2. Partial Combinatory Algebras

To make the paper self-contained, we briefly review the basic definitions from partial combinatory algebra. Our presentation follows van Oosten (Reference van Oosten2008).

A partial applicative structure (pas) is a set $\mathcal{A}$ together with a partial map $\cdot$ from $\mathcal{A}\times \mathcal{A}$ to $\mathcal{A}$ . We usually write $ab$ instead of $a\cdot b$ , and think of this as “ $a$ applied to $b$ .” If this is defined, we denote this by $ab\!\downarrow$ . By convention, application associates to the left, so we write $abc$ instead of $(ab)c$ . Terms over $\mathcal{A}$ are built from elements of $\mathcal{A}$ , variables, and application. If $t_1$ and $t_2$ are terms, then so is $t_1t_2$ . If $t(x_1,\ldots, x_n)$ is a term with variables $x_i$ , and $a_1,\ldots, a_n \in \mathcal{A}$ , then $t(a_1,\ldots, a_n)$ is the term obtained by substituting the $a_i$ for the $x_i$ . For closed terms (i.e. terms without variables) $t$ and $s$ , we write $t \simeq s$ if either both are undefined, or both are defined and equal. Here application is strict in the sense that for $t_1t_2$ to be defined, it is required that both $t_1$ and $t_2$ are defined.

Combinatory completeness of pcas can be characterized by the existence of combinators $k$ and $s$ , just as in classical ca and lambda calculus. In van Oosten (Reference van Oosten2008), it is stated that the following theorem is “essentially due to Feferman (Reference Feferman and Crossley1975).”

In the following, we will always assume that our pcas are nontrivial, that is, have more than one element. This automatically implies that they are infinite and have $k\neq s$ .

The prime example of a pca is Kleene’s first model $\mathcal{K}_1$ that was already mentioned in the introduction. This is a model defined on the natural numbers, with application

\begin{equation*} n \cdot m = \varphi _n(m). \end{equation*}

Thus $\mathcal{K}_1$ models the setting of classical computability theory. We can also relativize this to an arbitrary oracle $X$ , thus obtaining the relativized pca $\mathcal{K}_1^X$ .

Kleene’s second model $\mathcal{K}_2$ , from the book Kleene and Vesley (Reference Kleene and Vesley1965), is a pca defined on Baire space $\omega ^\omega$ . Application $\alpha \cdot \beta$ in this model can be informally described as applying the continuous functional with code $\alpha$ to the real $\beta$ . The original coding of $\mathcal{K}_2$ is a bit cumbersome, but it is essentially equivalent to

\begin{equation*} \alpha \cdot \beta = \Phi _{\alpha (0)}^{\alpha \oplus \beta }, \end{equation*}

where $\Phi _e$ is the $e$ -th Turing functional, and the application is understood to be defined if the RHS is total. This coding, used in Shafer and Terwijn (Reference Shafer and Terwijn2021), is easier to work with. See the appendix of Golov and Terwijn (Reference Golov and Terwijn2023) for a proof (and precise statement) of the equivalence with the original coding.

An interesting variant of $\mathcal{K}_2$ , called the van Oosten model, is obtained by extending the domain to include partial functions, cf. van Oosten (Reference van Oosten, Cooper and Truss1999). $\mathcal{K}_2$ is uncountable, but restricting attention to computable sequences gives a countable pca $\mathcal{K}_2^{\mathrm{eff}}$ . Similarly, restricting to $X$ -computable sequences gives a pca $\mathcal{K}_2^X$ for every $X$ . In Golov and Terwijn (Reference Golov and Terwijn2023), the relations between these and other pcas are studied using embeddings.

Many other examples of pcas can be found in the literature. For example, pcas have been extensively used in constructive mathematics, see Beeson (Reference Beeson1985), Troelstra and van Dalen (Reference Troelstra and van Dalen1988). In particular, they have been used as a basis for models of constructive set theory, as in McCarty (Reference McCarty1986), Rathjen (Reference Rathjen2006), and Frittaion and Rathjen (Reference Frittaion and Rathjen2021). Pcas are also pivotal in the categorical treatment of realizability, cf. van Oosten (Reference van Oosten2008, Chapter 2) where they serve as a basis for realizability toposes. In particular, Hyland’s famous effective topos is the realizability topos of $\mathcal{K}_1$ . For a categorical characterization of pcas, see Cockett and Hofstra (Reference Cockett and Hofstra2008) (extending early work of Longo and Moggi Reference Longo, Moggi, Chytil and Koubek1984); for a discussion of pcas in the context of oracles see Kihara (Reference Kihara2022).

A pca that has been particularly important in connection with ca and lambda calculus is Scott’s graph model (Scott Reference Scott and Kanger1975). This pca is a model of the lambda calculus (see Barendregt Reference Barendregt1984), and it is also closely related to the enumeration degrees in computability theory, cf. Odifreddi (Reference Odifreddi1999). We will discuss this model in Section 5, where we also explain how the restriction of this model to the c.e. sets can be seen as a completion of $\mathcal{K}_1$ .

3. Effective Presentations of Pcas

Below, we will call a ca $\mathcal{A}$ $Y$ -computable if $\mathcal{A}$ has a representation such that the application $\cdot$ in $\mathcal{A}$ is $Y$ -computable. We will also require that equality on $\mathcal{A}$ is $Y$ -decidable. The following definition (similar to notions used in Golov and Terwijn Reference Golov and Terwijn2023) makes this precise, using numberings to represent $\mathcal{A}$ . Recall that a numbering is a surjective function $\gamma :\omega \to \mathcal{A}$ . We think of $n\in \omega$ as a code for $\gamma (n)\in \mathcal{A}$ .

Definition 3.1. Let $\mathcal{A}$ be a pca and $Y\subseteq \omega$ . We call $\mathcal{A}$ partial $Y$ -computable if there exist a numbering $\gamma :\omega \to \mathcal{A}$ and a partial $Y$ -computable function $\psi$ such that for all $n$ and $m$ , $\gamma (n)\cdot \gamma (m)\!\downarrow$ in $\mathcal{A}$ if and only if $\psi (n,m)\!\downarrow$ , and

(1) \begin{equation} \gamma (n)\cdot \gamma (m) = \gamma (\psi (n,m)). \end{equation}

We also require that equality on $\mathcal{A}$ is $Y$ -decidable, meaning that the set $\{(n,m)\mid \gamma (n)=\gamma (m)\}$ is $Y$ -computable. If $\mathcal{A}$ is total, i.e., a ca, then $\psi$ is total, and we simply call $\mathcal{A}$ $Y$ -computable. (This is consistent with Definition 3.2 below.)

Notice that the numbering $\gamma$ in Definition 3.1 is not required to be computable in any way. Also, nontrivial combinatorial algebras are never computable, cf. Barendregt (Reference Barendregt1984, 5.1.15).

Definition 3.1 focuses on representing application in a pca as a p.c. function. For the record, we also mention another way to define effective representations.

Definition 3.2. We call a pca $\mathcal{A}$ $Y$ -c.e. if there exist a numbering $\gamma :\omega \to \mathcal{A}$ such that the set

(2) \begin{equation} \big \{ (n,m,k)\mid \gamma (n)\cdot \gamma (m)\!\downarrow = \gamma (k) \big \} \end{equation}

is $Y$ -c.e. Again we also require that equality on $\mathcal{A}$ is $Y$ -decidable, meaning that the set $\{(n,m)\mid \gamma (n)=\gamma (m)\}$ is $Y$ -computable. We call $\mathcal{A}$ $Y$ -computable if the set (2) is $Y$ -computable. Footnote ¹

As an example, note that $\mathcal{K}_1$ is p.c. in the sense that $a\cdot b$ is a p.c. function on $\omega$ , and that $\mathcal{K}_1$ is c.e. in the sense that $a\cdot b\!\downarrow = c$ is a c.e. relation. We note here that the two definitions are equivalent:

Note that the above definitions are in the spirit of the c.e. structures in Selivanov (Reference Selivanov, Cooper and Goncharov2003), where they are called positive structures. These are defined as structures in which the predicates are c.e., and the functions are computable. The latter makes sense for total functions, but in the case of pcas, we are dealing with a partial application operator, in which case it is natural to have this as a c.e. function.

For pcas on $\omega$ (i.e. with $\gamma :\omega \to \mathcal{A}$ the identity), we have that equality on $\mathcal{A}$ is decidable, so the two notions of pca are equivalent. This is the type of pca that was used in Fokina and Terwijn (Reference Fokina and Terwijn2024). Without the condition that equality on $\mathcal{A}$ is decidable (or c.e.), it is not clear that the two definitions are equivalent in general, though we do not know of an example of a pca that is p.c. but not c.e.

Finally, in the case of a $Y$ -computable pca or ca $\mathcal{A}$ (which is what we will mostly use below), the requirement that equality on $\mathcal{A}$ is $Y$ -decidable actually follows from the fact that Equation (2) is $Y$ -computable, namely let $n$ be such that $\gamma (n)$ is the identity (which exists in any pca).

4. Embeddings and Isomorphisms

There are at least three notions of embedding for pcas, depending on what structure is required to be preserved. For example, the choice of the combinators $k$ and $s$ from Theorem 2.2 can be regarded as part of the structure or not. For instance, Zoethout (Reference Zoethout2022, p. 33) does not consider $k$ and $s$ to be part of the structure of a pca. We have the following notions of embedding of pcas:

• • Only preserve applications. This notion was studied in Bethke (Reference Bethke1988), Asperti and Ciabattoni (Reference Asperti and Ciabattoni1997), Shafer and Terwijn (Reference Shafer and Terwijn2021), and Golov and Terwijn (Reference Golov and Terwijn2023).

• • Besides applications, also preserve $k$ and $s$ , for a particular choice of these combinators. This stronger notion was studied in Bethke et al. (Reference Bethke, Klop and de Vrijer1996, Reference Bethke, Klop and de Vrijer1999).

• • There is an even weaker notion of embedding, using the notion of applicative morphism, that was introduced in Longley (Reference Longley1994), see also Longley and Normann (Reference Longley and Normann2015). Applicative morphisms do not have to preserve applications; instead, there have to be terms in the codomain that simulate applications in the domain. This notion is useful in realizability theory, see van Oosten (Reference van Oosten2008).

Our primary interest here is the notion of embedding where $k$ and $s$ are not considered part of the signature, but we will also be using the stronger notion of embedding, especially when we talk about completions. To distinguish the two, we will refer to them as weak and strong embeddings. (In Golov and Terwijn Reference Golov and Terwijn2023, weak embeddings were simply called embeddings.) To distinguish applications in different pcas, we also write $\mathcal{A}\models a\cdot b\!\downarrow$ if this application is defined in $\mathcal{A}$ .

Definition 4.1. For given pcas $\mathcal{A}$ and $\mathcal{B}$ , an injection $f: \mathcal{A} \to \mathcal{B}$ is a weak embedding if for all $a, b \in \mathcal{A}$ ,

(3) \begin{equation} \mathcal{A}\models ab\!\downarrow \; \Longrightarrow \; \mathcal{B}\models f(a)f(b)\!\downarrow \, = f(ab). \end{equation}

If $\mathcal{A}$ embeds into $\mathcal{B}$ , in this way we write $\mathcal{A}\hookrightarrow \mathcal{B}$ . If in addition to (3), for a specific choice of combinators $k$ and $s$ of $\mathcal{A}$ , $f(k)$ and $f(s)$ serve as combinators for $\mathcal{B}$ , we call $f$ a strong embedding.

A (total) ca $\mathcal{B}$ is called a weak completion of $\mathcal{A}$ if there exists a weak embedding $\mathcal{A} \hookrightarrow \mathcal{B}$ . If the embedding is strong, we call $\mathcal{B}$ a strong completion.

Two pcas $\mathcal{A}$ and $\mathcal{B}$ are isomorphic, denoted by $\mathcal{A}\cong \mathcal{B}$ , if there exists a bijection $f:\mathcal{A}\rightarrow \mathcal{B}$ such that for all $a,b\in \mathcal{A}$ , $ab\!\downarrow$ if and only if $f(a)f(b)\!\downarrow$ , and in this case

\begin{equation*} f(a)\cdot f(b) = f(ab). \end{equation*}

Besides the term completion, in the literature also the term extension is used. Bethke et al. (Reference Bethke, Klop and de Vrijer1999, Definition 1.5) call a pca $\mathcal{B}$ an extension of a pca $\mathcal{A}$ if $\mathcal{A} \subseteq \mathcal{B}$ , the application $\cdot _{\mathcal{A}}$ in $\mathcal{A}$ is the restriction of application $\cdot _{\mathcal{B}}$ in $\mathcal{B}$ to the domain of $\cdot _{\mathcal{A}}$ , and $\mathcal{B}$ and $\mathcal{A}$ both have the same combinators $k$ and $s$ as in Theorem 2.2.

Now suppose that $f:\mathcal{A}\hookrightarrow \mathcal{B}$ is a strong embedding. Then $f(\mathcal{A})\subseteq \mathcal{B}$ is an extension in the above sense, where both $f(\mathcal{A})$ and $\mathcal{B}$ have combinators $f(k)$ and $f(s)$ . Note that $\mathcal{A} \cong f(\mathcal{A})$ if we define application in $f(\mathcal{A})$ by $f(\mathcal{A})\models f(a)\cdot f(b) \!\downarrow =f(c)$ if and only if $\mathcal{A}\models a \cdot b\!\downarrow = c$ . So we see that total extensions and completions amount to the same thing, provided that in both cases we have to specify whether to also fix $s$ and $k$ or not.

In Terwijn (Reference Terwijn2023), it is shown that weak and strong embeddability and completions are different: There exists a pca that is weakly completable, but not strongly completable.Footnote ²

5. Complexity of Completions of $\mathcal{K}_1$

It was an important open question in the 1970s whether every pca has a strong completion. The question was raised by Barendregt, Mitschke, and Scott, and discussed at a meeting in Swansea in 1974, cf. Bethke et al. (Reference Bethke, Klop and de Vrijer1999). (Note that this predates Feferman’s paper Feferman Reference Feferman and Crossley1975.) A negative answer was obtained by Klop (Reference Klop1982), see also Bethke et al. (Reference Bethke, Klop and de Vrijer1999). Other examples of incompletable pcas can be found in Bethke (Reference Bethke1987) and Bethke and Klop (Reference Bethke, Klop, Dowek, Heering, Meinke and Möller1996).

In contrast to these examples, $\mathcal{K}_1$ does have strong completions. This follows from the criterion given in Bethke et al. (Reference Bethke, Klop and de Vrijer1996) about the existence of unique head-normal forms, which is satisfied in $\mathcal{K}_1$ . The completion of $\mathcal{K}_1$ resulting from this is a certain term model $T(\omega )/\!\!\sim$ . On the face of it, the equivalence relation $\sim$ is not computable, since it is essentially equivalence of terms in $\mathcal{K}_1$ . That indeed it cannot be computable follows from Theorem 5.2, and also from the fact that computable combinatorial algebras do not exist.

We now discuss how another famous pca can be seen as a completion of $\mathcal{K}_1$ . Scott’s graph model $\mathcal{G}$ is a pca defined on the power set $\mathcal{P}(\omega )$ , with application defined by

\begin{equation*} X \cdot Y = \big \{ x \mid \exists u (\langle x,u\rangle \in X \wedge D_u\subseteq Y) \big \}. \end{equation*}

Here $D_u$ as always denotes the finite set with canonical code $u$ , and $\langle \cdot, \cdot \rangle$ denotes an effective pairing function. $\mathcal{E}$ is defined as the restriction of $\mathcal{G}$ to the c.e. sets. That $\mathcal{G}$ and $\mathcal{E}$ are (total) cas is implicit in Scott (Reference Scott and Kanger1975). Note the close connection with enumeration reducibility (cf. Odifreddi Reference Odifreddi1999, XIV): For all sets $Y$ and $Z$ , $Z\leqslant _e Y$ is equivalent with $X\cdot Y = Z$ for some c.e. set $X$ .

In Golov and Terwijn (Reference Golov and Terwijn2023, Corollary 7.5, it was shown that $\mathcal{K}_1 \hookrightarrow \mathcal{E}$ , so that we can see $\mathcal{E}$ as a weak completion of $\mathcal{K}_1$ . Note that equality on $\mathcal{E}$ is equality of c.e. sets, which is $\Pi ^0_2$ -complete when we represent c.e. sets by their indices.Footnote ³ So this is more complicated than equality in the term model $T(\omega )/\!\!\sim$ .

We can see $\mathcal{E}$ as a combination of $\mathcal{K}_1$ and $\mathcal{K}_2$ . Indeed we have

\begin{equation*} \mathcal {K}_1 \hookrightarrow \mathcal {E} \hookrightarrow \mathcal {K}_2 \end{equation*}

(the latter by Golov and Terwijn Reference Golov and Terwijn2023, Corollary 6.2, so that we can view $\mathcal{E}$ as a kind of middle ground between Kleene’s models. This combination famously gives a model of the $\lambda$ -calculus, as shown in Scott (Reference Scott and Kanger1975), see Odifreddi (Reference Odifreddi1999, XIV.4).

Below, we use that for an embedding $f:\mathcal{K}_1\hookrightarrow \mathcal{A}$ of $\mathcal{K}_1$ into a pca $\mathcal{A}$ , it suffices to know the value of $f$ on finitely many elements. This observation was also used in Golov and Terwijn (Reference Golov and Terwijn2023, Theorem 4.1), and it can be used to bypass the fact that embeddings such as $f$ do not have to be computable. Below we give a somewhat simpler version of this trick, using the following lemma.

Lemma 5.1. There exist elements $t_n\in \mathcal{K}_1$ , $n\geqslant 1$ , such that for all $n$ and $m$ ,

\begin{align*} t_n \cdot m = \begin{cases} n & \textrm{if}\ m=0 \\ t_{n+1} & \textrm{if}\ m\gt 0. \end{cases} \end{align*}

Proof. By the recursion theorem, let $d\in \omega$ be a code such that

\begin{align*} \varphi _d(n,m) = \begin{cases} n & \text{if $m=0$} \\ S^1_1(d,n+1) & \text{if $m\gt 0$.} \end{cases} \end{align*}

Here $S^1_1$ is the primitive recursive function from the S-m-n-theorem. Define $t_n = S^1_1(d,n)$ . W.l.o.g. we may assume $t_n\gt 0$ for all $n$ . Then $\varphi _{t_n}(m) = \varphi _{S^1_1(d,n)}(m) = \varphi _d(n,m) = t_{n+1}$ for $m\gt 0$ and equal to $n$ otherwise.

In Golov and Terwijn (Reference Golov and Terwijn2023, Corollary 4.2, it was proved that if $\mathcal{K}_1^X\hookrightarrow \mathcal{A}$ is a weak embedding of $\mathcal{K}_1^X$ into a pca $\mathcal{A}$ with $Y$ -c.e. inequality, then $X\leqslant _T Y$ . We obtain a stronger conclusion when we assume that $\mathcal{A}$ is total and $Y$ -computable.

Proof. The successor function $S$ in $\mathcal{K}_1$ satisfies $S^n(0)=n$ , but we need an element $t$ such that the $n$ -fold application $t\cdot \ldots \cdot t\cdot 0$ equals $n$ , with the convention that application associates to the left, not to the right. Let $t = t_1$ be as in Lemma 5.1. Then for the $n$ -fold application, we have $t\cdot \ldots \cdot t\cdot 0 = t_n \cdot 0 = n$ for every $n\gt 0$ . Since $f$ is an embedding, we obtain from this

(4) \begin{equation} f(n) = f(t\cdot \ldots \cdot t\cdot 0) = f(t)\cdot \ldots \cdot f(t)\cdot f(0), \end{equation}

with the applications repeated $n$ times. So we see that the image of $f$ is completely determined by $f(t)$ and $f(0)$ .

To show that $Y\not \leqslant _T X$ , let $A$ , $B$ be a $X$ -computably inseparable pair of $X$ -c.e. sets, and let $e$ be a code such that for all $x$ ,

\begin{align*} \varphi _e^X(x) = \begin{cases} 0 & \text{if $x\in A$} \\ 1 & \text{if $x\in B$} \\ \uparrow & \text{otherwise}. \end{cases} \end{align*}

Then we have in particular that

\begin{align*} \mathcal{K}_1^X \models e \cdot x \!\downarrow = 0 &\Longrightarrow \mathcal{A} \models f(e)\cdot f(x) = f(0), \\ \mathcal{K}_1^X \models e \cdot x \!\downarrow = 1 &\Longrightarrow \mathcal{A} \models f(e)\cdot f(x) = f(1). \end{align*}

Since $\mathcal{A}$ is total, for every $x$ the application $f(e)\cdot f(x)$ is always defined in $\mathcal{A}$ , and by Equation (4), it is equal to a term containing only $f(t)$ , $f(0)$ , and application. Because $\mathcal{A}$ is $Y$ -computable, we can compute a code of $f(e)\cdot f(x)$ effectively from $x$ . (All we need is $e$ , and codes of $f(t)$ and $f(0)$ , all of which are fixed.) Furthermore, since the definition of $Y$ -computable pca entails that equality on $\mathcal{A}$ is $Y$ -decidable, we can decide with $Y$ whether $f(e)\cdot f(x)$ is equal to $f(0)$ or $f(1)$ or not. It follows that the set $C = \{x \mid \mathcal{A}\models f(e)\cdot f(x) = f(0)\}$ is $Y$ -computable and separates $A$ and $B$ , and since $A$ and $B$ have no $X$ -computable separation $Y$ is not $X$ -computable.

That $X\leqslant _T Y$ can be shown using a very similar argument. Instead of $\varphi _e^X$ above, use the characteristic function $\varphi _d^X(x)$ which is $0$ if $x\in X$ and $1$ if $x\notin X$ . Then the rest of the argument above, replacing $e$ with $d$ , shows that $X$ is $Y$ -computable. So we have $Y\not \leqslant _T X$ and $X\leqslant _T Y$ , hence $X\lt _T Y$ .

From Theorem 5.2, we see that, in particular, $\mathcal{K}_1$ does not have a computable weak completion, which also follows from the fact that combinatorial algebras are never computable, see Barendregt (Reference Barendregt1984, 5.1.15). We now show that this is optimal, namely that there exist completions of low Turing degree. (Recall that $Y$ is low if $Y'\leqslant _T \emptyset '$ ).

6. Complete Extensions of $\mathrm{PA}$

Following modern terminology, we call a Turing degree a PA degree if it is the degree of a complete extension of Peano arithmetic (cf. Downey and Hirschfeldt Reference Downey and Hirschfeldt2010, p. 84). We will simply call a set PA-complete if it has PA degree.

In this section, we show that every (strong or weak) completion of $\mathcal{K}_1$ computes a PA degree, and vice versa. Since there exist PA-complete sets of low degree (Downey and Hirschfeldt Reference Downey and Hirschfeldt2010, p. 87), Theorem 5.3 follows from the statement of this equivalence; however, this does not make the proof of Theorem 5.3 superfluous, since the tree $T$ from its proof is used in the proof of the equivalence.

The following result strengthens Theorem 5.2.

Putting Proposition 6.1 and Theorem 6.2 together, we obtain the following characterization:

In the case of PA degrees, more is known, namely that they are closed upwards. We do not know whether the degrees of (weak or strong) completions of $\mathcal{K}_1$ are also upwards closed.

7. Nonstandard Models of $\mathrm{PA}$

As mentioned in Beeson (Reference Beeson1985, VI.2.5) and van Oosten (Reference van Oosten2008), every model $M$ of Peano Arithmetic $\mathrm{PA}$ defines a pca on $M$ , with application defined by

(6) \begin{equation} a\cdot b \!\downarrow = c \text{ if } M \models \varphi _a(b)\!\downarrow = c \end{equation}

for all $a,b,c\in M$ . By restricting Equation (6) to standard numbers $a,b,c\in \omega$ , we obtain a pca on $\omega$ , which we will call $\mathcal{K}_1(M)$ . Note that $\mathcal{K}_1(M)$ is just Kleene’s first model $\mathcal{K}_1$ “inside $M$ .” It is a pca because we can pick combinators $k,s \in \mathcal{K}_1$ as in Theorem 2.2 such that $\mathrm{PA}$ proves that they have the required properties.

Note that for $a,b,c\in \omega$ we have that $a\cdot b\!\downarrow = c$ in $\mathcal{K}_1$ if and only if $\exists y(T(a,b,y) \wedge U(y)=c)$ , where $T$ and $U$ are the primitive recursive predicate and function from Kleene’s normal form theorem (cf. Odifreddi Reference Odifreddi1989). In a nonstandard model $M$ , this $y$ can be nonstandard, so that more computations converge than in reality. In particular, in general, we have

(7) \begin{equation} \mathcal{K}_1 \models a\cdot b \!\downarrow = c \; \Longrightarrow \; \mathcal{K}_1(M) \models a\cdot b \!\downarrow = c, \end{equation}

but not conversely. For example, consider a model $M$ of $\mathrm{PA} + \neg \mathrm{con}(\mathrm{PA})$ , where $\mathrm{con}(\mathrm{PA})$ expresses the consistency of $\mathrm{PA}$ . Such models exist by Gödel’s second incompleteness theorem. If we consider the p.c. function $\varphi _a$ that on input $b$ searches for a proof of an inconsistency in $\mathrm{PA}$ , then the computation $a\cdot b$ will converge in $\mathcal{K}_1(M)$ , but not in $\mathcal{K}_1$ (assuming $\mathrm{PA}$ is consistent).

Note that by Equation (7), we have an embedding $\mathcal{K}_1 \hookrightarrow \mathcal{K}_1(M)$ for every model $M$ of $\mathrm{PA}$ . This is in fact a strong embedding, as the same combinators $s$ and $k$ can be used in $\mathcal{K}_1(M)$ .

Since $\mathrm{PA}$ proves that certain p.c. functions are nontotal, any model of $\mathrm{PA}$ has nontotal p.c. functions. This remains true if we restrict to standard numbers, that is, there are standard numbers $e,n\in \omega$ such that $\mathrm{PA}$ proves that $\varphi _e(n)$ never halts. In particular, we see that $\mathcal{K}_1(M)$ is never a total pca (i.e. a ca). Therefore, we have:

By Tennenbaum’s theorem (cf. Boolos and Jeffrey Reference Boolos and Jeffrey1974), there are no computable nonstandard models of $\mathrm{PA}$ . More precisely, there are no nonstandard models in which $+$ is computable. (This is an extension of Tennenbaum’s theorem due to Kreisel.) It follows that there are also no nonstandard models that are c.e., because $+$ is a total operation, and in a c.e. model, it would actually be computable, contradicting Kreisel’s result. So it would seem that the pcas $\mathcal{K}_1(M)$ for nonstandard models $M$ cannot be used for the problems about c.e. pcas discussed in computable structure theory (see Fokina and Terwijn Reference Fokina and Terwijn2024). However, for models $M_0$ and $M_1$ of $\mathrm{PA}$ , we do not have in general that

\begin{equation*} M_0 \not \cong M_1 \Longrightarrow \mathcal {K}_1(M_0) \not \cong \mathcal {K}_1(M_1). \end{equation*}

To see this, let $M_0=\omega$ be the standard model, and let $M_1$ be a nonstandard model that has the same first-order theory as $\omega$ (which exists by the compactness theorem). Then $M_0 \not \cong M_1$ , but $\mathcal{K}_1(M_0) = \mathcal{K}_1(M_1) = \mathcal{K}_1$ .

In particular, we see that it is possible that $\mathcal{K}_1(M)$ is c.e. (in the sense of Definition 3.2) for a nonstandard model $M$ of $\mathrm{PA}$ . This prompts the following question:

In the following, we note that though $\mathcal{K}_1(M)$ is always noncomputable, it can have low degree (if we do not require that it is also c.e.).

Proof. Consider the computably inseparable pair of sets

\begin{align*} A &= \{x\in \omega \mid \varphi _x(x)\!\downarrow =0\} \\ B &= \{x\in \omega \mid \varphi _x(x)\!\downarrow =1\}. \end{align*}

Now consider the set $C = \{x\in \omega \mid \mathcal{K}_1(M) \models x\cdot x\!\downarrow =0\}$ . $C$ is computable from $\mathcal{K}_1(M)$ , and it follows from Equation (7) that it separates $A$ and $B$ , from which it follows immediately that $\mathcal{K}_1(M)$ cannot be computable.

Proof. We have to show that there exists a model $M$ and a low set $Y$ such that $\mathcal{K}_1(M)$ is $Y$ -computable (in the sense of Definition 3.2), that is, such that the set

\begin{equation*} Z= \{(a,b,c)\mid \mathcal {K}_1(M)\models a\cdot b\!\downarrow = c\} \end{equation*}

is $Y$ -computable. According to Jockusch and Soare, there exists a complete and consistent extension $X$ of $\mathrm{PA}$ of low Turing degree. Let $M$ be a model with theory $X$ . We can identify the set $Z$ with the set of sentences $a\cdot b\!\downarrow = c$ that hold in $M$ . Since $Z$ is then just a subset of $X$ consisting of sentences of a specific form, there is a computable set $R$ such that $Z = R\cap X$ , and we have $(R\cap X)' \leqslant _T X'\leqslant _T \emptyset '$ so that $Z$ is low.