Two possible motivations: 2 / 23 Introduction Hypothesis - PowerPoint PPT Presentation

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Let j : V ≺ M with crt( j ) = κ . We define the critical sequence � κ 0 , κ 1 , . . . � as κ 0 = κ and j ( κ n ) = κ n +1 . Definition (Kunen, 1972) Let κ be a cardinal. Then κ is n-huge iff there is a j : V ≺ M with crt( j ) = κ , κ n M ⊆ M . Definition (Reinhardt, 1970) Let κ be a cardinal. Then κ is ω -huge or Reinhardt iff there is a j : V ≺ M with crt( j ) = κ 0 , λ M ⊆ M , with λ = sup n ∈ ω κ n . Equivalently, if there is a j : V ≺ V , with κ = crt( j ). 7 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem (Kunen, 1971) There is no Reinhardt cardinal 8 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem (Kunen, 1971) There is no Reinhardt cardinal. Proof Let S ω = { α < λ + : cof( α ) = ω } 8 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem (Kunen, 1971) There is no Reinhardt cardinal. Proof Let S ω = { α < λ + : cof( α ) = ω } . By Solovay there exists � S ξ : ξ < κ � a partition of S ω in stationary sets 8 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem (Kunen, 1971) There is no Reinhardt cardinal. Proof Let S ω = { α < λ + : cof( α ) = ω } . By Solovay there exists � S ξ : ξ < κ � a partition of S ω in stationary sets. It’s a quick calculation that j ( λ ) = λ and j ( λ + ) = λ + 8 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem (Kunen, 1971) There is no Reinhardt cardinal. Proof Let S ω = { α < λ + : cof( α ) = ω } . By Solovay there exists � S ξ : ξ < κ � a partition of S ω in stationary sets. It’s a quick calculation that j ( λ ) = λ and j ( λ + ) = λ + . Let j ( � S ξ : ξ < κ � ) = � T ξ : ξ < κ 1 � 8 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem (Kunen, 1971) There is no Reinhardt cardinal. Proof Let S ω = { α < λ + : cof( α ) = ω } . By Solovay there exists � S ξ : ξ < κ � a partition of S ω in stationary sets. It’s a quick calculation that j ( λ ) = λ and j ( λ + ) = λ + . Let j ( � S ξ : ξ < κ � ) = � T ξ : ξ < κ 1 � . C = { α < λ + : j ( α ) = α } is an ω -club, therefore there exists α ∈ C ∩ T κ 8 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem (Kunen, 1971) There is no Reinhardt cardinal. Proof Let S ω = { α < λ + : cof( α ) = ω } . By Solovay there exists � S ξ : ξ < κ � a partition of S ω in stationary sets. It’s a quick calculation that j ( λ ) = λ and j ( λ + ) = λ + . Let j ( � S ξ : ξ < κ � ) = � T ξ : ξ < κ 1 � . C = { α < λ + : j ( α ) = α } is an ω -club, therefore there exists α ∈ C ∩ T κ . Let α ∈ S ξ . Then j ( α ) = α ∈ T j ( ξ ) ∩ T κ . 8 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Large cardinals are really large, but there is a trick to apply their properties to small cardinals 9 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Large cardinals are really large, but there is a trick to apply their properties to small cardinals. Generic large cardinals are a “virtual” version of large cardinals 9 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Large cardinals are really large, but there is a trick to apply their properties to small cardinals. Generic large cardinals are a “virtual” version of large cardinals. Definition (Jech, Prikry, 1976) Let κ be a cardinal, I an ideal on P ( κ ). Then P ( κ ) / I is a forcing notion 9 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Large cardinals are really large, but there is a trick to apply their properties to small cardinals. Generic large cardinals are a “virtual” version of large cardinals. Definition (Jech, Prikry, 1976) Let κ be a cardinal, I an ideal on P ( κ ). Then P ( κ ) / I is a forcing notion. If G is generic for P ( κ ) / I , then G is a V -ultrafilter on P ( κ ) and there exists j : V ≺ Ult( V , G ) 9 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Large cardinals are really large, but there is a trick to apply their properties to small cardinals. Generic large cardinals are a “virtual” version of large cardinals. Definition (Jech, Prikry, 1976) Let κ be a cardinal, I an ideal on P ( κ ). Then P ( κ ) / I is a forcing notion. If G is generic for P ( κ ) / I , then G is a V -ultrafilter on P ( κ ) and there exists j : V ≺ Ult( V , G ). I is precipitous iff Ult( V , G ) is well-founded, and in that case there exists j : V ≺ M ⊆ V [ G ] 9 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Large cardinals are really large, but there is a trick to apply their properties to small cardinals. Generic large cardinals are a “virtual” version of large cardinals. Definition (Jech, Prikry, 1976) Let κ be a cardinal, I an ideal on P ( κ ). Then P ( κ ) / I is a forcing notion. If G is generic for P ( κ ) / I , then G is a V -ultrafilter on P ( κ ) and there exists j : V ≺ Ult( V , G ). I is precipitous iff Ult( V , G ) is well-founded, and in that case there exists j : V ≺ M ⊆ V [ G ]. We say that κ is a generically measurable cardinal. 9 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation One can extend the definition to all the large cardinals above: generic γ -supercompact, generic huge, generic n -huge 10 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation One can extend the definition to all the large cardinals above: generic γ -supercompact, generic huge, generic n -huge. In fact, the Theorem above by Laver is in fact divided in two 10 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation One can extend the definition to all the large cardinals above: generic γ -supercompact, generic huge, generic n -huge. In fact, the Theorem above by Laver is in fact divided in two: Theorem (Laver) Con(huge cardinal) → Con( ℵ 1 is generic huge cardinal and j ( ℵ 2 ) = ℵ 3 ) 10 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation One can extend the definition to all the large cardinals above: generic γ -supercompact, generic huge, generic n -huge. In fact, the Theorem above by Laver is in fact divided in two: Theorem (Laver) Con(huge cardinal) → Con( ℵ 1 is generic huge cardinal and j ( ℵ 2 ) = ℵ 3 ). Proposition If j : V ≺ M ⊆ V [ G ], M closed under ℵ 3 -sequences, crt( j ) = ℵ 2 and j ( ℵ 2 ) = ℵ 3 , then ( ℵ 3 , ℵ 2 ) ։ ( ℵ 2 , ℵ 1 ). 10 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof Suppose not. Let U of type ( ℵ 3 , ℵ 2 ) be a counterexample 11 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof Suppose not. Let U of type ( ℵ 3 , ℵ 2 ) be a counterexample. Then j ( U ) is of tpye ( ℵ M 3 , ℵ M 2 ) 11 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof Suppose not. Let U of type ( ℵ 3 , ℵ 2 ) be a counterexample. Then j ( U ) is of tpye ( ℵ M 3 , ℵ M 2 ). But by hugeness j “ U is in M , and j ′′ U ≺ j ( U ) 11 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof Suppose not. Let U of type ( ℵ 3 , ℵ 2 ) be a counterexample. Then j ( U ) is of tpye ( ℵ M 3 , ℵ M 2 ). But by hugeness j “ U is in M , and j ′′ U ≺ j ( U ). Finally, j “ U is of type ( ℵ 3 , ℵ 2 ) = ( ℵ M 2 , ℵ M 1 ) 11 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof Suppose not. Let U of type ( ℵ 3 , ℵ 2 ) be a counterexample. Then j ( U ) is of tpye ( ℵ M 3 , ℵ M 2 ). But by hugeness j “ U is in M , and j ′′ U ≺ j ( U ). Finally, j “ U is of type ( ℵ 3 , ℵ 2 ) = ( ℵ M 2 , ℵ M 1 ). In the same way, Proposition If j : V ≺ M ⊆ V [ G ], M closed under ℵ n +1 -sequences, crt( j ) = ℵ 1 and j ( ℵ 1 ) = ℵ 2 , j ( ℵ 2 ) = ℵ 3 , . . . , then ( ℵ n +1 , . . . , ℵ 2 , ℵ 1 ) ։ ( ℵ n , . . . , ℵ 1 , ℵ 0 ). 11 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition κ is J´ onsson iff every structure for a countable language with domain of cardinality κ has a proper elementary substructure with domain of the same cardinality 12 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition κ is J´ onsson iff every structure for a countable language with domain of cardinality κ has a proper elementary substructure with domain of the same cardinality. Then ℵ ω is J´ onsson is ( . . . , ℵ 2 , ℵ 1 ) → ( . . . , ℵ 1 , ℵ 0 ) 12 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition κ is J´ onsson iff every structure for a countable language with domain of cardinality κ has a proper elementary substructure with domain of the same cardinality. Then ℵ ω is J´ onsson is ( . . . , ℵ 2 , ℵ 1 ) → ( . . . , ℵ 1 , ℵ 0 ). Open Problem What about Con( ℵ ω is J´ onsson)? 12 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition κ is J´ onsson iff every structure for a countable language with domain of cardinality κ has a proper elementary substructure with domain of the same cardinality. Then ℵ ω is J´ onsson is ( . . . , ℵ 2 , ℵ 1 ) → ( . . . , ℵ 1 , ℵ 0 ). Open Problem What about Con( ℵ ω is J´ onsson)? There is no ω -huge (and Shelah proved there is no generic ω -huge)! What can we do? 12 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Kunen proved in fact ¬∃ j : V λ +2 ≺ V λ +2 13 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Kunen proved in fact ¬∃ j : V λ +2 ≺ V λ +2 . This leaves space for the following definitions: 13 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Kunen proved in fact ¬∃ j : V λ +2 ≺ V λ +2 . This leaves space for the following definitions: Definition I3 iff there exists λ s.t. ∃ j : V λ ≺ V λ ; 13 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Kunen proved in fact ¬∃ j : V λ +2 ≺ V λ +2 . This leaves space for the following definitions: Definition I3 iff there exists λ s.t. ∃ j : V λ ≺ V λ ; I2 iff there exists λ s.t. ∃ j : V λ +1 ≺ 1 V λ +1 ; 13 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Kunen proved in fact ¬∃ j : V λ +2 ≺ V λ +2 . This leaves space for the following definitions: Definition I3 iff there exists λ s.t. ∃ j : V λ ≺ V λ ; I2 iff there exists λ s.t. ∃ j : V λ +1 ≺ 1 V λ +1 ; I1 iff there exists λ s.t. ∃ j : V λ +1 ≺ V λ +1 ; 13 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Kunen proved in fact ¬∃ j : V λ +2 ≺ V λ +2 . This leaves space for the following definitions: Definition I3 iff there exists λ s.t. ∃ j : V λ ≺ V λ ; I2 iff there exists λ s.t. ∃ j : V λ +1 ≺ 1 V λ +1 ; I1 iff there exists λ s.t. ∃ j : V λ +1 ≺ V λ +1 ; I0 For some λ there exists a j : L ( V λ +1 ) ≺ L ( V λ +1 ) , with crt( j ) < λ 13 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Kunen proved in fact ¬∃ j : V λ +2 ≺ V λ +2 . This leaves space for the following definitions: Definition I3 iff there exists λ s.t. ∃ j : V λ ≺ V λ ; I2 iff there exists λ s.t. ∃ j : V λ +1 ≺ 1 V λ +1 ; I1 iff there exists λ s.t. ∃ j : V λ +1 ≺ V λ +1 ; I0 For some λ there exists a j : L ( V λ +1 ) ≺ L ( V λ +1 ) , with crt( j ) < λ . With the ”right“ forcing, generic I* implies ℵ ω is J´ onsson. 13 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Disclaimer: it is still not clear how strong this is 14 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Disclaimer: it is still not clear how strong this is: Theorem (Foreman,1982) Con(2-huge cardinal) → Con( ℵ 1 is generic 2-huge cardinal and . . . ) 14 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Disclaimer: it is still not clear how strong this is: Theorem (Foreman,1982) Con(2-huge cardinal) → Con( ℵ 1 is generic 2-huge cardinal and . . . ). Open Problem What about Con( ℵ 1 is generic 3-huge cardinal and . . . )? 14 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition (GCH) Generic I0 at ℵ ω is true 15 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition (GCH) Generic I0 at ℵ ω is true if there exists a forcing notion P such that for any generic G there exists j : L ( P ( ℵ ω )) ≺ L ( P ( ℵ ω )) V [ G ] and P is reasonable. Examples: P = Coll( ℵ 3 , ℵ 2 ) 15 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition (GCH) Generic I0 at ℵ ω is true if there exists a forcing notion P such that for any generic G there exists j : L ( P ( ℵ ω )) ≺ L ( P ( ℵ ω )) V [ G ] and P is reasonable. Examples: P = Coll( ℵ 3 , ℵ 2 ), P = product of P n , where P n = Coll( ℵ n +1 , ℵ n ). 15 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )) 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )). Theorem Suppose generic I0 at ℵ ω 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )). Theorem Suppose generic I0 at ℵ ω . Then in L ( P ( ℵ ω )) 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )). Theorem Suppose generic I0 at ℵ ω . Then in L ( P ( ℵ ω )): 1. ℵ ω +1 is measurable 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )). Theorem Suppose generic I0 at ℵ ω . Then in L ( P ( ℵ ω )): 1. ℵ ω +1 is measurable; 2. Θ is weakly inaccessible 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )). Theorem Suppose generic I0 at ℵ ω . Then in L ( P ( ℵ ω )): 1. ℵ ω +1 is measurable; 2. Θ is weakly inaccessible; 3. Θ is limit of measurable cardinals 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )). Theorem Suppose generic I0 at ℵ ω . Then in L ( P ( ℵ ω )): 1. ℵ ω +1 is measurable; 2. Θ is weakly inaccessible; 3. Θ is limit of measurable cardinals. Confront this with: Theorem (Shelah) If ℵ ω is strong limit, then 2 ℵ 0 < ℵ ω 4 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Definition Θ = sup { α : ∃ π : P ( ℵ ω ) ։ α, π ∈ L ( P ( ℵ ω )). Theorem Suppose generic I0 at ℵ ω . Then in L ( P ( ℵ ω )): 1. ℵ ω +1 is measurable; 2. Θ is weakly inaccessible; 3. Θ is limit of measurable cardinals. Confront this with: Theorem (Shelah) If ℵ ω is strong limit, then 2 ℵ 0 < ℵ ω 4 . (From now on, let’s suppose crt( j ) = ℵ 2 and j ( ℵ 2 ) = ℵ 3 ). 16 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (1) It is practically the same proof as Kunen’s Theorem 17 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (1) It is practically the same proof as Kunen’s Theorem. Suppose � S ξ : ξ < ℵ 2 � is an ω -stationary partition 17 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (1) It is practically the same proof as Kunen’s Theorem. Suppose � S ξ : ξ < ℵ 2 � is an ω -stationary partition. Now, j ↾ L α ( P ) ∈ L ( P ( ℵ ω ))[ G ], so C = { α < ℵ ω +1 : j ( α ) = α } ∈ L ( P ( ℵ ω ))[ G ] 17 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (1) It is practically the same proof as Kunen’s Theorem. Suppose � S ξ : ξ < ℵ 2 � is an ω -stationary partition. Now, j ↾ L α ( P ) ∈ L ( P ( ℵ ω ))[ G ], so C = { α < ℵ ω +1 : j ( α ) = α } ∈ L ( P ( ℵ ω ))[ G ]. As before, then there exists α ∈ T ξ ∩ T ℵ 2 . In L ( P ( ℵ ω )) we have some choice, namely DC ℵ ω ... 17 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation 18 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation For points (2) and (3) we need more choice than DC ℵ ω 19 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation For points (2) and (3) we need more choice than DC ℵ ω : Coding Lemma ∀ η < Θ ∀ ρ : P ( ℵ ω ) ։ η ∃ γ < Θ ∀ A ⊆ P ( ℵ ω ) ∃ B ⊆ P ( ℵ ω ) B ∈ L γ ( P ( ℵ ω )) B ⊆ A and { ρ ( a ) : a ∈ B } = { ρ ( a ) : a ∈ A } . 19 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ) 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } . Apply the Coding Lemma to this, to find B ∈ L γ ( P ( ℵ ω )) such that { ρ ( a ) : a ∈ B } = A 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } . Apply the Coding Lemma to this, to find B ∈ L γ ( P ( ℵ ω )) such that { ρ ( a ) : a ∈ B } = A . Therefore P ( α ) ⊆ L γ ( P ( ℵ ω )) 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } . Apply the Coding Lemma to this, to find B ∈ L γ ( P ( ℵ ω )) such that { ρ ( a ) : a ∈ B } = A . Therefore P ( α ) ⊆ L γ ( P ( ℵ ω )). Proof of (3) The measurable cardinals will be the first γ ’s such that L γ ( P ( ℵ ω )) ≺ 1 L ( P ( ℵ ω )) above a fixed point 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } . Apply the Coding Lemma to this, to find B ∈ L γ ( P ( ℵ ω )) such that { ρ ( a ) : a ∈ B } = A . Therefore P ( α ) ⊆ L γ ( P ( ℵ ω )). Proof of (3) The measurable cardinals will be the first γ ’s such that L γ ( P ( ℵ ω )) ≺ 1 L ( P ( ℵ ω )) above a fixed point. Prove the Coding Lemma inside L γ ( P ( ℵ ω )) 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } . Apply the Coding Lemma to this, to find B ∈ L γ ( P ( ℵ ω )) such that { ρ ( a ) : a ∈ B } = A . Therefore P ( α ) ⊆ L γ ( P ( ℵ ω )). Proof of (3) The measurable cardinals will be the first γ ’s such that L γ ( P ( ℵ ω )) ≺ 1 L ( P ( ℵ ω )) above a fixed point. Prove the Coding Lemma inside L γ ( P ( ℵ ω )). One can prove, as before, that the ω -club filter on γ is ℵ ω +1 -complete 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } . Apply the Coding Lemma to this, to find B ∈ L γ ( P ( ℵ ω )) such that { ρ ( a ) : a ∈ B } = A . Therefore P ( α ) ⊆ L γ ( P ( ℵ ω )). Proof of (3) The measurable cardinals will be the first γ ’s such that L γ ( P ( ℵ ω )) ≺ 1 L ( P ( ℵ ω )) above a fixed point. Prove the Coding Lemma inside L γ ( P ( ℵ ω )). One can prove, as before, that the ω -club filter on γ is ℵ ω +1 -complete. Change the filter with the ω -club filter generated by the fixed points of k : N ≺ P ( ℵ ω ) 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Proof of (2) One has to prove that if there exists ρ : P ( ℵ ω ) ։ α , then there exists π : P ( ℵ ω ) ։ P ( α ). Let A ⊆ α , and consider { a : ρ ( a ) ∈ A } . Apply the Coding Lemma to this, to find B ∈ L γ ( P ( ℵ ω )) such that { ρ ( a ) : a ∈ B } = A . Therefore P ( α ) ⊆ L γ ( P ( ℵ ω )). Proof of (3) The measurable cardinals will be the first γ ’s such that L γ ( P ( ℵ ω )) ≺ 1 L ( P ( ℵ ω )) above a fixed point. Prove the Coding Lemma inside L γ ( P ( ℵ ω )). One can prove, as before, that the ω -club filter on γ is ℵ ω +1 -complete. Change the filter with the ω -club filter generated by the fixed points of k : N ≺ P ( ℵ ω ). Pick � A ξ : ξ < γ � and choose inside each one the sets of fixed points that witness the non-empty intersection. 20 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Having just ℵ ω +1 measurable is nothing new 21 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Having just ℵ ω +1 measurable is nothing new: Theorem (Apter, 1985) Suppose κ is 2 λ -supercompact, with λ measurable. Then there is a model of ZF+ ℵ ω +1 is measurable 21 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Having just ℵ ω +1 measurable is nothing new: Theorem (Apter, 1985) Suppose κ is 2 λ -supercompact, with λ measurable. Then there is a model of ZF+ ℵ ω +1 is measurable. It’s the rest that it is interesting 21 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Having just ℵ ω +1 measurable is nothing new: Theorem (Apter, 1985) Suppose κ is 2 λ -supercompact, with λ measurable. Then there is a model of ZF+ ℵ ω +1 is measurable. It’s the rest that it is interesting: Definition Define D ( λ ) as the following 21 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Having just ℵ ω +1 measurable is nothing new: Theorem (Apter, 1985) Suppose κ is 2 λ -supercompact, with λ measurable. Then there is a model of ZF+ ℵ ω +1 is measurable. It’s the rest that it is interesting: Definition Define D ( λ ) as the following: in L ( P ( λ )): 1. λ + is measurable; 2. Θ is a weakly inaccessible limit of measurable cardinals 21 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Having just ℵ ω +1 measurable is nothing new: Theorem (Apter, 1985) Suppose κ is 2 λ -supercompact, with λ measurable. Then there is a model of ZF+ ℵ ω +1 is measurable. It’s the rest that it is interesting: Definition Define D ( λ ) as the following: in L ( P ( λ )): 1. λ + is measurable; 2. Θ is a weakly inaccessible limit of measurable cardinals. Therefore, the Theorem proves that if we have generic I0 at ℵ ω , then D ( ℵ ω ). 21 / 23

Introduction Hypothesis Motivation Generic I0 Thesis Motivation Theorem L ( R ) � AD → L ( R ) � D ( ω ) 22 / 23