Main Goals. The Computably Enumerable Sets. The Computably Enumerable Sets, Creative Sets

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1 Main Goals The Computably Enumerable Sets A Tutorial Peter Cholak University of Notre Dame Department of Mathematics Peter.Cholak.1@nd.edu Supported by NSF Division of Mathematical Science To understand, as best we can, the local and global structure of c.e. (r.e.) sets under inclusion. What do we know now? We provide an incomplete answer. What are the good open questions? The story is rather complex. September 2008 The Computably Enumerable Sets, Creative Sets W e is the domain of the eth Turing machine. ({W e : e ω}, ) are the c.e. (r.e.) sets under inclusion,. These sets are the same as the Σ 0 1 sets, {x : (N, +,, 0, 1) ϕ(x)}, where ϕ is Σ 0 1. W e,s is the domain of the eth Turing machine at stage s. For safety, all sets are c.e., infinite, and coinfinite, unless otherwise noted. 0, 1,,, and (disjoint union) are definable from in. Let K, the jump, be the set {e : ϕ e (e) }. Let H, the halting set, be the set { e, x : ϕ e (x) }. W 1 X iff there is a computable one-to-one function on the set of natural numbers, ω, such that x W iff f (x) X. X is 1-complete iff for all c.e. sets W, W 1 X. Both H and K are 1-complete.

2 Computably Isomorphic Sets Some Automorphism Results X and Y are computably isomorphic iff there is a computable permutation, p, of ω such that p(x) = Y. Assume X and Y are computably isomorphic which is witnessed via a computable permutation p. Then Φ(W ) = p(w ) is an automorphism of. If W is c.e. then so is p(w ). X Y iff p(x) p(y ). The first clause depends on the fact that p is computable. The second depends on the fact that p is a permutation. Theorem (Myhill) X is 1-complete iff X and K are computably isomorphic. All 1-complete sets are in the same (effective) orbit. The infinite coinfinite computable sets are in the same effective orbit. There is a computable permutation p such that p(r) = R and p(r) = R. Some Definability Results Automorphisms vs. Definability R is computable iff Y [R Y = ω]. The infinite coinfinite computable sets form an effective orbit. Theorem (Harrington 84) A c.e. set A is 1-complete iff ( C A)( B C)( R)[R is computable & R C is noncomputable & R A = R B]. The 1-complete sets form a definable orbit. The type realized by a 1-complete set is a principal type. The same is true for the infinite coinfinite computable sets (with a just little more upcoming work). X is automorphic to Y, X Y, iff there is an automorphism of such that Φ(X) = Y. If the 1-complete sets fail to form an orbit then there must be a c.e. set which not 1-complete but yet is automorphic to the 1-complete sets. The failure to find these automorphisms lead Leo to the property defining the 1-complete sets. How?

3 Turing Reductions More Related Results X is Turing reducible to Y, X T Y, iff there is a Turing machine M such that M Y = χ X. X is complete iff, for all c.e. W, W T X. K, H are complete. X jump, X, is the set K X = {e : ϕ X e (e) }. X(α) is α iterations of the jump (at limit level take the disjoint union of the previous levels.) Theorem (Harrington-Soare 91) There is a realizable definable property Q(A) such that if Q(A) then A < T K and A is noncomputable. The(?) solution to Post s Program. Theorem (Harrington-Soare 96) There is a realizable definable property T (A) such that if T (A) then K A and A is not 1-complete (A is promptly simple). Related s The Maximal Sets M is maximal if for all W either W M or M W = ω. Conjecture (Harrington) The 1-complete sets are the only orbit of which remains an orbit when we restrict the allowable automorphisms those induced by a computable permutations. (Completeness) Which c.e. sets are automorphic to complete sets? Theorem (Friedberg) Maximal sets exist. Make sure M is not split into two infinite pieces by any W e. Use markers Γ e to mark the eth element of M s. Let σ (e, x, s) = {i e : x W i,s }; the eth state of x at stage s. Allow Γ e to pull at any stage to increase its e-state. Let M be maximal. If M W = ω then there is a computable set R W such that R W M and W R W = ω. So W = R W (W R W ).

4 Building Automorphisms The Filter of Finite Sets Theorem (Soare) If A and  are both noncomputable then there is an isomorphism, Φ, between the c.e. subsets of A, (A), to the c.e. subsets of  sending computable subsets of A to computable subsets of  and dually. False Let p be a computable 1-1 map from A to Â. Let Φ(W ) = p(w ). Let A = A 1 A 2, where A 1 is computable and infinite. Let  =  1  2 where  1,  2 is a nontrivial split of A. Let p i be a computable 1-1 map from A i to  i and p = p 1 p 2. F is the filter of finite sets (within ). is modulo F. X is the equivalence class of X in. X is computable iff X and X are both c.e. sets. X is finite iff every subset of X is computable. Theorem (Soare 74) If Φ Aut( ), Φ(A ) =  and A is infinite and coinfinite then there is a Ψ Aut() such that Ψ(A) = Â. Idea: find a permutation p such that Ψ(W ) = p(w ). We can use p to classify the Turing complexity of Ψ. Another option is to use a f such that Ψ(W e ) = W f (e). Blur the two structures and into one. Orbits of Maximal Sets Theorem (Soare 74) The maximal sets form an orbit. Define if W M then let Ψ(W ) = Φ(W ). If M W = ω then let Ψ(W ) = Φ(R W ) Φ(R W W ). The type realized by a maximal set is a principal type in. Friedberg Splits of Maximal Sets S 1 S 2 = M is a Friedberg split of M, for all W, if W M is infinite then W S i is also. (Downey, Stob) A nontrivial split of a maximal set (a hemimaximal set) is a Friedberg Split. Theorem (Downey, Stob) The Friedberg splits of the maximal sets form a definable orbit. Let M 1 M 2 = M. If W M = ω then W M is infinite. Ψ(W ) = ( Φ 1 (R W M 1 ) Φ 2 (R W M 2 ) ) Φ 1 (R W W M 1 ) Φ 1 (R W W M 2 ).

5 Interaction with the C.E. Turing Degrees A class D of degrees is invariant if there is a class S of (c.e.) sets such that 1. d D implies there is a W in S and d. 2. W S implies deg(w ) D and 3. S is closed under automorphic images (but need not be one orbit). Theorem (Martin) A c.e. degree h is high (h = O ) iff there is a maximal set M such that M h. Like True Arithmetic Theorem (Harrington-Nies 98) True arithmetic can be interpreted in with parameters. The degree of the theory of is 0 ω. The natural numbers can be coded with parameters into. There is a chance that the definable sets in can be easily understood. The high degrees are invariant by a single orbit, the maximal sets. Theorem (Harrington 94) Not like True Arithmetic True arithmetic cannot be interpreted in without parameters. (Not Necessarily Standard) is biinterpretable with true arithmetic (with parameters) iff there is an interpretable (with parameters) model of true arithmetic, M, and a definable (with parameters), one-to-one, function C, from c.e. sets to elements of M. Assume that is biinterpretable with true arithmetic with parameters. Then interprets true arithmetic without parameters, has a finite automorphism base, and is atomic. Scott Analysis Fix a countable L-structure A. We work in L ω1,ω. This language allows countable disjunctions and conjunctions in the natural way. The goal is to find a countable fragment L A such that A is the atomic model of T A, the complete theory of A in L A. So in L E the 1-type of a c.e. set A completely describes A s orbit. Moreover if two c.e. sets are not in the same orbit then they have different 1-types and hence there is a definable difference between these sets. The version of the Scott analysis presented is strongly influenced by Sacks. True arithmetic is not biinterpretable in with parameters.

6 Finding L A Bounding δ < ω 1 Via induction on the ordinals. Let L A 0 = L. For limit λ, let L A λ = δ<λ L A δ. Let T A δ be the complete theory of A in LA δ. The induction step: Let L A δ+1 be the least fragment of L ω 1,ω extending L A δ such that for all n, if t( x) is a non-principal n-type of T A δ in LA δ+1 realized in A then {ϕ( x) : ϕ( x) t( x)} is in L A δ+1. If there are no non-principal types let L A = L A δ. For some δ < ω 1 all the n-types of T A δ principal. realized in A are Assume some non-principal type t δ+1 of T A δ+1 is realized in A. Let t δ be the restriction of t δ+1 to L A δ. Since t δ+1 is non-principal over T A δ+1, there is a ϕ( x) LA δ+1 such that the formulas x[t δ ( x) ϕ( x)] and x[t δ ( x) ϕ( x)] are in T A δ+1. These formulas are realized in A by two different n-tuples, a and b. The definable difference between a and b is made by a formula of L A δ+1 which is not in LA δ. As there are only countable many such pairs there are at most countable many such definable differences. Homogeneous and Atomic Models If A is a homogeneous model of T A δ model of T A δ+1. then A is an atomic Theorem For all arithmetic L-structures A, A is a homogeneous model of T A ω CK. 1 Let p( x) be a n-type and q( x, y) be a n + 1-type of T A ω CK 1 containing p( x). Suppose there are n-tuples, a and b, such that A realizes p( a), p( b), and yq( a, y). Assume we cannot find a d such that A realizes q( b, d). Then for each d A, there is a δ d such that A realizes q δd ( b, d). Let δ = lim δ d. It is hyperarithmetic to find δ and hence δ < ω CK 1. But then A realizes y q( b, y). The Scott Rank Let the Scott Rank be the least δ such that A is an atomic model of T A δ. The Scott rank of an arithmetic structure is ω CK Theorem (Cholak, Downey, Harrington) The Scott rank of is ω CK

7 Getting the Scott Rank at least ω CK 1 Getting the Scott Rank to ω CK If has Scott Rank δ < ω CK 1 then the set I = { i, j : W i is automorphic to W j } is hyperarithmetical. We will show that I is Σ 1 1-complete. This was conjectured by Slaman and Woodin in If has Scott Rank ω CK 1 then, for each A the set I A = {i : A is automorphic to W i } is hyperarithmetical. We will show that I A is Σ 1 1-complete, for some A. The Isomorphism Problem Theorem (Kleene) The set of computable not well-founded infinite branching trees is Π 1 1 -complete. Theorem There is a computable tree T such that the collection of computable trees S which are isomorphic to T is Σ 1 1-complete. (The Isomorphism Problem for T.) Theorem For all finite α, there is a computable tree T α such that for computable trees S, T α and S are isomorphic iff T α and S are isomorphic via an isomorphism computable in 0 (α) and this is not the case for 0 (β), for any β < α. The same can be done for Boolean Algebras and Linear Orderings. Coding Trees into Orbits Coding: For each T build a A T such that the tree coded by A, which we denote as T (A T ), is isomorphic to T via an isomorphism Λ T 0 (2). Coding is preserved under automorphic images: If  A T via an automorphism Φ then T (Â) exists and T (Â) T via an isomorphism Λ Φ, where Λ Φ T Φ 0 (2). Sets coding isomorphic trees belong to the same orbit: If T T via isomorphism Λ then A T A T via an automorphism Φ Λ where Φ Λ T Λ 0 (2). (Here isomorphism refers to isomorphism between trees while automorphisms are of the structure.

8 Some Byproducts of Our Work Theorem (Cholak, Downey, Harrington) For all finite α > 8 there is a properly 0 α orbit. Theorem (Cholak, Downey, Harrington) If  and A T are automorphic via Ψ and T T (Â) via Λ then A T  via Φ Λ where Φ Λ T Λ 0 (8). (In this situation T is an invariant.) Can we find an invariant (in terms of trees?) which works for all A? Theorem The A T can have any hemimaximal degree. Moreover there is an A T whose orbit contains a representative of every hemimaximal degree. The Conjectured Approach For a c.e. set A, L (A) is {W A : W a c.e. set} under modulo the ideal of finite sets (F). Theorem (Lachlan (1968)) For each computable Boolean Algebra B i, there is c.e. set hhsimple H i such that L (H i ) B i. The set { i, j : L (H i ) L (H j )} is Σ 1 1 -complete. So replace L (H i ) L (H j ) with H i H j. But this fails! D-hhsimple Sets (The sets disjoint from A) D(A) = {B : W (B A W and W A = )} under inclusion. Let D(A) be modulo D(A). If A is simple then D(A) 0 3 L (A). A is D-hhsimple iff D(A) is a Boolean algebra. Except for the creative sets, until recently all known orbits were orbits of D-hhsimple sets. Complexity Restrictions If A is D-hhsimple and A and  are in the same orbit then D(A) 0 3 D(Â). Theorem (Maass 84) If A is D-hhsimple and simple (i.e., hhsimple) if L (A) 0 3 L (Â) then A Â. (Cholak, Harrington) If A is D-hhsimple and simple (i.e., hhsimple) then A  iff L (A) 0 3 L (Â). (Cholak, Harrington) The set { i, j : W i W j and W i is hhsimple} is Σ 0 5. Hence the Slaman-Woodin plan of attack fails.

9 Related Complexity Restrictions The Structure S R (A) If A and  are automorphic then D(A) and D(Â) are 0 6 -isomorphic. (Cholak, Harrington) The set { i, j : W i W j and W e is D-hhsimple} is Σ 0 7. Hence must code a Σ 1 1-complete set into D(A). If A is simple then A  iff A 0 6 Â. Let S(A) = {B : C(B C = A)}. Let S (A) be the quotient structure S(A) modulo F. Let R(A) = {R : R A and R is computable}. Let S R (A) be the quotient structure S(A) modulo R(A). If B is a Σ 3 set of indices of splits of A, then B generates an Σ 3 Boolean algebra of S R (A). If A and  are both promptly simple then A  iff A 0 3 Â. The Restriction Theorem The Pattern P 1 If A and  are automorphic via Φ then the structures S R (A) and S R (Â) are 0 3-isomorphic via an isomorphism Ψ induced by Φ. Given a split T of A we would like to find in a 3 fashion an index for Φ(T ). We cannot do this. But we can find an index for set X such that X R(Â) Φ(T ). Code up the splits of A via movements on different patterns. b 5 b b 2 4 R b 1 U 0 r 2 4 r 4 1 U b 4 b 3 U 2 4 U 3

10 Movement on the Pattern P The Formula ϕ P (A, U, B, S) is Defined to be There is a D such that a ball x is at node n at stage s iff x D n,s. Only balls in S enter the pattern P. So if x enters S at stage s then x also enters D b0.) If a ball later enters A it must leave the pattern. If x is at node labelled by a U i at stage s then x must be in U i at stage s. There are RED moves: If a ball x is at node labelled by R, (In P 3, b 2 is only such node and hence x must be in D b2 ) then x must move downward in the pattern. There are BLUE moves determined by the sets B. If x is at node n not labelled by R and there is a downward move to n, x moves downward at stage s iff x enters B n at stage s. The Complexity of the formula ϕ P (A, U, B, S) 1. ϕ P (A, U, B, S) can be written as Σ 3 formula in the language L = { }. 2. Moreover if ϕ P (A, U, B, S) holds is Σ 0 3 question in arithmetic. Computable Conditions on S An Universal Interpretation Looking for lots of balls moving though the patterns. X is computable modulo A iff there is a Y such that X Y A and X Y A = ω. In another application we will want S not to be a computable modulo A split of W. For the Restriction Theorem A will be empty and we will use a different unnamed Σ 3 computability condition on S. Uniformly in U, P, A and W (think of all of these items as indexed by e), there is a B e such that the following are equivalent: There is a S such that S is a split of W, S is not computable modulo A, and ϕ P (A, U, B e, S). for all B, there is a S such that S is a split of W, S is not computable modulo A, and ϕ P (A, U, B, S). (Here we say U, A and W realize P.) Whether U, A and W realize P is Σ 0 4 (in arithmetic). Similar results hold with the other unnamed condition on S.

11 The pattern P 3 Realizing and Not Realizing Patterns b 5 b b 1 U 0 4 U 3 Effectively in i,c (these two are coded by e), S, and B e, there are U and D such that D witnesses that ϕ Pi (, U, B e, S) holds (and all the components of U and D are contained in S). (S is chosen so that the computable condition holds) Furthermore, for all j i, effectively in j there are B j (each component being completely contained in S) such that for all splits S S of C, either S is computable or, for all X, X does not witness ϕ Pj (, U, B j, S) (for the same set U as above). So U, A, and C realizes P i via the set S but does not realize any other P j (via S or any split of C contained in S.) The boxes refer to the block in P 0. Coding the Splits of A The Decoding Needed for Restriction Theorem Using specially constructed parameters E and F which are subsets of A, there is 3, one to one (but onto), listing of nontrivial splits of A. Moreover for each nontrivial split T i of A there is a corresponding unique split S of F such that S i T i and S i T i are nontrivial splits of F. Apply the last theorem to 2i, F (both coded by e), S i T i, B e. Also apply the last theorem to 2i + 1, F (both coded by e), S i T i, B e. So U,, and F realizes P 2i (P 2i+1 ) via S i T i (S i T ). Moreover, using the extra parameters F, F, if U,, and F realizes P 2i and P 2i+1 via X T i and X T i respectively then X R(A) S i. The collection of the splits S i of F is 3 ideal, I, of S R (F). Theorem (Harrington s Ideal Definability Theorem) For n 1, there is a formula ψ n (X, F, Y ), where Y = n, such that, for each Y (all c.e.), {X : ψ n (X, F, Y )} is a Σ 0 2n+1 -ideal of S R(F) and, for each Σ 0 2n+1 -ideal I of S R(F), there is a Y such that I = {X : ψ n (X, F, Y )}. Given T look for a X such that there an Ŝ in Î, determined by ψ 1, Φ(F), Φ( Y ), such that Φ( U),, Φ(E) realize P 2i via Ŝ X, Φ( U),, Φ(E) realize P 2i+1 via Ŝ X, and the unnamed Σ 3 computability condition holds on Ŝ relative to Φ(E) and Φ(F). It is the case that X R(Â) Φ(T ).

12 Effective Listing of Splits A listing of splits of A, {S i : i ω}, is an effective listing of splits of A iff there are computable function f and f such that, for all i, S i = W f (i) and if S i = W f (i) then S i S i = A. Let S e = W e A = {x : s t[t > s, x (W e,s A s ), x A t } and S e = A\W = {x : s[x (A s W e,s )]}. {S e : e ω} is an effective listing of splits. (W e A) (A\W e ) = A. Every effective listing of splits can be written as in the above lemma (change the enumeration). But not every listing of splits is effective. Extension Theorem Extendible subalgebras A subalgebra B of S R (A) is extendible iff there is an effective listing of splits, {S i : i ω}, of A and a 3 set B such that the splits {S i : i B} generate B. The trivial subalgebra of S R (A) is extendible. The subalgebra A generated by the entry sets is extendible (this is what we call an entry set Boolean algebra for A). We will want two extendible subalgebras to be isomorphic. But we will want only consider isomorphisms which maps nicely between the given representations. In this case we say these extendible subalgebras are extendibly isomorphic. Supports Let B S R (A) and B S R (Â) be two extendible Boolean algebras (witnessed the lists {S i : i ω} and { S i : i ω} respectively) which are 0 3 extendibly isomorphic via Θ. Then there is a Φ such that Φ is an 0 3 isomorphism between (A) and (Â), for all i B, Φ(S i ) = R Θ(S i ), and for all i B, Φ 1 (Ŝ i ) = R Θ 1 (Ŝ i ). The proof is dynamic. The proof uses Cholak s Translation Theorem and Soare s Original Extension Theorem. S supports X iff S X and (X A) S is a computably enumerable set. W A supports W. W = (W A) (W A) (A W ) and (W A) (W A) is the computably enumerable set W \A. An extendible subalgebra B supports L if for all W L there a i B such that S i supports W. (A support must come from the chosen representation of B).

13 Extensions to Automorphisms Assume that 1. L (A) and L (Â) are isomorphic via Ψ. 2. B and B are extendible algebras which are extendibly 0 3 isomorphic via Θ. 3. B supports L (A). 4. B supports L (Â). 5. Ψ and Θ preserves supports. Then there is an automorphism Λ of such that Λ(A) = Â, Λ L (A) = Ψ and Λ (A) is 0 3. Computable to Computable A map Ξ from a substructure of G (A) to Ĝ (Â) preserves the computable subsets if R R(A) G iff Ξ(R) R(Â) Ĝ. C(A) is the set of W e such that either A W e or W e A. Theorem (Soare s Automorphism Theorem 74) Let A and  be two noncomputable computably enumerable sets. 1. Then is a 0 3 isomorphism Λ between (A) C(A) and (Â) C(Â). Furthermore a 0 3-index for Λ can be found uniformly from indexes for A and Â. 2. In addition, Λ preserves the computable sets. Automorphisms to Automorphisms Theorem (The Restriction Theorem) If A and  are automorphic via Ψ then the structures S R (A) and S R (Â) are 0 3-isomorphic structures via an isomorphism Γ induced by Ψ. Assume A and  are automorphic via Ψ. Let B be an extendible algebra (of S R (A)). Then there are extendible B (of S R (Â)) and Θ such that 1. B and B are extendibly 0 3-isomorphic via Θ. 2. If i B and S i supports W then Θ(S i ) supports Ψ(W ). Theorem (The Conversion Theorem, Cholak, Harrington) If A and  are automorphic via Ψ then they are automorphic via Λ where Λ L (A) = Ψ and Λ (A) is 0 3. (Completeness) Main s Which c.e. sets are automorphic to complete sets? Explore the relationship between orbits and Turing degree and Turing degree (jump) classes. By Harrington and Soare we know this is related to dynamic properties. (Cone Avoidance) Given an incomplete c.e. degree d and an incomplete c.e. set A, is there a  automorphic to A such that d T Â?

14 Prompt Sets A is promptly simple iff there is computable function p such that for all e, if W e is infinite then there is a stage s with x (W e,at s A p(s) ). A c.e. set A is prompt iff there is computable function p such that for all e, if W e is infinite then there is a stage s with x W e,at s and A s x A p(s) x. Theorem (Maass, Shore, Stob) There is a definable property P(A) which implies A is prompt and furthermore for all prompt degree, d, there is set A such that P(A) and A d. Theorem (Cholak, Downey, Stob) All prompt simple sets are automorphic to a complete set. Tardy Sets D is 2-tardy iff for every computable nondecreasing function p(s) there is an e such that X 2 e = D and ( x)( s)[if x X2 e,s then x D p(s) ]. Theorem (Harrington, Soare) There are realizable definable properties Q(D) and P(D, C) such that Q(D) implies that D is 2-tardy (so not Turing complete), if there is a C such that P(D, C) and D is 2-tardy then Q(D) (and D is high), Almost Prompt Sets X = (W e1 W e2 ) (W e1 W e2 )... (W e2n 1 W e2n ) iff X is 2n-c.e. and X is 2n + 1-c.e. iff X = Y W e, where Y is 2n-c.e. Let Xe n be the eth n-c.e. set. A is almost prompt iff there is a computable nondecreasing function p(s) such that for all e and n if Xe n = A then ( x)( s)[x Xn e,s and x A p(s)]. Prompt implies almost prompt. So every Turing complete set is almost prompt. Theorem (Harrington, Soare) All almost prompt sets are automorphic to a complete set. s about Tardiness How do the following sets of degrees compare: the high n hemimaximal degrees, the tardy degrees, for each n, {d : there is a n-tardy D such that d T D}, {d : there is a 2-tardy D such that Q(D) and d T D}, {d : there is a A d which is not automorphic to a complete set}. Theorem (Harrington, Soare) There is a maximal 2-tardy set. Is there a nonhigh 2-tardy set which is automorphic to a complete set?

15 Main Open s, Again On the Complexity of the Orbits Look at the index set of all  in the orbit of A with the hopes of finding some answers. The index set of such  is in Σ 1 1. (Completeness) Which c.e. sets are automorphic to complete sets? (Cone Avoidance) Given an incomplete c.e. degree d and an incomplete c.e. set A, is there a  automorphic to A such that d T Â? Are these arithmetical questions? If A is hhsimple then {e : W e is automorphic to A} is Σ 0 5. If A is simple then {e : W e is automorphic to A} is Σ 0 8. Is every low 2 simple set automorphic to a complete set? At least this is an arithmetic question. (Kjos-Hanssen) Is there a simple set such that the index set of its orbit is Σ 0 8-complete? Similarly for hhsimple. d-simple Sets (Lerman, Soare) A coinfinite set A is d-simple if for all X there is a Y X such that 1. X A = Y A and 2. ( Z)[Z X infinite (Z Y ) A ]. (Lerman, Soare) If A and B are d-simple and low is A automorphic to B? Are all prompt (low 2 ) d-simple sets automorphic? (Harrington, Soare) Is every d-simple (low 2 ) set automorphic to a complete set? Are these arithmetic questions? Invariant Classes A class D of degrees is invariant if there is a class S of (c.e.) sets such that 1. d D implies there is a W in S and d. 2. W S implies deg(w ) D and 3. S is closed under automorphic images (but need not be one orbit). The high degrees are invariant by a single orbit. Theorem (Maass) The prompt degrees are invariant by a single orbit.

16 Restrictions on Invariant Classes Theorem (Cholak 95) All incomplete c.e. sets are automorphic to a high set. All invariant jump degree classes must be upward closed. Theorem (Downey, Harrington) There is a property S(A), a prompt low degree d 1, a prompt high 2 degree d 2 greater than d 1, and tardy high 2 degree e such that for all E T e, S(E) and if d 1 T D T d 2 then S(D). So no single orbit can show all noncomputable degrees are invariant. There is no fat orbit. Orbits Containing Prompt or Tardy High Sets Let A be incomplete. If the orbit of A contains a set of high prompt degree must the orbit of A contain a set from all high prompt degrees? If the orbit of A contains a set of high tardy degree must the orbit of A contain a set from all high tardy degrees? A positive answer to both questions would answer the cone avoidance question. But not the completeness question. Are the 2-tardy (high) degrees invariant? via Q(A)? Is the class of degrees containing sets not automorphic to a complete set invariant? How Fat is Possible? More Invariant Degree Classes Is there a tardy (2-tardy) (L n, H n ) set A whose orbit contains a set of every prompt (L n, H n ) degree? For every degree a there is a set A a whose orbit contains every high degree. Theorem (Shoenfield 76) Atomless sets are nonlow 2 and every nonlow 2 degree contains an atomless set. The nonlow 2 degrees are invariant. Conjecture (Harrington, Soare) The nonlow degrees are not invariant. There is properly low 2 degree d such that if A d then A is automorphic to a low set. There is a low 2 set which is not automorphic to a low set. Theorem (Cholak, Harrington 02) For n 2, nonlow n and high n degrees are invariant.

17 Coding the Double Jump into Same Approach Fails for Single Jumps Theorem (Cholak, Harrington 02) Let C = {a : a is the Turing degree of a Σ 3 set greater than 0 }. Let D C such that D is upward closed. Then there is a non-elementary (L ω1,ω) L(A) property ϕ D (A) such that D D iff there is an A where A D and ϕ D (A). If a > b then there is a A a such that for all B b, A is not automorphic to B (in fact, L (A) L (B)). Theorem (Cholak) For all A and for all high degrees h there is a  h such that L (A) L (Â). Conjecture (Harrington) For all A and degrees d if A T d is there a  d such that L (A) L (Â). s on Coding Single Jump and Degrees (Can single jumps be coded into?) Let J be C.E.A. in 0 but not of degree 0. Is there a degree a such that a T J and, for all A a, there is an  with A automorphic to  and  < T a or  T a? (Can a single Turing degree be coded into?) Is there a incomplete degree d and an incomplete set A such that, for all  automorphic to A, d Â? A d?