Minimal collapse maps at arbitrary projective level

Transcription

1 Minimal collapse maps at arbitrary projective level Vladimir Kanovei 1 Vassily Lyubetsky 2 1 IITP RAS and MIIT, Moscow. Supported by RFBR grant IITP RAS, Moscow. Support of RSF grant acknowledged Descriptive Set Theory in Turin September 6 8, 2017 Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

2 Generic collapse maps Back Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

3 Generic collapse maps Back 1 Cohen : there exist generic cofinal maps a : ω ω 1 (in fact, for any two cardinals) Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

4 Generic collapse maps Back 1 Cohen : there exist generic cofinal maps a : ω ω 1 (in fact, for any two cardinals) 2 Prikry : there exist minimal cofinal maps a : ω ω 1, Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

5 Generic collapse maps Back 1 Cohen : there exist generic cofinal maps a : ω ω 1 (in fact, for any two cardinals) 2 Prikry : there exist minimal cofinal maps a : ω ω 1, the minimality means that if b V[a], b : ω ω1 V is cofinal then a V[b] Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

6 Generic collapse maps Back 1 Cohen : there exist generic cofinal maps a : ω ω 1 (in fact, for any two cardinals) 2 Prikry : there exist minimal cofinal maps a : ω ω 1, the minimality means that if b V[a], b : ω ω1 V is cofinal then a V[b] 3 Uri Abraham 1984 : if V = L is the ground model then there exists a minimal cofinal map a : ω ω1 V such that a is (coded by) a lightface Π2 1 real singleton in V[a]. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

7 Generic collapse maps Back 1 Cohen : there exist generic cofinal maps a : ω ω 1 (in fact, for any two cardinals) 2 Prikry : there exist minimal cofinal maps a : ω ω 1, the minimality means that if b V[a], b : ω ω1 V is cofinal then a V[b] 3 Uri Abraham 1984 : if V = L is the ground model then there exists a minimal cofinal map a : ω ω1 V such that a is (coded by) a lightface Π2 1 real singleton in V[a]. 4 VK + VL, the main result : if V = L is the ground model and n 3 then there exists a minimal cofinal map a : ω ω1 V such that it is true in V[a] that Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

8 Generic collapse maps Back 1 Cohen : there exist generic cofinal maps a : ω ω 1 (in fact, for any two cardinals) 2 Prikry : there exist minimal cofinal maps a : ω ω 1, the minimality means that if b V[a], b : ω ω1 V is cofinal then a V[b] 3 Uri Abraham 1984 : if V = L is the ground model then there exists a minimal cofinal map a : ω ω1 V such that a is (coded by) a lightface Π2 1 real singleton in V[a]. 4 VK + VL, the main result : if V = L is the ground model and n 3 then there exists a minimal cofinal map a : ω ω1 V such that it is true in V[a] that 1 a is (coded by) a lightface Πn 1 real singleton, but Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

9 Generic collapse maps Back 1 Cohen : there exist generic cofinal maps a : ω ω 1 (in fact, for any two cardinals) 2 Prikry : there exist minimal cofinal maps a : ω ω 1, the minimality means that if b V[a], b : ω ω1 V is cofinal then a V[b] 3 Uri Abraham 1984 : if V = L is the ground model then there exists a minimal cofinal map a : ω ω1 V such that a is (coded by) a lightface Π2 1 real singleton in V[a]. 4 VK + VL, the main result : if V = L is the ground model and n 3 then there exists a minimal cofinal map a : ω ω1 V such that it is true in V[a] that 1 a is (coded by) a lightface Πn 1 real singleton, but 2 every Σn 1 real is constructible. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

10 Cohen-style collapse Back Definition (Cohen-style collapse forcing) The forcing ω <ω 1 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 / 9

11 Minimal cofinal map Back Definition (Minimal cofinal map forcing, Prikry + folclore) The forcing P consists of all trees T ω 1 <ω such that 1 every node of T has a branching node above it; 2 every branching node of T is an ω 1 -branching node. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

12 Minimal cofinal map Back Definition (Minimal cofinal map forcing, Prikry + folclore) The forcing P consists of all trees T ω 1 <ω such that 1 every node of T has a branching node above it; 2 every branching node of T is an ω 1 -branching node. The Laver-style version P Laver requires that in addition 3 any node of T above a branching node is branching itself. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

13 Minimal cofinal map Back Definition (Minimal cofinal map forcing, Prikry + folclore) The forcing P consists of all trees T ω 1 <ω such that 1 every node of T has a branching node above it; 2 every branching node of T is an ω 1 -branching node. The Laver-style version P Laver requires that in addition 3 any node of T above a branching node is branching itself. P Laver is more difficult to deal with. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

14 Minimal cofinal map Back Definition (Minimal cofinal map forcing, Prikry + folclore) The forcing P consists of all trees T ω 1 <ω such that 1 every node of T has a branching node above it; 2 every branching node of T is an ω 1 -branching node. The Laver-style version P Laver requires that in addition 3 any node of T above a branching node is branching itself. P Laver is more difficult to deal with. Both P and P Laver naturally adjoin a cofinal map a : ω ω1 V, 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 / 9

15 Uri Abraham cofinal map Back Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

16 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 1 each U ξ P is a set of cardinality ℵ 1 ; 2 U adds a single generic map, so U is very non-homogeneous; 3 being U-generic is Π2 1. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

17 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 1 each U ξ P is a set of cardinality ℵ 1 ; 2 U adds a single generic map, so U is very non-homogeneous; 3 being U-generic is Π2 1. There is also a P Laver -version, actually used by Abraham. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

18 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 1 each U ξ P is a set of cardinality ℵ 1 ; 2 U adds a single generic map, so U is very non-homogeneous; 3 being U-generic is Π2 1. 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 / 9

19 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 1 each U ξ P is a set of cardinality ℵ 1 ; 2 U adds a single generic map, so U is very non-homogeneous; 3 being U-generic is Π2 1. 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 / 9

20 Π 1 n -singleton cofinal map Back Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

21 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 / 9

22 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. Definition (Π 1 n -singleton cofinal map forcing) Let n 3. In L, we define U n using a 1 n path through P, generic so it meets all dense subsets of P of boldface class Σ 1 n 1. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

23 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. Definition (Π 1 n -singleton cofinal map forcing) Let n 3. In L, we define U n using a 1 n path through P, generic so it meets all dense subsets of P of boldface class Σ 1 n 1. The genericity condition makes the forcing properties of U n to be very close to those of the whole homogeneous forcing notion P up to the nth level of the projective hierarchy. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

24 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. Definition (Π 1 n -singleton cofinal map forcing) Let n 3. In L, we define U n using a 1 n path through P, generic so it meets all dense subsets of P of boldface class Σ 1 n 1. The genericity condition makes the forcing properties of U n to be very close to those of the whole homogeneous forcing notion P up to the nth level of the projective hierarchy. In particular U n forces all lightface Σ 1 n reals to be constructible. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

25 A problem Back Problem In the context of the Namba forcing, define a generic extension L[a] of L by a cofinal map a : ω ω2 L, such that ω1 L is not collapsed and a is definable in L[a]. Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

26 Acknowledgements Titlepage The speaker thanks the organizers for support Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9

27 Acknowledgements Titlepage The speaker thanks everybody for patience Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September / 9