Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Sealing the Universally Baire sets Nam Trang (joint with G.Sargsyan) University of North Texas Luminy Workshop in Set Theory September 22–27, 2019 Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 1 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Sealing A set of reals A is γ -universally Baire if there are trees T , U on ω × λ for some λ such that A = p [ T ] = R \ p [ U ] and whenever g is a < Hom -generic, in V [ g ] , p [ T ] = R \ p [ U ] . We write A g for p [ T ] V [ g ] ; this is the canonical interpretation of A in V [ g ] . A is universally Baire if A is γ -universally Baire for all γ . Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 2 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Sealing A set of reals A is γ -universally Baire if there are trees T , U on ω × λ for some λ such that A = p [ T ] = R \ p [ U ] and whenever g is a < Hom -generic, in V [ g ] , p [ T ] = R \ p [ U ] . We write A g for p [ T ] V [ g ] ; this is the canonical interpretation of A in V [ g ] . A is universally Baire if A is γ -universally Baire for all γ . Let Hom ∞ be the set of universally Baire sets. Given a generic g , we let Hom ∞ = ( Hom ∞ ) V [ g ] g and R g = R V [ g ] . Also, if A = p [ T ] for some tree T , then let A g = p [ T ] ∩ V [ g ] . Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 2 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Sealing A set of reals A is γ -universally Baire if there are trees T , U on ω × λ for some λ such that A = p [ T ] = R \ p [ U ] and whenever g is a < Hom -generic, in V [ g ] , p [ T ] = R \ p [ U ] . We write A g for p [ T ] V [ g ] ; this is the canonical interpretation of A in V [ g ] . A is universally Baire if A is γ -universally Baire for all γ . Let Hom ∞ be the set of universally Baire sets. Given a generic g , we let Hom ∞ = ( Hom ∞ ) V [ g ] g and R g = R V [ g ] . Also, if A = p [ T ] for some tree T , then let A g = p [ T ] ∩ V [ g ] . Definition (Woodin) Sealing is the conjunction of the following statements. g , R g ) � AD + and P ( R g ) ∩ L ( Hom ∞ 1 For every set generic g , L ( Hom ∞ g , R g ) = Hom ∞ g . 2 For every set generic g over V , for every set generic h over V [ g ] , there is an elementary embedding j : L ( Hom ∞ g , R g ) → L ( Hom ∞ h , R h ) . such that for every A ∈ Hom ∞ g , j ( A ) = A h . Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 2 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Tower Sealing Definition (Woodin) Tower Sealing is the conjunction of: 1 For any set generic g , L ( Hom ∞ g ) � AD + , and Hom ∞ = P ( R ) ∩ L ( Hom ∞ g , R g ) . g 2 For any set generic g , in V [ g ] , suppose δ is Woodin. Whenever G is V [ g ] -generic for either the P <δ -stationary tower or the Q <δ -stationary tower at δ , then j ( Hom ∞ g ) = Hom ∞ g ∗ G , where j : V [ g ] → M ⊂ V [ g ∗ G ] is the generic elementary embedding given by G . Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 3 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Tower Sealing Definition (Woodin) Tower Sealing is the conjunction of: 1 For any set generic g , L ( Hom ∞ g ) � AD + , and Hom ∞ = P ( R ) ∩ L ( Hom ∞ g , R g ) . g 2 For any set generic g , in V [ g ] , suppose δ is Woodin. Whenever G is V [ g ] -generic for either the P <δ -stationary tower or the Q <δ -stationary tower at δ , then j ( Hom ∞ g ) = Hom ∞ g ∗ G , where j : V [ g ] → M ⊂ V [ g ∗ G ] is the generic elementary embedding given by G . Theorem (Woodin, [Lar04]) Suppose there is a proper class of Woodin cardinals. Let δ be a supercompact cardinal and G be V -generic such that in V [ G ] , V δ + 1 is countable. Then Sealing and Tower Sealing hold in V [ G ] . Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 3 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Tower Sealing Definition (Woodin) Tower Sealing is the conjunction of: 1 For any set generic g , L ( Hom ∞ g ) � AD + , and Hom ∞ = P ( R ) ∩ L ( Hom ∞ g , R g ) . g 2 For any set generic g , in V [ g ] , suppose δ is Woodin. Whenever G is V [ g ] -generic for either the P <δ -stationary tower or the Q <δ -stationary tower at δ , then j ( Hom ∞ g ) = Hom ∞ g ∗ G , where j : V [ g ] → M ⊂ V [ g ∗ G ] is the generic elementary embedding given by G . Theorem (Woodin, [Lar04]) Suppose there is a proper class of Woodin cardinals. Let δ be a supercompact cardinal and G be V -generic such that in V [ G ] , V δ + 1 is countable. Then Sealing and Tower Sealing hold in V [ G ] . Remark: Assume there is a proper class of Woodin cardinals. For any A ∈ Hom ∞ , one can seal the model L ( A , R ) , i.e. for any set generic extension V [ g ] , there is an elementary embedding j : L ( A , R ) → L ( A g , R g ) such that j ( A ) = A g . Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 3 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines LSA A cardinal κ is OD-inaccessible if for every α < κ there is no surjection f : P ( α ) → κ that is definable from ordinal parameters. Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 4 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines LSA A cardinal κ is OD-inaccessible if for every α < κ there is no surjection f : P ( α ) → κ that is definable from ordinal parameters. Definition (Woodin, Sargsyan) The Largest Suslin Axiom, abbreviated as LSA, is the conjunction of the following statements: 1 AD + . 2 There is a largest Suslin cardinal. 3 The largest Suslin cardinal is OD-inaccessible. Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 4 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines LSA A cardinal κ is OD-inaccessible if for every α < κ there is no surjection f : P ( α ) → κ that is definable from ordinal parameters. Definition (Woodin, Sargsyan) The Largest Suslin Axiom, abbreviated as LSA, is the conjunction of the following statements: 1 AD + . 2 There is a largest Suslin cardinal. 3 The largest Suslin cardinal is OD-inaccessible. Clause (3) of LSA is equivalent to “the largest Suslin cardinal is a member of the Solovay sequence". Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 4 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines LSA A cardinal κ is OD-inaccessible if for every α < κ there is no surjection f : P ( α ) → κ that is definable from ordinal parameters. Definition (Woodin, Sargsyan) The Largest Suslin Axiom, abbreviated as LSA, is the conjunction of the following statements: 1 AD + . 2 There is a largest Suslin cardinal. 3 The largest Suslin cardinal is OD-inaccessible. Clause (3) of LSA is equivalent to “the largest Suslin cardinal is a member of the Solovay sequence". Prior to [ST], LSA was not known to be consistent. [ST] shows that it is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. Nowadays, the axiom plays a key role in many aspects of inner model theory, and features prominently in Woodin’s Ultimate L framework (see [Woo17, Definition 7.14] and Axiom I and Axiom II on page 97 of [Woo17]). Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 4 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines LSA − over − UB Definition (Sargsyan-T.) Let LSA − over − uB be the statement: For all V -generic g , in V [ g ] , there is A ⊆ R g such that L ( A , R g ) � LSA and Hom ∞ is the Suslin co-Suslin sets of L ( A , R g ) . g Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 5 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Equiconsistency results Theorem (Sargsyan-T., 2018-2019, [ST19b]) Sealing , Tower Sealing , and LSA − over − uB are equiconsistent over “there exists a proper class of Woodin cardinals and the class of measurable cardinals is stationary". Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 6 / 27
Sealing LSA − over − UB Results Sealing and Inner Model Theory Open problems Proof outlines Equiconsistency results Theorem (Sargsyan-T., 2018-2019, [ST19b]) Sealing , Tower Sealing , and LSA − over − uB are equiconsistent over “there exists a proper class of Woodin cardinals and the class of measurable cardinals is stationary". In the above theorem, one can add to the list of equiconsistencies the following statements: LSA − over − uB − be the statement: For all V -generic g , in V [ g ] , there is A ⊆ R g such that 1 L ( A , R g ) � LSA and Hom ∞ is contained in the Suslin co-Suslin sets of L ( A , R g ) . g Nam Trang (joint with G.Sargsyan) Sealing the Universally Baire sets 6 / 27
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