Minimal collapse maps at arbitrary projective level Vladimir Kanovei - - PowerPoint PPT Presentation

Minimal collapse maps at arbitrary projective level

Vladimir Kanovei 1 Vassily Lyubetsky 2

1IITP RAS and MIIT, Moscow. Supported by RFBR grant 17-01-00757 2IITP 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 ⇒

1

Cohen : there exist generic cofinal maps a : ω → ω1 (in fact, for any two cardinals)

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9

Generic collapse maps Back ⇒

1

Cohen : there exist generic cofinal maps a : ω → ω1 (in fact, for any two cardinals)

2

Prikry : there exist minimal cofinal maps a : ω → ω1,

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9

Generic collapse maps Back ⇒

1

Cohen : there exist generic cofinal maps a : ω → ω1 (in fact, for any two cardinals)

2

Prikry : there exist minimal cofinal maps a : ω → ω1, the minimality means that if b ∈ V[a], b : ω → ωV

1 is cofinal

then a ∈ V[b]

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9

Generic collapse maps Back ⇒

1

Cohen : there exist generic cofinal maps a : ω → ω1 (in fact, for any two cardinals)

2

Prikry : there exist minimal cofinal maps a : ω → ω1, the minimality means that if b ∈ V[a], b : ω → ωV

1 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 → ωV

1 such that

a is (coded by) a lightface Π1

2 real singleton in V[a].

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9

Generic collapse maps Back ⇒

1

Cohen : there exist generic cofinal maps a : ω → ω1 (in fact, for any two cardinals)

2

Prikry : there exist minimal cofinal maps a : ω → ω1, the minimality means that if b ∈ V[a], b : ω → ωV

1 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 → ωV

1 such that

a is (coded by) a lightface Π1

2 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 → ωV

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 ⇒

1

Cohen : there exist generic cofinal maps a : ω → ω1 (in fact, for any two cardinals)

2

Prikry : there exist minimal cofinal maps a : ω → ω1, the minimality means that if b ∈ V[a], b : ω → ωV

1 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 → ωV

1 such that

a is (coded by) a lightface Π1

2 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 → ωV

1 such

that it is true in V[a] that

1

a is (coded by) a lightface Π1

n real singleton, but

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9

Generic collapse maps Back ⇒

1

Cohen : there exist generic cofinal maps a : ω → ω1 (in fact, for any two cardinals)

2

Prikry : there exist minimal cofinal maps a : ω → ω1, the minimality means that if b ∈ V[a], b : ω → ωV

1 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 → ωV

1 such that

a is (coded by) a lightface Π1

2 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 → ωV

1 such

that it is true in V[a] that

1

a is (coded by) a lightface Π1

n real singleton, but

2

every Σ1

n real is constructible.

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 2 / 9

Cohen-style collapse Back ⇐ Definition (Cohen-style collapse forcing) The forcing ω1<ω consists of all strings (finite sequences) of

• rdinals α < ω1.

The forcing ω1<ω naturally adjoins a map a : ω

• nto

− → ω1.

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 ⊆ ω1<ω such that

1

every node of T has a branching node above it;

2

every branching node of T is an ω1-branching node.

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 ⊆ ω1<ω such that

1

every node of T has a branching node above it;

2

every branching node of T is an ω1-branching node. The Laver-style version PLaver 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 2017 4 / 9

Minimal cofinal map Back ⇐ Definition (Minimal cofinal map forcing, Prikry + folclore) The forcing P consists of all trees T ⊆ ω1<ω such that

1

every node of T has a branching node above it;

2

every branching node of T is an ω1-branching node. The Laver-style version PLaver requires that in addition

3

any node of T above a branching node is branching itself. PLaver 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 ⊆ ω1<ω such that

1

every node of T has a branching node above it;

2

every branching node of T is an ω1-branching node. The Laver-style version PLaver requires that in addition

3

any node of T above a branching node is branching itself. PLaver is more difficult to deal with. Both P and PLaver naturally adjoin a cofinal map a : ω → ωV

1 , but

such a map a is not definable in V[a] since the forcing notions P and PLaver 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<ω2 Uξ ⊆ P , such that

1

each Uξ ⊆ P is a set of cardinality ℵ1;

2

U adds a single generic map , so U is very non-homogeneous;

3

“being U-generic” is Π1

2 .

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<ω2 Uξ ⊆ P , such that

1

each Uξ ⊆ P is a set of cardinality ℵ1;

2

U adds a single generic map , so U is very non-homogeneous;

3

“being U-generic” is Π1

2 .

There is also a PLaver -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<ω2 Uξ ⊆ P , such that

1

each Uξ ⊆ P is a set of cardinality ℵ1;

2

U adds a single generic map , so U is very non-homogeneous;

3

“being U-generic” is Π1

2 .

There is also a PLaver -version, actually used by Abraham. The forcing U adjoins a cofinal map a : ω → ω1 to L, and a is a Π1

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<ω2 Uξ ⊆ P , such that

1

each Uξ ⊆ P is a set of cardinality ℵ1;

2

U adds a single generic map , so U is very non-homogeneous;

3

“being U-generic” is Π1

2 .

There is also a PLaver -version, actually used by Abraham. The forcing U adjoins a cofinal map a : ω → ω1 to L, and a is a Π1

2 -singleton in the extension.

The single generic object construction goes back to Jensen 1970 minimal-Π1

2 -singleton forcing .

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 5 / 9

Π1

n -singleton cofinal map

Back ⇐

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 6 / 9

Π1

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 ≤ ℵ1.

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 6 / 9

Π1

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 ≤ ℵ1. Definition (Π1

n -singleton cofinal map forcing)

Let n ≥ 3. In L , we define Un using a ∆1

n path through P,

generic so it meets all dense subsets of P of boldface class Σ1

n−1 .

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 6 / 9

Π1

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 ≤ ℵ1. Definition (Π1

n -singleton cofinal map forcing)

Let n ≥ 3. In L , we define Un using a ∆1

n path through P,

generic so it meets all dense subsets of P of boldface class Σ1

n−1 .

The genericity condition makes the forcing properties of Un 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 2017 6 / 9

Π1

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 ≤ ℵ1. Definition (Π1

n -singleton cofinal map forcing)

Let n ≥ 3. In L , we define Un using a ∆1

n path through P,

generic so it meets all dense subsets of P of boldface class Σ1

n−1 .

The genericity condition makes the forcing properties of Un to be very close to those of the whole homogeneous forcing notion P up to the nth level of the projective hierarchy. In particular Un forces all lightface Σ1

n reals to be constructible.

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 6 / 9

A problem Back Problem In the context of the Namba forcing , define a generic extension L[a]

• f L by a cofinal map a : ω → ωL

2 , such that ωL 1 is not collapsed and

a is definable in L[a].

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 7 / 9

Acknowledgements Titlepage

The speaker thanks the organizers for support

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 8 / 9

Acknowledgements Titlepage

The speaker thanks everybody for patience

Kanovei Lyubetsky Minimal collapse maps at projective levels Torino September 2017 9 / 9