Diagonal extender based Prikry forcing Dima Sinapova University of - PowerPoint PPT Presentation

Prikry type forcing The following are some variations: 1. Magidor forcing: ◮ start with an increasing sequence � U α | α < λ � of normal measures on κ ; ◮ force to add a club set of order type λ in κ . 2. Supercompact Prikry: ◮ start with a supercompactness measure U on P κ ( η ); ◮ force to add an increasing ω -sequence of sets x n ∈ ( P κ ( η )) V , with η = � n x n . 3. Gitik-Sharon’s diagonal supercompact Prikry: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Prikry type forcing The following are some variations: 1. Magidor forcing: ◮ start with an increasing sequence � U α | α < λ � of normal measures on κ ; ◮ force to add a club set of order type λ in κ . 2. Supercompact Prikry: ◮ start with a supercompactness measure U on P κ ( η ); ◮ force to add an increasing ω -sequence of sets x n ∈ ( P κ ( η )) V , with η = � n x n . 3. Gitik-Sharon’s diagonal supercompact Prikry: ◮ start with a sequence � U n | n < ω � of supercompactness measures on P κ ( κ + n ); Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Prikry type forcing The following are some variations: 1. Magidor forcing: ◮ start with an increasing sequence � U α | α < λ � of normal measures on κ ; ◮ force to add a club set of order type λ in κ . 2. Supercompact Prikry: ◮ start with a supercompactness measure U on P κ ( η ); ◮ force to add an increasing ω -sequence of sets x n ∈ ( P κ ( η )) V , with η = � n x n . 3. Gitik-Sharon’s diagonal supercompact Prikry: ◮ start with a sequence � U n | n < ω � of supercompactness measures on P κ ( κ + n ); ◮ force to add an increasing ω -sequence of sets x n ∈ P κ (( κ + n ) V ) with ( κ + ω ) V = � n x n . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Prikry type forcing The following are some variations: 1. Magidor forcing: ◮ start with an increasing sequence � U α | α < λ � of normal measures on κ ; ◮ force to add a club set of order type λ in κ . 2. Supercompact Prikry: ◮ start with a supercompactness measure U on P κ ( η ); ◮ force to add an increasing ω -sequence of sets x n ∈ ( P κ ( η )) V , with η = � n x n . 3. Gitik-Sharon’s diagonal supercompact Prikry: ◮ start with a sequence � U n | n < ω � of supercompactness measures on P κ ( κ + n ); ◮ force to add an increasing ω -sequence of sets x n ∈ P κ (( κ + n ) V ) with ( κ + ω ) V = � n x n . The strategy: add subsets to a large cardinal, then singularize it. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Extender based forcing Alternative way: start with a singular κ and blow up its powerset in a Prikry fashion via extender based forcing . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Extender based forcing Alternative way: start with a singular κ and blow up its powerset in a Prikry fashion via extender based forcing . ◮ Developed by Gitik-Magidor. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Extender based forcing Alternative way: start with a singular κ and blow up its powerset in a Prikry fashion via extender based forcing . ◮ Developed by Gitik-Magidor. ◮ Large cardinal hypothesis: λ > κ , κ = sup n κ n , each κ n is λ + 1 strong. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Extender based forcing Alternative way: start with a singular κ and blow up its powerset in a Prikry fashion via extender based forcing . ◮ Developed by Gitik-Magidor. ◮ Large cardinal hypothesis: λ > κ , κ = sup n κ n , each κ n is λ + 1 strong. n κ n , and so 2 κ becomes λ . ◮ Adds λ sequences through � Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Extender based forcing Alternative way: start with a singular κ and blow up its powerset in a Prikry fashion via extender based forcing . ◮ Developed by Gitik-Magidor. ◮ Large cardinal hypothesis: λ > κ , κ = sup n κ n , each κ n is λ + 1 strong. n κ n , and so 2 κ becomes λ . ◮ Adds λ sequences through � ◮ Preserves κ + , and adds a weak square sequence at κ . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Extender based forcing Alternative way: start with a singular κ and blow up its powerset in a Prikry fashion via extender based forcing . ◮ Developed by Gitik-Magidor. ◮ Large cardinal hypothesis: λ > κ , κ = sup n κ n , each κ n is λ + 1 strong. n κ n , and so 2 κ becomes λ . ◮ Adds λ sequences through � ◮ Preserves κ + , and adds a weak square sequence at κ . ◮ No need to add subsets of κ in advance, so can keep GCH below κ (as opposed to the above forcings). Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Extender based forcing Alternative way: start with a singular κ and blow up its powerset in a Prikry fashion via extender based forcing . ◮ Developed by Gitik-Magidor. ◮ Large cardinal hypothesis: λ > κ , κ = sup n κ n , each κ n is λ + 1 strong. n κ n , and so 2 κ becomes λ . ◮ Adds λ sequences through � ◮ Preserves κ + , and adds a weak square sequence at κ . ◮ No need to add subsets of κ in advance, so can keep GCH below κ (as opposed to the above forcings). ◮ Allows more flexibility when interleaving collapses in order to make κ a small cardinal (e.g. ℵ ω ). Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. ◮ In the ground model κ is supercompact and GCH holds. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. ◮ In the ground model κ is supercompact and GCH holds. The κ n ’s will be chosen generically. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. ◮ In the ground model κ is supercompact and GCH holds. The κ n ’s will be chosen generically. ◮ No bounded subsets of κ are added. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. ◮ In the ground model κ is supercompact and GCH holds. The κ n ’s will be chosen generically. ◮ No bounded subsets of κ are added. ◮ In the final model, GCH holds below κ , and 2 κ > κ + . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. ◮ In the ground model κ is supercompact and GCH holds. The κ n ’s will be chosen generically. ◮ No bounded subsets of κ are added. ◮ In the final model, GCH holds below κ , and 2 κ > κ + . In particular, SCH fails at κ . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. ◮ In the ground model κ is supercompact and GCH holds. The κ n ’s will be chosen generically. ◮ No bounded subsets of κ are added. ◮ In the final model, GCH holds below κ , and 2 κ > κ + . In particular, SCH fails at κ . ◮ Collapses ( κ + ) V . More precisely, ( κ + ω +1 ) V becomes the successor of κ in the generic extension. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry Theorem (S.) Starting from a supercompact cardinal κ , there is a forcing which simultaneously singularizes κ and increases its powerset. ◮ Combine extender based forcing with diagonal supercompact Prikry. ◮ In the ground model κ is supercompact and GCH holds. The κ n ’s will be chosen generically. ◮ No bounded subsets of κ are added. ◮ In the final model, GCH holds below κ , and 2 κ > κ + . In particular, SCH fails at κ . ◮ Collapses ( κ + ) V . More precisely, ( κ + ω +1 ) V becomes the successor of κ in the generic extension. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - Preliminaries Let σ : V → M witness that κ is κ + ω +2 + 1 - strong and let E = � E α | α < κ + ω +2 � be κ complete ultrafilters on κ , where: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - Preliminaries Let σ : V → M witness that κ is κ + ω +2 + 1 - strong and let E = � E α | α < κ + ω +2 � be κ complete ultrafilters on κ , where: 1. each E α = { Z ⊂ κ | α ∈ σ ( Z ) } Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - Preliminaries Let σ : V → M witness that κ is κ + ω +2 + 1 - strong and let E = � E α | α < κ + ω +2 � be κ complete ultrafilters on κ , where: 1. each E α = { Z ⊂ κ | α ∈ σ ( Z ) } 2. for α ≤ E β , π β,α : κ → κ are such that σπ β,α ( β ) = α . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - Preliminaries Let σ : V → M witness that κ is κ + ω +2 + 1 - strong and let E = � E α | α < κ + ω +2 � be κ complete ultrafilters on κ , where: 1. each E α = { Z ⊂ κ | α ∈ σ ( Z ) } 2. for α ≤ E β , π β,α : κ → κ are such that σπ β,α ( β ) = α . 3. if α ≤ E β , then E α is the projection of E β by π βα Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - Preliminaries Let σ : V → M witness that κ is κ + ω +2 + 1 - strong and let E = � E α | α < κ + ω +2 � be κ complete ultrafilters on κ , where: 1. each E α = { Z ⊂ κ | α ∈ σ ( Z ) } 2. for α ≤ E β , π β,α : κ → κ are such that σπ β,α ( β ) = α . 3. if α ≤ E β , then E α is the projection of E β by π βα 4. the π β,α ’s commute. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - Preliminaries Let σ : V → M witness that κ is κ + ω +2 + 1 - strong and let E = � E α | α < κ + ω +2 � be κ complete ultrafilters on κ , where: 1. each E α = { Z ⊂ κ | α ∈ σ ( Z ) } 2. for α ≤ E β , π β,α : κ → κ are such that σπ β,α ( β ) = α . 3. if α ≤ E β , then E α is the projection of E β by π βα 4. the π β,α ’s commute. 5. for a ⊂ κ + ω +2 , with | a | < κ , there are unboundedly many β ∈ κ + ω +2 , such that for all α ∈ a , α ≤ E β . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a ◮ for all α ≤ β ≤ E γ in a , ν ∈ π max a ,γ ” A , π γ,α ( ν ) = π β,α ( π γ,β ( ν )). Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a ◮ for all α ≤ β ≤ E γ in a , ν ∈ π max a ,γ ” A , π γ,α ( ν ) = π β,α ( π γ,β ( ν )). ◮ for all α < β in a , ν ∈ A , π max a ,α ( ν ) < π max a ,β ( ν ) Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a ◮ for all α ≤ β ≤ E γ in a , ν ∈ π max a ,γ ” A , π γ,α ( ν ) = π β,α ( π γ,β ( ν )). ◮ for all α < β in a , ν ∈ A , π max a ,α ( ν ) < π max a ,β ( ν ) ◮ � b , B , g � ≤ 0 � a , A , f � if b ⊃ a , π max b , max a ” B ⊂ A , and g ⊃ f . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a ◮ for all α ≤ β ≤ E γ in a , ν ∈ π max a ,γ ” A , π γ,α ( ν ) = π β,α ( π γ,β ( ν )). ◮ for all α < β in a , ν ∈ A , π max a ,α ( ν ) < π max a ,β ( ν ) ◮ � b , B , g � ≤ 0 � a , A , f � if b ⊃ a , π max b , max a ” B ⊂ A , and g ⊃ f . ◮ g ≤ � a , A , f � if: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a ◮ for all α ≤ β ≤ E γ in a , ν ∈ π max a ,γ ” A , π γ,α ( ν ) = π β,α ( π γ,β ( ν )). ◮ for all α < β in a , ν ∈ A , π max a ,α ( ν ) < π max a ,β ( ν ) ◮ � b , B , g � ≤ 0 � a , A , f � if b ⊃ a , π max b , max a ” B ⊂ A , and g ⊃ f . ◮ g ≤ � a , A , f � if: ◮ g ⊃ f , dom ( g ) ⊃ a , Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a ◮ for all α ≤ β ≤ E γ in a , ν ∈ π max a ,γ ” A , π γ,α ( ν ) = π β,α ( π γ,β ( ν )). ◮ for all α < β in a , ν ∈ A , π max a ,α ( ν ) < π max a ,β ( ν ) ◮ � b , B , g � ≤ 0 � a , A , f � if b ⊃ a , π max b , max a ” B ⊂ A , and g ⊃ f . ◮ g ≤ � a , A , f � if: ◮ g ⊃ f , dom ( g ) ⊃ a , ◮ g (max a ) ∈ A , Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Q = Q 0 ∪ Q 1 is defined as follows: ◮ Q 1 = { f : κ + ω +2 → κ | | f | < κ + ω +1 } ◮ Q 0 has conditions of the form p = � a , A , f � such that: ◮ f ∈ Q 1 ◮ a ⊂ κ + ω +2 , | a | < κ , and a ∩ dom ( f ) = ∅ ◮ a has an ≤ E − maximal element and A ∈ E max a ◮ for all α ≤ β ≤ E γ in a , ν ∈ π max a ,γ ” A , π γ,α ( ν ) = π β,α ( π γ,β ( ν )). ◮ for all α < β in a , ν ∈ A , π max a ,α ( ν ) < π max a ,β ( ν ) ◮ � b , B , g � ≤ 0 � a , A , f � if b ⊃ a , π max b , max a ” B ⊂ A , and g ⊃ f . ◮ g ≤ � a , A , f � if: ◮ g ⊃ f , dom ( g ) ⊃ a , ◮ g (max a ) ∈ A , ◮ for all β ∈ a , g ( β ) = π max a ,β ( g (max a )). Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Properties of Q : Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Properties of Q : ◮ Q is equivalent to Q 1 , which is equivalent to the Cohen poset for adding κ + ω +2 many subsets to κ + ω +1 . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Properties of Q : ◮ Q is equivalent to Q 1 , which is equivalent to the Cohen poset for adding κ + ω +2 many subsets to κ + ω +1 . ◮ Q has the Prikry property. I.e. for p ∈ Q 0 and a formula φ , there is q ≤ p with q ∈ Q 0 such that q � Q φ or q � Q ¬ φ . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Properties of Q : ◮ Q is equivalent to Q 1 , which is equivalent to the Cohen poset for adding κ + ω +2 many subsets to κ + ω +1 . ◮ Q has the Prikry property. I.e. for p ∈ Q 0 and a formula φ , there is q ≤ p with q ∈ Q 0 such that q � Q φ or q � Q ¬ φ . ◮ Q has the κ + ω +2 chain condition. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the basic modules Properties of Q : ◮ Q is equivalent to Q 1 , which is equivalent to the Cohen poset for adding κ + ω +2 many subsets to κ + ω +1 . ◮ Q has the Prikry property. I.e. for p ∈ Q 0 and a formula φ , there is q ≤ p with q ∈ Q 0 such that q � Q φ or q � Q ¬ φ . ◮ Q has the κ + ω +2 chain condition. ◮ Q is < κ closed. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Conditions in P are of the form p = � x 0 , f 0 , ..., x l − 1 , f l − 1 , A l , F l , ... � where l = length ( p ) and: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Conditions in P are of the form p = � x 0 , f 0 , ..., x l − 1 , f l − 1 , A l , F l , ... � where l = length ( p ) and: 1. For n < l , ◮ x n ∈ P κ ( κ + n ), and for i < n , x i ≺ x n , Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Conditions in P are of the form p = � x 0 , f 0 , ..., x l − 1 , f l − 1 , A l , F l , ... � where l = length ( p ) and: 1. For n < l , ◮ x n ∈ P κ ( κ + n ), and for i < n , x i ≺ x n , ◮ f n ∈ Q 1 Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Conditions in P are of the form p = � x 0 , f 0 , ..., x l − 1 , f l − 1 , A l , F l , ... � where l = length ( p ) and: 1. For n < l , ◮ x n ∈ P κ ( κ + n ), and for i < n , x i ≺ x n , ◮ f n ∈ Q 1 2. For n ≥ l , Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Conditions in P are of the form p = � x 0 , f 0 , ..., x l − 1 , f l − 1 , A l , F l , ... � where l = length ( p ) and: 1. For n < l , ◮ x n ∈ P κ ( κ + n ), and for i < n , x i ≺ x n , ◮ f n ∈ Q 1 2. For n ≥ l , ◮ A n ∈ U n , and x l − 1 ≺ y for all y ∈ A l . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Conditions in P are of the form p = � x 0 , f 0 , ..., x l − 1 , f l − 1 , A l , F l , ... � where l = length ( p ) and: 1. For n < l , ◮ x n ∈ P κ ( κ + n ), and for i < n , x i ≺ x n , ◮ f n ∈ Q 1 2. For n ≥ l , ◮ A n ∈ U n , and x l − 1 ≺ y for all y ∈ A l . ◮ F n is a function with domain A n , for y ∈ A n , F n ( y ) ∈ Q 0 . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Conditions in P are of the form p = � x 0 , f 0 , ..., x l − 1 , f l − 1 , A l , F l , ... � where l = length ( p ) and: 1. For n < l , ◮ x n ∈ P κ ( κ + n ), and for i < n , x i ≺ x n , ◮ f n ∈ Q 1 2. For n ≥ l , ◮ A n ∈ U n , and x l − 1 ≺ y for all y ∈ A l . ◮ F n is a function with domain A n , for y ∈ A n , F n ( y ) ∈ Q 0 . 3. For x ∈ A n , denote F n ( x ) = � a n x , A n x , f n x � . Then for l ≤ n < m , y ∈ A n , z ∈ A m with y ≺ z , we have a n y ⊂ a m z . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Properties of P : Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Properties of P : ◮ P has the Prikry property. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Properties of P : ◮ P has the Prikry property. ◮ P has the κ + ω +2 chain condition. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Properties of P : ◮ P has the Prikry property. ◮ P has the κ + ω +2 chain condition. ◮ Cardinals ≤ κ and ≥ κ + ω +1 are preserved. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Properties of P : ◮ P has the Prikry property. ◮ P has the κ + ω +2 chain condition. ◮ Cardinals ≤ κ and ≥ κ + ω +1 are preserved. ◮ ( κ + ω +1 ) V becomes the successor of κ in the generic extension. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

The hybrid Prikry - the main forcing Properties of P : ◮ P has the Prikry property. ◮ P has the κ + ω +2 chain condition. ◮ Cardinals ≤ κ and ≥ κ + ω +1 are preserved. ◮ ( κ + ω +1 ) V becomes the successor of κ in the generic extension. ◮ P blows up the powerset of κ to ( κ + ω +2 ) V . And so, in the generic extension SCH fails at κ . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: p ∈ G , l ( p ) ≤ n a p F n = � n ( x n ), Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: p ∈ G , l ( p ) ≤ n a p F n = � n ( x n ), and F = � n F n Proposition Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: p ∈ G , l ( p ) ≤ n a p F n = � n ( x n ), and F = � n F n Proposition 1. t α / ∈ V iff α ∈ F. Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: p ∈ G , l ( p ) ≤ n a p F n = � n ( x n ), and F = � n F n Proposition 1. t α / ∈ V iff α ∈ F. 2. If α < β are both in F, then t α < ∗ t β . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: p ∈ G , l ( p ) ≤ n a p F n = � n ( x n ), and F = � n F n Proposition 1. t α / ∈ V iff α ∈ F. 2. If α < β are both in F, then t α < ∗ t β . 3. F is unbounded in ( κ + ω +2 ) V . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: p ∈ G , l ( p ) ≤ n a p F n = � n ( x n ), and F = � n F n Proposition 1. t α / ∈ V iff α ∈ F. 2. If α < β are both in F, then t α < ∗ t β . 3. F is unbounded in ( κ + ω +2 ) V . Then in the generic extension, 2 κ = ( κ + ω +2 ) V = ( κ ++ ) V [ G ] . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Blowing up the powerset of κ . Let G be P -generic. G adds: � x n | n < ω � , such that setting κ n = def x n ∩ κ , κ = sup n κ n , and functions f n : ( κ + ω +2 ) V → κ , n < ω . ◮ Set t α ( n ) = f n ( α ); each t α ∈ � n κ . (With some more work we can actually make t α ∈ � n κ n ) ◮ In V [ G ], setting F p n ( x ) = � a p n ( x ) , A p n ( x ) , f p n ( x ) � , define: p ∈ G , l ( p ) ≤ n a p F n = � n ( x n ), and F = � n F n Proposition 1. t α / ∈ V iff α ∈ F. 2. If α < β are both in F, then t α < ∗ t β . 3. F is unbounded in ( κ + ω +2 ) V . Then in the generic extension, 2 κ = ( κ + ω +2 ) V = ( κ ++ ) V [ G ] . We can also interleave collapses in the usual way to make κ = ℵ ω Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Applications and questions Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing

Applications and questions ◮ This construction increases the powerset of κ while preserving GCH below κ and collapsing κ + . Dima Sinapova University of California Irvine Diagonal extender based Prikry forcing