Proof Given a c.e. set A, we have to split A into disjoint low c.e. sets $A_0$ and $A_1$ meeting the global requirements

$$ \begin{align*} \mathcal{R}^{global}_{i,e}: \; \text{ If } \Phi^{A_i}_e \text{ is total then } \Phi^{A_i}_e \text{ is } \omega^{\omega}\text{-c.a.} \end{align*} $$

for $e \geq 0$ and $i \leq 1$ . We split $\mathcal {R}^{global}_{i,e}$ into local requirements $\mathcal {R}_{2\langle e,x\rangle +i}$ ( $ x \geq 0$ ) where requirement $\mathcal {R}_{2\langle e,x\rangle +i}$ attempts to preserve the computation $\Phi ^{A_{i,s}}_{e,s}(x)$ (whenever this computation is defined) by restraining numbers $< \varphi ^{A_{i,s}}_{e}(x)$ from $A_i$ . I.e., if $\Phi ^{A_{i,s}}_{e,s}(x) \downarrow $ and a number $< \varphi ^{A_{i,s}}_{e}(x)$ enters A at stage $s+1$ then $\mathcal {R}_{2\langle e,x\rangle +i}$ requires that this number is put into $A_{1-i}$ . We will argue that if we give a requirement higher priority if its index is lesser, then this strategy suffices to meet the global requirements hence to make $A_0$ and $A_1$ totally $\omega ^{\omega }$ -c.a. In order to show this, we first describe the construction more formally.

W.l.o.g. assume that A is infinite, fix a 1–1 computable function a enumerating A, and let $A_{s} = \{a(t): t < s\}$ . Note that here, $A_s$ refers to the set of elements enumerated into A by stage s and is not to be confused with the sets $A_0$ and $A_1$ . The sets $A_0$ and $A_1$ are enumerated in stages where at stage $s+1$ we decide whether $a(s)$ is put into $A_0$ or $A_1$ . So $A_{i,s}= \{a(t): t < s \; \& \; a(t)\in A_i\}$ . The restraint imposed by requirement $\mathcal {R}_{2\langle e,x\rangle +i}$ on $A_i$ at stage $s+1$ is defined by

$$ \begin{align*}r(2\langle e,x\rangle +i,s)=\begin{cases} \varphi^{A_{i,s}}_{e}(x), & \text{if } \Phi^{A_{i,s}}_{e,s}(x) \downarrow, \\ 0, & \text{otherwise}. \end{cases} \end{align*} $$

Then, at stage $s+1$ , $a(s)$ is put into $A_1$ if the least n such that $a(s) < r(n,s)$ is even (or if no such n exists), and $a(s)$ is put into $A_0$ otherwise. Moreover, we call $\mathcal {R}_{2\langle e,x\rangle +i}$ an i-requirement and an i-e-requirement, and we say that requirement $\mathcal {R}_{2\langle e,x\rangle +i}$ is injured at stage $s+1$ if $a(s) < r(2\langle e,x\rangle +i,s)$ and $a(s)$ is enumerated into $A_i$ . (Note that i-requirements impose restraint on $A_i$ .)

This completes the construction.

Obviously, the sets $A_0$ and $A_1$ are disjoint and c.e. and $A = A_0 \cup A_1$ . So $A = A_0\sqcup A_1$ . Moreover, just as in the standard proof of Sacks’ Splitting Theorem, it follows by a straightforward induction on $n \geq 0$ that any requirement $\mathcal {R}_{n}$ is injured at most finitely often. So we may fix $s_n$ minimal such that $\mathcal {R}_{n}$ is not injured after stage $s_n$ . Then, for $n = 2\langle e,x\rangle +i$ , any computation $\Phi ^{A_i,t}_{e,t}(x)$ existing at a stage $t \geq s_n$ is preserved, hence

(1) $$ \begin{align} \Phi^{A_i}_e(x) \uparrow \; \Leftrightarrow \; \forall \; t \geq s_{2\langle e,x\rangle+i} \; (\Phi^{A_i,t}_{e,t}(x) \uparrow). \end{align} $$

So, in particular, the requirements $\mathcal {R}_{2\langle e,e\rangle +i}$ ensure that the standard lowness requirements

$$ \begin{align*}\mathcal{Q}_{2e+i}: \; \exists^{\infty} s (\Phi^{A_i,s}_{e,s}(e) \downarrow) \; \Rightarrow \Phi^{A_i}_e(e) \downarrow\end{align*} $$

are satisfied ( $e \geq 0, i \leq 1$ ). Hence the sets $A_0$ and $A_1$ are low.

It remains to show that the global $\omega ^{\omega }$ -c.a. requirements $\mathcal {R}^{global}_{i,e}$ are met. Fix $i \leq 1$ and $e \geq 0$ such that $\Phi ^{A_i}_e$ is total. Define the canonical computable approximation $\psi $ of $\Phi ^{A_i}_e$ induced by $\{ \Phi ^{A_{i,s}}_{e,s}\}_{s \geq 0}$ by letting

$$ \begin{align*} \psi(x,s) = \Phi^{A_{i,s'}}_{e,s'}(x) \; \text{ for the least } s' \geq s \text{ such that } \Phi^{A_{i,s'}}_{e,s'}(x) \downarrow. \end{align*} $$

Then it suffices to define a computable function $c: \omega \times \omega \to \omega ^{\omega }$ such that, for $x, s \geq 0$ ,

(2) $$ \begin{align} c(x,s+1) \leq c(x,s) \end{align} $$

and

(3) $$ \begin{align} \psi(x,s+1) \not= \psi(x,s) \; \Rightarrow \; c(x,s+1) \not= c(x,s). \end{align} $$

The definition of c is based on the following observations where we let $n_x = 2\langle e, x \rangle +i$ .

Claim 1. (a) If $\psi (x,s+1) \not = \psi (x,s)$ then $\mathcal {R}_{n_x}$ is injured at stage $s+1$ .

(b) If a requirement $\mathcal {R}_n$ is injured at stage $s+1$ then there is a number $n' < n$ such that $r(n',s+1)=r(n',s)$ and $a(s) < r(n',s)$ , hence

(4) $$ \begin{align} |\overline{A_{s+1}} \upharpoonright r(n',s+1)| < |\overline{A_{s}} \upharpoonright r(n',s)|. \end{align} $$

(c) If $r(n,s+1) \not = r(n,s)$ then $r(n,s)=0$ or $\mathcal {R}_n$ is injured at stage $s+1$ .

Proof (a) Assume $\psi (x,s+1) \not = \psi (x,s)$ . Then, by definition of $\psi $ , $\psi (x,s) = \Phi ^{A_{i,s}}_{e,s}(x) \downarrow $ and either $\Phi ^{A_{i,s+1}}_{e,s+1}(x) \uparrow $ or $\psi (x,s+1) = \Phi ^{A_{i,s+1}}_{e,s+1}(x) \downarrow $ . So, in either case, $\Phi ^{A_{i,s}}_{e,s}(x) \downarrow \not = \Phi ^{A_{i,s+1}}_{e,s+1}(x)$ . This implies that $r(n_x,s) = \varphi ^{A_{i,s}}_{e}(x)$ and $A_{i,s+1} \upharpoonright \varphi ^{A_{i,s}}_{e}(x) \not = A_{i,s} \upharpoonright \varphi ^{A_{i,s}}_{e}(x)$ . So $\mathcal {R}_{n_x}$ is injured at stage $s+1$ .

(b) Assume that $\mathcal {R}_{n}$ is injured at stage $s+1$ . Fix $i'$ such that $\mathcal {R}_{n}$ is an $i'$ -requirement. Then, by construction, $a(s)$ is put into $A_{i'}$ and there is a $(1-i')$ -requirement $\mathcal {R}_{n'}$ such that $n' < n$ and $a(s) < r(n',s)$ . Finally, by $A_{1-i',s+1} = A_{1-i',s}$ and $r(n',s)> 0$ , it holds that $r(n',s) = \varphi ^{A_{1-i',s}}_{e'}(x') = \varphi ^{A_{1-i',s+1}}_{e'}(x') = r(n',s+1)$ for the unique numbers $e',x' \geq 0$ such that $n'=2 \langle e',x' \rangle +(1-i')$ .

(c) Assume that $r(n,s+1) \not = r(n,s)$ and $r(n,s)> 0$ , and fix $e',x' \geq 0$ and $i' \leq 1$ such that $n=2 \langle e',x' \rangle +i'$ . Then, by $r(n,s)> 0$ , $\Phi ^{A_{i',s}}_{e',s}(x') \downarrow $ and $r(n,s) = \varphi ^{A_{i',s}}_{e'}(x')$ . By $r(n,s+1) \not = r(n,s)$ , this implies that either $\Phi ^{A_{i',s+1}}_{e',s+1}(x')$ is undefined or $\Phi ^{A_{i',s+1}}_{e',s+1}(x')$ is defined but $\varphi ^{A_{i',s+1}}_{e',s+1}(x') \downarrow \not = \varphi ^{A_{i',s}}_{e',s}(x')$ . It follows that $a(s) < \varphi ^{A_{i',s}}_{e'}(x') = r(n,s)$ and $a(s)$ is put into $A_{i'}$ at stage $s+1$ . So $\mathcal {R}_{n}$ is injured at stage $s+1$ .

This completes the proof of Claim 1.

Now, for the definition of the computable function c, we represent the ordinals $< \omega ^{\omega }$ by nonempty finite tuples of nonnegative integers where the $(k+1)$ -tuple $(a_k, \dots , a_0)$ represents the ordinal

$$ \begin{align*}\sum_{i=0}^k a_i \omega^i = a_k \omega^k + \dots + a_2 \omega^2 + a_1 \omega + a_0.\end{align*} $$

Then $c(x,s)$ is defined by

$$ \begin{align*}\begin{array}{lll} c(x,s) & = & (c_0(0,s), c_1(0,s), c_0(1,s), c_1(1,s), \dots, c_0(n_x-1,s), c_1(n_x-1,s)) \\ & = & \sum_{n < n_x} \bigg(c_0(n,s) \cdot \omega^{2(n_x-n)-1} +c_1(n,s) \cdot \omega^{2(n_x-n)-2}\bigg), \end{array} \end{align*} $$

where

$$ \begin{align*}c_0(n,s)=\begin{cases} 0, & \text{if } r(n,s)>0, \\ 1, & \text{otherwise}, \end{cases} \text{ } \; \; \text{ and } \; \; \text{ } c_1(n,s) = |\overline{A_s} \upharpoonright r(n,s)|. \end{align*} $$

(Here and in the following we assume that $n_x> 0$ . If $n_x=0$ then, by Claim 1(a), $\psi (x,s+1) = \psi (x,s)$ for all stages $s \geq 0$ . So (2) and (3) will hold if we let $c(x,s) =0$ for all $s \geq 0$ .)

Obviously, $c(x,s)$ is computable. So it only remains to establish (2) and (3).

For a proof of (2) fix x and s such that $c(x,s+1) \not = c(x,s)$ . Choose $2n+i'$ minimal such that $i' \leq 1$ , $n < n_x$ , and $c_{i'}(n,s+1) \not = c_{i'}(n,s)$ . It suffices to show that $c_{i'}(n,s+1) < c_{i'}(n,s)$ . Note that requirement $\mathcal {R}_n$ is not injured at stage $s+1$ (namely, otherwise, it would follow by Claim 1(b) that there is a number $n' < n$ such that $c_1(n',s+1) < c_1(n',s)$ contradicting minimality of $2n+i'$ ). Now distinguish the following two cases. First assume that $r(n,s+1)=r(n,s)$ . Then $c_0(n,s+1)=c_0(n,s)$ , hence $i'=1$ . By assumption this implies that

$$ \begin{align*}|\overline{A_{s+1}} \upharpoonright r(n,s)| = c_{i'}(n,s+1) \not= c_{i'}(n,s) = |\overline{A_{s}} \upharpoonright r(n,s)|.\end{align*} $$

As $\overline {A_{s+1}} \subseteq \overline {A_s}$ it follows that $c_{i'}(n,s+1) < c_{i'}(n,s)$ . Finally, assume that $r(n,s+1)\not =r(n,s)$ . Since $\mathcal {R}_n$ is not injured at stage $s+1$ , it follows by Claim 1(c) that $r(n,s)=0$ . So, by case assumption, $r(n,s+1)> 0$ , and we may conclude that $i'=0$ and $c_{0}(n,s+1)=0 < 1 = c_{0}(n,s)$ . This completes the proof of (2).

Finally, for a proof of (3), fix x and s such that $\psi (x,s+1) \not = \psi (x,s)$ . Then, by Claim 1(a) and (b), there is a number $n' < n_x$ such that (4) holds whence $c_1(n',s+1) < c_1(n',s)$ . By definition of c, this implies $c(x,s+1) \not = c(x,s)$ . So (3) holds.

This completes the proof of Theorem 2.1.

Proof We note that the proof does not make $A_0$ and $A_1$ Turing incomparable. If A is totally $\omega ^2$ -c.a. then we can use the usual Sacks’ splitting theorem to obtain Turing incomparable $A_0$ and $A_1$ . The proof is an infinite injury, although we will not be using a tree to organize the construction. Rather, we will be using a mechanism similar to e-states used in the maximal set construction.

Proof In order to avoid technicalities, we give the proof for $\alpha = \omega ^2$ . The general case is obtained by a similar argument using the fact that any ordinal $\alpha < \varepsilon _0$ has an effective Cantor normal form (see Reference).

Call a triple $(f,k,p)$ of total computable functions of type $\omega \times \omega \to \omega $ a computable $\omega ^2$ -approximation (of g) if (f converges to g and), for any numbers x and s, the following hold.

1. (i) If $f(x,s+1) \not = f(x,s)$ then $(k(x,s+1),p(x,s+1)) \not = (k(x,s),p(x,s))$ , and

2. (ii) if $(k(x,s+1),p(x,s+1)) \not = (k(x,s),p(x,s))$ then either $k(x,s+1) < k(x,s)$ or $k(x,s+1) = k(x,s)$ and $p(x,s+1) < p(x,s)$ .

By viewing $k(x,s)$ and $p(x,s)$ as the coefficients of the Cantor normal form of the ordinal $o(x,s) = k(x,s) \cdot \omega + p(x,s) < \omega ^2$ , the functions k and p assign an ordinal $o(x,s) < \omega ^2$ to each value $f(x,s)$ such that whenever $f(x,s)$ changes, i.e., $f(x,s+1) \not = f(x,s)$ , then the corresponding ordinal decreases, i.e., $o(x,s+1) < o(x,s)$ . So f is a convergent approximation and, by definition, a function g is $\omega ^2$ -c.a. if and only if there is a computable $\omega ^2$ -approximation $(f,k,p)$ of g. Moreover, given a computable $\omega ^2$ -approximation $(f,k,p)$ of a function g, by slowing down the approximation we obtain a primitive recursive $\omega ^2$ -approximation of g, i.e., a computable $\omega ^2$ -approximation $(\hat {f},\hat {k},\hat {p})$ of g where the functions $\hat {f},\hat {k},\hat {p}$ are primitive recursive (see Reference). Finally, since there is a computable numbering of the primitive recursive functions, we easily obtain a computable sequence $\{({f}_i, {k}_i, {p}_i)\}_{i \geq 0}$ of all primitive recursive $\omega ^2$ -approximations. So, by dropping the parameters ${k}_i$ and ${p}_i$ , this gives the desired computable sequence $\{f_i\}_{i \geq 0}$ of computable convergent approximations providing approximations of all total $\omega ^2$ -c.a. functions.

Proof of Theorem 4.1

We construct the desired c.e. sets A and C by a tree argument. It suffices to meet the noncomputability requirements

$$ \begin{align*}P_n:\overline{C}\ne W_n\end{align*} $$

and the (global) splitting requirements

$$ \begin{align*}R^{global}_{e}: \text{ If } X_e \cup Y_e=A \text{ and } X_e \text{ is totally } \alpha\text{-c.a. then } C \le_T Y_e \end{align*} $$

for $n \geq 0$ and $e \geq 0$ , respectively, where $\{(X_e,Y_e)\}_{e \geq 0}$ is a computable numbering of all disjoint pairs of c.e. sets.

In order to meet the global requirement $R^{global}_{e}$ we define a total function $g_e \leq _T X_e$ and ensure that $C \le _T Y_e$ provided that $X_e \cup Y_e=A$ and $g_e$ is $\alpha $ -c.a. In fact, since this task is too complex to be handled directly, we break it up into the local splitting requirements

$$ \begin{align*} R_{\langle e, i \rangle}: \text{ If } X_e \cup Y_e=A \text{ and if } f_i \text{ converges to } g_e \text{ then } C \le_T Y_e \end{align*} $$

( $i \geq 0$ ) where $\{f_i\}_{i \geq 0}$ is a computable sequence of computable convergent approximations as in Lemma 4.2.

Here the reduction $C\leq _T Y_e$ will be ensured via delayed simple permitting. During the construction we will ensure that for all but finitely many y, if $Y_e\upharpoonright y$ is stable at s then $C\upharpoonright y$ is stable at $\theta (s)$ for a computable function $\theta $ . This delay is due to the working of the priority tree and also the need to synchronize between different R requirements. This will be discussed below in detail.

Let $A_s$ and $C_s$ be the finite parts of A and C, respectively, enumerated by the end of stage s (where $A_0=C_0=\emptyset $ ), and fix uniformly computable enumerations $\{X_{e,s}\}_{s \geq 0}$ and $\{Y_{e,s}\}_{s \geq 0}$ of the c.e. sets $X_e$ and $Y_e$ , respectively ( $e \geq 0$ ). We define uniformly computable approximations $g_e(z)[s]$ of the functions $g_e$ ( $e \geq 0$ ), where $g_e(z)[s]$ is the value of $g_e(z)$ at the end of stage s, and where we obey the following rules (for $e,z,s \geq 0$ ).

1. (g₀) $g_e(z)[s] = z$ for $s \leq z$ .

2. (g₁) If $g_e(z)[s+1] \not = g_e(z)[s]$ then $X_{e,s+1} \upharpoonright g_e(z)[s]+1 \not = X_{e,s} \upharpoonright g_e(z)[s]+1$ .

3. (g₂) If $g_e(z)[s+1] \not = g_e(z)[s]$ then $g_e(z)[s+1] =s+1$ (hence $g_e(z)[s] < g_e(z)[s+1]$ .

4. (g₃) There are at most finitely many s such that $g_e(z)[s+1] \not = g_e(z)[s]$ .

So, intuitively, we may view $g_e(z)$ as (the final position of) a movable marker, where $g_e(z)[s]$ is the position of the marker at the end of stage s. The marker $g_e(z)$ is moved only finitely often ((g $_3$ )), it is not moved prior to stage $z+1$ , and its initial position is z ((g $_0$ )), and if it is moved then it is moved to a higher position, namely the current stage ((g $_2$ )), and the move is permitted by a change of $X_e$ at or below the current position ((g $_1$ )). So, obviously, the rules for defining $g_e(z)[s]$ guarantee that, $g_e(z)[s]$ is nondecreasing in s, $g_e(z)[s] \geq z$ and, for $s \geq z$ , $g_e(z)[s] \leq s$ , $g_e(z) = \lim _{s \to \infty } g_e(z)[s] < \omega $ exists for all $z \geq 0$ , and $g_e \leq _T X_e$ ( $e \geq 0$ ). (In the construction below we tacitly assume that, for $s \leq z$ , $g_e(z)[s]$ is defined according to (g $_0$ ), and that $g_e(z)[s+1] = g_e(z)[s]$ unless explicitly stated otherwise.)

We call a splitting requirement $R_{\langle e, i \rangle }$ infinitary if its premises are correct, i.e., if $X_e \cup Y_e =A$ and $\lim _{s} f_i(z,s) = g_e(z)$ for all numbers z, and we call $R_{\langle e, i \rangle }$ finitary otherwise. The priority tree of the construction is the full binary tree $T = \{0,1\}^{< \omega }$ . A node (i.e., binary string) $\alpha $ of length n codes a guess which of the first n R-requirements are infinitary, where $\alpha (m)=0$ denotes that $R_m$ is infinitary and $\alpha (m)=1$ denotes that $R_m$ is finitary ( $m <n$ ). At the end of any stage s of the construction we define a string $\delta _s$ of length s as follows, where $\delta _s$ is the guess at the type of the first s splitting requirements $R_0, \dots , R_{s-1}$ with which we work at stage $s+1$ .

First, define the length function $\ell $ by letting $\ell (\langle e, i \rangle ,s)$ be the greatest $\ell \leq s$ such that

$$ \begin{align*}\forall \; z < \ell \; (g_e(z)[s] = f_i(z,s) \; \& \; A_s \upharpoonright g_e(z)[s]+1 = (X_{e,s} \cup Y_{e,s}) \upharpoonright g_e(z)[s]+1).\end{align*} $$

Since, for any number z, $\lim _{s \to \infty } g_e(z)[s] < \omega $ and $\lim _{s \to \infty } f_i(z,s)< \omega $ exist (by construction of $g_e$ and by choice of $f_i$ , respectively), it follows that $\lim _{s \to \infty } \ell (\langle e, i \rangle ,s) = \omega $ if $R_{\langle e, i \rangle }$ is infinitary, and $\lim _{s \to \infty } \ell (\langle e, i \rangle ,s) < \omega $ otherwise. (For the construction it will be crucial that, for infinitary $R_{\langle e, i \rangle }$ and for any numbers y and z such that $y \leq g_e(z)[s]$ and $z < \ell (\langle e, i \rangle ,s)$ where $y \not \in A_s$ , it holds that $y \not \in X_{e,s} \cup Y_{e,s}$ and, by enumerating y into A at stage $s+1$ we can force y to enter either $X_e$ or $Y_e$ at a stage $\geq s+1$ .)

Next, for each node $\alpha $ , inductively define $\alpha $ -stages as follows. Each stage $s\geq 0$ is a $\lambda $ -stage. If s is an $\alpha $ -stage and if $\ell (|\alpha |,s)>\ell (|\alpha |,t)$ for all $\alpha $ -stages $t<s$ , then call s $\alpha $ -expansionary. Then an $\alpha $ -stage s is an $\alpha 0$ -stage if s is $\alpha $ -expansionary and s is an $\alpha 1$ -stage otherwise.

Finally, for any $s \geq 0$ , let $\delta _s \in T$ be the unique node $\alpha $ of length s such that s is an $\alpha $ -stage (and call a node $\beta $ accessible at stage $s+1$ if $\beta \sqsubseteq \delta _s$ ), and let $\delta $ be the left most path through T, such that, for any number $m \geq 0$ , $\delta \upharpoonright m \sqsubset \delta _s$ for infinitely many stages s. The path $\delta $ is the true path through T, i.e., for any $m \geq 0$ , $\delta (m) = 0$ if and only if the splitting requirement $R_m$ is infinitary. This fact, to which we refer as the True Path Lemma in the following, is proved by a straightforward induction on m using the above observations on the length function $\ell $ .

For each node $\beta $ of length n there is a strategy $P_{\beta }$ for meeting the noncomputability requirement $P_n$ which is based on the guess that a higher priority R-requirement $R_m$ $(m < n)$ is infinitary iff $\beta (m) = 0$ . (For notational convenience, we call $R_m$ —as well as m and $\beta \upharpoonright m$ — $\beta $ -infinitary if $\beta (m)=0$ and $\beta $ -finitary otherwise ( $m < |\beta |$ ). So, for $\beta = \delta \upharpoonright n$ , $R_m$ is $\beta $ -infinitary iff $R_m$ is infinitary.) We will guarantee that the strategy $P_{\beta }$ on the true path, $\beta = \delta \upharpoonright n$ , meets the requirement $P_n$ .

Before we explain the strategies $P_{\beta }$ , we make some general remarks on the format of the construction. The strategies $P_{\beta }$ are finitary. Numbers are enumerated into the sets A and C under construction only by these strategies. At any stage $s+1> 0$ there is a unique $\beta $ such that $P_{\beta }$ becomes active at stage $s+1$ , and, for this $\beta $ , $\beta \leq \delta _s$ , i.e., either $\beta $ is to the left of $\delta _s$ ( $\beta <_L \delta _s$ ) or $\beta $ is an initial segment of $\delta _s$ ( $\beta \sqsubseteq \delta _s$ ). Moreover, if $P_{\beta }$ acts at stage $s+1$ then all strategies $P_{\beta '}$ with $\beta < \beta '$ are initialized at stage $s+1$ . (So, in particular, all strategies to the right of the true path $\delta $ are initialized infinitely often.) Finally, if $P_{\beta }$ is in its initial state at stage s and acts at stage $s+1$ then $\beta $ is accessible at stage $s+1$ and $P_{\beta }$ may enumerate only numbers $\geq s+1$ into A or C at any later stage. (For technical convenience, we let this first action be vacuous and start with the proper actions for the sake of $P_n$ only afterwards.) Note that this framework ensure that any strategy $P_{\beta }$ with $\beta \sqsubset \delta $ is initialized only finitely often, and, in order to ensure that an infinitary splitting requirement $R_{\langle e,i \rangle }$ is met, it suffices to ensure that $Y_e$ can compute the numbers enumerated into C by strategies $P_{\beta }$ with $(\delta \upharpoonright \langle e,i \rangle )0 = \delta \upharpoonright (\langle e,i \rangle +1) \sqsubseteq \beta $ .

The strategy $P_{\beta }$ ( $|\beta |=n$ ) is a refinement of the usual noncomputability strategy: at some stage $s+1$ , appoint $x = s+1$ as follower. Then wait for a stage $s' \geq s$ such that $x \in W_{n,s'}$ , i.e., such that the follower x is realized at stage $s'$ (and all larger stages). If there is such a stage $s'$ then enumerate x into C at stage $s'+1$ , and keep x out of C otherwise. (Followers will be the only numbers which may be enumerated into C.)

Now, if $\langle e,i \rangle < n$ is $\beta $ -infinitary then the strategy $P_{\beta }$ has to ensure that the set $Y_e$ “knows” whether or not the follower x is put into C (provided that $(\beta \upharpoonright \langle e,i \rangle )0 \sqsubset \delta $ ). (If there is no $\beta $ -infinitary R-requirement then $P_{\beta }$ is just the basic strategy and we call $P_{\beta }$ trivial.) If $R_{\langle e,i \rangle }$ , $\langle e,i \rangle < n$ , is the unique $\beta $ -infinitary requirement of higher priority then this can be achieved by the following basic module. First, at some stage $s+1$ , we appoint an unused number $z \geq s+1$ as tracker and an unused number $y \geq s+1$ such that $y \leq g_e(z)[s]$ as agitator. (For simplicity, by (g $_0$ ), we let $z = s+1$ and $y= g_e(z)[s]=g_e(s+1)[s]=s+1$ .) Next, at any stage $s'+1> s+1$ , we appoint an unused number $x \geq g_e(z)[s] (=g_e(z)[s'])$ as follower (for simplicity, we let $s'=s+1$ and $x = s'+1 = s+2$ ). Now, if there is a stage $s">s'$ such that x is realized at stage $s"$ then we further wait for a stage $s"'> s"$ such that $\ell (\langle e,i \rangle ,s"') \geq z$ . (We say, we wait for confirmation. Note that, for $x \in W_n$ , such a stage $s"'$ must exist if $\beta $ is on the true path $\delta $ , i.e., if $P_{\beta }$ ’s guess that $R_{\langle e,i \rangle }$ is infinitary is correct.) Then $y \leq g_e(z)[s"'] \leq x$ (since, for the tracker z, $g_e(z)$ may be changed only by $P_{\beta }$ hence $g_e(z)[s"'] = g_e(z)[s]$ ) and

$$ \begin{align*}g_e(z)[s"'] = f_i(z,s"') \; \& \; A_{s"'} \upharpoonright g_e(z)[s"']+1 = (X_{e,s"'} \cup Y_{e,s"'}) \upharpoonright g_e(z)[s"']+1).\end{align*} $$

So enumerating the agitator y into A at stage $s"'+1$ forces y to enter either $X_{e}$ or $Y_{e}$ at a stage $>s"'$ (still assuming $\beta \sqsubseteq \delta $ ). If $s""$ is the least stage $> s"'$ at which this happens and $Y_e(y)$ changes then, by $y \leq x$ , this permits the enumeration of the follower x into C thereby meeting $P_n$ (recall that $C\leq _T Y_e$ via the identity use). Otherwise, the enumeration of $y \leq g_e(z)[s"']$ into $X_e$ allows us to redefine $g_e(z)$ at stage $s""+1$ . In this case, we replace the agitator and the follower by new unused numbers $y'$ and $x'$ , respectively, adjust the value of $g_e(z)[s""+1]$ in such a way that $y' \leq g_e(z)[s""+1] \leq x'$ (for simplicity, at stage $s""+1$ we let $y' = g_e(z)[s""+1] = s""+1$ and at stage $s""+2$ we let $x'=s""+2$ ), and iterate the above process for the parameters $z, y', x'$ . Note that, though we have replaced the agitator and follower, the tracker z is fixed. Since at any stage $t+1$ at which we raise the value of $g_e(z)$ there has been a lesser stage $t'$ such that $g_e(z)[t] = g_e(z)[t'] = f_i(z,t')$ , this process must stop after finitely many rounds since any change of $g_e(z)[t]$ is mirrored by a change of $f_i(z,t)$ , and $f_i$ is a convergent approximation. So (assuming $\beta \sqsubseteq \delta $ ), eventually, there will be a follower which either is never realized or is realized and put into C whence $P_n$ is met. So the above action of the strategy $P_{\beta }$ for $P_n$ is finitary, and—assuming $\beta \sqsubset \delta $ —it ensures that $P_n$ is met.

If there is more than one $\beta $ -infinitary R-requirement then some synchronization is needed. While in the basic module the required reduction of C to $Y_e$ is obtained by direct permitting, in the general case we achieve this by delayed permitting. In order to demonstrate this we consider the case of two $\beta $ -infinitary R-requirements, say $R_{\langle e_0,i_0 \rangle }$ and $R_{\langle e_1,i_1 \rangle }$ where $\langle e_0,i_0 \rangle < \langle e_1,i_1 \rangle $ (and where $\alpha _0$ and $\alpha _1$ are the corresponding nodes expanded by $\beta $ , i.e., $\alpha _0 0 \sqsubset \alpha _1 0 \sqsubseteq \beta $ ). There will be a 0-module related to $R_{\langle e_0,i_0 \rangle }$ and a 1-module related to $R_{\langle e_1,i_1 \rangle }$ . We start by appointing a $0$ -tracker $z_{0}$ and a $0$ -agitator $y_{0}$ (corresponding to $\langle e_0,i_0 \rangle $ ), a $1$ -tracker $z_{1}$ and a $1$ -agitator $y_{1}$ (corresponding to $\langle e_1,i_1 \rangle $ ), and a (common) follower x with the required properties as in the basic module where we have to ensure that all numbers are unused and the parameters for $\alpha _0$ differ from the corresponding parameters for $\alpha _1$ (for simplicity, at some stage $s+1$ we let $z_0 = s+1$ and $y_0 = g_{e_0}(z_0)[s]=s+1$ , at stage $s+2$ we let $z_1= s+2$ and $y_1 = g_{e_1}(z_1)[s+1]=s+2$ , and at stage $s+3$ we let $x = s+3$ ). Then we start the basic module for $\langle e_1,i_1 \rangle $ , i.e., the $1$ -module, with the parameters $z_1, y_1,x$ , which involves waiting for x to be confirmed and enumerating the agitator $y_1$ . Now it may happen that there will be a stage $s'$ such that the current follower $x[s']$ is permitted by $Y_{e_1}$ to enter C at stage $s'+1$ (if this is not the case and $\beta \sqsubset \delta $ then, as in the basic module, we may argue that $P_n$ is met). Now, at stage $s'+1$ , instead of enumerating $x[s']$ into C, we activate the basic module for $\langle e_0,i_0\rangle $ , i.e., the 0-module, with the previously defined agitator and tracker and the current follower (note that $x[s+2] \leq x[s']$ ), wait for getting (0-)confirmation, if so enumerate $y_0$ into A, and wait for the corresponding change of $X_{e_0}$ or $Y_{e_0}$ (say at stage $s"$ ; note that, assuming that $\alpha _0 0 \sqsubset \delta $ , such a stage must exist). Now, if $Y_{e_0}$ permits $x[s']$ , then the attack is completed by enumerating $x[s']$ into C. Otherwise, i.e., if $X_{e_0}$ allows us to raise the value of $g_{e_0}(z_0)$ , then—as in the basic module—we replace the $0$ -agitator and raise the value of $g_{e_0}(z_0)$ at stage $s"+1$ (by setting $y_0[s"+1] = g_{e_0}(z_0)[s"+1] = s"+1$ ) and, at the next stages, we reset the parameters of the 1-module including the follower (by setting $z_1[s"+2] = s"+2$ , $y_{1}[s"+2] = g_{e_1}(s"+2)[s"+2]$ and $x[s"+3]=s"+3$ ) and start the basic module for $\langle e_1, i_1\rangle $ with these new parameters.