Corollary 10.3.5. If A, B ∈ 2ω with A ≤T B and A00 ≤T B0 then B computes a universal indifferent set for the class of open dense sets enumerable in A.
Proof. The restriction of A to ∅nfor some n is not necessary.
Theorem 10.3.3 tells us that the class of 1-generic sets contains a comea-ger subset with a universal indifferent set. Theorem 10.3.4 provides a specific example, namely the class of all weakly 2-generic sets. However, is there a uni-versal indifferent set for class of 1-generic sets itself? Miller, in unpublished work, has proved that there is not.
Theorem 10.3.6(Miller). Let I ⊆ ω be infinite. There are sets G, A ⊆ ω such that Gis 1-generic, A is not 1-generic, and G∆A ⊆ I. In other words, I is not universally indifferent for the class of 1-generic sets.
Proof. We define the characteristic function of G in stages. At stage 0 we pick some element i0 ∈ I greater than 0. We define σ0 = 0n10∅0(0)where n is chosen so that |0n1| = i0 i.e. i0 is the location of the 0 in σ0 before the coding location of ∅0(0).
At stage s + 1, we define σs+1extending σsas follows. If σsavoids Ss, then define τs= λ. If σsdoes not avoid Ss then let τs be the first string enumerated into Ss that extends σs. We take is+1 ∈ I such that is+1 > |τs| and we now define σs+1 = τs0n10∅0(s + 1)where n is chosen so that |τs0n1| = is+1. Let G = S
nσn. The construction ensures that the set G is 1-generic. Define X = {in: σn
avoids Sn}. Define A = G[X]. From A and σn, we can determine whether or not σnmeets or avoids Sn. Thus we can determine σn+1 and so we can determine
∅0(n+1). This shows that A ≥T ∅0and consequently A cannot be 1-generic.
10.4 Indifference and 1-genericity
We know that there is no universal indifferent set for the class of all 1-generic sets. However we will establish that all 1-generic sets have an indifferent set with respect to the class of all generic sets. It is known that there exists 1-generic sets with indifferent sets.
Theorem 10.4.1 (Jockusch and Posner [43]). If A ∈ GL2, then there exists G, I Turing below A such that G is 1-generic and I is an indifferent set for G.
The terminology used by Jockusch and Posner is different. They con-structed a function f : ω → {0, 1, 2} such that f is 1-generic and any char-acteristic function obtained from f by replacing 2’s with 0’s and 1’s is also 1-generic. Given such an f , let G = {x : f (x) = 1} and I = {x : f (x) = 2}.
Clearly I is an indifferent set for G. This result can be strengthened by the work of Cai and Shore.
Theorem 10.4.2 (Jockusch and Posner; Cai and Shore [10, 43] ). If A ∈ ANC then there exists G, I Turing below A such that G is 1-generic and I is an indifferent set for G.
We will strengthen Jockusch and Posner’s result in a different direction.
Theorem 10.4.5 establishes that any X ∈ GL2 computes an indifferent set for any 1-generic set it bounds. We make use of the following well-known lemma.
Lemma 10.4.3. If G is a 1-generic set and X ⊆ ω is finite, then G[X] is 1-generic.
Given a 1-generic set G and e, n ∈ ω we can find some point m such that no matter how we change G on the first n bits, the resulting set meets or avoids Seafter m bits.
Lemma 10.4.4. Let G be a 1-generic set. There is a function fG : ω2 → ω with fG ≤T G ⊕ ∅0 such that for all n, e ∈ ω, for all X ⊆ {0, 1, . . . , n − 1}, we have that G[X] fG(n, e)meets or avoids Se.
Proof. Take any n and e and let X1, X2, . . . , X2n be a list of all subsets of the set {0, 1, . . . , n − 1}. For all i, 1 ≤ i ≤ 2n, we have that G[Xi] is 1-generic as Xi is finite. Hence there is some mi such that G[Xi] mi meets or avoids Se. This mi is computable in G ⊕ ∅0 because we can query whether successive initial segments of G[Xi] meet or avoid Seuntil we find one that does. Define fG(n, e) = max{mi : 1 ≤ i ≤ 2n}.
We are now ready to give our basic existence result for indifferent sets for 1-generic sets.
Theorem 10.4.5. Given A ≥T G where A ∈ GL2 and G is a 1-generic set, there exists I ≤T Asuch that I is an indifferent set for G with respect to 1-genericity and I is 1-generic.
Proof. Let G be 1-generic. We can make I both 1-generic and an indifferent set for G by satisfying the following requirements for all e ∈ ω.
Qe: I meets or avoids Se.
Re: For all X ⊆ I we have that G[X]meets or avoids Se.
These requirements can be met by constructing an I that meets the follow-ing dense sets:
Ce = {σ ∈ 2<ω : σmeets or avoids Se}, and
10.4. INDIFFERENCE AND 1-GENERICITY 177
De= {σ ∈ 2<ω : ∀X ⊆ σ, G[X] |σ| meets or avoids Se}.
We say X ⊆ σ if X ⊆ {i : i < |σ| ∧ σ(i) = 1}.
In this proof the notion of forcing that we use is just Cohen forcing so P = 2<ω and σ ≤P τ if σ τ . There is a function computable in ∅0 that uni-formly witnesses the density of the sequence of sets hCni: let c(σ, e) be the first extension of σ to enter Se if such an extension exists, or σ otherwise. There is also a function computable in G ⊕ ∅0that witnesses the density of the sequence of sets hDni. We define d(σ, e) = σ0fG(|σ|,e)−|σ|.
If I d(σ, e) and X ⊆ I, then G[X] |d(σ, e)| can only differ from G on the first |σ| bits. As fG(|σ|, e) = |d(σ, e)|by Lemma 10.4.4, G[X] |d(σ, e)| meets or avoids Se.
By applying Theorem 10.2.3, there is an A-computable ≤P decreasing se-quence hσii that meets all sets in the sequences hCei and hDei. Hence taking I = limiσi, we have that I is 1-generic and I is an indifferent set for G.
Corollary 10.4.6. If {Gi}i∈ωis a countable family of 1-generic sets and A ≥T ⊕i∈ωGi with A ∈ GL2, then there is an I ≤T Asuch that for all i ∈ ω, I is an indifferent set for Gi.
Proof. We simply replace the sequence of sets hDei with
De,i= {σ ∈ 2<ω : ∀X ⊆ σ, Gi,[X] |σ| meets or avoids Se}.
Now because the sequence {Gi}i∈ω is uniformly computable in A there is a function computable in A ⊕ ∅0that witnesses the density of this sequence.
One possible use of indifferent sets for 1-generic sets is as coding locations.
We can take a 1-generic G, and then form another 1-generic ˆGby changing G on some bits of an indifferent set. We can recover these changes from the join of G and ˆG. The following theorem is an application of this idea.
Corollary 10.4.7. Given A ≥T G where A ∈ GL2 and G is a 1-generic set, there exists a 1-generic ˆGsuch that G ⊕ ˆG ≡T A.
Proof. G has an A-computable indifferent set I that is also 1-generic. As I is 1-generic there are infinitely many even numbers in I and infinitely many odd numbers in I. Define the following A computable subset of I. If 0 ∈ A, then let x0be the first even element of I, otherwise let x0 be the first odd element of I. We inductively define xi+1to be the first even element of I greater than xi
if i + 1 ∈ A and the first odd element of I greater than xi otherwise. Let X = {xi : i ∈ ω}. As G, X ≤T Awe have that G[X] ≤T A. Further, A ≤T G ⊕ G[X]
because x ∈ A if and only if the ith position where G and G[X] differ is even.
Thus A ≡T G ⊕ G[X].
In our final result for this section, we will show that any ∆02 1-generic set has a co-c.e. indifferent set. This was originally shown by Fitzgerald using a full approximation argument. We give an alternative proof.
Theorem 10.4.8(Fitzgerald). If G is a ∆02 1-generic set, then there exists a co-c.e.
set I such that I is an indifferent set for G.
Proof. Let G ∈ ∆02be a 1-generic set. We will show that there is a function f ≤T
∅0 such that for any set I such that pI, the principal function of I, majorizes f we have that I is an indifferent set for G. Because for any ∆02 function f , there is a co-c.e. set whose principal function majorizes f we are done.
Define f (e) as follows. Set ne,0 = 0and then ne,i+1 = fG(ne,i, e) + 1. Now set f (e) = ne,e+1. The function f is computable in ∅0 because fG ≤T G ⊕ ∅0 = ∅0. Let I be a co-c.e. set such that for all x, pI(x) ≥ f (x). Take any e. Now consider the pairwise disjoint sets [ne,i, ne,i+1− 1] for i such that 0 ≤ i < e + 1. Because ne,e+1 = f (e) ≤ pI(e), There must be some i such that [ne,i, ne,i+1− 1] ∩ I = ∅.
Now because ne,i+1− 1 = fG(ne,i), for any X ⊆ I, G[X] fG(ne,i, e)only differs from G on the first ne,i many bits. Thus G[X] meets or avoids Se and so I is a co-c.e. indifferent set for G.