Minimal collapse maps at arbitrary projective level Vladimir Kanovei 1 Vassily Lyubetsky 2 1 IITP RAS and MIIT, Moscow. Supported by RFBR grant 17-01-00757 2 IITP RAS, Moscow. Support of RSF grant 14-50-00150 acknowledged Descriptive Set Theory in Turin September 6–8, 2017 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 1 / 9
Generic collapse maps Back ⇒ Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Generic collapse maps Back ⇒ Cohen : there exist generic cofinal maps a : ω → ω 1 (in fact, 1 for any two cardinals) Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Generic collapse maps Back ⇒ Cohen : there exist generic cofinal maps a : ω → ω 1 (in fact, 1 for any two cardinals) Prikry : there exist minimal cofinal maps a : ω → ω 1 , 2 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Generic collapse maps Back ⇒ Cohen : there exist generic cofinal maps a : ω → ω 1 (in fact, 1 for any two cardinals) Prikry : there exist minimal cofinal maps a : ω → ω 1 , 2 the minimality means that if b ∈ V [ a ], b : ω → ω V 1 is cofinal then a ∈ V [ b ] Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Generic collapse maps Back ⇒ Cohen : there exist generic cofinal maps a : ω → ω 1 (in fact, 1 for any two cardinals) Prikry : there exist minimal cofinal maps a : ω → ω 1 , 2 the minimality means that if b ∈ V [ a ], b : ω → ω V 1 is cofinal then a ∈ V [ b ] Uri Abraham 1984 : if V = L is the ground model then 3 there exists a minimal cofinal map a : ω → ω V 1 such that a is (coded by) a lightface Π 1 2 real singleton in V [ a ]. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Generic collapse maps Back ⇒ Cohen : there exist generic cofinal maps a : ω → ω 1 (in fact, 1 for any two cardinals) Prikry : there exist minimal cofinal maps a : ω → ω 1 , 2 the minimality means that if b ∈ V [ a ], b : ω → ω V 1 is cofinal then a ∈ V [ b ] Uri Abraham 1984 : if V = L is the ground model then 3 there exists a minimal cofinal map a : ω → ω V 1 such that a is (coded by) a lightface Π 1 2 real singleton in V [ a ]. VK + VL, the main result : if V = L is the ground model and 4 n ≥ 3 then there exists a minimal cofinal map a : ω → ω V 1 such that it is true in V [ a ] that Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Generic collapse maps Back ⇒ Cohen : there exist generic cofinal maps a : ω → ω 1 (in fact, 1 for any two cardinals) Prikry : there exist minimal cofinal maps a : ω → ω 1 , 2 the minimality means that if b ∈ V [ a ], b : ω → ω V 1 is cofinal then a ∈ V [ b ] Uri Abraham 1984 : if V = L is the ground model then 3 there exists a minimal cofinal map a : ω → ω V 1 such that a is (coded by) a lightface Π 1 2 real singleton in V [ a ]. VK + VL, the main result : if V = L is the ground model and 4 n ≥ 3 then there exists a minimal cofinal map a : ω → ω V 1 such that it is true in V [ a ] that a is (coded by) a lightface Π 1 n real singleton, but 1 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Generic collapse maps Back ⇒ Cohen : there exist generic cofinal maps a : ω → ω 1 (in fact, 1 for any two cardinals) Prikry : there exist minimal cofinal maps a : ω → ω 1 , 2 the minimality means that if b ∈ V [ a ], b : ω → ω V 1 is cofinal then a ∈ V [ b ] Uri Abraham 1984 : if V = L is the ground model then 3 there exists a minimal cofinal map a : ω → ω V 1 such that a is (coded by) a lightface Π 1 2 real singleton in V [ a ]. VK + VL, the main result : if V = L is the ground model and 4 n ≥ 3 then there exists a minimal cofinal map a : ω → ω V 1 such that it is true in V [ a ] that a is (coded by) a lightface Π 1 n real singleton, but 1 every Σ 1 n real is constructible. 2 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9
Cohen-style collapse Back ⇐ Definition ( Cohen-style collapse forcing ) The forcing ω 1 <ω consists of all strings (finite sequences) of ordinals α < ω 1 . The forcing ω 1 <ω naturally adjoins a map a : ω onto → ω 1 . − Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 3 / 9
Minimal cofinal map Back ⇐ Definition ( Minimal cofinal map forcing, Prikry + folclore ) The forcing P consists of all trees T ⊆ ω 1 <ω such that every node of T has a branching node above it; 1 every branching node of T is an ω 1 -branching node. 2 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 4 / 9
Minimal cofinal map Back ⇐ Definition ( Minimal cofinal map forcing, Prikry + folclore ) The forcing P consists of all trees T ⊆ ω 1 <ω such that every node of T has a branching node above it; 1 every branching node of T is an ω 1 -branching node. 2 The Laver-style version P Laver requires that in addition any node of T above a branching node is branching itself. 3 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 4 / 9
Minimal cofinal map Back ⇐ Definition ( Minimal cofinal map forcing, Prikry + folclore ) The forcing P consists of all trees T ⊆ ω 1 <ω such that every node of T has a branching node above it; 1 every branching node of T is an ω 1 -branching node. 2 The Laver-style version P Laver requires that in addition any node of T above a branching node is branching itself. 3 P Laver is more difficult to deal with. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 4 / 9
Minimal cofinal map Back ⇐ Definition ( Minimal cofinal map forcing, Prikry + folclore ) The forcing P consists of all trees T ⊆ ω 1 <ω such that every node of T has a branching node above it; 1 every branching node of T is an ω 1 -branching node. 2 The Laver-style version P Laver requires that in addition any node of T above a branching node is branching itself. 3 P Laver is more difficult to deal with. Both P and P Laver naturally adjoin a cofinal map a : ω → ω V 1 , but such a map a is not definable in V [ a ] since the forcing notions P and P Laver are too homogeneous . Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 4 / 9
Uri Abraham cofinal map Back ⇐ Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 5 / 9
Uri Abraham cofinal map Back ⇐ Definition ( Uri Abraham forcing, in L) In L , the forcing U is a subset U = � ξ<ω 2 U ξ ⊆ P , such that each U ξ ⊆ P is a set of cardinality ℵ 1 ; 1 U adds a single generic map , so U is very non-homogeneous; 2 “being U -generic” is Π 1 2 . 3 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 5 / 9
Uri Abraham cofinal map Back ⇐ Definition ( Uri Abraham forcing, in L) In L , the forcing U is a subset U = � ξ<ω 2 U ξ ⊆ P , such that each U ξ ⊆ P is a set of cardinality ℵ 1 ; 1 U adds a single generic map , so U is very non-homogeneous; 2 “being U -generic” is Π 1 2 . 3 There is also a P Laver -version, actually used by Abraham. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 5 / 9
Uri Abraham cofinal map Back ⇐ Definition ( Uri Abraham forcing, in L) In L , the forcing U is a subset U = � ξ<ω 2 U ξ ⊆ P , such that each U ξ ⊆ P is a set of cardinality ℵ 1 ; 1 U adds a single generic map , so U is very non-homogeneous; 2 “being U -generic” is Π 1 2 . 3 There is also a P Laver -version, actually used by Abraham. The forcing U adjoins a cofinal map a : ω → ω 1 to L , and a is a Π 1 2 -singleton in the extension. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 5 / 9
Uri Abraham cofinal map Back ⇐ Definition ( Uri Abraham forcing, in L) In L , the forcing U is a subset U = � ξ<ω 2 U ξ ⊆ P , such that each U ξ ⊆ P is a set of cardinality ℵ 1 ; 1 U adds a single generic map , so U is very non-homogeneous; 2 “being U -generic” is Π 1 2 . 3 There is also a P Laver -version, actually used by Abraham. The forcing U adjoins a cofinal map a : ω → ω 1 to L , and a is a Π 1 2 -singleton in the extension. The single generic object construction goes back to Jensen 1970 minimal- Π 1 2 -singleton forcing . Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 5 / 9
Π 1 n -singleton cofinal map Back ⇐ Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 6 / 9
Π 1 n -singleton cofinal map Back ⇐ Observation The Uri Abraham forcing U is essentially a ∆ 1 2 path through a certain POset P of sets U ⊆ P of cardinality card U ≤ ℵ 1 . Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 6 / 9
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