2.1 True stage relations

The true stage relations allow us to computably approximate transfinite iterations of the Turing jump.

A concrete computable ordinal is a computable well-ordering of a computable subset of $\mathbb N$ , in which the successor relation and collection of limit points are both computable. For concrete computable ordinals $\alpha $ and $\beta $ we write $\alpha <\beta $ if $\alpha $ is an initial segment of $\beta $ . For every concrete computable ordinal $\alpha $ we obtain a partial ordering $\leqslant _\alpha $ on $\omega +1$ with a variety of pleasing properties. In particular:

1. (i) $\leqslant _0$ is the usual ordering $\leqslant $ on $\omega +1$ .

2. (ii) If $\alpha <\beta $ then $s\leqslant _\beta t$ implies $s\leqslant _\alpha t$ .

3. (iii) $(\omega +1,\leqslant _\alpha )$ is a tree, with root 0.

4. (iv) $\{s\,:\, s<_\alpha \omega \}$ is the unique infinite path of $(\omega ,\leqslant _\alpha )$ .

5. (v) The restriction of $\leqslant _\alpha $ to $\omega $ is computable, uniformly in $\alpha $ .

6. (vi) A set $A\subseteq \mathbb N$ is $\Sigma ^0_{1+\alpha }$ if and only if its characteristic function $1_A$ has a computable $\alpha $ -enumeration: a computable function $f\colon \mathbb N\times \omega \to \{0,1\}$ satisfying, for all $x\in \mathbb N$ :

   • • If $s\leqslant _\alpha t$ and $f(x,s)=1$ then $f(x,t)=1$ .

   • • $1_A(x) = \lim \{f(x,s)\,:\, s<_\alpha \omega \}$ .

In (vi), and below, we write $\mathbb N\times \omega $ (rather than $\mathbb N^2$ ) to indicate that the first input x is an element of the “space” $\mathbb N$ , and the second represents a stage of the approximation.

For Propositions 2.1 and 2.3 below, recall that a function $h\colon \mathbb N\to \mathbb N$ is $\Delta ^0_\alpha $ -measurable if for all n, $h^{-1}[\{n\}]$ is $\Delta ^0_{\alpha }$ , uniformly in n. This is equivalent to being $\Sigma ^0_\alpha $ -measurable.

We will use the following corollary of property (vi).

That is, for a while, $f(x,s)$ could be ?, indicating that we are not yet sure what $h(x)$ is; but once some value is guessed for $h(x)$ , we never change our mind again. Along the $\alpha $ -true stages (the stages $s<_\alpha \omega $ ), we eventually guess the correct value.

We remark that (vi) is uniform: given a $\Sigma ^0_{1+\alpha }$ index of A, we can effectively compute an $\alpha $ -enumeration of A. As a result, Proposition 2.1 is also uniform: from a $\Delta ^0_{1+\alpha }$ -index for the sets $h^{-1}(\{n\})$ , we can effectively compute a computable index of an $\alpha $ -decision procedure f.

The following is a “higher limit lemma.” It is proved in [Reference Day, Greenberg, Harrison-Trainor and Turetsky2, Proposition 3.6].

We will require a particular type of $\alpha $ -approximations, that generalises the notion of an $\alpha $ -enumeration. Let $n\geqslant 1$ . An $(\alpha ,n)$ -enumeration is an $\alpha $ -approximation f such that for all x,

• • $f(x,0)=0$ ;

• • for all s,

  $$\begin{align*}\# \left\{ t\leqslant_{\alpha} s \,:\, f(x,t)\ne f(x,t^{-\alpha}) \right\} \leqslant n, \end{align*}$$
  where $t^{-\alpha }$ is t’s predecessor in the tree $(\omega ,<_\alpha )$ .

An $(\alpha ,1)$ -enumeration is simply an $\alpha $ -enumeration (with the added requirement that $f(x,0)=0$ , which is an easy modification).

2.2 Class descriptions

A (computable) class description $\Gamma $ consists of a well-founded computable tree $T_\Gamma \subset \omega ^{<\omega }$ , computably labelled as follows:

1. (i) If $\sigma \in T_\Gamma $ is not a leaf of $T_\Gamma $ , then $\sigma $ is labelled by a pair $(\xi _\sigma ,\eta _\sigma )$ where $\xi _\sigma $ and $\eta _\sigma $ are concrete computable ordinals, and $\eta _\sigma \geqslant 1$ .

2. (ii) If $\sigma \in T_\Gamma $ is a leaf of $T_\Gamma $ then $\sigma $ is labelled by a value $\Gamma (\sigma )\in \{0,1\}$ .

We use the term internal node of a tree T to denote a node of T that is not a leaf of T. We require that if $\sigma \preccurlyeq \tau $ are both internal nodes of $T_\Gamma $ then $\xi _\sigma \leqslant \xi _\tau $ . We let $o(\Gamma ) = \xi _{{\left \langle {}\right \rangle }}$ be the $\xi $ -label of the root ${\left \langle {}\right \rangle }$ of $T_\Gamma $ , unless $T_\Gamma $ consists only of the root, in which case we set $o(\Gamma )=\omega _1$ (where $\omega _1$ is treated as a formal symbol). We similarly set $\xi _\sigma = \omega _1$ for a leaf $\sigma $ of $T_\Gamma $ . We let .

The idea is that at an internal node $\sigma $ we need to choose one of its children. The leftmost child is considered a default, the one we choose initially. We can change our mind about the child we are choosing, but the “number” of mind changes is bounded by $\eta _\sigma $ : every time we change our mind, we need to decrease the ordinal. The ordinal $\xi _\sigma $ tells us at what level we conduct this approximation. Informally, this means that the approximation is computable from $\emptyset ^{(\xi _\sigma )}$ . Technically, we use the true stage relations. Our intention is formalised using the notion of a $\Gamma $ -name.

If $\Gamma $ is a computable class description, then a (computable) $\Gamma $ -name N consists of a choice, for each internal $\sigma \in T_\Gamma $ , of a pair of functions $(f_\sigma ,\beta _\sigma )$ , uniformly computable given $\sigma $ , both defined on $\mathbb N\times \omega $ , satisfying the following:

1. (i) For each $(x,s)\in \mathbb N\times \omega $ , $f_\sigma (x,s)$ is a child of $\sigma $ on $T_\Gamma $ , and $\beta _\sigma (x,s) \leqslant \eta _\sigma $ .

2. (ii) For each $x\in \mathbb N$ and $s,t\in \omega $ , if $s\leqslant _{\xi _\sigma }t$ then:

   • • $\beta _\sigma (x,t)\leqslant \beta _\sigma (x,s)$ ;

   • • if $f_\sigma (x,t)\ne f_\sigma (x,s)$ then $\beta _\sigma (x,t)< \beta _\sigma (x,s)$ .

3. (iii) For each $(x,s)\in \mathbb N\times \omega $ , if $\beta _\sigma (x,s) = \eta _\sigma $ then $f_\sigma (x,s)$ is the leftmost child of $\sigma $ on $T_\Gamma $ (the default outcome of $\sigma $ ).

Let N be a $\Gamma $ -name. The definition ensures that for all internal $\sigma \in T_\Gamma $ , for all $x\in \mathbb N$ , the limit $\lim \left \{ f_\sigma (x,s) \,:\, s<_{\xi _{\sigma }} \omega \right \}$ exists, and we denote it by $f_\sigma (x)$ . Since $T_\Gamma $ is well-founded, for each $x\in \mathbb N$ , the sequence $\sigma _0(x) = {\left \langle {}\right \rangle }$ , $\sigma _1(x) = f_{\sigma _0(x)}(x)$ , $\sigma _2(x) = f_{\sigma _1(x)}(x)$ , …, terminates in a leaf of $T_\Gamma $ that we denote by $\ell ^N(x)$ . We then let $N(x)$ be the value $\Gamma (\ell ^N(x))$ assigned by $\Gamma $ to this leaf. The subset of $\mathbb N$ named by N is the set whose characteristic function is $x\mapsto N(x)$ .

We will abuse notation and use $\Gamma $ to denote both the description and the class that it described, even though a given class may have many different descriptions. A collection of subsets of $\mathbb N$ is a described class if it is the class described by some class description.

The simplest examples are the class descriptions $\Gamma $ consisting only of the root ${\left \langle {}\right \rangle }$ , labelled either 0 or $1$ . The former gives the class $\{\emptyset \}$ , and the latter the class $\{\mathbb N\}$ . The next simplest example is a tree consisting of the root and two children. The root is labelled with some $\xi $ and $\eta = 1$ , the leftmost child is labelled 0 and the other one 1 (see Figure 1). The resulting class is $\Sigma ^0_{1+\xi }$ . Replacing $\eta =1$ with any $\eta $ , the resulting class is $D_{\eta }(\Sigma ^0_{1+\xi })$ of iterated differences of $\Sigma ^0_{1+\xi }$ sets.

Figure 1

The simplest descriptions of $\Sigma ^0_{1+\xi }$ and $\Pi ^0_{1+\xi }$ .

2.3 Described classes are principal pointclasses

Every described class is a lightface (effective) pointclass. The following is essentially proved in [Reference Day, Greenberg, Harrison-Trainor and Turetsky1], but in the setting of $\mathbb N$ is particularly simple.

Similarly, we have the following proposition.

Note that it is not the case that a described class $\Gamma $ is always closed under taking unions or intersections, the simplest counter-example being $D_2(\Sigma ^0_1)$ .

In [Reference Day, Greenberg, Harrison-Trainor and Turetsky1], it is shown that every described boldface class has a universal set. The same construction holds in the discrete setting.

As a result, the effective pointclass $\Gamma $ is principal: there is a set $B\in \Gamma $ such that $\Gamma = \left \{ A \,:\, A\leqslant _m B \right \}$ .

2.4 Definition by cases

The analogue of the following proposition is proved in [Reference Day, Greenberg, Harrison-Trainor and Turetsky1].

2.5 Ordinal invariance

To define the true stage relations, and in general, to use ordinals in computability, we need concrete ordinals. If $\alpha $ and $\alpha '$ are two concrete computable ordinals of the same order-type, then they may fail to be computably isomorphic. Nonetheless, we can computably translate between the true stage relations involved.

The reason is that even if $\alpha $ and $\alpha '$ are not computably isomorphic, the iterated jumps $\emptyset ^{(\alpha )}$ and $\emptyset ^{(\alpha ')}$ are Turing equivalent, uniformly. As a result, the $\Sigma ^0_{1+\alpha }$ sets are the same as the $\Sigma ^0_{1+\alpha '}$ sets, again uniformly. For more details see [Reference Day, Greenberg, Harrison-Trainor and Turetsky1, Proposition 2.20].

The uniformity shows that the choice of concrete copies of the ordinals $\xi _\sigma $ does not affect the class defined by a description.

Note that we cannot relax condition (iii) to . Here the presentation matters, as the names use the particular copies of the $\eta $ -ordinals, rather than the associated true stage relations. To translate names effectively, we would need uniformly computable isomorphisms between and .

2.6 $\Sigma $ and $\Pi $ classes

To differentiate between classes within a dual pair $\{\Gamma ,\check {\Gamma }\}$ , we use the following definition.

The leftmost leaf of $T_\Gamma $ is the “ultimate default” (the default outcome of the default outcome of the default outcome…) The notation generalises that for the classes $\Sigma ^0_{1+\xi }$ and $\Pi ^0_{1+\xi }$ (Figure 1).

In [Reference Greenberg and Turetsky9], it is shown that restricted to “efficient” descriptions (to be discussed later), all descriptions of a particular class have the same type, thus we can talk about a described class having type $\Pi $ or type $\Sigma $ . It is also shown that a described class has the separation property if and only if it is a $\Pi $ -class. A more complicated condition characterises the classes with the reduction property: those are the classes that have hereditarily $\Sigma $ -type descriptions (for all internal $\sigma \in T_\Gamma $ , $\Gamma _\sigma $ has $\Sigma $ -type).

2.7 Finite descriptions

Note that we do not require that the ordinals $\xi _\sigma $ be finite. A class is finitely described if it has a finite description.

In fact, the classes in the fine hierarchy are precisely those classes that have a finite description in which every $\xi _\sigma $ -ordinal is finite as well. We let the extended fine hierarchy denote the collection of all finitely described classes, partially ordered by inclusion. We prove Theorem 2.16 in the following section.

3.1 The tree $S_\Gamma $

In [Reference Greenberg and Turetsky9], the authors define the containment game $G_{\texttt {cont}}(\Gamma ,\Lambda )$ that characterises containment between the described boldface classes. It is a clopen game. They use determinacy of such games to show that the described Wadge classes are semi-linearly ordered. The arguments in that paper can be carried over to the current setting, provided that the games and the winning strategies are computable. When restricted to finite classes, all games are finite, and so have finite winning strategies, and therefore, computable ones.

In the current article we present a simplification of the argument for the setting of finite class descriptions. The games we present are not technically finite, but we will observe that they are essentially finite, with finite positional strategies.

We remark that much of what we do here can be extended to computable class descriptions that are not finite. However, the semi-linear-ordering principle will fail in general.Footnote ²

Definition 3.2. For a class description $\Gamma $ with $o(\Gamma )<\omega _1$ , let

where $\tau ^-$ is the predecessor of $\tau $ on $T_{\Gamma }$ . This is a subtree of $T_\Gamma $ . The internal nodes of $S_\Gamma $ are precisely those nodes $\sigma \in T_\Gamma $ with $\xi _\sigma = o(\Gamma )$ . The leaves of $S_\Gamma $ are those nodes $\sigma \in T_\Gamma $ (internal or not) that are minimal with respect to $\xi _\sigma> o(\Gamma )$ . Again recall that for leaves $\sigma $ of $T_\Gamma $ we set $\xi _\sigma = \omega _1$ , so every leaf of $T_\Gamma $ has a predecessor which is a leaf of $S_\Gamma $ , possibly itself.

Lemma 3.3. Let $\Gamma $ be a computable class description with $o(\Gamma )<\omega _1$ . Let N be a computable $\Gamma $ -name. For a leaf $\sigma $ of $S_\Gamma $ , the set

$$\begin{align*}\left\{ x\in \mathbb N \,:\, \ell^N(x)\succcurlyeq \sigma \right\} \end{align*}$$

is $\Delta ^0_{1+o(\Gamma )+1}$ , uniformly in $\sigma $ .

The game characterisation of containment also yields information about containment and subclasses (see Proposition 3.12 below). For now, we observe the following.

The containment being uniform means that given $\sigma $ and a $\Gamma _\sigma $ -name M we can compute a $\Lambda $ -name M equivalent to $\Lambda $ (one that names the same set). For example, Lemma 3.1 is uniform in $\sigma $ .

For this lemma and its proof, and similarly below, we appeal to Proposition 2.13 and therefore blur the distinction between concrete computable ordinals and their order-types. That is, the lemma also holds when $\operatorname {\mathrm {otp}}(o(\Gamma ))< \operatorname {\mathrm {otp}}(o(\Lambda ))$ .

3.2 The leaf selection game

The main tool for comparing classes at the same ordinal level is a “leaf selection” game. The game presented here is simpler than the one presented in [Reference Greenberg and Turetsky9], as we do not need to worry about passing and the termination of the game.

Let $\Gamma $ be a computable class description with $o(\Gamma )<\omega _1$ . An $S_\Gamma $ -position p consists of a choice, for each internal node $\sigma $ of $S_\Gamma $ , of

1. (i) a child $c_\sigma ^p$ of $\sigma $ on $S_\Gamma $ ;

2. (ii) an ordinal ,

subject to the following restrictions:

• • For all internal $\sigma \in S_\Gamma $ , if then $c_\sigma ^p$ is the default child of $\sigma $ ;.

• • For all but finitely many internal $\sigma \in S_\Gamma $ , .

Of course the latter condition always holds if $\Gamma $ is a finite class description. Its purpose is to ensure that there are only countably many $S_\Gamma $ -positions when $S_\Gamma $ is infinite. If $\Gamma $ is a finite class description, then there are only finitely many $S_\Gamma $ -positions (again note that this holds even if the $\xi _\sigma $ -ordinals are infinite).

For two $S_\Gamma $ -positions p and q, we let $q \leqslant p$ if for every internal node $\sigma $ of $S_\Gamma $ ,

1. (iii) $\eta _\sigma ^q \leqslant \eta _\sigma ^p$ , and further, if $c_\sigma ^q\ne c_\sigma ^p$ then $\eta _\sigma ^q < \eta _\sigma ^p$ .

The initial $S_\Gamma $ -position is the position p determined by setting for all internal $\sigma \in S_\Gamma $ .

Every $S_\Gamma $ -position p determines a leaf $\tau ^p$ of $S_\Gamma $ , by following the choices from the root upwards, much like the definition of the leaf $\ell ^N(x)$ of $T_\Gamma $ used to compute the value $N(x)$ . Namely, $\tau ^p$ is the unique leaf $\tau $ of $S_\Gamma $ determined by $c_\sigma ^p\prec \tau $ for all $\sigma \prec \tau $ .

We let $\mathcal {P}_\Gamma $ denote the collection of all $S_\Gamma $ -positions, ordered by $\leqslant $ .

Let $\Gamma $ and $\Lambda $ be two class descriptions, and suppose that $\xi = o(\Gamma ) = o(\Lambda ) <\omega _1$ . In the game $G_{\texttt {leaf}}(\Gamma ,\Lambda )$ , two players, 1 and 2, take turns choosing positions:

$$\begin{align*}\begin{array}{ccccccccc} p[1] & & p[2] & & p[3] & & \dots \\ & q[1] & & q[2] & & q[3] & & \dots \end{array} \end{align*}$$

(so player 1 plays $p[1], p[2],\dots $ and player 2 plays $q[1], q[2],\dots $ ), satisfying:

• • each $p[k]$ is an $S_\Gamma $ -position, and each $q[k]$ is an $S_\Lambda $ -position;

• • for each $k\geqslant 1$ , $p[k+1]\leqslant p[k]$ and $q[k+1]\leqslant q[k]$ .

We write $\sigma [k] = \tau ^{p[k]}$ and $\rho [k] = \tau ^{q[k]}$ . By Lemma 3.5, the sequences $(\sigma [k])$ and $(\rho [k])$ both stabilise at a pair of leaves $(\sigma ^{\ast },\rho ^{\ast })$ of $S_\Gamma $ and $S_\Lambda $ . This is the outcome of the play of the game.

Note that for computable $\Gamma $ and $\Lambda $ , the game $G_{\texttt {leaf}}(\Gamma ,\Lambda )$ is computable (the partial orderings $\mathcal {P}_\Gamma $ and $\mathcal {P}_\Lambda $ are computable). However, neither player may have a useful computable strategy. We will show that when $\Gamma $ and $\Lambda $ are finite, such strategies exist.

Proof. This is the main part of the proof of [Reference Greenberg and Turetsky9, Proposition 3.5]. We simplify the argument. Let N be a computable $\Gamma $ -name; we devise a $\Lambda $ -name M naming the same set. For a leaf $\sigma $ of $S_\Gamma $ , let $X_\sigma $ be the set of $x\in \mathbb N$ such that $\ell ^N(x)\succcurlyeq \sigma $ .

Fix a leaf $\rho $ of $S_\Lambda $ . Since , by Proposition 2.10 and Lemma 3.3, there is a $\Lambda _\rho $ -name $M_\rho $ such that for all $\sigma $ such that $\Gamma _\sigma \subseteq \Lambda _\rho $ , for all $x\in X_\sigma $ , $M_\rho (x) = N_\sigma (x) = N(x)$ . Note that we are using the finiteness of $S_\Gamma $ to obtain uniformity of containment (and being able to “tell” if $\Gamma _\sigma \subseteq \Lambda _\rho $ or not); we will use the finiteness of $S_\Lambda $ to get that the names $M_\rho $ are uniformly computable.

The names $M_\rho $ define the approximations for M on all nodes that are not internal on $S_\Lambda $ . Thus, to define the name M, it remains to define $f_\sigma ^M$ and $\beta _\sigma ^M$ for all internal $\sigma \in S_\Lambda $ . This is done using a computable containment strategy $\mathfrak {S}$ for player 2. Fix $x\in \mathbb N$ . For each $s<\omega $ let $p_s$ be the $S_\Gamma $ -position defined, for all internal $\sigma \in S_\Gamma $ , by $c_\sigma ^{p_s} = f_\sigma ^N(x,s)$ and $\eta _\sigma ^{p_s} = \beta _\sigma ^N(x,s)$ . The notion of position and ordering of positions ensures that each $p_s$ is indeed an $S_\Gamma $ -position (if $\eta _\sigma ^p = \eta _\sigma $ then $c_\sigma ^p$ is the default outcome), and that $p_t\leqslant p_s$ when $s\leqslant _{o(\Gamma )} t$ .

For each s, consider the partial play $p_{s_0},p_{s_1}, \dots , p_{s_k}$ , where $0= s_0\leqslant _{o(\Gamma )} s_1 \leqslant _{o(\Gamma )} \dots \leqslant _{o(\Gamma )} s_k = s$ is the enumeration of the stages $r\leqslant _{o(\Gamma )} s$ . Thus, $k = |s|_{o(\Gamma )}+1$ , where $|s|_{o(\Gamma )}$ is the height of s on the tree $(\omega ,\leqslant _{o(\Gamma )})$ . The strategy $\mathfrak {S}$ gives a response $q_{s_0}, q_{s_1}, \dots , q_{s_k}$ ; note that $q_{s_i}$ only depends on $p_{s_0},\dots , p_{s_i}$ . For internal $\sigma \in S_\Lambda $ we let $f_\sigma ^M(x,s) = c_\sigma ^{q_s}$ and $\beta _\sigma ^M(x,s) = \eta _\sigma ^{q_s}$ . The fact that $\mathfrak {S}$ always responds with a legal play implies that $(f_\sigma ^M(x,-),\beta _\sigma ^M(x,-))$ obey the rules for properly defining a $\Lambda $ -name. The fact that it is a successful strategy implies that $\ell ^M(x)$ extends a leaf $\rho (x)$ of $S_\Lambda $ such that $\Gamma _{\sigma (x)}\subseteq \Lambda _{\rho (x)}$ ; by the definition of $M_{\rho (x)}$ we then get

$$\begin{align*}N(x) = N_{\sigma(x)} = M_{\rho(x)} = M(x) \end{align*}$$

as required.

Next, we show that for finite class descriptions, $G_{\texttt {leaf}}(\Gamma ,\Lambda )$ is effectively determined.

We now obtain the semi-linear-ordering principle for the extended fine hierarchy.

Note that this proposition, together with $\Gamma \nsubseteq \check {\Gamma }$ , imply the familiar pattern for the fine hierarchy: for any two finite $\Gamma $ and $\Lambda $ , either $\Gamma <\Lambda $ , or $\Lambda <\Gamma $ , or $\Gamma = \Lambda $ , or $\Gamma = \check {\Lambda }$ .

We summarise our findings.

Example 3.13. The two class descriptions in Figure 2 describe the same class $D_2(\Sigma ^0_1)$ of differences of c.e. sets. Let $\Gamma $ denote the description on the left and $\Lambda $ the description on the right. Note that $S_\Gamma = T_\Gamma $ and $S_\Lambda = T_\Lambda $ . Thus, a containment strategy for player 2 is one which guarantees an outcome in which both leaves have the same label, $\mathbf {0}$ or $\mathbf {1}$ . In $G_{\texttt {leaf}}(\Gamma ,\Lambda )$ , both players start with the defaults, labelled $\mathbf {0}$ , and player 1 cannot change labels more than twice, showing how player 2 can always move to match the label of the leaf chose by player 1. In $G_{\texttt {leaf}}(\Lambda ,\Gamma )$ , if player 1 shifts to the $\mathbf {1}$ -outcome, player 2 chooses the rightmost leaf labelled $\mathbf {1}$ on $T_\Gamma $ (choosing the other one would be a bad move, since no further changes would then be allowed).

Figure 2

Two descriptions of $D_2(\Sigma ^0_1)$ .

3.3 The successor class

Definition 3.14. Let $\Gamma $ be a class description. We let $\Gamma ^+$ denote the class description obtained by letting the root have three children, 0, 1, and 2; the first is a leaf of $T_{\Gamma ^+}$ , labelled $\mathbf {0}$ ; we set $\Gamma ^+_1 = \Gamma $ and $\Gamma ^+_2 = \check {\Gamma }$ (see Figure 3).

Figure 3

The successor class $\Gamma ^{+}$ .

The following proposition says that for a finite description $\Gamma $ , the pair consisting of $\Gamma ^+$ and its dual is the successor of the pair $\{\Gamma ,\check {\Gamma }\}$ in the extended fine hierarchy.Footnote ³

3.4 Efficient descriptions, and the ordinal level of a class

Let $\Gamma $ be a class description, with $o(\Gamma )<\omega _1$ . Suppose that there is some leaf $\sigma $ of $S_\Gamma $ such that for every leaf $\tau $ of $S_\Gamma $ we have $\Gamma _\tau \subseteq \Gamma _\sigma $ . Then as $o(\Gamma _\sigma )> o(\Gamma )$ , by Proposition 3.12 (and Lemma 3.1), $\Gamma \equiv \Gamma _\sigma $ . Such a class description is “wasteful.” In [Reference Greenberg and Turetsky9, Definition 4.1], the authors introduce the notion of an efficient description, one which is not wasteful. The definition is based on the following observation. Let $\mathcal {C}$ be a collection of finitely described classes. Proposition 3.11 implies that exactly one of the following holds: (1) $\mathcal {C}$ contains a $\subseteq $ -greatest element and (2) for all $\Gamma \in \mathcal {C}$ there is some $\Lambda \in \mathcal {C}$ with $\Gamma \subseteq \check {\Lambda }$ .

In the context of finite descriptions, we define the following.

The pleasing properties of efficient descriptions hold in the current context as well. For example, they determine the ordinal level of a finitely described class.

The proof is the same as that of the analogous [Reference Greenberg and Turetsky9, Proposition 4.2]: suppose that $o(\Gamma )<o(\Lambda )$ . Since $\Lambda \subseteq \Gamma $ , by Proposition 3.12, there is a leaf $\sigma $ of $S_\Gamma $ such that $\Lambda \subseteq \Gamma _\sigma $ . Since $\Gamma $ is efficient, there is a leaf $\tau $ of $S_\Gamma $ such that $\Gamma _\sigma \subseteq \check {\Gamma }_\tau $ , and $\check {\Gamma }_\tau \subseteq \check {\Gamma }$ ; so $\Lambda \subseteq \check {\Gamma }$ , contradicting $\Gamma \subseteq \Lambda $ .

This allows us to unambiguously define the ordinal level of a finitely described class. As in [Reference Greenberg and Turetsky9, Proposition 4.3], the ordinal level has a characterisation due to Louveau and Saint Raymond [Reference Louveau and Saint-Raymond15]: it is the greatest $\xi $ such that $\Gamma $ is closed under definitions by cases at level $\xi $ (Proposition 2.10); the proof is identical.

3.5 Well-foundedness of the extended fine hierarchy

To prove Theorem 2.16, it remains to show that the extended fine hierarchy is well-founded. We will give two arguments. For the second, we will define a “normal form” for descriptions, and will be able to directly calculate the ordinal rank of each class from these special descriptions, called “admissible.”

The first proof is a direct proof, using Proposition 3.12. We first remark that in fact, we can deduce well-foundedness directly from the results in [Reference Greenberg and Turetsky9], and the fact that the Wadge hierarchy is well-founded: using the containment characterisation in [Reference Greenberg and Turetsky9] for described Wadge classes, and an equivalent characterisation in the effective setting, we see that if $\dots < \Gamma _2< \Gamma _1 < \Gamma _0$ is an infinite descending sequence in the extended fine hierarchy, then the associated boldface classes are an infinite descending sequence $\dots < \boldsymbol {\Gamma }_2 < \boldsymbol {\Gamma }_1 < \boldsymbol {\Gamma }_0$ in the Wadge degrees, which is impossible.

This argument is unsatisfying. The proof of well-foundedness for Borel Wadge classes relies on heavy tools, such as Borel determinacy, and universal sets for the boldface classes. It seems that there should be a direct, “local” argument. We give such a proof. The same proof can be also used to show that the Wadge hierarchy is well-founded. The final part of the proof is similar to the Martin–Monk argument, in its use of Baire category. However, the bulk of the argument seems different.

First proof that the extended fine hierarchy is well-founded. Suppose, for a contradiction, that there is a sequence $\Gamma ^0,\Gamma ^1,\dots $ of finite class descriptions such that $\Gamma ^{n+1}<\Gamma ^n$ for all n. (We use superscripts to accommodate nodes/stage numbers as subscripts.)

For all n, we let $\Gamma ^{n,1} = \Gamma ^n$ and $\Gamma ^{n,0} = \check {\Gamma }^n$ .

Since this proof is not effective, we blur the distinction between a concrete ordinal and its order-type. We therefore let

which is a countable ordinal.

Fix $X\in 2^\omega $ . We define an array of nodes $\sigma ^n_\alpha \in T_{\Gamma ^n}$ for $n\in \mathbb N$ and $\alpha \leqslant \xi ^{\ast }$ by transfinite recursion on $\alpha $ . To simplify notation, we let $\Gamma ^{n}_\alpha = \Gamma ^n_{\sigma ^n_\alpha }$ and . We similarly define $\Gamma ^{n,i}_\alpha = \Gamma ^{n,i}_{\sigma ^n_\alpha }$ for $i=0,1$ .

We will ensure that for all n and $\alpha \leqslant \beta \leqslant \xi ^{\ast }$ ,

1. (i) $\sigma ^n_\alpha \preccurlyeq \sigma ^n_{\beta }$ ;

2. (ii) $\xi ^n_\alpha \geqslant \alpha $ ;

3. (iii) $\Gamma ^{n+1}_\alpha \subseteq \Gamma ^{n,X(n)}_\alpha $ .

We start with $\sigma ^n_0$ being the root of $T_{\Gamma ^n}$ . The containment property (iii) follows from the assumption that $\Gamma ^{n+1}\subseteq \Delta (\Gamma ^n)$ .

If $\beta \leqslant \xi ^{\ast }$ is a limit ordinal, and $\sigma ^n_\alpha $ were defined for all $\alpha <\beta $ , then for each n, by (i), and by the fact that $T_{\Gamma ^n}$ is finite (well-founded would be enough), the sequence $(\sigma ^n_{\alpha })_{\alpha <\beta }$ stabilises; we let $\sigma ^n_\beta $ be that stable value. Note that (iii) holds at $\beta $ since for all n there is some $\alpha <\beta $ such that both $\sigma ^{n+1}_\beta = \sigma ^{n+1}_\alpha $ and $\sigma ^n_\beta = \sigma ^n_\alpha $ . Also (ii) holds since it holds at each $\alpha <\beta $ .

Let $\alpha < \xi ^{\ast }$ and suppose that $\sigma ^n_\alpha $ have been defined for all n. We define $\sigma ^n_{\alpha +1}$ .

If $\xi ^n_\alpha>\alpha $ then we let $\sigma ^n_{\alpha +1} = \sigma ^n_\alpha $ . It thus remains to define $\sigma ^n_{\alpha +1}$ for all n such that $\xi ^n_\alpha = \alpha $ . For such n we will choose $\sigma ^n_{\alpha +1}$ to be a leaf of $S_{\Gamma ^n_\alpha }$ ; this will ensure that (ii) holds for n and $\alpha +1$ , as we will have $\xi ^n_{\alpha +1}>\xi ^n_\alpha $ .

Let I be any maximal interval of $\mathbb N$ such that $\xi ^n_\alpha = \alpha $ for all $n\in I$ . There are two cases.

If I is finite, let $m = \max I$ . We choose $\sigma ^n_{\alpha +1}$ for $n\in I$ by reverse recursion on n. The maximality of I and m implies that $\xi ^{m+1}_\alpha> \alpha $ . Hence by Proposition 3.12(b), we can choose $\sigma ^m_{\alpha +1}\succ \sigma ^m_\alpha $ to be a leaf of $S_{\Gamma ^m_\alpha }$ such that $\Gamma ^{m+1}_{\alpha }\subseteq \Gamma ^{m,X(m)}_{\alpha +1}$ . If $n\in I$ , $n<m$ , and $\sigma ^{n+1}_{\alpha +1}$ was defined, then by Proposition 3.12(c), we can find a leaf $\sigma ^n_{\alpha +1}$ of $S_{\Gamma ^n_\alpha }$ such that $\Gamma ^{n+1}_{\alpha +1}\subseteq \Gamma ^{n,X(n)}_{\alpha +1}$ .

The more complicated case is when I is infinite, i.e., a final segment of $\mathbb N$ . In this case we need to play the leaf selection games. By (iii) and Proposition 3.12(c), fix, for each $n\in I$ , a containment strategy $\mathfrak {S}_n$ for player 1 in the game $G_{\texttt {leaf}}(\Gamma ^{n,X(n)}_\alpha ,\Gamma ^{n+1}_\alpha )$ .

For each n in I, we define a descending sequence of $S_{\Gamma ^\alpha _n}$ positions, $p^n_0\geqslant p^n_1\geqslant p^n_2\geqslant \dots $ starting with the first move given by $\mathfrak {S}_n$ . We then “steal moves”: we let $p^n_1$ be the move given by $\mathfrak {S}_n$ in response to the move $p^{n+1}_0$ , regarded as a move for player 2 in the game $G_{\texttt {leaf}}(\Gamma ^{n,X(n)}_\alpha , \Gamma ^{n+1}_\alpha )$ . In general, if $p^n_k$ is defined for all $n\in I$ , then we let $p^n_{k+1}$ be the move given by $\mathfrak {S}_n$ in response to $p^{n+1}_0,\dots , p^{n+1}_k$ .

For each n, the sequence of leaves $(\tau ^{p^n_k})$ stabilises to a leaf $\sigma ^n_{\alpha +1}$ of $S_{\Gamma ^n_\alpha }$ .

We check that (iii) holds at $\alpha +1$ . Let $n\in \mathbb N$ . There are four cases:

• • If $\xi ^n_\alpha ,\xi ^{n+1}_\alpha $ are both $>\alpha $ , then this follows from (iii) holding at stage $\alpha $ .

• • If $\xi ^n_\alpha = \xi ^{n+1}_{\alpha } = \alpha $ then n and $n+1$ belong to the same interval I. If I is finite, then (iii) follows by construction. If I is infinite, this follows from the success of the strategy $\mathfrak {S}_n$ .

• • If $\alpha = \xi ^n_\alpha < \xi ^{n+1}_\alpha $ then again this is by construction, as in this case n is the maximal element of its interval I.

• • If $\xi ^n_\alpha> \xi ^{n+1}_\alpha = \alpha $ then this follows from (iii) holding at stage $\alpha $ , together with the choice of $\xi ^{n+1}_{\alpha +1}$ to be a leaf of $S_{\Gamma ^{n+1}_\alpha }$ , and Proposition 3.12(a).

At the end of this construction we obtain nodes $\sigma ^n_{\xi ^{\ast }}\in T_{\Gamma ^n}$ with $\xi _{\sigma ^n_{\xi ^{\ast }}}\geqslant \xi ^{\ast }$ ; by choice of $\xi ^{\ast }$ , this means that $\xi _{\sigma ^n_{\xi ^{\ast }}} = \omega _1$ . Let $\ell (X,n)$ be the label of $\sigma ^n_{\xi ^{\ast }}$ on $T_{\Gamma ^n}$ . By (iii), $\ell (X,n) = \ell (X,n+1)$ if and only if $X(n)=0$ .

We observe that in the construction above, each choice $\sigma ^n_\alpha $ depended only on $\Gamma ^n$ and $\Gamma ^{n+1}$ , and $X(n)$ . This implies:

• (*): If $X,Y\in 2^\omega $ , $n\in \mathbb N$ , and $X\! \upharpoonright \!{[n,\infty )} = Y\! \upharpoonright \!{[n,\infty )}$ then $\ell (X,n) = \ell (Y,n)$ .

For $i = 0,1$ let $B_i$ be the set of $X\in 2^\omega $ such that $\ell (X,0)=i$ . The sets $B_0$ and $B_1$ are Borel: essentially, they can be decided by roughly $\xi ^{\ast }$ many Turing jumps of X (and the sequence of descriptions $(\Gamma _n)$ ). On the other hand, they fail the property of Baire, as in the Martin–Monk argument. If $X,Y\in 2^\omega $ and $X\triangle Y = \{n\}$ then $l(X,n+1) = l(Y,n+1)$ (by ( $*$ )), but by reverse induction on $m\leqslant n$ we see that $l(X,m) = 1-l(Y,m)$ . Hence, $X\in B_0\leftrightarrow Y\in B_1$ .

4.1 Containment and equivalence between admissible descriptions

Admissibility allows us to easily characterise containment between classes. For a finite class description $\Gamma $ , let

$$\begin{align*}\mathcal{C}(\Gamma) = \bigcup \{\Gamma_\sigma\,:\, \sigma \text{ is a leaf of }S_\Gamma\} \end{align*}$$

be the union of the classes $\Gamma _\sigma $ for the leaves $\sigma $ of $S_\Gamma $ . If $\Gamma $ is efficient then $\mathcal {C}(\Gamma ) = \Gamma _\sigma \cup \check {\Gamma }_\sigma $ for some leaf $\sigma $ . For two finite efficient class descriptions $\Gamma $ and $\Lambda $ we write $\mathcal {C}(\Gamma )< \mathcal {C}(\Lambda )$ if $\mathcal {C}(\Gamma )\subsetneq \mathcal {C}(\Lambda )$ ; this is equivalent to having some leaf $\tau $ of $S_\Lambda $ such that for all leaves $\sigma \in S_\Gamma $ , $\Gamma _\sigma \subseteq \Lambda _\tau $ , equivalently, $\Gamma _\sigma < \Lambda _\tau $ . Note that for efficient $\Gamma $ we have $\mathcal {C}(\Gamma ) = \mathcal {C}(\check {\Gamma })$ . Below we use the fact that if $\Gamma $ is admissible, then the maximal classes $\Gamma _\sigma $ appear at non-default children of the root: there is some n such that $\mathcal {C}(\Gamma ) = \Gamma _n \cup \check {\Gamma }_n$ (and there is some $n'$ such that $\Gamma _{n'}=\check {\Gamma }_{n}$ ).

4.2 Ubiquity of admissible descriptions

Let $\Theta $ be a finite class description. How do we go about finding an admissible description equivalent to $\Theta $ ? The main idea is the following. Consider all the classes $\Theta _\sigma $ , where $\sigma $ is a leaf of $S_\Theta $ . Among these classes we can identify a maximal pair: some $\Gamma $ such that $\Theta _\sigma \subseteq \Gamma $ or $\Theta _\sigma \subseteq \check {\Gamma }$ for all such $\sigma $ . In most cases we will show that $\Theta \equiv \mathrm {SU}_{\xi ,n}(\Gamma ,\Lambda )$ , where $\xi = o(\Gamma )$ . What is n? In light of Proposition 3.12, we will let n be the length of the longest descending sequence of $S_\Gamma $ positions $p_1,p_2,\dots , p_n$ that alternates between $\Gamma $ and $\check {\Gamma }$ . There will be three possibilities:

1. (1) Only $\Gamma $ appears as $\Theta _\sigma $ (and $\check {\Gamma }$ does not): in this case $\Theta \equiv \Gamma $ .

2. (2) For some $n>1$ , there is an $S_\Theta $ -play alternating between $\Gamma $ and $\check {\Gamma }$ of length n, starting with $\Gamma $ , but all such alternating plays starting with $\check {\Gamma }$ have length at most $n-1$ . In this case $\Theta \equiv \mathrm {SU}_{\xi ,n-1}(\Gamma ,\Gamma )$ .

3. (3) There is no such preference for $\Gamma $ over $\check {\Gamma }$ (or the other way round): there is a $\Gamma /\check {\Gamma }$ alternating sequence of length n starting with $\Gamma $ , and one of the same length starting with $\check {\Gamma }$ , but no alternating sequences (of either kind) of length $n+1$ . In this case $\Theta \equiv \mathrm {SU}_{\xi ,n}(\Gamma ,\Lambda )$ for some $\Lambda <\Gamma $ .

The main challenge will be to identify the class $\Lambda $ in the third case. Naively, we would think that we should take the class obtained from $\Theta $ by removing all leaves $\tau $ of $S_\Theta $ with $\Theta _\tau $ equivalent to one of $\Gamma $ and $\check {\Gamma }$ . Consider, however, trying to show that $\mathrm {SU}_{\xi ,n}(\Lambda ,\Gamma )\subseteq \Theta $ . In the leaf-selection game, while player 1 extends the default at the root, player 2 can copy his moves. But once player 1 moves away, player 2 still needs to be able to produce a $\Gamma /\check {\Gamma }$ alternating play of length n (by choosing appropriate leaves of $S_\Theta $ ). We know that there is such a play; but there is no reason to believe that such a play is still available to us after the moves made at the first part of the game. Ideally, we would need to restrict ourselves to a class $\Lambda $ determined by some collection of leaves of $S_\Theta $ such that every $S_\Lambda $ -play can be extended to a $\Gamma /\check {\Gamma }$ alternating play of length n. It is not clear, though, how to identify such a collection of leaves, and further, why player 2 would be able to win the game in the other direction (showing $\Theta \subseteq \mathrm {SU}_{\xi ,n}(\Gamma ,\Lambda )$ ).

The solution is to look first not for a collection of leaves of $S_\Theta $ , but at the collection of $S_\Theta $ -positions that can be extended by maximal alternating sequences. Among the classes appearing in these, we can identify a maximal pair of classes $\Gamma ', \check {\Gamma }'$ , and the process can repeat.

For this reason, we will need to extend the leaf-selection game, to accommodate restrictions on the kind of positions we allow in the game.

To take the first case above into account (where the ordinal level of the admissible description increases), we need to extend the notation $S_\Theta $ . Let $\xi $ be a computable ordinal. For a class description $\Theta $ with $o(\Theta )\geqslant \xi $ , define $S_{\Theta ,\xi }$ as follows:

• • If $o(\Theta )=\xi $ then $S_{\Theta ,\xi } = S_\Theta $ .

• • If $o(\Theta )>\xi $ then $S_{\Theta ,\xi }$ consists only of the root of $T_\Theta $ .

Note that both cases can be defined together as in the original definition of $S_\Theta $ , replacing $o(\Theta )$ by $\xi $ .

$S_{\Theta ,\xi }$ -positions are defined as in Definition 3.2; when $o(\Theta )>\xi $ , there is just one $S_{\Theta ,\xi }$ position p, determined by taking $\tau ^p$ to be the root of $T_\Theta $ . Note that these notions apply even when $o(\Theta )=\omega _1$ .

Fixing $\xi $ , in this proof, we let $\mathcal {P}$ and $\mathcal {Q}$ denote nonempty collections of $S_{\Theta ,\xi }$ -positions, for some $\Theta $ , that are upwards closed: if $p\in \mathcal {P}$ and $q\geqslant p$ then $q\in \mathcal {P}$ .

Let $\Theta $ and $\Xi $ be class descriptions with ordinal levels $\geqslant \xi $ ; let $\mathcal {P}$ be a nonempty, upwards closed collection of $S_{\Theta ,\xi }$ -positions, and let $\mathcal {Q}$ be such a collection of $S_{\Xi ,\xi }$ -positions. The game $G_{\texttt {leaf}}(\mathcal {P},\mathcal {Q})$ is defined as the game $G_{\texttt {leaf}}(\Theta ,\Xi )$ , except that the trees used are $S_{\Theta ,\xi }$ and $S_{\Xi ,\xi }$ , and further, player 1 is only allowed to choose positions from $\mathcal {P}$ , while player 2 must choose positions from $\mathcal {Q}$ . We write

$$\begin{align*}\mathcal{P}\leqslant \mathcal{Q} \end{align*}$$

if player 2 has a computable containment strategy in the game $G_{\texttt {leaf}}(\mathcal {P},\mathcal {Q})$ : one which guarantees an outcome $(\sigma ,\rho )$ satisfying $\Theta _\sigma \subseteq \Xi _\rho $ . We write $\mathcal {P}\equiv \mathcal {Q}$ if $\mathcal {P}\leqslant \mathcal {Q}$ and $\mathcal {Q}\leqslant \mathcal {P}$ .