The Global Structure of the Turing Degrees · PDF fileThe Global Structure of the Turing Degrees W. Hugh. Woodin University of California, Berkeley C ... Odifreddi-Shore Theorem (Odifreddi

The Global Structure of the Turing Degrees

W. Hugh. Woodin

University of California, Berkeley

CJanuary 9 , 2012

The Turing Degreesbasic definitions

Definition

I D denotes the partial order of the Turing degrees.I a+ b denotes the join of two degrees.

I A⊕B denotes the recursive join of two sets.

I a′ denotes the Turing jump of a,I a(n) denotes the n-th Turing jump of a:

I a(n+1) = (a(n))′.

I A subset I of D is an ideal if and only ifI x ∈ I and y ≤T x implies y ∈ I,I x ∈ I and y ∈ I implies x+ y ∈ I.

I A ideal I is a jump ideal if for all x ∈ I, x′ ∈ I.

Two fundamental structures

1. D = (D,≤T ).

2. (P(ω), ω,+, ·,∈).

meta-question

What information if any is lost in passing from the structure

(P(ω), ω,+, ·,∈)

to the structure D?

To what extent are these two structures the same?

1. Do these structures have the same logical theory?

2. What are the relations on D which are logically de�nable inD with or without parameters?

3. Is there an automorphism of D?


1. D = (D,≤T ).

2. (P(ω), ω,+, ·,∈).

meta-question


(P(ω), ω,+, ·,∈)

to the structure D?






1. D = (D,≤T ).

2. (P(ω), ω,+, ·,∈).

meta-question


(P(ω), ω,+, ·,∈)

to the structure D?





The Coding Theorem

Theorem (Slaman-Woodin, 1986)

For every n there is a �rst order formula

ϕ(x1, . . . , xn, y1, . . . , ym)

such that for every countable set R ⊂ Dn there exists−→p = (p1, . . . , pm) such that

R = {(d1, . . . , dn) ∈ Dn | D |= ϕ[p1, . . . , pn, d1, . . . , dm]}

I The method of proof is quite general and is based onCohen generics.

The Coding Theorem

Theorem (Slaman-Woodin, 1986)

For every n there is a �rst order formula

ϕ(x1, . . . , xn, y1, . . . , ym)

such that for every countable set R ⊂ Dn there exists−→p = (p1, . . . , pm) such that

R = {(d1, . . . , dn) ∈ Dn | D |= ϕ[p1, . . . , pn, d1, . . . , dm]}

I The method of proof is quite general and is based onCohen generics.

Applications: The theory of DSimpson’s theorem

Theorem (Simpson, 1977)

There is a recursive interpretation of the second order theory ofarithmetic in the �rst order theory of D.

Proof.

Specifying a standard model of arithmetic involves specifying

I a countable set N ,

I a distinguished element “0”,

I operations + and ·,such that N = (N, 0,+, ·) satisfies finitely many first orderproperties together with second order induction.

I By the coding theorem, any and all such specifications areuniformly definable in D.




Proof.









Proof.






Effective Coding and Decoding Theorems

Theorem (Effective Coding Theorem)

Suppose that R ⊂ Dn is countable and R is recursive in the setX. Then there are parameters −→p below degree(X ′) which codeR in D and this coding is absolute to any interval

([0, y]T ,≤T )

such that −→p ∈ [0, y]T .

Theorem (Decoding Theorem)

Suppose that −→p is a sequence of degrees which lie below y, −→pcodes R ⊂ Dn, and y = degree(Y ).

Then R has a presentation which is Σ05(Y ).

Effective Coding and Decoding Theorems

Theorem (Effective Coding Theorem)

Suppose that R ⊂ Dn is countable and R is recursive in the setX. Then there are parameters −→p below degree(X ′) which codeR in D and this coding is absolute to any interval

([0, y]T ,≤T )

such that −→p ∈ [0, y]T .

Theorem (Decoding Theorem)

Suppose that −→p is a sequence of degrees which lie below y, −→pcodes R ⊂ Dn, and y = degree(Y ).

Then R has a presentation which is Σ05(Y ).

Representatives and the Nerode-Shore Theorem

Theorem

For any degree x and representative X of x, there areparameters −→p such that

(1) −→p codes an isomorphic copy of N with a unary predicatefor X;

(2) the elements of −→p are recursive in x+ 0′.

Theorem

Suppose that −→p is a sequence of degrees below y, and −→p codesan isomorphic copy of N together with a unary predicate U .Then for Y ∈ y, U is Σ0

5(Y ).

Theorem (Nerode and Shore, 1980)

Suppose that π : D∼→D. For every degree x, if x is greater than

π−1(0′) then π(x) is arithmetic in x.


Theorem




Theorem


5(Y ).





Theorem




Theorem


5(Y ).




Applications to Aut(D)


Suppose π : D∼→D is an automorphism of D and

x ≥T

(π−1(0′)

)(5)+ π−1(π(0′)(5)).

Then π(x) = x. Consequently, π is the identity on a cone.

Proof.

Using only x ≥T

(π−1(0′)

)(5), there exist y1 and y2 such that

1. y1 ∨ y2 = x;

2. π(y1) and π(y2) are greater than 0′;

3. y(5)1 ≤T x and y

(5)2 ≤T x.

Thus π(yi) ≤T x and so π(x) ≤T x.

Since x ≥T π−1(π(0′)(5)), π(x) ≥T π(0′)(5) and so applying the

above to π−1 at π(x), x = π−1(π(x)) ≤T π(x).




x ≥T

(π−1(0′)

)(5)+ π−1(π(0′)(5)).


Proof.

Using only x ≥T

(π−1(0′)


1. y1 ∨ y2 = x;


3. y(5)1 ≤T x and y

(5)2 ≤T x.







x ≥T

(π−1(0′)

)(5)+ π−1(π(0′)(5)).


Proof.

Using only x ≥T

(π−1(0′)


1. y1 ∨ y2 = x;


3. y(5)1 ≤T x and y

(5)2 ≤T x.




Local automorphismsOdifreddi-Shore

Theorem (Odifreddi and Shore, 1991)

Suppose that π is an automorphism of D, 0′ ∈ I and that πrestricts to an automorphism of I.

I Suppose that there is a presentation of I which is recursivein A.

Then the restriction of π to I has a presentation which isarithmetic in A.

Proof.

Code an enumeration f : ω → I using parameters −→p which arearithmetic in A.

I The action of π on I is determined by the action of π on −→p .

Since π−1(0′) ∈ I, π−1(0′) is recursive in A. Therefore by theNerode and Shore Theorem, π(−→p ) is arithmetic in A.

Local automorphismsOdifreddi-Shore

Theorem (Odifreddi and Shore, 1991)

Suppose that π is an automorphism of D, 0′ ∈ I and that πrestricts to an automorphism of I.

I Suppose that there is a presentation of I which is recursivein A.

Then the restriction of π to I has a presentation which isarithmetic in A.

Proof.

Code an enumeration f : ω → I using parameters −→p which arearithmetic in A.

I The action of π on I is determined by the action of π on −→p .

Since π−1(0′) ∈ I, π−1(0′) is recursive in A. Therefore by theNerode and Shore Theorem, π(−→p ) is arithmetic in A.

Outline of what will follow

Survey of joint work with T. Slaman from 20 years ago.

I the idea is to exploit the Shore-Odifreddi Theorem usingmeta-mathematical methods from Set Theory.

Focus will be on D.

I The methods apply to a wide class of generalizations of DI since everything is just based on the Coding Theorem

I and that proof is just based on Cohen generics.

Outline of what will follow

Survey of joint work with T. Slaman from 20 years ago.

I the idea is to exploit the Shore-Odifreddi Theorem usingmeta-mathematical methods from Set Theory.

Focus will be on D.

I The methods apply to a wide class of generalizations of DI since everything is just based on the Coding Theorem

I and that proof is just based on Cohen generics.

Persistent Automorphisms

Definition

An automorphism ρ of a countable ideal I is persistent if forevery degree x there is a countable ideal I1 such that

I x ∈ I1 and I ⊆ I1;I there is an automorphism ρ1 of I1 such that the restriction

of ρ1 to I is equal to ρ.

Theorem

Suppose that π : D∼→D and I ⊆ D is an ideal closed under π

and π−1.

Then π � I is persistent.


Definition

An automorphism ρ of a countable ideal I is persistent if forevery degree x there is a countable ideal I1 such that

I x ∈ I1 and I ⊆ I1;I there is an automorphism ρ1 of I1 such that the restriction

of ρ1 to I is equal to ρ.

Theorem

Suppose that π : D∼→D and I ⊆ D is an ideal closed under π

and π−1.

Then π � I is persistent.


Theorem

Suppose that ρ : I ∼→I, that J ⊆ I is a jump ideal, and that

ρ(0′) + ρ−1(0′) ∈ J .

Then ρ � J is an automorphism of J .

Proof.

Follows from the effective coding and decoding theorems.

Theorem

Suppose I, J are ideals in D, ρ : I → J is an isomorphism,and that (

ρ−1(0′))(5)

+ ρ−1((ρ(0′)

)(5)) ∈ I ∩ J .Then I = J .


Theorem

Suppose that ρ : I ∼→I, that J ⊆ I is a jump ideal, and that

ρ(0′) + ρ−1(0′) ∈ J .

Then ρ � J is an automorphism of J .

Proof.

Follows from the effective coding and decoding theorems.

Theorem

Suppose I, J are ideals in D, ρ : I → J is an isomorphism,and that (

ρ−1(0′))(5)

+ ρ−1((ρ(0′)

)(5)) ∈ I ∩ J .Then I = J .


Corollary

Suppose that I is an ideal, 0′ ∈ I, and that ρ is a persistentautomorphism of I.

Suppose J is the jump-closure of I and that I∗ is an ideal suchthat J ⊆ I∗.

Then ρ extends to an automorphism of I∗.

Proof.

Let π extend ρ to an ideal extending I∗ and let I∗∗ be therange of π � I∗. Then since π � J is an automorphism of J ,(

π−1(0′))(5)

+ π−1(

(π(0′))(5))∈ I∗ ∩ I∗∗.

But π � I∗ : I∗ ∼→I∗∗ and so I∗ = I∗∗.

Question

Does ρ extend to a persistent automorphism of I∗?


Corollary




Proof.


π−1(0′))(5)

+ π−1(

(π(0′))(5))∈ I∗ ∩ I∗∗.

But π � I∗ : I∗ ∼→I∗∗ and so I∗ = I∗∗.

Question



Corollary




Proof.


π−1(0′))(5)

+ π−1(

(π(0′))(5))∈ I∗ ∩ I∗∗.

But π � I∗ : I∗ ∼→I∗∗ and so I∗ = I∗∗.

Question


Persistent Automorphismsjump ideals

Theorem

Suppose that I is an ideal in D such that 0′ ∈ I and that thereis a presentation of I which is recursive in A.

I Suppose that J is a jump ideal which includes A and ρ isan automorphism of J that restricts to an automorphismof I.

Then the restriction ρ � I of ρ to I has a presentation which isarithmetically de�nable from A.

Proof.

Apply the Odifreddi-Shore argument. There is a code −→p for acounting of I which is arithmetically definable from A. Lettinga = degree(A), ρ(a) is arithmetic in A since ρ−1(0′) ≤T a.Therefore ρ(−→p ) is arithmetic in A.

Persistent Automorphismsjump ideals

Theorem

Suppose that I is an ideal in D such that 0′ ∈ I and that thereis a presentation of I which is recursive in A.

I Suppose that J is a jump ideal which includes A and ρ isan automorphism of J that restricts to an automorphismof I.

Then the restriction ρ � I of ρ to I has a presentation which isarithmetically de�nable from A.

Proof.

Apply the Odifreddi-Shore argument. There is a code −→p for acounting of I which is arithmetically definable from A. Lettinga = degree(A), ρ(a) is arithmetic in A since ρ−1(0′) ≤T a.Therefore ρ(−→p ) is arithmetic in A.

Persistent Automorphismspersistent extensions

Theorem


I Suppose J is a jump ideal which extends I.

Then ρ extends to a persistent automorphism of J .

Proof.

Fix a set A and a presentation of J which is recursive A.Suppose ρ∗ is an extension of ρ to jump ideal J ∗ such thatdegree(A) ∈ J ∗.I Then ρ∗ � J is an automorphism of J which is

arithmetically definable from A.

There are only countably many such automorphisms of J andone of these must be persistent.

Persistent Automorphismspersistent extensions

Theorem


I Suppose J is a jump ideal which extends I.

Then ρ extends to a persistent automorphism of J .

Proof.

Fix a set A and a presentation of J which is recursive A.Suppose ρ∗ is an extension of ρ to jump ideal J ∗ such thatdegree(A) ∈ J ∗.I Then ρ∗ � J is an automorphism of J which is

arithmetically definable from A.

There are only countably many such automorphisms of J andone of these must be persistent.

Persistent Automorphismscounting

Corollary


Then the following hold.

(1) ρ is arithmetically de�nable in any presentation of I.

(2) If J is a jump ideal which contains I then ρ extends to apersistent automorphism of J which is arithmeticallyde�nable in any presentation of J .

Two key consequences:

I Persistent automorphisms of I are locally presented.

I There are only countably many persistent automorphismsof I.

Persistent Automorphismscounting

Corollary


Then the following hold.

(1) ρ is arithmetically de�nable in any presentation of I.

(2) If J is a jump ideal which contains I then ρ extends to apersistent automorphism of J which is arithmeticallyde�nable in any presentation of J .

Two key consequences:

I Persistent automorphisms of I are locally presented.

I There are only countably many persistent automorphismsof I.

Persistent Automorphismsabsoluteness

Theorem

The property I is a representation of a countable ideal I,0′ ∈ I, and R is a presentation of a persistent automorphism ρof I is Π1

1.

Proof.

ρ is persistent if and only if for every presentation J of a jumpideal J extending I, there is an arithmetic in J extension of ρto J . This property is Π1

1.

Corollary

The properties R is a presentation of a persistent automorphismand There is a countable map ρ : I ∼→I such that 0′ ∈ I, ρ ispersistent and not equal to the identity are absolute betweenwell-founded models of ZFC.


Theorem


1.

Proof.


1.

Corollary



Theorem


1.

Proof.


1.

Corollary


Persistent Automorphismsmodel theoretically

I Fix T to be ZFC\Replacement + Σ1-Replacement.

Definition

Suppose that M = (M,∈M) is a model of T .

1. M is an ω-model if NM is isomorphic to the standardmodel of arithmetic.

2. M is well-founded if the binary relation ∈M iswell-founded:

I there is no infinite sequence (mi : i ∈ N) of elements of Msuch that for all i, mi+1 ∈M mi.

I if M = (M,∈M) is wellfounded then

M∼= (X,∈)

for (a unique) transitive set X.


Theorem

Suppose that M is an ω-model of T . Let I be an element of Msuch that

M |= I is a countable ideal in D such that 0′ ∈ I.

Then every persistent automorphism of I is also an element ofM.

Proof.

M is closed under arithmetic definability.


Theorem

Suppose that M is an ω-model of T . Let I be an element of Msuch that

M |= I is a countable ideal in D such that 0′ ∈ I.

Then every persistent automorphism of I is also an element ofM.

Proof.

M is closed under arithmetic definability.


Corollary

Suppose that M is an ω-model of T and that ρ and I areelements of M such that 0′ ∈ I, ρ : I ∼→I, and I is countable inM. Then

ρ is persistent =⇒M |= ρ is persistent.

Proof.

Persistent automorphisms extend persistently to any jump idealJ which contains I. For any such J ∈M, these extensionsbelong to M.


Corollary

Suppose that M is an ω-model of T and that ρ and I areelements of M such that 0′ ∈ I, ρ : I ∼→I, and I is countable inM. Then

ρ is persistent =⇒M |= ρ is persistent.

Proof.

Persistent automorphisms extend persistently to any jump idealJ which contains I. For any such J ∈M, these extensionsbelong to M.

Generic Persistence

Definition

Suppose that I is an ideal in D and ρ is an automorphism of I.We say that ρ is generically persistent if there is a genericextension V [G] of V in which I is countable and ρ is persistent.

Theorem

Suppose that ρ : I ∼→I is generically persistent. If V [G] is ageneric extension of V in which I is countable then ρ ispersistent in V [G].

Proof.

Absoluteness.

Generic Persistence

Definition


Theorem


Proof.

Absoluteness.

Generic Persistence

Definition


Theorem


Proof.

Absoluteness.


Theorem

Suppose that π : D∼→D. Then π is generically persistent.

Proof.

Suppose not. Then the failure of π to be generically persistentreflects to a countable well-founded model M |= T .Let G be an M-generic counting of DM. Thus

M[G] |= π � DM is not persistent.

But π � DM is persistent and this contradicts the absolutenessof persistence.


Theorem

Suppose that π : D∼→D. Then π is generically persistent.

Proof.

Suppose not. Then the failure of π to be generically persistentreflects to a countable well-founded model M |= T .Let G be an M-generic counting of DM. Thus

M[G] |= π � DM is not persistent.

But π � DM is persistent and this contradicts the absolutenessof persistence.

Applications to Aut(D)definability of automorphisms

Theorem

Suppose that V [G] is a generic extension of V . Suppose that

π : DV → DV

is an automorphism, π ∈ V [G], and π is generically persistentin V [G].

Then π ∈ L(RV ).

Proof.

π is generically persistent, so π is arithmetically definablerelative to any V [G]-generic counting of DV . Consequently, πmust belong to the ground model for such countings, namelyL(RV ).

Applications to Aut(D)definability of automorphisms

Theorem

Suppose that V [G] is a generic extension of V . Suppose that

π : DV → DV

is an automorphism, π ∈ V [G], and π is generically persistentin V [G].

Then π ∈ L(RV ).

Proof.

π is generically persistent, so π is arithmetically definablerelative to any V [G]-generic counting of DV . Consequently, πmust belong to the ground model for such countings, namelyL(RV ).

Applications to Aut(D)global extension of persistent automorphisms

Theorem

Suppose that 0′ ∈ I and that ρ : I ∼→I is persistent. Then ρ canbe extended to a global automorphism π : D

∼→D.

Proof.

ρ can be persistently extended to DV in a generic extension ofV . This extension belongs to L(RV ).

Corollary

The statement

I There is a non-trivial automorphism of the Turing degrees

is equivalent to a Σ12 statement. It is therefore absolute between

well-founded models of ZFC.


Theorem


∼→D.

Proof.


Corollary

The statement





Theorem


∼→D.

Proof.


Corollary

The statement




Applications to Aut(D)lifting automorphisms to generic extensions

Theorem

Suppose that π is an automorphism of D and that V [G] is ageneric extension of V .

Then there exists an automorphism

π∗ : DV [G] → DV [G]

such that π∗ ∈ V [G] and π = π � D.

Proof.

There is a persistent extension of π to an automorphism,

π∗ : DV [G] → DV [G]

in any generic extension of V [G] in which DV [G] is countable.Necessarily π∗ belongs to V [G].

Applications to Aut(D)lifting automorphisms to generic extensions

Theorem

Suppose that π is an automorphism of D and that V [G] is ageneric extension of V .

Then there exists an automorphism

π∗ : DV [G] → DV [G]

such that π∗ ∈ V [G] and π = π � D.

Proof.

There is a persistent extension of π to an automorphism,

π∗ : DV [G] → DV [G]

in any generic extension of V [G] in which DV [G] is countable.Necessarily π∗ belongs to V [G].

Representing Automorphisms

Definition

Suppose τ : D→ D and F : 2ω → 2ω. Then F represents τ if

degree(F (X)) = τ(degree(X))

for all X ∈ 2ω.

I D ⊆ 2<ω is dense if for all s ∈ 2<ω there exists t ∈ D suchthat t extends s.

I Suppose X is a set of dense subsets of 2<ω. Then G ∈ 2ω isX-generic if for all D ∈ X, there exists n such thatG � n ∈ D.

Theorem

Suppose that π : D∼→D. There is a countable set X of dense

subsets of 2<ω such that π is represented by a continuousfunction F on the set X-generic reals.


Definition



for all X ∈ 2ω.



Theorem




Definition



for all X ∈ 2ω.



Theorem



Images of generic degrees

Corollary

If π : D∼→D then π has a Borel representation; in fact, π has a

representation that is arithmetic in a real parameter.

Theorem

Suppose that π : D∼→D and that X is a countable set of dense

subsets of 2<ω.

Then there countable set Y of dense subsets of 2<ω such that forall Y-generic reals G there exist G0 and G1 such that

(1) G0 and G1 are each X-generic,

(2) degree(G0) ≤T π(degree(G)) ≤T degree(G1).

Images of generic degrees

Corollary

If π : D∼→D then π has a Borel representation; in fact, π has a

representation that is arithmetic in a real parameter.

Theorem

Suppose that π : D∼→D and that X is a countable set of dense

subsets of 2<ω.

Then there countable set Y of dense subsets of 2<ω such that forall Y-generic reals G there exist G0 and G1 such that

(1) G0 and G1 are each X-generic,

(2) degree(G0) ≤T π(degree(G)) ≤T degree(G1).

Bounding π(z) by z′′

Theorem

For every Z ⊆ ω, there is a countable family of dense open setsX such that for all X-generic reals G,

π(degree(Z ⊕G))′′ ≥T degree(Z)

I The proof of the Coding Theorem just gives:

π(degree(Z ⊕G))(5) ≥T degree(Z)

Theorem

For every z ∈ D, π(z)′′ ≥T z.

Corollary

For every z ∈ D, z′′ ≥T π(z).


Theorem





Theorem


Corollary



Theorem





Theorem


Corollary


The cone above 0′′

Corollary

For any 2-generic set G,

degree(G) + 0′′ ≥T π(degree(G)).

Theorem

Suppose that π : D∼→D.

(1) For all x ∈ D, x+ 0′′ ≥T π(x).

(2) For all x ∈ D, if x ≥T 0′′ then x = π(x).

Proof.

(1): A degree above 0′′ can be written as a join of 2-genericdegrees.

(2): By (1) applied to both π and π−1, x ≥T π(x) andx ≥T π

−1(x). Apply π to conclude π(x) ≥T π(π−1(x)) = x.


Corollary



Theorem


(1) For all x ∈ D, x+ 0′′ ≥T π(x).


Proof.





Corollary



Theorem


(1) For all x ∈ D, x+ 0′′ ≥T π(x).


Proof.




Representing Aut(D) by arithmetic functions

Theorem


(1) There is a recursive functional {e} such that for all G, if Gis 5-generic, then π(degree(G)) is represented by {e}(G, ∅′′).

(2) There is an arithmetic function F : 2ω → 2ω such that forall X ∈ 2ω, π(degree(X)) is represented by F (X).

Proof.

There is a countable set X of dense subsets of 2ω such that π iscontinuously represented on the X-generic reals. Use thatπ(degree(G)) ≤ G′′ to get {e} which works for all Y-generic Gfor some countable set Y extending X.

I Conclude that {e} works for all G which are 5-generic.

Since the 5-generics generate D, the representation on5-generics propagates to a representation everywhere.

Representing Aut(D) by arithmetic functions

Theorem


(1) There is a recursive functional {e} such that for all G, if Gis 5-generic, then π(degree(G)) is represented by {e}(G, ∅′′).

(2) There is an arithmetic function F : 2ω → 2ω such that forall X ∈ 2ω, π(degree(X)) is represented by F (X).

Proof.

There is a countable set X of dense subsets of 2ω such that π iscontinuously represented on the X-generic reals. Use thatπ(degree(G)) ≤ G′′ to get {e} which works for all Y-generic Gfor some countable set Y extending X.

I Conclude that {e} works for all G which are 5-generic.

Since the 5-generics generate D, the representation on5-generics propagates to a representation everywhere.

Consequences

Theorem

Aut(D) is countable.

Theorem

If g is 5-generic and π : D∼→D, then π is determined by its

action on g.

Proof.

Fix a recursive functional {e} such that π(G) = {e}(G, 0′′) forall 5-generic reals G.

I If G is 5-generic, then {e}(G, ∅′′) ≡T G iff the same is truefor all 5-generics.

Consequences

Theorem


Theorem


action on g.

Proof.



Consequences

Theorem


Theorem


action on g.

Proof.



Interpreting Aut(D) within Dassignments

I T is the theory: ZFC\Replacement + Σ1-Replacement.

Definition

An assignment of reals consists of

I A countable ω-model M of T .

I A function f and a countable ideal I in D such that

f : DM ∼→I.

Definition

For assignments (M0, f0, I0) and (M1, f1, I1), (M1, f1, I1)extends (M0, f0, I0) if and only if

I DM0 ⊆ DM1 ,

I I0 ⊆ I1,I and f1 � DM0 = f0.

Interpreting Aut(D) within Dassignments

I T is the theory: ZFC\Replacement + Σ1-Replacement.

Definition

An assignment of reals consists of

I A countable ω-model M of T .

I A function f and a countable ideal I in D such that

f : DM ∼→I.

Definition

For assignments (M0, f0, I0) and (M1, f1, I1), (M1, f1, I1)extends (M0, f0, I0) if and only if

I DM0 ⊆ DM1 ,

I I0 ⊆ I1,I and f1 � DM0 = f0.

Interpreting Aut(D) within Dn-extendable assignments

Definition

By induction on n:

1. An assignment (M0, f0, I0) is 1-extendable if for all z thereexists an assignment (M1, f1, I1) extending (M0, f0, I0)such that z ∈ I1.

2. An assignment (M0, f0, I0) is (n+1)-extendable if for all zthere exists an n-extendable assignment (M1, f1, I1)extending (M0, f0, I0) such that z ∈ I1.

Interpreting Aut(D) within D

Lemma

Suppose (M, f, I) is an assignment such that(f−1(0′)

)(5)+ f−1

((f(0′)

)(5)) ∈ I.Then f is an automorphism of I.

Proof.

DM is a jump ideal and so(f−1(0′)

)(5)+ f−1

((f(0′))(5)

)∈ DM ∩ I.

Corollary

If (M, f, I) is a 3-extendable assignment, then there is aπ : D

∼→D such that for all x ∈ DM, π(x) = f(x).


Lemma


)(5)+ f−1

((f(0′)


Proof.


)(5)+ f−1

((f(0′))(5)

)∈ DM ∩ I.

Corollary




Lemma


)(5)+ f−1

((f(0′)


Proof.


)(5)+ f−1

((f(0′))(5)

)∈ DM ∩ I.

Corollary



Theorem (Definability in D with parameters)

If g is the Turing degree of an arithmetically de�nable 5-genericset, then the relation R(−→c , d) given by

R(−→c , d) ⇐⇒ −→c codes a real D and D has degree d

is de�nable in D from g.

Theorem (Definability in D without parameters)

Suppose that R is a relation on D which is invariant underAut(D). Then following are equivalent.

(1) R is induced by a degree invariant relation R̂ on 2ω whichis de�nable without parameters in

(P(ω), ω,+, ·,∈) .

(2) R is de�nable in D without parameters.

Theorem (Definability in D with parameters)

If g is the Turing degree of an arithmetically de�nable 5-genericset, then the relation R(−→c , d) given by


is de�nable in D from g.

Theorem (Definability in D without parameters)

Suppose that R is a relation on D which is invariant underAut(D). Then following are equivalent.

(1) R is induced by a degree invariant relation R̂ on 2ω whichis de�nable without parameters in

(P(ω), ω,+, ·,∈) .

(2) R is de�nable in D without parameters.

Invariance of the double-jump

Theorem

For every Z ⊆ ω, there is a countable family of dense open setsX such that such that for all X-generic G,

π(degree(Z ⊕G))′′ ≥T degree(Z ′′).

I The earlier theorem was that:

π(degree(Z ⊕G))′′ ≥T degree(Z).

Theorem

Suppose that π : D∼→D. For all z ∈ D, π(z)′′ = z′′.

Corollary

The relation y = x′′ is invariant under π.

Invariance of the double-jump

Theorem

For every Z ⊆ ω, there is a countable family of dense open setsX such that such that for all X-generic G,

π(degree(Z ⊕G))′′ ≥T degree(Z ′′).

I The earlier theorem was that:

π(degree(Z ⊕G))′′ ≥T degree(Z).

Theorem

Suppose that π : D∼→D. For all z ∈ D, π(z)′′ = z′′.

Corollary

The relation y = x′′ is invariant under π.

Defining the double-jump and then (later) the jump

Theorem

The function x 7→ x′′ is de�nable in D.

Proof.

We have already shown that the relation y = x′′ is invariantunder all automorphisms of D. It is clearly degree invariant anddefinable in second order arithmetic. Therefore, it is definablein D.

Theorem (Shore and Slaman, 1999)

For all x ∈ D, the following conditions are equivalent.

(1) x 6≤T 0′.

(2) There exists y ∈ D such that x+ y ≥T y′′.


The function x 7→ x′ is de�nable in D.


Theorem


Proof.




(1) x 6≤T 0′.





Theorem


Proof.




(1) x 6≤T 0′.





Theorem


Proof.




(1) x 6≤T 0′.




Biinterpretability

Definition

D is biinterpretable with second order arithmetic if and only ifthe relation on −→c and d given by


is definable in D.

Theorem

The following are equivalent.

(1) D is biinterpretable with second order arithmetic.

(2) D is rigid.

Conjecture

D is biinterpretable with second order arithmetic.

Biinterpretability

Definition



is definable in D.

Theorem



(2) D is rigid.

Conjecture


Biinterpretability

Definition



is definable in D.

Theorem



(2) D is rigid.

Conjecture


Documents

The Global Structure of the Turing Degrees · PDF fileThe Global Structure of the Turing Degrees W. Hugh. Woodin University of California, Berkeley C ... Odifreddi-Shore Theorem (Odifreddi