The Global Structure of the Turing Degrees · PDF fileThe Global Structure of the Turing...
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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?
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?
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?
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.
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.
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.
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.
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.
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.
Applications to Aut(D)
Theorem (Nerode and Shore, 1980)
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).
Applications to Aut(D)
Theorem (Nerode and Shore, 1980)
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).
Applications to Aut(D)
Theorem (Nerode and Shore, 1980)
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).
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.
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.
Persistent Automorphisms
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 .
Persistent Automorphisms
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 .
Persistent Automorphisms
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∗?
Persistent Automorphisms
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∗?
Persistent Automorphisms
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∗?
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
Suppose that I is an ideal, 0′ ∈ I, and that ρ is a persistentautomorphism of I.
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
Suppose that I is an ideal, 0′ ∈ I, and that ρ is a persistentautomorphism of I.
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
Suppose that I is an ideal, 0′ ∈ I, and that ρ is a persistentautomorphism of I.
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
Suppose that I is an ideal, 0′ ∈ I, and that ρ is a persistentautomorphism of I.
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.
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.
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.
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.
Persistent Automorphismsmodel theoretically
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.
Persistent Automorphismsmodel theoretically
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.
Persistent Automorphismsmodel theoretically
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.
Persistent Automorphismsmodel theoretically
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
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
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.
Applications to Aut(D)
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)
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.
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.
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.
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.
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.
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.
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).
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).
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).
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.
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.
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.
Representing Aut(D) by arithmetic functions
Theorem
Suppose that π : D∼→D.
(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
Suppose that π : D∼→D.
(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
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
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.
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).
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).
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).
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
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.
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′′.
Theorem (Shore and Slaman, 1999)
The function x 7→ x′ is de�nable in D.
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′′.
Theorem (Shore and Slaman, 1999)
The function x 7→ x′ is de�nable in D.
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′′.
Theorem (Shore and Slaman, 1999)
The function x 7→ x′ is de�nable in D.
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′′.
Theorem (Shore and Slaman, 1999)
The function x 7→ x′ is de�nable in D.
Biinterpretability
Definition
D is biinterpretable with second order arithmetic if and only ifthe relation on −→c and d given by
R(−→c , d) ⇐⇒ −→c codes a real D and D has degree d
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
D is biinterpretable with second order arithmetic if and only ifthe relation on −→c and d given by
R(−→c , d) ⇐⇒ −→c codes a real D and D has degree d
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
D is biinterpretable with second order arithmetic if and only ifthe relation on −→c and d given by
R(−→c , d) ⇐⇒ −→c codes a real D and D has degree d
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.