2.1 Computations, strings, and degrees

We largely adopt standard notation from [Reference Odifreddi9] which we briefly review. Set difference is denoted by $X \setminus Y$ , powerset by $\mathcal {P}\left (X\right )$ , the e-th set r.e. in X is $W^{X}_{e}$ , and the e-th computable functional applied to X by $\Phi _{i}\left (X\right )$ . We denote convergence and divergence by $\Phi _{i}\left (X;y\right )\downarrow $ and $\Phi _{i}\left (X;y\right )\uparrow $ respectively. Convergence in s-steps is denoted by $\Phi _{i}\left (X;y\right )\downarrow _s \iff \Phi _{i,s}\left (X;y\right )\downarrow $ and its negation by .

We write elements of $2^{<\omega }$ and $\omega ^{<\omega }$ (referred to as strings) like $\langle {x_0, x_1, \ldots , x_{n-1}}\rangle $ with ${\langle \rangle }$ denoting the empty string. For elements in $2^{<\omega }, \omega ^{<\omega }, \omega ^{\omega }, 2^{\omega }$ (identifying sets with their characteristic functions) we denote that $\sigma $ is (non-strictly) extended by $\tau $ by $\sigma \prec \tau $ , incompatibility by $\mid $ , compatibility by ${\,\not |\,}$ and use $<_L$ to denote the lexicographic ordering. For elements of $2^{<\omega }, \omega ^{<\omega }$ we denote the length of $\sigma $ by $\lvert \sigma \rvert $ and write $\sigma ^{-}$ to indicate the immediate predecessor of $\sigma $ under $\prec $ and write $\alpha \widehat {\phantom {x}} \beta $ to denote $\alpha $ concatenated with $\beta $ and write ${\langle i\rangle }^n$ to denote $\langle i\rangle $ concatenated with itself $n \leq \omega $ times. We let $\Phi _{e}\left (\sigma \right )$ denote the longest string $\tau , \lvert \tau \rvert \leq \lvert \sigma \rvert $ such that $\tau (n) = \Phi _{e,\lvert \sigma \rvert }\left (\sigma ;n\right )\downarrow $ and relativize to define $\Phi _{e}\left (\sigma \oplus X\right )$ in the obvious manner.

We let $\left \ulcorner \alpha \right \urcorner $ denote the canonical bijection of $\omega ^{< \omega }$ with $\omega $ where $\alpha \prec \beta \implies \left \ulcorner \alpha \right \urcorner < \left \ulcorner \beta \right \urcorner $ , $i < j \implies \left \ulcorner \alpha \widehat {\phantom {x}}\langle i\rangle \right \urcorner < \left \ulcorner \alpha \widehat {\phantom {x}}\langle j\rangle \right \urcorner $ , and $\left \ulcorner {\langle \rangle }\right \urcorner = 0$ . We regularly gloss over the distinction between strings and their codes as that between sets and their characteristic functions. We let $\langle x,y\rangle = \frac {1}{2}(x+y)(x+y+1)+y$ (this is a bijection of $\omega ^2$ with $\omega $ ) and we let ${\left ({\langle a,b\rangle }\right )_{0}} = a$ and ${\left ({\langle a,b\rangle }\right )_{1}} = b$ . We define $A \oplus B$ , $\oplus _{n \in S} X_n$ , $ {X}^{\left [n\right ]}$ , and ${X}^{\left [< n\right ]}$ standardly and extend these operations to strings in the obvious fashion (preserving the identification of $2^{\omega }$ with ${\mathcal {P}}\left (\omega \right )$ ).

A set X is arithmetic in Y (written $X \leq _{\mathbf {a}} Y$ ) just if there is a formula in the language of arithmetic (with a designated set constant) $\psi _e$ such that $Y \models \psi _e(z) \iff z \in X$ (see [Reference Odifreddi9] for details). In this case we write $\psi _e(Y) = X$ . Recall that an equivalent characterization of arithmetic reducibility is given by $X \leq _{\mathbf {a}} Y \iff \left (\exists n \right )\!\left (X \leq _{\mathbf {T}} {{Y}^{(n)}} \right )$ . We denote the arithmetic degree of $\emptyset $ by ${\mathbf {0}}_{\mathbf {a}}$ and, as the arithmetic jump of X is defined to be $X^{\omega }$ , that of ${\mathbf {0}}^{(\omega )}$ by ${{\mathbf {0}}_{\mathbf {a}}}'$ . An arithmetic degree is minimal just if it has exactly one predecessor under $\leq _{\mathbf {a}}$ .

2.2 Trees and forcing

A tree T is a $\prec $ closed set of strings and $[T]$ is the set of paths through T. We define ${T^{\langle \infty \rangle }} = {\left \{{\sigma \in T}\mid {\left (\exists f \in [T] \right )\!\left (\sigma \prec f \right )}\right \}}$ and call a tree pruned if $T = {T^{\langle \infty \rangle }}$ . A node $\sigma \in T$ is terminal if $\sigma $ has no extensions in T, branching if it has more than one immediate extension, $\omega $ -branching if it has infinitely many immediate extensions and the root of T if it is the unique $\prec $ least branching node in T. A tree is $\omega $ -branching if every branching node is $\omega $ -branching. A subtree of T is a tree $\hat {T} \subset T$ .

We abuse notation and write $T{{\restriction _{l}}}$ for ${\left \{{\sigma \in T}\mid {\lvert \sigma \rvert \leq l}\right \}}$ . We write $\sigma {\mathbin {\ast }} T$ for ${\left \{{\sigma \widehat {\phantom {x}}\tau }\mid {\tau \in T}\right \}}$ , for $ {\left \{{\tau }\mid {\sigma \widehat {\phantom {x}}\tau \in T}\right \}}$ (with the unique element of ) and called the subtree of T above $\sigma $ . We recall that the standard topology on $\omega ^{\omega }$ is induced by basic open sets $[\sigma ] = {\left \{f\mid f \succ \sigma \right \}}$ for $\sigma \in \omega ^{<\omega }$ and likewise for $2^{\omega }$ . A set is perfect if it contains no isolated points and we call a tree T perfect if $[T]$ is perfect.

When working with unpruned trees in $\omega ^{<\omega }$ we can’t identify trees ( $\prec $ closed sets) with $\prec $ respecting functions on strings as we’ll do overFootnote ² $2^{<\omega }$ . We use the term f-tree for a $\prec , <_L$ respecting partial function $T{\mathop {:}{\omega ^{<\omega }}\mapsto {\omega ^{<\omega }}}$ with a $\prec $ closed domain that preserves longest common initial segments, i.e., $T(\sigma \widehat {\phantom {x}}\langle n\rangle )(\left \lvert T(\sigma )\right \rvert )$ is strictly monotonic in n (on its domain). A f-tree $\hat {T}$ is a subtree of an f-tree T if $\mathrm {rng} \hat {T} \subset \mathrm {rng} T$ . An f-tree T is branching if every non-terminal element in $\mathrm {dom} T$ has multiple immediate successors, (weakly) $ \omega $ -branching if every non-terminal element in $\mathrm {dom} T$ has infinitely many immediate successors, and $\omega $ -branching if whenever $\sigma \in \mathrm {dom} T$ is non-terminal than $\sigma \widehat {\phantom {x}}\langle n\rangle \in \mathrm {dom} T$ for all n. Unless otherwise stated, we assume every f-tree is a branching f-tree and generalize the notions of being pruned, ${T^{\langle \infty \rangle }}$ , , $\sigma {\mathbin {\ast }} T$ , and $T^{\left \langle \tau \right \rangle }$ to f-trees in the obvious way.

We write $\sigma \Vdash \psi $ to denote forcing over $2^{<\omega }$ or $\omega ^{<\omega }$ and $\sigma \Vdash _T \psi $ to denote local forcing on the (pruned) tree T (see [Reference Odifreddi9] for details). A set/function is $\kappa $ -generic iff it forces either $\psi $ or $\lnot \psi $ for every $\Sigma ^1_\lambda $ sentence with $\lambda < 1 + \kappa $ and weakly $\kappa $ -generic (note the application at limit ordinals) iff it meets every dense $\Sigma ^1_\lambda $ set of strings. Recall that a set of strings W is dense if every string is extended by an element of W.

2.3 $ \prod ^0_{n}$ classes and $\omega $ -REA sets

A $\prod ^0_{n}$ set class is the set of elements in ${\mathcal {P}}\left (\omega \right )$ (identified with $2^{\omega }$ ) that satisfy some $\prod ^0_{n}$ formula with a free set variable. A $\prod ^0_{n}$ function class is defined likewise for elements in $\omega ^{\omega }$ . We will use the term $\prod ^0_{n}$ class without further specification to refer to a $\prod ^0_{n}$ set class. We note that if $n> 0$ then ${\mathscr {F} \subset \omega ^{\omega }}$ is a $\prod ^0_{n}$ function class iff there is a computable relation R and a quantifier block $\forall x \ldots Qy$ containing n alternations such that $f \in \mathscr {F}$ iff $\forall x \ldots Qy R(f{{\restriction _{y}}}, x, \ldots , y)$ (and likewise for a $\prod ^0_{n}$ set class).

An immediate consequence of this fact is that $\prod ^0_{1}$ classes can be identified with the set of paths through a computable tree. Interestingly, up to degree, $\prod ^0_{2}$ classes and $\prod ^0_{1}$ function classes are equivalent in the following sense.

As the name would suggest, a $\prod ^0_{n}$ singleton is a set/function that’s the only element in a $\prod ^0_{n}$ class. Recall that every $\omega \text {-REA}$ set is a $\prod ^0_{2}$ singleton but not vice-versa. For the interested reader unfamiliar with the $\omega \text {-REA}$ sets we refer them to [Reference Jockusch and Shore6] but as these sets will primarily play a motivating role in this paper it’s enough for the reader to understand that they are the result of effectively iterating the operation $X \mapsto X \oplus W^(X)_{i}$ $\omega $ many times (so the ( $n +1$ )-st component must be uniformly r.e. in the first n components).

2.4 Ordinal notations

We will generalize our main theorem past $\omega $ to arbitrary ordinal notations. The reader interested in only claims about arithmetic reductions (such as the headline corollary) can assume that notations only range over $\omega \cup {\left \{\omega \right \}}$ as the lemmas required for larger ordinal notations have been exiled to Appendix B. However, some notational conventions, inspired by [Reference Sacks12], are still necessary in either case.

Kleene’s set of ordinal notations is $\mathcal {O}$ with ordering $<_{\mathcal {O}}$ . The height of $\kappa $ is $\left \lVert \kappa \right \rVert _{\mathcal {O}}$ . $\overrightarrow {\mathcal {O}}$ is the set of limit notations, ${}^{+}{\mathcal {O}}$ the set of successor notations. For $ \lambda $ a limit notation we denote the n-th element of the effectively given increasing sequence defining $\lambda $ by ${\left \{ \lambda \right \}}^{\mathcal {O}}(n)$ . We elide the differences between finite notations and elements of $ \omega $ as well as that between $\omega $ and some canonical notation for it.

2.5 Rates of growth

We say that $g \in \omega ^{\omega }$ is C-escaping if g isn’t dominated by any $f \in C$ , X-escaping for $X \subset \omega $ if it escapes from ${\left \{{f}\mid {f \equiv _{\mathbf {T}} X}\right \}}$ and arithmetically escaping if it escapes from the set of arithmetic functions. Recall that f majorizes (dominates) g iff $f(x) \geq g(x)$ for all x (all but finitely many x). Following [Reference Odifreddi9] we draw on the fact that a Turing degree is hyperimmune just if it contains a $\mathbf {0}$ -escaping function and say that an arithmetic degree is arithmetically hyperimmune just if it contains an arithmetically escaping function. As there is no notion of a set being arithmetically hyperimmune to cause confusion, we call X arithmetically hyperimmune just if it is of arithmetically hyperimmune degree.

It’s a well-known fact that if A is an r.e. set then A is uniformly computable in any g majorizing $m_A(x) = \min {\left \{{s}\mid {A{{\restriction _{x}}} = A_s{{\restriction _{x}}}}\right \}}$ . Thus, the degree of an r.e. set can be characterized in terms of the rate of growth of a function computable in that degree. However, it’s slightly less well-known that this isn’t just true of r.e. sets but of all $\prod ^0_{2}$ singletons.

3.1 Motivation

One of the most powerful methods to prove results about r.e. sets is to threaten to code one set into another (see, e.g., Sack’s proof of the Density theorem [Reference Sacks10]). However, translating this approach to the $\omega \text {-REA}$ sets under $\leq _{\mathbf {a}}$ faces two serious barriers. First, the fact we can only place elements into the ( $n + 1$ )-th component of an $\omega \text {-REA}$ set if they are enumerated in an r.e. fashion from the n-th component. This makes it very difficult to threaten to code X into Y without following through. Second, the coding would somehow have to control/react to facts about arbitrarily many jumps of Y.

One potential way to avoid these difficulties with coding is to ignore the details about what elements enter a set and just focus on rate of growth/domination. Every non-arithmetic $\prod ^0_{2}$ singleton (and hence $ \omega \text {-REA}$ set) computes an arithmetically escaping function and we know that non-domination strength is often a good way to build sets of smaller degree, e.g., Kurtz’s proof that every hyperimmune degree computes a weak $1$ -generic [Reference Kurtz7].

Also, [Reference Andrews, Gerdes and Miller1] showed that every ${\mathbf {0}}'$ -escaping function computes a weak $2$ -generic (and is thus not of minimal Turing degree). While further non-domination strength won’t ensure we can compute a weak $3$ -generic an examination of the proof of this claim [Reference Andrews, Gerdes and Miller1] suggests that this is more about the ease of avoiding genericity not necessarily a limitation on the computational power of non-domination strength. This leaves open the possibility that arithmetically escaping functions compute $\omega $ -generics under some local forcingFootnote ³ or other forcing notion. Besides, it simply seems intuitively unlikely that a minimal arithmetic degree, a degree which should have the very least amount of computational power, could include an arithmetically escaping function.

This possibility is rendered more plausible by the fact that the standard construction of a minimal arithmetic degree [Reference Sacks11] naturally produces a set which doesn’t compute any arithmetically escaping functions.

Proof Suppose $f \in \omega ^{\omega }$ and $X \models \psi (x,y) \iff f(x) = y$ for some $\Sigma ^1_n$ sentence $\psi $ . We now construct an arithmetic g majorizing f as follows.

Let $m = n + 2$ and consider the sentence that asserts $\psi (x,y)$ defines a total function

$$\begin{align*}\varphi \overset{{\rm def}}{=} \left(\forall x\right)\!\left(\exists y\right)\!\left(\psi(x,y)\right) \land \left(\forall x\right)\!\left(\forall y\right)\!\left(\forall y'\right)\!\left(\psi(x,y) \land \psi(x,y') \implies y = y'\right). \end{align*}$$

By assumption, $X \models \varphi $ and therefore there is some $\sigma \prec X, \sigma \in T_m$ with $\sigma \Vdash _{T_m} \varphi $ . Let $\hat {T}$ be the strings in $T_m$ compatible with $\sigma $ .

We compute $g(x)$ by searching for a finite set of pairs $y_i, \tau _i$ with $\tau _i \Vdash _{T_m} \psi (x,y_i)$ such that every path through $[\hat {T}]$ extends some $\tau _i$ . Since all $Y \in [\hat {T}]$ are m locally generic every such $Y \models \varphi $ . Thus, for all $Y \in [\hat {T}]$ there are $x, y$ such that $Y \Vdash _{T_m} \psi (x,y)$ . Thus, there is an set of pairs $y_i, \tau _i$ as described and the $ \tau _i$ may be taken to be incompatible. As $T_m$ is finitely branching by König’s lemma, there can only be finitely many elements in $\hat {T}$ extending no $\tau _i$ ensuring that we can always find a finite collection of such pairs. Now let $g(x)$ be larger than all the $y_i$ in our set of pairs.

If $Y \in [\hat {T}]$ then g majorizes the function $f_Y$ where $f_Y(x) = y \iff Y \models \psi (x,y)$ and as $X \in [\hat {T}]$ it follows that g majorizes f. As $T_m \geq _{\mathbf {T}} \hat {T}$ is arithmetic it follows that g is an arithmetic function majorizing f.

The trees $T_n$ in the above proposition track the trees used in the construction of a minimal arithmetic degree. Thus, the usual construction of a minimal arithmetic degree produces an arithmetically hyperimmune-free degree.

3.2 An arithmetically hyperimmune minimal degree

Unfortunately, despite the reasons to conjecture that arithmetically hyperimmune functions couldn’t be of minimal arithmetic degree, it turns out not to be the case. Indeed, it turns out that any amount of non-domination strength is compatible with being of minimal arithmetic degree. This contrasts with the situation in the Turing degrees where no ${\mathbf {0}}'$ escaping function can be of minimal Turing degree.

We will break the proof of this theorem up into a sequence of lemmas. However, before we do that let’s first verify that the theorem actually provides the desired (arguably undesired) arithmetically minimal, arithmetically hyperimmune degree.

Recall from the construction of a minimal Turing degree that

and that in that construction we built a sequence of computable trees $T_e \subset 2^{<\omega }$ with $T_{e+1}$ a subtree of $T_e$ such that one of the following obtains (here we identify pruned, perfect binary trees and the corresponding f-trees).

1. 1. (Partiality) $\left (\forall f \in [T_e]\right )\!\left (\Phi _{e}\left (f\right )\uparrow \right ).$

2. 2. (Non e-splitting) For all $\tau , \tau ' \in T_e$ , $\tau , \tau '$ don’t e-split.

3. 3. (e-splitting) For all $\sigma \in T_e$ , $T_e(\sigma \widehat {\phantom {x}}\langle 0\rangle )$ and $T_e(\sigma \widehat {\phantom {x}}\langle 1\rangle )$ e-split.

We then build $f \in \bigcap_{e \in \omega } [T_e]$ ensuring that either $\Phi _{e}\left (f\right )$ is partial (1), computable (2), or computes f (3). We adopt the same general approach, but, to handle arithmetic reductions rather than Turing reductions we’ll replace the notion of e-splitting with an analogue based on local forcing (as in the construction of a minimal arithmetic degree from [Reference Sacks11]). We’ll then adjust this construction to allow us to build $\omega $ -branching trees. First, however, we introduce notation to represent an analogue of partial application of a functional for forcing.

In other words, $\psi ^{\Vdash }(\sigma )$ represents the initial segment of $\psi (A)$ whose values have been determined for $A \succ \sigma $ (assuming A is sufficiently generic).

We now prove the key lemma we’ll use to establish Theorem 3.2.

We can now complete the proof of the theorem.

3.3 Fast growth and definability

In retrospect, perhaps we shouldn’t be too surprised by the result in the last section. After all, there are minimal Turing degrees of hyperimmune degree (e.g., any minimal degree below ${\mathbf {0}}'$ ). And maybe we don’t have to completely give up the idea of using non-domination strength to show that no $\omega \text {-REA}$ set can be of minimal arithmetic degree. While, surprisingly, unlike a true $1$ -generic, a weak $1$ -generic can be of minimal Turing degree (a minimal degree below ${\mathbf {0}}'$ can’t be hyperimmune-free and thus computes a weak $1$ -generic which, by virtue of being non-computable, must be of that very minimal degree), we were still able to use non-domination strength to demonstrate the non-existence of minimal r.e. degrees by identifying a property (weak $1$ -genericity) that enough non-domination strength would let us satisfy (compute) but which couldn’t hold of any set of r.e. degree. Perhaps we could similarly show that every arithmetically escaping function computes a non-arithmetic set with a property that guarantees it’s not of $\omega \text {-REA}$ degree.

What might play the role of this property in the arithmetic degrees? The result in [Reference Andrews, Gerdes and Miller1] tells us that it can’t be $\omega $ -genericity but the following lemma suggests a different way of generalizing the idea that more non-domination strength should somehow allow us to compute less definable sets.

In this light, we can think of the results from [Reference Andrews, Gerdes and Miller1] as showing us that any $\mathbf {0}$ -escaping function computes a set that’s not a $\prod ^0_{1}$ singleton and any ${\mathbf {0}}'$ -escaping function computes a set that’s not a $\prod ^0_{2}$ singleton.Footnote ⁴ We leave it as an exercise to demonstrate that the techniques in [Reference Andrews, Gerdes and Miller1] show that sufficient non-domination strength allows us to compute a set that’s not a $\prod ^0_{3}$ singleton. Thus, a plausible conjecture is that an arithmetically escaping function computes a set that’s not an arithmetic singleton (i.e., not a $\prod ^0_{n}$ singleton for any n). If true, this would prove that no $\omega \text {-REA}$ set is on minimal arithmetic degree.

5.1 Verifying the main theorem

We now prove a utility lemma which will let us show that Theorem 4.2 follows from the claim above.

We can now show that Theorem 4.2 follows from Proposition 5.2.

5.2 Constructing the tower

We now prove Proposition 5.2 follows from Proposition 4.1. To deal with notations for ordinals above $\omega $ we use some results proved in Appendices A and B. We start by defining $ l^{\Diamond }(\beta ), {\beta }^{\Diamond }$ via Definition B.3. As mentioned above, readers only interested in the case $\alpha = \omega $ can skip the material on ordinal notations in Appendices A and B. They will, however, still need to know that we can replace $T_\omega $ with a tree $T^{\prime }_\omega $ with $[T^{\prime }_\omega ] = [T_\omega ]$ in which the membership of $\sigma \in T^{\prime }_\omega $ for $\lvert \sigma \rvert = 2n$ can be computed uniformly in ${\mathbf {0}}^{(2n)}$ (a special case of the general result proved in Lemma B.6).

We construct a computable functional $\Upsilon $ such that for all even notations $\beta $ with $\beta \leq _{\mathcal {O}} \alpha $ , $\Upsilon (\beta , {\mathbf {0}}^{(\beta )}, \cdot )$ gives the characteristic function for $T_\beta $ . Regarding functionals as r.e-sets of axioms we can define a functional $\Upsilon $ assuming that we already have access to some partial functional $\hat {\Upsilon }$ and then use the recursion theorem to yield a single functional satisfying $\Upsilon = \tilde {\Upsilon }$ . Formally speaking, $\Upsilon $ and $\tilde {\Upsilon }$ are defined to be sets of axioms; however, we will only specify those sets implicitly by instead defining the trees $T_\beta $ in terms of the trees $\tilde {T}_\beta $ where we understand that $\tilde {T}_\beta $ represents the set defined by $\tilde {\Upsilon }(\beta , X, \cdot )$ on the guess that $X = {\mathbf {0}}^{(\beta )}$ .

Specifically, we define $T_\alpha $ to be $T_\alpha $ . That is, regardless of the behaviour of $\tilde {\Upsilon }$ , $\Upsilon (\alpha , X, \cdot )$ gives the computation that yields the characteristic function for $ T_\alpha $ when $X = {\mathbf {0}}^{(\alpha )}$ . Given $\beta $ an even notation with $\beta <_{\mathcal {O}} \alpha $ we define the tree $T_\beta $ as follows (once $\Upsilon $ has verified the $\Sigma ^1_1$ fact that $\beta <_{\mathcal {O}} \alpha $ is an even notation).

Let $\lambda = {\beta }^{\Diamond }$ . If $\lvert \sigma \rvert \leq l^{\Diamond }(\beta )$ then $\sigma \in T_\beta $ iff $\sigma \in \tilde {T}^{\prime }_\lambda $ where if $\lambda $ is a limit notation, $\tilde {T}^{\prime }_\lambda $ is the result of applying Lemma B.6 to $\tilde {T}_\lambda $ so that $\tilde {T}^{\prime }_\lambda {{\restriction _{l^{\Diamond }(\beta )}}}$ has membership uniformly computable in ${\mathbf {0}}^{(\beta )}$ and $[\tilde {T}^{\prime }_\lambda ]$ = $ [\tilde {T}_\lambda ]$ . If $\lambda $ is a successor notation then $\tilde {T}^{\prime }_\lambda = \tilde {T}_\lambda $ .

For $\sigma '$ with $\left \lvert \sigma '\right \rvert> l^{\Diamond }(\beta )$ let $\sigma ' = \sigma \widehat {\phantom {x}} \tau $ where $\lvert \sigma \rvert = l^{\Diamond }(\beta )$ . Set $\sigma ' \notin T_\beta $ if $\sigma \notin \tilde {T}^{\prime }_\lambda $ . If $\sigma \in \tilde {T}^{\prime }_\lambda $ then apply Proposition 4.1 to to yield some $T^{\sigma }_\beta $ and place $\sigma ' \in T_\beta $ just if $\tau \in T^{\sigma }_\beta $ . Note that, we can enforce the fact that $\tilde {T}^{\sigma }_{\beta +2}$ is always a tree by only allowing strings into this set when all their predecessors have been seen to be in the set.

We now define $\Upsilon $ to be the fixed point rendering $\tilde {\Upsilon } = \Upsilon $ . Thus, for an even notation $\beta \leq _{\mathcal {O}} \alpha $ we define $T_\beta = \Upsilon (\beta , {\mathbf {0}}^{(\beta )})$ . Note that, by part 3 of Proposition B.5 we can be sure that our fixed point will satisfy $T_\beta {{\restriction _{l^{\Diamond }(\beta )}}} \subset T_{\beta +2}{{\restriction _{{l^{\Diamond }(\beta )}}}}$ so that $ \tilde {T}^{\sigma }_{\beta +2}$ will be defined for $\sigma \in T_{\beta }, \lvert \sigma \rvert = l^{\Diamond }(\beta )$ .

We define the functional $\Gamma $ in a similar fashion assuming we have some $\tilde {\Gamma }$ and applying the recursion theorem. Note that, the argument above produces two indexes i and $q(i)$ for the functional $\Upsilon $ where q is the computable functional implicitly defined by the construction above and i is the index given by the recursion theorem applied to q. While i and $q(i)$ result in the same set of axioms, we can’t guarantee that the application of Proposition 4.1 produces the same $T, \hat {T}$ for different indexes for the same tree S. To handle this issue, we use $\tilde {T}_{\beta + 2}$ to indicate the definition in terms of i and apply Proposition 4.1 to not to ensure we get the f-tree $\hat {T}$ associated with the construction of .

With this in mind, we define $\Gamma ^{\beta +2}_\beta $ for $\beta \leq _{\mathcal {O}} \alpha $ an even notation as follows. If ${\lvert \sigma \rvert \leq l^{\Diamond }(\beta )}$ then $\Gamma ^{\beta +2}_\beta (\sigma ) = \sigma $ if $\sigma \in \tilde {T}_{\beta + 2}$ and undefined otherwise. If $\left \lvert \sigma '\right \rvert> l^{\Diamond }(\beta )$ then let $\sigma = \sigma '{{\restriction _{{l^{\Diamond }(\beta )}}}}$ and if $\sigma \notin \tilde {T}_{\beta + 2}$ then $\Gamma ^{\beta +2}_\beta (\sigma ')\uparrow $ . Otherwise, we define $\Gamma ^{\beta +2}_\beta (\sigma ') = \sigma \widehat {\phantom {x}} \hat {T}(\tau )$ where, $\sigma ' = \sigma \widehat {\phantom {x}} \tau $ and $\hat {T}$ is the f-tree produced by Proposition 4.1 as applied to . For $\beta ' <_{\mathcal {O}} \beta $ we define $ \Gamma ^{\beta +2}_{\beta '}$ to be $\Gamma ^{\beta +2}_{\beta } \circ \tilde {\Gamma }^{\beta }_{\beta '}$ .

Finally, for $\lambda $ a limit and $\beta <_{\mathcal {O}} \lambda \leq _{\mathcal {O}} \alpha $ , $\beta $ an even notation we define $\Gamma ^{\lambda }_{\beta }(\sigma )$ as follows. Let $\beta _n$ be the sequence of notations given by part 4 of Proposition B.5. Let m be the least with $\lvert \sigma \rvert \leq m$ and $\beta \leq _{\mathcal {O}} \beta _m$ and define $\Gamma ^{\lambda }_{\beta }(\sigma )$ to be equal to $ \tilde {\Gamma }^{\beta _m}_{\beta }(\sigma )$ . Finally, we define $\Gamma $ via the recursion theorem so that $\Gamma = \tilde {\Gamma }$ .

To verify this construction produces the desired result, we suppose that $\beta , \gamma $ is the lexicographically least pair of even notations $\gamma <_{\mathcal {O}} \beta \leq _{\mathcal {O}} \alpha $ such that $\Gamma ^{\beta }_\gamma $ isn’t an f-tree that’s a homeomorphism of $[T_\beta ]$ with $[T_\gamma ]$ and then observe that if $\beta = \beta ' + 2$ then since $\Gamma ^{\beta '+2}_{\beta '}$ is a homeomorphism we get a contradiction. Similarly, if $ \lambda = \beta $ is a limit then by Proposition B.5, 6 of Proposition 4.1, and the inductive assumption regarding $\Gamma ^{\beta _n}_\beta $ we can derive that $\Gamma ^{\lambda }_\beta $ is an f-tree that’s a homeomorphism of $[T_\lambda ]$ with its image contained in $[T_\beta ]$ . Now suppose that some $f \in [T_\beta ]$ isn’t equal to $\Gamma ^{\lambda }_\beta (g)$ for any $g \in [T_\lambda ]$ . Then, for some n we have that $f \succ \Gamma ^{\beta _n}_\beta (\sigma )$ for no $\sigma \in T^{\prime }_\lambda $ with $\lvert \sigma \rvert = l^{\Diamond }(\beta _n)$ and thus by the inductive assumption $f \notin [T_\beta ]$ . Thus, the claim holds for $\Gamma ^{\lambda }_\beta $ as well. This suffices to prove Proposition 5.2.

6.1 Machinery

The construction will build T via a stagewise approximation $T_s$ with $T_{s+1} \supset T_s$ and $\sigma \in T$ iff $\sigma \in T_{\left \ulcorner \sigma \right \urcorner }$ . We will also maintain a stagewise approximation $\hat {T}_{s}$ to $\hat {T}$ . We will organize the construction of by drawing on ideas from $\prod ^0_{2}$ constructions in the r.e. sets. In particularly, we will arrange modules on a priority tree, denoted $ \mathbb {T} \subset \omega ^{<\omega }$ , but the differences between constructing a set and a tree require we make some modifications.

Usually, in a $\prod ^0_{2}$ tree construction we assign modules to elements of $\omega ^{<\omega }$ , denoting an arbitrary module assigned to $\alpha $ by ${\mathcal {M}}_{\alpha }$ (for a module of type R, ${\mathcal {R}}_{\alpha }$ ) and use only the outcome of that module at stage s to determine where next to visit at stage s. However, rather than working on meeting a requirement globally for the entire tree T we will assign modules to work on meeting a requirement for the path extending $\hat {T}(\sigma )$ for some $\sigma $ . Thus, rather than working on $\mathbb {T} \subset \omega ^{<\omega }$ we work on $\mathbb {T} \subset \omega ^{<\omega } \times \omega ^{<\omega }$ . More specifically, we identify modules in our construction with pairs $(\alpha , \sigma )$ where $\alpha $ is the string built up out of the outcomes of prior modules and $\sigma $ indicates the element in $\mathrm {dom} \hat {T}$ we are working to define. The idea is that at certain points in our construction, rather than following a single outcome of a module, we will simultaneously work both above (our approximation to) $\hat {T}(\sigma \widehat {\phantom {x}}\langle m\rangle )$ and above/to define $ \hat {T}(\sigma \widehat {\phantom {x}}\langle m'\rangle )$ .

When we specify a module ${\mathcal {M}}_{\xi }$ we also specify a set $\mathfrak {O}(\xi )$ of pairs of potential successors $(o, \delta )$ where o is a potential outcome of the module (identified with elements in $\omega $ but written more suggestively) and $\delta \in \omega ^{<\omega }$ . As usual, at each stage any module we visit will have a single outcome o but as there may be multiple pairs $(o, \delta ) \in \mathfrak {O}(\xi )$ there may be multiple immediate successors of this module which both get visited at this stage. In our construction most modules will only allow $\delta = \langle \rangle $ but some modules will have successors both of the form $(o, \langle \rangle )$ and $(o, \langle n\rangle )$ (for some particular value n). Thus, we will only work on finitely many modules at any stage. With this in mind, we define our priority tree $\mathbb {T}$ inductively as follows. In what follows, keep in mind that ${\left ({\langle a,b\rangle }\right )_{0}} = a$ and ${\left ({\langle a,b\rangle }\right )_{1}} = b$ .

The careful reader might note the possibility that $\xi ^{-}$ could fail to be unique if we aren’t careful. To avoid this, we assume that the outcomes of $\xi = \langle \alpha ,\sigma \rangle $ are modified to be of the form $ \langle o,\sigma \rangle $ . This ensures that $\xi ^{-}$ is uniquely defined and $\prec $ is always a linear order when restricted to the predecessors of $\xi $ on $\mathbb {T}$ . As it won’t cause any confusion, we will assume this happens in the background and will present our outcomes untransformed.

We now define what it means for a node on this tree to be to the left of another node (we retain the terminology ‘left of’ even though its not longer visually accurate) and extend this to a set of nodes as follows.

Our truepath, and its approximations, will no longer be single paths but sets of nodes. Informally speaking, we define to be the set of nodes visited at stage s following the rules described above but not visiting any extensions of a node $\xi $ being visited for the first time at stage s. Formally speaking, we give the following definition.

Note that our priority construction will never reinitialize any nodes. That is, our construction will satisfy the following condition.

With the action of the priority tree defined we need to specify how the modules are able to control the construction. As described above, modules will directly enumerate elements into $T = \bigcup _{s \in \omega } T_s$ with a deadline of stage $s = \left \ulcorner \sigma \right \urcorner $ to place $\sigma $ into T. We use a bit more machinery to specify our approximation to $\hat {T}$ . Each module ${\mathcal {M}}_{\xi }$ receives a string $\delta ^{\xi }$ and specifies a string $\delta ^{\nu }$ for each successor to $\xi $ . If $\xi = \langle \alpha ,\sigma \rangle $ then we understand the module $\xi $ to be executing on the guess that $\delta ^{\xi } \prec \hat {T}(\sigma )$ .

This is sufficient for modules that only need to manipulate a single path but some modules will need to manipulate the collection of potential branches of $\hat {T}(\sigma )$ . To this end, some modules will also define an infinite set $\Theta ^{\xi }$ of branches with the n-th element (ordered lexicographically) indicated by ${\theta ^{\xi }_{n}}$ . In our construction, we will ensure that our definition of $\delta ^{\xi }$ and $\Theta ^{\xi }$ satisfy the following condition.

These are mostly straightforward demands that what the module at $\xi $ does is compatible with what $\xi ^{-}$ does and defines its output promptly. However, a few points deserve mentioning. The requirement that $ \delta ^{\xi } \in T_{s+1}$ will ensure that $\mathrm {rng} \hat {T} \subset T$ . The final condition will enable multiple modules who all want to ensure the leftmost branch extending $\hat {T}(\sigma )$ has some property to cooperate. Without this condition, a module that only ensured ${\theta ^{\xi }_{0}}$ has some property might find their work erased by the next module leaving all extensions of ${\theta ^{\xi }_{0}}$ out of the set of branches it specifies.

We also impose the following condition on the construction to (help) ensure that if then the modules above $\xi $ get to control whether $\delta ^{\xi }$ extends to a path through T.

6.2 Requirements

With an understanding of how our $\mathbb {T}$ operates we are now in a position to present the requirements our construction will meet and arrange the modules we will use to meet them on the tree. Recall that we seek to prove the following result.

We work to meet the following requirements during the construction. We state the requirements in unrelativized form. Unlike requirements in the construction of a single set, we work to ensure that the requirement of the form ${\mathscr {R}}_{e}$ is satisfied for all e and all $\sigma \in {S^{\langle \infty \rangle }}$ with $\lvert \sigma \rvert = e$ . For the purposes of stating the requirements, we use $\hat {T}^-(\sigma )$ to denote an extension of $ \hat {T}(\sigma ^{-})$ that would be extended by $\hat {T}(\sigma )$ if the later were defined. We won’t actually define this function but just use it in the requirements to stand in for some string to be defined later.Footnote ⁵

$$ \begin{align*} {\mathscr{P}}_{e}:\quad &\hat{T}(\sigma) \mid \Phi_{e}\left(X\right) \lor \Phi_{e}\left(X\right)\uparrow\\ {\mathscr{L}}_{e}:\quad & \left(\forall {f \kern1.3pt{\in}\kern1.3pt [T], f \kern1.2pt{\succ}\kern1.2pt \hat{T}(\sigma)}\right)\!\left(\Phi_{e}\left(f \kern1.2pt{\oplus}\kern1.2pt X\right)\uparrow\right) \kern1.2pt{\lor}\kern1.2pt \left(\forall {f \kern1.2pt{\in}\kern1.2pt [T], f \kern1.3pt{\succ}\kern1.3pt \hat{T}(\sigma)}\right)\!\left(\Phi_{e}\left(f \kern1.3pt{\oplus}\kern1.3pt X\right)\kern-1pt\downarrow\right)\\{\mathscr{H}}_{\sigma}:\quad& S(\sigma)\downarrow = 0 \implies \hat{T}(\sigma)\downarrow \land \left(\forall m\right)\!\left(\hat{T}(\sigma)\widehat{\phantom{x}}\langle m\rangle \notin T \right) \\ &S(\sigma)\downarrow = 1 \implies \hat{T}(\sigma)\downarrow \in {T^{\langle \infty\rangle}} \iff \sigma \in {S^{\langle\infty\rangle}} \\ &\qquad S(\sigma)\uparrow \implies \hat{T}(\sigma)\uparrow \land \hat{T}^-(\sigma) \notin {T^{\langle\infty\rangle}}\\{\mathscr{S}}^n_{e}:\quad& T^{\left\langle \hat{T}(\sigma\widehat{\phantom{x}}\langle n\rangle) \right\rangle} \text{ is totally non-}e\text{-splitting } \lor m> n \implies \\ &\qquad\qquad\!\!\hat{T}^-(\sigma\widehat{\phantom{x}}\langle n\rangle) \text{ and } \hat{T}^-(\sigma\widehat{\phantom{x}}\langle m\rangle) e-\text{split} \end{align*} $$

Note that $\Phi _{e}\left (f\right )\downarrow $ means $\left (\forall n\right )\!\left (\Phi _{e}\left (f;n\right )\downarrow \!\right )$ (and similarly for $\Phi _{e}\left (f\right )\uparrow $ ) and that, when we speak of e-splittings we mean the notion relativized to X. The statement of ${\mathscr {H}}_{\sigma }$ is a bit odd in the case where $\sigma \in S$ since in that we do nothing except don’t try to stop $\hat {T}(\sigma )$ from potentially extending to a full path should $\sigma $ extend to a full path through S.

Each requirement gets its own module to assist in meeting it; however, some modules get helper modules. For instance, we break up meeting the requirement ${\mathscr {H}}_{\sigma }$ into a module ${\mathcal {H}}^+_{\sigma ^{-}}$ responsible for creating an $\omega $ -branching above $\hat {T}(\sigma ^{-})$ and a module ${\mathcal {H}}_{\sigma }$ responsible for ensuring $\hat {T}(\sigma )$ doesn’t extend to a path through T if $\sigma \notin S$ . Similarly, we supplement ${\mathcal {L}}_{e}$ with submodules ${\mathcal {L}}^n_{e}$ responsible for checking if we can extend $\Phi _{e}\left (f\right )$ to converge on $\Phi _{e}\left (f;n\right )$ . Finally, we use the module $ {\mathcal {S}}^n_{-1}$ as a helper to split off those modules who will work above $\hat {T}(\sigma \widehat {\phantom {x}}\langle n\rangle )$ from those modules working to define $\hat {T}(\sigma \widehat {\phantom {x}}\langle m\rangle ), m> n$ .

Before we get into any further details, we give a high-level overview of how this is all supposed to work. If we suppose that we’ve just defined $\hat {T}(\sigma )$ and wish to define $\hat {T}(\sigma \widehat {\phantom {x}}\langle m\rangle )$ we start with the module ${\mathcal {H}}^+_{\sigma }$ which will specify a bunch of immediate extensions of $\hat {T}(\sigma )$ (placing them in T). We start by executing ${\mathcal {S}}^0_{e}, \lvert \sigma \rvert = e$ then ${\mathcal {S}}^0_{e -1 }$ and so forth all of which work to ensure the leftmost potential extension of $\hat {T}(\sigma )$ e-split or $e -1$ split or so on with all the remaining potential extensions. When we finally get to the module $ {\mathcal {S}}^0_{-1}$ it specifies that $\hat {T}(\sigma \widehat {\phantom {x}}\langle 0\rangle )$ extends the leftmost branch as extended by all the modules ${\mathcal {S}}^0_{e'}, e' \leq e$ and the construction now splits into one part which works on the next module ${\mathcal {P}}_{e}$ along that path specified as an initial segment of $\hat {T}(\sigma \widehat {\phantom {x}}\langle 0\rangle )$ and another part where ${\mathcal {S}}^1_e$ starts working to define the node which will be extended by $ \hat {T}(\sigma \widehat {\phantom {x}}\langle 1\rangle )$ . After the module of the form ${\mathcal {P}}_{e}$ we work on the next module of the form ${\mathcal {L}}_{e}$ and then, if we are above the total outcome any ${\mathcal {L}}_{e'}, e' \leq e$ we implement the next helper module of the form ${\mathcal {L}}^m_{e'}$ . Finally, after those modules, comes the module ${\mathcal {H}}_{\sigma \widehat {\phantom {x}}\langle m\rangle }$ (assuming we took the path working on $\hat {T}(\sigma \widehat {\phantom {x}}\langle m\rangle )$ ) which guesses whether or not $\sigma \widehat {\phantom {x}}\langle m\rangle \in S$ . If it determines $\sigma \widehat {\phantom {x}}\langle m\rangle $ is in S then we go on to ${\mathcal {H}}^+_{\sigma \widehat {\phantom {x}}\langle m\rangle }$ . If it determines that $\sigma \widehat {\phantom {x}}\langle m\rangle $ is not in S then no module is assigned above that outcome preventing any path from being constructed.

With this in mind, we can now give a formal definition of $\hat {T}$ and its stagewise approximation.

6.3 Modules

We now describe the operation of the modules. For this subsection, we describe the operation of the module assuming it is located at the node $\xi \in \mathbb {T}, \xi = (\alpha , \sigma )$ and executing at stage s. As we only define the outcome of the module at $\xi $ when $s_\xi> 0$ we understand the previous outcome of the module to be undefined when $s_\xi \leq 1$ .

6.4 Verification

Before we verify the individual requirements, we verify that the construction controls the paths through T in the manner desired.

Note that this covers the case where ${\mathcal {H}}_{\sigma }$ doesn’t have any extension $\nu $ in the truepath because S doesn’t converge on $\sigma $ .