ferent set. This contrasts strongly with indifferent sets for Martin-L ¨of random-ness which must be complete [34]. The fact that indifferent sets for 1-generic sets can be computationally weak raises the possibility that a 1-generic set might be able to compute its own indifferent set. We investigate this ques-tion in Secques-tion 10.6 for ∆02 1-generic sets. We establish that some but not all
∆02 generic sets have this property. We consider which c.e. sets bound a 1-generic set with this property. We show that any c.e. set which is not of totally ωω-c.a. degree bounds such a 1-generic set. On the other hand no c.e. set of totally ω-c.a. degree bounds such a 1-generic set. These results are presented in Theorems 10.6.8 and 10.6.16 respectively.
In Section 10.7, we consider similar questions for weakly 1-generic sets.
These results offer interesting contrasts with those for 1-generic sets. First in Theorem 10.7.1 we show that any hyperimmune set I computes a weakly 1-generic set G that I is an indifferent set for. This tells us that I is an indiffer-ent set for some weakly 1-generic set if and only if I is hyperimmune (Corol-lary 10.7.2). In Theorem 10.7.4 we show that if a set G is weakly 1-generic and I is a set whose principal function escapes domination by any G-computable function then I is an indifferent set for G. Further, we show in Theorem 10.7.5, just as is the case for 1-generic sets, that if A ∈ GL2 then A computes an in-different set for any weakly 1-generic set it bounds. A difference to the case of 1-generic sets is provided in Theorem 10.7.6. In this theorem we show that any ∆02weakly 1-generic set computes a set it is indifferent to.
We conclude in Section 10.8 with some open questions.
10.2 Background and notation
Given A and X ⊆ ω, we will write both A[X]and A∆X to denote the symmetric difference of A and X, i.e. the set (A \ X) ∪ (X \ A). This is the set which differs from A at precisely the elements of X. If σ, τ ∈ 2<ω, then by σ∆τ we mean σ0ω∆τ 0ω. Our central definition is the following.
Definition 10.2.1. Let A ⊆ 2ωand I ⊆ ω.
(i). Take A ∈ A. If for all X ⊆ I we have that A[X] ∈ A then we call I an indifferent set for A with respect to A.
(ii). We call I a universal indifferent set for A if I is an indifferent set for all A ∈ Awith respect to A.
We will be interested in the case in which A is comeager, i.e. contains the intersection of countably many open dense subsets. We denote the eth c.e. set
of finite strings by Se.
Let A ⊆ ω and S ⊆ 2<ω. We will say that A meets S if (∃σ ≺ A)(σ ∈ S), and that A avoids S if (∃σ ≺ A)(∀τ ∈ S) (σ 6 τ ). We will also say that a string σ meets S if for some τ σ, τ ∈ S, and σ avoids S if no string comparable with σ is in S. We call a set of strings S dense if for all σ ∈ 2<ω there exists τ ∈ S such that τ σ.
Definition 10.2.2. If G ⊆ ω meets or avoids all sets of finite strings computably enumerable in ∅n−1, then G is n-generic. If G meets all dense sets computably enumerable in ∅n−1, then G is weakly n-generic.
For an introduction to n-generic sets see survey papers by Jockusch and Kumabe [42, 49] and Kumabe’s thesis [48]. Weakly n-generic sets were intro-duced by Kurtz [51]. In this chapter we will focus our study on 1-generic and weakly 1-generic sets.
A function f is ω-c.a. if f (x) = limsg(x, s)for some computable g and there is some computable h such that for all x,
|{s : g(x, s + 1) 6= g(x, s)}| ≤ h(x).
A Turing degree is a is array non-computable, or ANC if for any ω-c.a. function g, there is a function f ≤T asuch that f escapes domination by g. This class of degrees was introduced by Downey, Jockusch and Stob [32, 33]. A degree a is in GLn if an= (a ∨ ∅0)n−1. We write GLn for the complement of GLn. A well-known fact is that the degree of any 1-generic set is GL1.
The use of GL2 degrees and ANC degrees to perform Cohen forcing con-structions was noted by Jockusch and Posner [43], and by Downey, Jockusch and Stob [33] respectively. Forcing using GL2 degrees makes use of the fol-lowing characterisation of Martin: a ∈ GL2 if and only if for any function g ≤T a ∨ ∅0 there is a function f ≤T a such that f escapes domination by g [62]. The following theorem of Cai and Shore extends these ideas and helps us understand the computational power required to undertake the forcing con-structions used in this chapter [10]. An A-computable notion of forcing P, is a set P of forcing conditions with a partial order ≤Pon P which contains a greatest element 1, such that P is computable in A. Let C be a sequence of dense sub-sets of P. A sequence hpii is C-generic if it meets each element of C and for all i, pi ≥P pi+1.
Theorem 10.2.3(Cai, Shore). Suppose that P is an A-computable notion of forcing, that C = hDni is a sequence of sets dense in P, and that there is a function d(x, y) = Φ(A⊕∅0; x, y)witnessing their density, i.e. ∀p ∈ P ∀n(d(p, n) ≤P p∧d(p, n) ∈ Dn).
10.2. BACKGROUND AND NOTATION 171
(i). If A ∈ GL2 then there is a C-generic sequence computable in A.
(ii). If A ∈ ANC and the use from ∅0in the computation of Φ(A⊕∅0; x, y)is bounded by a function computable in A, then there is also a C-generic sequence com-putable in A.
A function computable in ∅0 with use bounded by a computable func-tion is called wtt-reducible to ∅0. A c.e. degree a is of totally ω-c.a. degree if for all f ≤T a, f is ω-c.a. The class of totally ω-c.a. sets was introduced by Downey, Greenberg and Weber [27] and also studied by Barmpalias, Downey and Greenberg [5]. A forthcoming monograph of Downey and Greenberg gen-eralises this concept [26]. The terminology below follows that monograph.
Let R = (R, ≤R) be a computable well-ordering of a computable set R.
An R-computable approximation of a function f is a computable approximation hfsis<ω of f , equipped with a uniformly computable sequence hosis<ωof func-tions from ω to R such that for all x and s:
• os+1(x) ≤Ros(x).
• If fs+1(x) 6= fs(x), then os+1(x) <R os(x).
The sequence hosis<ω, together with the well-foundedness of R, witnesses the fact that the approximation hfsis<ωindeed reaches a limit.
Definition 10.2.4. A function f : ω → ω is computably approximable (or R-c.a.) if it has an R-computable approximation.
It is possible to use this definition to establish a hierarchy in the Turing degrees by restricting ourselves to certain well-orderings. Every ordinal α has a unique expression as the sum
ωα1n1+ ωα2n2 + · · · + ωαknk
where ni < ω are nonzero and α1 > α2 > · · · > αkare ordinals. This is called the Cantor normal form of α. Further,
ε0 = supn
ω, ωω, ωωω, ωωωω, . . .o
is the least ordinal γ such that ωγ = γ, so for all α < ε0, every ordinal appearing in the Cantor normal form of α is strictly smaller than α.
Let R = (R, <R)be a computable well-ordering, and let | · | : R → otp(R) be the unique isomorphism between R and its order-type. The pullback to R of the Cantor normal form function is the function nfR whose domain is R and is defined by letting
nfR(z) = h(z1, n1), (z2, n2), . . . , (zk, nk)i
where ni < ωare nonzero, zi ∈ R, z1 >R z2 >R· · · >Rzk, and
|z| = ω|z1|n1+ ω|z2|n2+ · · · + ω|zk|nk.
Definition 10.2.5. A computable well-ordering R is canonical if its associated Cantor normal form function nfRis also computable.
Downey and Greenberg have established that for every ordinal α ≤ ε0
there is a canonical well-ordering of order-type α and further that any two canonical well-orderings of order-type α are computably isomorphic. Hence we can define a function f as being α-c.a. if it is R-c.a. for some canonical well-ordering of order-type α.
Definition 10.2.6. If α ≤ ε0, then a Turing degree a is totally α-c.a. if every function f ∈ a is α-c.a.
It is not difficult to show that a is totally α-c.a. if and only if every function f ≤T ais α-c.a. The following theorem establishes that the α-c.a. degrees do indeed form a hierarchy.
Theorem 10.2.7(Downey, Greenberg [26]). Let α ≤ ε0. There is a totally α-c.a.
degree which is not totally γ-c.a. for any γ < α if and only if α is a power of ω. If α is a power of ω then in fact there is a c.e. degree which is totally α-c.a. but not totally γ-c.a. for any γ < α.