Goodness and Jump Inversion in the Enumeration Degrees. Charles M. - - PowerPoint PPT Presentation

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Goodness and Jump Inversion in the Enumeration Degrees.

Charles M. Harris

Department Of Mathematics University of Leeds

CiE Sofia 2011

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Outline

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Section Guide

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

7

Other Results

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

What is enumeration reducibility?

Definition (Intuitive)

A≤eB if there exists an effective procedure that, given any enumeration of B, computes an enumeration A.

Definition (Formal)

A≤eB if there exists a c.e. set We such that for all x ∈ ω x ∈ A iff ∃u [ x, u ∈ We & Du ⊆ B ] This is written A = Φe(B).

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Enumeration Degrees

• For X, Y ⊆ ω, X ≡e Y iff X ≤eY and Y ≤eX.
• This is an equivalence relation (whose equivalence classes

we call enumeration degrees).

Notation

• De, ≤, DT, ≤ denote the structures of the enumeration

degrees and Turing degrees. (Or more simply De and DT).

• dege(A) denotes the degree of the set A.
• We say, for Γ ∈ {Σ, Π, ∆} that the degree x is Γ0

n if it

contains a Γ0

n set.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Enumeration Degrees

• For X, Y ⊆ ω, X ≡e Y iff X ≤eY and Y ≤eX.
• This is an equivalence relation (whose equivalence classes

we call enumeration degrees).

Notation

• De, ≤, DT, ≤ denote the structures of the enumeration

degrees and Turing degrees. (Or more simply De and DT).

• dege(A) denotes the degree of the set A.
• We say, for Γ ∈ {Σ, Π, ∆} that the degree x is Γ0

n if it

contains a Γ0

n set.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Enumeration Degrees

• 0e is the class of c.e. sets.
• dege(X ⊕ Y) is the least upper bound of dege(X) and

dege(Y).

• There exist enumeration degrees x and y which have no

greatest lower bound.

• Consequence: De is an upper semilattice.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Enumeration Degrees (De).

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Totality

• Y is total if Y ≤eY.

(X ⊕ X is total for any X and if f is total then Gf is total.)

• If Y is total then, for any X: X ≤eY iff X is c.e. in Y.
• Consequence: A≤TB

iff A ⊕ A≤eB ⊕ B.

• ι : DT → De induced by Z → Z ⊕ Z is an u.s.l.

embedding.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Jumps

Notation

We assume a standard effective listing of Turing machines {ϕe}e∈ω, c.e. sets {We}e∈ω (where We = { x | ϕe(x) ↓ }), and enumeration operators {Φe}e∈ω. (i) K denotes the halting set { e | ϕe(e) ↓ }. (ii) For any Z, define : IZ = { e | e ∈ ΦZ

e }.

• JZ = IZ ⊕ IZ is total.
• If Z is Σ0

2 then IZ is total.

(In fact, if Z has a good approximation, as defined below, then IZ is total).

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Totality and Jumps IX = { x | x ∈ Φx(X) }

• z′ = dege(JZ)

some/any Z ∈ z.

• K ≡1 I∅.

So K ≡e J∅ (=def I∅ ⊕ I∅).

• 0′

e = dege(J∅) = dege(K).

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Totality, Jumps and Local Enumeration Degrees

• x is Σ0

2 iff x ≤ 0′ e (Cooper).

• In fact: X is Σ0

2 iff X ≤eK.

• Equivalently: X is Σ0

2 iff X is c.e. in K.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Local Enumeration Degrees

Notation

De, ≤ 0′

e denotes the structure of the Σ0 2 enumeration

degrees.

• De, ≤ 0′

e is an upper semilattice.

• 0e and 0′

e are the bottom and top elements of De, ≤ 0′ e .

• De, ≤ 0′

e is dense (Cooper 1984).

• DT, ≤ 0′

T is a substructure (u.s.l.) of De, ≤ 0′ e .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Canonical Embedding ι.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Jump Classes

• x is lown if xn = 0n

e .

• x is highn if xn = 0n+1

e

.

• We also say that x is low / high in the case when n = 1.
• Ln = { x | x is lown }.
• Hn = { x | x is highn }.
• I = { x | ∀n[ 0n < xn < 0n+1 ] }.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The High/Low Jump Hierarchy

Lemma

For every n ≥ 0 there exist total degrees x, y ≤ 0′

e such that

x ∈ Hn+1 − Hn and y ∈ Ln+1 − Ln. There also exist total z ≤ 0′

e such that z ∈ I (the class of intermediate degrees).

Proof.

Apply the equivalent results (Sacks 1963, 1967) proved in the context of the Σ0

1 Turing degrees in conjunction with the jump

preservation properties of the embedding ι : DT → De.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Lowness and proper Σ0

2-ness

Proposition (Cooper and McEvoy)

x is low iff every X ∈ x is ∆0

2.

Definition

x < 0′

e is properly Σ0 2 if x contains no ∆0 2 set and downwards

properly Σ0

2 if every y ∈ { z | 0e < z ≤ y } is properly Σ0 2.

Definition

x < 0′

e is cuppable if there exists y < 0′ e such that x ∪ y = 0′ e .

Otherwise x is noncuppable.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Good Approximations

Definition (Lachlan and Shore 1992)

A uniformly computable enumeration of finite sets {Xs}s∈ω is said to be a good approximation to the set X if: (1) ∀s (∃t ≥ s)[ Xt ⊆ X ] (2) ∀x [ x ∈ X iff ∃t (∀s ≥ t)[ Xs ⊆ X ⇒ x ∈ Xs ] ].

Lemma (Jockusch 1968)

X is Σ0

2 iff X has a good Σ0 2 approximation. I.e. a good

approximation with(2 ′) replacing (2). (2 ′) ∀x [ x ∈ X iff ∃t (∀s ≥ t)[ x ∈ Xs ] ].

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Some Essential Notation.

Notation

X [e] denotes the set { e, x | e, x ∈ X }.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Section Guide

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

7

Other Results

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Jump Uniformity

Definition

A set B is said to be jump uniform under ≤e if, for any set A, A≤eJB ⇔ ∃X[ X ≤eB & A = { e | X [e] is finite } ] (1) where JB is notation for the enumeration jump of B and X [e] notation for the eth column of X.

Lemma (Griffith 2003, H 2010)

Every good approximable set is jump uniform under ≤e.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Jump Interpolation

Theorem (Griffith 2003, H 2010)

For any set B which has a good approximation and any set A<eB there exists a set A≤eC<eB such that JC ≡e JB.

Proof.

Construct X ≤eB, such that X satisfies , for all e ∈ ω, the following requirements. Ne : B = ΦA⊕X

e

He : e ∈ ΦB

e

⇔ X [e] infinite ( = ω[e]).

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Characterising the Double Jump.

Notation

For any set X we use the notation Inf =def { e | We infinite }, whereas InfSet(X) =def { e | ΦX

e infinite }.

Lemma (H 2010)

If the set X has a good approximation, then InfSet(X) ≡e J2

X.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Section Guide

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

7

Other Results

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

A Special Σ0

2 Enumeration Degree

Theorem (Cooper and Copestake 1988)

For every high a ≤ 0′

e there exists 0e < b < a such that, for all

∆0

2 degrees 0e < c < a, b ⊥ c.

Corollary

There exists 0e < b < 0′

e such that, for all ∆0 2 degrees

0e < c < 0′

e , b ⊥ c.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Background for the proof.

We will be using the following type of computable listing.

• A listing of the ∆0

2 sets {Ce}e∈ω with associated non

decreasing Σ0,K

1

approximation of finite sets {Ce,s}s∈ω such that, for all e, Ce =

• s∈ω

Ce,s .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

A Special Σ0

2 Enumeration Degree

Theorem (H 2011)

There exists a high enumeration degree 0e < b < 0′

e such that,

for all ∆0

2 degrees 0e < c < 0′ e , b ⊥ c.

Proof.

We construct A c.e. in K such that (for all e ∈ ω) the following requirements are satisfied. (Using {We, Φe, Ce}e∈ω.) He : We infinite ⇔ A[2e] finite Ne : A = ΦCe

e

⇒ K≤eCe Pe : Ce = ΦA

e

⇒ Ce is c.e.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

A Special Degree.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Section Guide

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

7

Other Results

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Cuppability

Proposition (Cooper, Sorbi and Yi 1996)

If 0e < x < 0′

e is ∆0 2, then x is cuppable.

Corollary (Cooper, Sorbi and Yi 1996)

Every noncuppable 0e < x < 0′

e is downwards properly Σ0 2.

Proposition (Cooper, Sorbi and Yi 1996)

There exists a noncuppable degree 0e < x < 0′

e .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Cuppability.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Noncuppability.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Cuppability

Proposition (Giorgi 2008)

There exists a high noncuppable degree x < 0′

e .

Proposition (Giorgi, Sorbi and Yang 2006)

For every nonlow total degree c ≤ 0′

e there exists a

noncuppable degree 0e < x < c.

Corollary

Hence there exist low2 noncuppable x < 0′

e .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Background for the proof.

We will be using the following type of computable listings.

• A listing of the Σ0

2 sets {Be}e∈ω with associated Σ0,K 1

approximation of finite sets {Be,s}s∈ω such that, for all e, Be =

• s∈ω

Be,s .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

A low2 noncuppable degree

Theorem (Giorgi, Sorbi and Yang 2006)

There exists a non cuppable Σ0

2 enumeration degree a > 0e

such that a′′ = 0′′

e (i.e. such that a is low2).

Proof.

We construct sets A and C c.e. in K such that (for all e ∈ ω) the following requirements are satisfied. Ne : A = We Le : ΦA

e infinite

⇔ C[e] finite Pe : K = ΦBe⊕A

e

⇒ K≤eBe .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Low2 noncuppability. There exists low2 degree

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Noncuppability and the High/Low Jump Hierarchy

Theorem

For every Σ0

2 enumeration degree b there exists a noncuppable

Σ0

2 degree a > 0e such that b′ ≤ a′ and a′′ ≤ b′′.

Proof.

Given a Σ0

2 set B we construct A c.e. in K and C c.e. in IB such

that (for all e ∈ ω) the following requirements are satisfied. He : e ∈ ΦB

e

⇔ A[e] infinite ( = ω[e]) Le : ΦA

e infinite

⇔ C[e] finite Pe : K = ΦBe⊕A

e

⇒ K≤eBe .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Noncuppability and the High/Low Jump Hierarchy.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Noncuppability and the High/Low Jump Hierarchy

Corollary

For every n > m ≥ 0 there exist noncuppable enumeration degrees x, y ≤ 0′

e such that x ∈ Hm+1 − Hm and

y ∈ Ln+1 − Ln. There also exists noncuppable z ≤ 0′

e such

that z ∈ I. Finally note that the property of being noncuppable can be replaced by that of being downwards properly Σ0

2, in the above

Corollary.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Section Guide

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

7

Other Results

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Total Incomparability

Question

Does there exist a ∆0

2 enumeration degree 0e < a < 0′ e such

that, for any total enumeration degree 0e < c < 0′

e , a ⊥ c?

Answer (Arslanov, Cooper and Kalimullin 2003)

• No. For any ∆0

2 degree a < 0′ e there exists a total degree

a ≤ c < 0′

e .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

High Quasiminimality

Notation

We say that a set Y is characteristic if Y = X ⊕ X for some set X.

Note

For any set X, χX ≡1 X ⊕ X. Also an enumeration degree is total iff it contains a characteristic function. Hence an enumeration degree is total iff it contains a characteristic set.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

High Quasiminimality

Theorem

There is a ∆0

2 degree c such that: (a) c < 0′ e , (b) for all total x,

if x ≤ c, then x = 0e, and (c) c′ = 0′′

e .

Proof.

The overall strategy is to construct a set C such that the following requirements are satisfied (for all e ∈ ω). R : C is ∆0

2 ,

He : We is infinite ⇔ C[e] is finite, Ne : ΦC

e is a characteristic set ⇒ ΦC e is c.e.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

High ∆0

2 Quasiminimality.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Section Guide

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

7

Other Results

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Goodness and Badness

Notation

An enumeration degree is said to be good if it contains a good approximable set. Otherwise it is said to be bad.

Reminder

Every Σ0

2 degree is good. In fact every Σ0 2 set has a good Σ0 2

approximation.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Badness in the Π0

2 Enumeration Degrees

Fact 1.

We can deduce, from the density of the class of good enumeration degrees (Lachlan and Shore 1992) and the proof by Calhoun and Slaman (1996) of the existence of a generalised low minimal cover in the Π0

2 enumeration degrees

that there exists a bad Π0

2 enumeration degree x such that

every X ∈ x is ∆0

3, i.e. x′ ≤ 0′′ e .

Fact 2.

Since every Σ0

2 set is good approximable, if A is ∆0 2 then both A

and A are good approximable.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Jump Complexity of Badness

Question

Suppose that A ∈ a is Π0

2 and that a is bad. Given Fact 2,

dege(A) cannot be low. Se can we find a bad degree a and Π0

2 set A ∈ a such that

dege(A) is low2 (i.e. dege(A)′′ = 0′′

e ) ?

Reminder (with A as above)

If dege(A)′′ = 0′′

e then dege(A)′ ≤ 0′′ e .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Jump Complexity of Badness

Proposition

There exists a Π0

2 set A such that J(2) A ≤eJ(2) ∅

and satisfying: (∀e)(∃y)[ y ∈ A iff (ΦA

e)[y] is infinite ].

Proof.

Enumerate the set A as also an auxiliary set C using K as Turing oracle so as to satisfy, for all e ∈ ω, Le : ΦA

e is infinite

iff C[e] is finite, Pe : ∃y[ y ∈ A iff (ΦA

e)[y] is infinite ] .

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The Jump Complexity of Badness

Corollary

There exists a Π0

2 enumeration degree a and Π0 2 set A ∈ a such

that dege(A)′′ = 0′′

e, so that a′≤e0′′ e—and thus every x ≤ a only

contains ∆0

3 sets—and such that a is not jump uniform, and

hence is also bad.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

A Bad Π0

2 Degree.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Section Guide

1

The local enumeration degrees

2

Jump Interpolation

3

A special Σ0

2 enumeration degree

4

Cuppable and noncuppable enumeration degrees

5

A special ∆0

2 enumeration degree

6

The Boundaries of Goodness

7

Other Results

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

Other Results

Theorem (Cooper et al. In preparation.)

If b ≤ 0′

e is low2, then there exists low2 0e < a ≤ 0′ e such that

a ∩ b = 0e.

Proposition

If A is Σ0

2 then CoInfSet(A) =def { e | ΦA e is infinite } ≡e J(3) A .

Conjecture

If a < 0′

e is upwards properly Σ0 2 then a is high.

Goodness and Jump Inversion. Charles M. Harris

Local Degrees Interpolation A Special Σ0

2 Degree

Cuppable Degrees A Special ∆0

2 Degree

Goodness Other

The End

Goodness and Jump Inversion. Charles M. Harris