The tree property at the double successor of Ajdin Halilovi c a - PowerPoint PPT Presentation

“The tree property” at the double successor of a measurable cardinal κ with 2 κ large “The tree property” at the double successor of Ajdin Halilovi´ c a measurable cardinal κ with 2 κ large (joint work with Sy Friedman) Definitions Theorem Ajdin Halilovi´ c Motivation (joint work with Sy Friedman) The proof The end The University of South East Europe - Lumina Bucharest, Romania Sy David Friedman’s 60th-Birthday Conference Vienna, 11.7.2013 Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Definitions “The tree property” at the double successor of a measurable cardinal κ with 2 κ large Ajdin Halilovi´ c (joint work with Sy Friedman) Definitions Theorem Motivation The proof The end Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Definitions “The tree property” at the double successor Definition of a measurable cardinal κ with 2 κ large A tree is a strict partial ordering ( T , < ) with the property that Ajdin Halilovi´ c for each x ∈ T , { y : y < x } is well-ordered by < . (joint work with Sy Friedman) The α th level of a tree T consists of all x such that { y : y < x } has order-type α . Definitions Theorem The height of T is the least α such that the α th level of T is Motivation empty. The proof A branch in T is a maximal linearly ordered subset of T . The end We say that a branch is cofinal if it hits every level of T . Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Definitions “The tree property” at the double successor Definition of a measurable cardinal κ with 2 κ large A tree is a strict partial ordering ( T , < ) with the property that Ajdin Halilovi´ c for each x ∈ T , { y : y < x } is well-ordered by < . (joint work with Sy Friedman) The α th level of a tree T consists of all x such that { y : y < x } has order-type α . Definitions Theorem The height of T is the least α such that the α th level of T is Motivation empty. The proof A branch in T is a maximal linearly ordered subset of T . The end We say that a branch is cofinal if it hits every level of T . Definition An infinite cardinal κ has the tree property if every tree of height κ whose levels have size < κ has a cofinal branch. Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Theorem “The tree property” at the double successor of a measurable cardinal κ with 2 κ large Ajdin Halilovi´ c (joint work with Sy Friedman) Definitions Theorem Motivation The proof The end Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Theorem “The tree Definition property” at the double successor of a measurable We say that a cardinal κ is γ -hypermeasurable if there is an cardinal κ with 2 κ large elementary embedding j : V → M with crit( j ) = κ such that H ( γ ) V = H ( γ ) M . Ajdin Halilovi´ c (joint work with Sy Friedman) Definitions Theorem Motivation The proof The end Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Theorem “The tree Definition property” at the double successor of a measurable We say that a cardinal κ is γ -hypermeasurable if there is an cardinal κ with 2 κ large elementary embedding j : V → M with crit( j ) = κ such that H ( γ ) V = H ( γ ) M . Ajdin Halilovi´ c (joint work with Sy Friedman) Theorem (Friedman, H.) Definitions Theorem Assume that V is a model of ZFC and κ is λ + -hypermeasurable 1 Motivation in V , where λ is the least weakly compact cardinal greater than The proof κ . Then there exists a forcing extension of V in which κ is still The end measurable, κ ++ has the tree property and 2 κ = κ +++ . Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Theorem “The tree Definition property” at the double successor of a measurable We say that a cardinal κ is γ -hypermeasurable if there is an cardinal κ with 2 κ large elementary embedding j : V → M with crit( j ) = κ such that H ( γ ) V = H ( γ ) M . Ajdin Halilovi´ c (joint work with Sy Friedman) Theorem (Friedman, H.) Definitions Theorem Assume that V is a model of ZFC and κ is λ + -hypermeasurable 1 Motivation in V , where λ is the least weakly compact cardinal greater than The proof κ . Then there exists a forcing extension of V in which κ is still The end measurable, κ ++ has the tree property and 2 κ = κ +++ . If the assumption is strengthened to the existence of a 2 θ -hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ ) then the proof can be generalized to get 2 κ = θ . Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Theorem “The tree Definition property” at the double successor of a measurable We say that a cardinal κ is γ -hypermeasurable if there is an cardinal κ with 2 κ large elementary embedding j : V → M with crit( j ) = κ such that H ( γ ) V = H ( γ ) M . Ajdin Halilovi´ c (joint work with Sy Friedman) Theorem (Friedman, H.) Definitions Theorem Assume that V is a model of ZFC and κ is λ + -hypermeasurable 1 Motivation in V , where λ is the least weakly compact cardinal greater than The proof κ . Then there exists a forcing extension of V in which κ is still The end measurable, κ ++ has the tree property and 2 κ = κ +++ . If the assumption is strengthened to the existence of a 2 θ -hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ ) then the proof can be generalized to get 2 κ = θ . By forcing with the Prikry forcing over the above models one 3 gets Con( cof ( κ ) = ω , TP( κ ++ ), 2 κ large). Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Where did such a theorem come from? “The tree property” at the double successor of a measurable cardinal κ with 2 κ large Ajdin Halilovi´ c (joint work with Sy Friedman) Definitions Theorem Motivation The proof The end Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Where did such a theorem come from? “The tree property” at the double successor Natasha Dobrinen and Sy used a generalization of Sacks forcing to of a measurable cardinal κ with reduce the large cardinal strength required to obtain the tree property 2 κ large at the double successor of a measurable cardinal κ from a Ajdin Halilovi´ c (joint work with supercompact to a weakly compact hypermeasurable cardinal. In Sy Friedman) their model 2 κ = κ ++ . Definitions Theorem Motivation The proof The end Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Where did such a theorem come from? “The tree property” at the double successor Natasha Dobrinen and Sy used a generalization of Sacks forcing to of a measurable cardinal κ with reduce the large cardinal strength required to obtain the tree property 2 κ large at the double successor of a measurable cardinal κ from a Ajdin Halilovi´ c (joint work with supercompact to a weakly compact hypermeasurable cardinal. In Sy Friedman) their model 2 κ = κ ++ . Definitions On the other hand, TP( ℵ 2 ) is consistent with large continuum (a Theorem detailed proof was given by Spencer Unger). So, the idea was to Motivation prove the analogous result for TP( κ ++ ) with κ measurable, using The proof The end Mitchell’s forcing together with a ”surgery” argument. Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with

Where did such a theorem come from? “The tree property” at the double successor Natasha Dobrinen and Sy used a generalization of Sacks forcing to of a measurable cardinal κ with reduce the large cardinal strength required to obtain the tree property 2 κ large at the double successor of a measurable cardinal κ from a Ajdin Halilovi´ c (joint work with supercompact to a weakly compact hypermeasurable cardinal. In Sy Friedman) their model 2 κ = κ ++ . Definitions On the other hand, TP( ℵ 2 ) is consistent with large continuum (a Theorem detailed proof was given by Spencer Unger). So, the idea was to Motivation prove the analogous result for TP( κ ++ ) with κ measurable, using The proof The end Mitchell’s forcing together with a ”surgery” argument. As in Dobrinen-Friedman paper, the consistency of a cardinal κ of Mitchell order λ + , where λ is weakly compact and greater than κ , is a lower bound on the consistency strength of TP( κ ++ ) with κ measurable and 2 κ = κ +++ . Therefore our result is in fact almost an equiconsistency result. Ajdin Halilovi´ c(joint work with Sy Friedman) “The tree property” at the double successor of a measurable cardinal κ with