ ) is λ + -closed and ≤ = ≤ ⋆ , then ◮ The E a are κ -complete normal measures. ◮ It is possible to get M λ E ⊆ M E . ◮ j ( κ ) ∈ C ( n ) .
Draft Second strategy: Defining extenders in a generic extension A natural candidate is the extender defined in the following way: → ∃ p ∈ G ∃ q ≤ j ( p ) \ κ ( p ⌢ q � M a ∈ τ ( ˙ X ∈ E a ← j ( P ) ˙ X )) . If ( M κ , ≤ , ≤ ⋆ ) is a Magidor iteration such that ( j ( M κ ) / M κ , ≤ ⋆ ) is λ + -closed and ≤ = ≤ ⋆ , then ◮ The E a are κ -complete normal measures. ◮ It is possible to get M λ E ⊆ M E . ◮ j ( κ ) ∈ C ( n ) . Issue How can we make sure j E ( κ ) ∈ C ( n ) ? As there is no combinatorial characterization for the class C ( n ) , a natural strategy is to make sure that j E ( κ ) = j ( κ ) .
Draft Second strategy: Defining extenders in a generic extension Issue How can we make sure j E ( 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 j E ( κ ) = j ( κ ) as j ( κ ) is still a C ( n ) -cardinal in the generic extension.
Draft 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.
Draft 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 ) = � κ α : α < κ � which are measurable not limit of the previous measurables ( κ α > sup β<α κ β ).
Draft 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 M κ = � M α , ˙ Q β : β < α ≤ κ � be a κ -stage iteration of forcings. We will say that � M κ , ≤ M κ , ≤ ∗ M κ � is a κ -stage Magidor iteration of Prikry forcings if the following conditions holds: 1 For all α < κ , � M α � ˙ Q α , ≤ ∗ Q α , ≤ ˙ Q α � has the Prikry property ˙ 2 For all p , q ∈ M κ , p ≤ M κ q iff For all α < κ , p ↾ α ≤ M α q ↾ α , 1 There is b ∈ [ κ ] <ω such that for every α ∈ κ \ b , 2 p ( α ) ≤ ∗ q ( α ) . p ↾ α � M α ˙ M α ˙ 3 For all p , q ∈ M κ , p ≤ ∗ M κ q iff p ≤ M κ q and the witness b for the condition 2.1 is the empty set.
Draft For each α < κ let U α be a normal measure over κ α . Denote by P U α the corresponding Prikry forcing. Magidor iteration of Prikry forcings with respect to ran ( ℓ ) Let M κ be the Magidor iteration where M 0 is the trivial forcing and for every ordinal α < κ if � M α “ ˇ U α is a normal measure over κ α ” then � M α ˙ U α , and � M α ˙ Q α = P ˇ Q α = { 1 } , otherwise. ◮ Since our measurables are not limit of the previous ones, for all α < κ , � M α ˙ Q α = P ˇ U α . ◮ � M κ , ≤ , ≤ ∗ � satisfies the Prikry property. ◮ Our forcing will be M κ / � 1 α : α < κ � . For the ease of clarity, let us also denote it by M κ .
Draft Proof of Main Theorem 2 By the previous comments, M κ is essentially a product . Formally, M κ is isomorphic to M ∗ ran ( ℓ ) , κ , where Definition (Magidor Product) The κ -Magidor product with respect to A = � κ α : α < κ � , M ∗ A , κ , is the set of all sequences p = �� s ( α ) , A α � : α < κ � such that (a) For every α < κ , ( s ( α ) , A α ) ∈ P U α , where P U α stands for the Prikry forcing with respect some normal measure U α over κ α ∈ A . (b) { α < κ : s ( α ) � = ∅} ∈ [ κ ] < ℵ 0 . Given two conditions p , q ∈ M ∗ A , κ , p ≤ q ( p is stronger than q ) if for every α < κ , p ( α ) ≤ P Uα q ( α ) . We will also say that p is a direct extension of q , p ≤ ⋆ q if for every α < κ , p ( α ) ≤ ⋆ P Uα q ( α )
Draft On the sequel we will denote by M κ the κ -Magidor product with respect to ran ( ℓ ) . A typical condition p of this forcing is of the form � ( ∅ , A 0 ) , · · · ( s ( α 0 ) , A α 0 ) , · · · , ( s ( α n ) , A α n ) , ( ∅ , A α n + 1 ) , · · · )
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