2.1 Theories

A theory is given as a set of axioms in a given signature. We take the signature to be part of the data of the theory. Note that we do not take a formula representing the axiom set or an index of the axiom set as part of the data. The same theory can have different enumerations. Moreover, these enumerations are enumerations of the axiom set and not of the theorems.

We write $U_{{\mathfrak {p}}}$ for the set of theorems of U and $U_{{\mathfrak {r}}}$ for the U-refutable sentences or anti-theorems of U, i.e., $U_{{\mathfrak {r}}} = \{ \varphi \mid U \vdash \neg \, \varphi \}$ .

We will also consider mono-consequence: $U \vdash _{{\mathfrak {m}}} \varphi $ iff there is a $\psi \in U$ such that $\psi \vdash \varphi $ . We have the corresponding notion of mono-consistency, which was developed by Lindström (cf. [Reference Lindström10]). We write $U_{{\mathfrak {m}}}$ for the set of mono-theorems of U and $U_{{\mathfrak {n}}}$ for the set of mono-refutable sentences.

Strictly speaking, there is no disjoint notion of mono-theory. A mono-theory is just a theory. However, sometimes we will still use the word to indicate that we intend to use the given set of sentences of the given signature with the notion of mono-consequence.

The notion of mono-theory plays a role in Theorem 2.9. Below we show the notion of mono-theory is connected to the idea of computable sequence of c.e. theories. There are heuristic differences between the notions. Sometimes it is better to think in terms of the ‘flatter’ notion of mono-theory, sometimes it is pleasant to visualise a sequence of theories.

We define $\widehat U$ as the set of all non-empty finite conjunctions of elements of U. We have the following convenient insight.

We note that we have $V \dashv U$ iff $V \dashv _{{\mathfrak {m}}} \widehat U$ . So, $\widehat {(\cdot )}$ is the right adjoint of the projection functor of theories (of a given signature) with $\dashv $ into theories (of the given signature) with $\dashv _{{\mathfrak {m}}}$ . We also note that, of course, $(\cdot )_{{\mathfrak {p}}}$ gives us an isomorphic functor. The advantage of $\widehat {(\cdot )}$ is the fact that it does not raise the complexity of the given set for most measures of complexity.

A computable sequence $(T_i)_{i\in \omega }$ of c.e. theories is given by a c.e. relation $\mathcal {T}(i,\varphi )$ . Here, of course, $T_i := \{ \varphi \mid \mathcal {T}(i,\varphi ) \}$ . We demand that all $T_i$ are in the same signature. We take this signature as part of the data of the sequence. A sequence of theories is consistent if each of its theories is.

Consider a computable sequence of c.e. theories, given by $\mathcal {T}(i,\varphi )$ . We define $\mathcal {T}^{{\mathfrak {m}}} := \bigcup _{i\in \omega } \widehat T_i$ . Clearly, $\mathcal {T}^{{\mathfrak {m}}}$ is a c.e. mono-theory. It is easy to see that $\mathcal {T}^{{\mathfrak {m}}} \vdash _{{\mathfrak {m}}}\varphi $ iff $T_i \vdash \varphi $ , for some i.

Conversely, given a c.e. mono-theory U we can associate a computable sequence $(T_i)_{i\in \omega }$ of c.e. theories as follows. Enumerate U in stages and, if, at stage i, a sentence $\varphi $ is enumerated, put $\varphi $ in $T_i$ . Clearly, $U \vdash _{{\mathfrak {m}}}\varphi $ iff $T_i \vdash \varphi $ , for some i.

Since metamathematical results on sequences of theories are mostly concerned with the relation $\exists i\in \omega \;T_i \vdash \varphi $ , we can usually replace sequences of theories given by $\mathcal {T}$ by mono-theories U and study $U \vdash _{{\mathfrak {m}}}\varphi $ .

2.2 Theory-extension

We may define various notions of theory-extension. The basic notion is simply $U\subseteq V$ : the V-axioms extend the U-axioms. Here the V-language may extend the U-language. We have three ‘dimensions’ of variation.

1. i. We can put restrictions on the V-language. We consider two possibilities. We use a superscript ${\mathfrak {s}}$ , for ‘same’ or for ‘signature’, to indicate that the U- and the V-language coincide. We use a superscript ${\mathfrak {c}}$ to indicate that the V-language extends the U-language by at most finitely many constants.

2. ii. We do not compare the axiom sets but appropriate closures of the axioms sets. When we compare the theorems, we indicate this by a subscript ${\mathfrak {p}}$ . We can also compare the mono-theorems. We indicate this by a subscript ${\mathfrak {m}}$ .

3. iii. We may put constraints on the cardinality of the extension. We use the subscript ${\mathfrak {f}}$ for finite extensions.

So we will use, e.g., $U\subseteq ^{{\mathfrak {s}}}_{{\mathfrak {f}}}V$ , for: V is a finite extension of U in the same language. If we use, e.g., $U\subseteq ^{{\mathfrak {s}}}_{{\mathfrak {pf}}}V$ , this is of course intended to mean that the theorems of V are theorems of a finite extension of U. We will use $U\dashv V$ for $U \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {p}}} V$ and $U\dashv _{{\mathfrak {m}}} V$ for $U \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} V$ .

We note that if $U_{{\mathfrak {p}}} = U^{\prime }_{{\mathfrak {p}}}$ and $V_{{\mathfrak {p}}} = V^{\prime }_{{\mathfrak {p}}}$ , then $U\subseteq ^{{\mathfrak {s}}}_{{\mathfrak {pf}}}V$ iff $U'\subseteq ^{{\mathfrak {s}}}_{{\mathfrak {pf}}}V'$ .

We will be looking at mono-extensions of non-mono theories. For this case the following notion of extension is a relevant one.

• • $U \Cup V := \{ \varphi \wedge \psi \mid \varphi \in U \text { and } \psi \in V \}$ .

• • $U \Subset V$ iff $U\Cup V \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} V$ .

Let $\mathcal {X}$ and $\mathcal {Y}$ be disjoint c.e. sets. Two sets $\mathcal {Z}$ and $\mathcal {W}$ weakly biseparate $\mathcal {X}$ and $\mathcal {Y}$ iff $\mathcal {X} \subseteq \mathcal {Z}$ , $\mathcal {Y} \subseteq \mathcal {W}$ , $\mathcal {Z} \cap \mathcal {Y} = \emptyset $ , and $\mathcal {W} \cap \mathcal {X} = \emptyset $ . We say that $\mathcal {Z}$ and $\mathcal {W}$ biseparate $\mathcal {X}$ and $\mathcal {Y}$ iff they weakly biseparate $\mathcal {X}$ and $\mathcal {Y}$ and $\mathcal {Z} \cap \mathcal {W} = \emptyset $ . We will not use the following theorem later, but we state it for the sake of understanding.

Proof (a). Suppose $U \Subset V$ . The inclusions $\widehat U \subseteq ^{{\mathfrak {s}}} U_{{\mathfrak {p}}} \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} U_{{\mathfrak {p}}} \Cup V$ are obvious. To prove $U_{{\mathfrak {p}}} \Cup V \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} V$ , it suffices to show that for any k, $\varphi _0, \ldots , \varphi _k \in U$ , and $\psi \in V$ , there exists a $\rho \in V$ such that

$$\begin{align*}\rho \vdash \varphi _0 \land \cdots \land \varphi _k \land \psi.\end{align*}$$

We prove the statement by induction on k, and the case of $k = 0$ is immediate from $U \Subset V$ . Assume that the statement holds for k and let $\varphi _0, \ldots , \varphi _k, \varphi _{k+1} \in U$ and $\psi \in V$ . By the induction hypothesis, there exists a $\rho \in V$ such that $\rho \vdash \varphi _0 \land \cdots \land \varphi _k \land \psi $ . Since $U \Subset V$ , there exists a $\rho ' \in V$ such that $\rho ' \vdash \varphi _{k+1} \land \rho $ . We obtain $\rho ' \vdash \varphi _0 \land \cdots \land \varphi _k \land \varphi _{k+1} \land \psi $ .

(b). Suppose $U \Subset V$ and V is mono-consistent. Since $U_{{\mathfrak {p}}} \subseteq _{{\mathfrak {m}}} V$ by (a), we have $U_{{\mathfrak {p}}} \subseteq V_{{\mathfrak {m}}}$ and $U_{{\mathfrak {r}}} \subseteq V_{{\mathfrak {n}}}$ . If $\xi \in U_{{\mathfrak {p}}} \cap V_{{\mathfrak {n}}}$ for some sentence $\xi $ , then there would be a $\psi \in V$ such that $\psi \vdash \neg \xi $ . Since $U_{{\mathfrak {p}}} \Cup V \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} V$ , there would be a $\rho \in V$ such that $\rho \vdash \xi \land \psi $ . Then, $\rho $ is inconsistent. This contradicts the mono-consistency of V. Therefore, $U_{{\mathfrak {p}}} \cap V_{{\mathfrak {n}}} = \emptyset $ . In a similar way, we can prove $U_{{\mathfrak {r}}} \cap V_{{\mathfrak {m}}} = \emptyset $ .

(c). By (a), $U \Subset U$ is equivalent to $U_{{\mathfrak {p}}} \Cup U \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} U$ , and to $U_{{\mathfrak {p}}} \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} U$ . Also, $U_{{\mathfrak {p}}} \subseteq ^{{\mathfrak {s}}}_{{\mathfrak {m}}} U$ is equivalent to $U_{{\mathfrak {m}}} = U_{{\mathfrak {p}}}$ .

2.3 Interpretability

An interpretation K of a theory U in a theory V is based on a translation of the U-language into the V-language. This translation commutes with the propositional connectives. In some broad sense, it also commutes with the quantifiers but here there are a number of extra features.

• • Translations may be more-dimensional: we allow a variable to be translated to an appropriate sequence of variables.

• • We may have domain relativisation: we allow the range of the translated quantifiers to be some domain definable in the V-language.

• • We may even allow the new domain to be built up from pieces of, possibly, different dimensions.

A further feature is that identity need not be translated to identity but can be translated to a congruence relation. Finally, we may also allow parameters in an interpretation. To handle these the translation may specify a parameter-domain. For details on the various kinds of translation, we refer the reader to [Reference Visser26].

We can define the obvious identity translation of a language in itself and composition of translations.

An interpretation is a triple ${\langle U,\tau ,V \rangle }$ , where $\tau $ is a translation of the U-language in the V-language such that, for all $\varphi $ , if $U \vdash \varphi $ , then $V \vdash \varphi ^\tau $ .Footnote ¹

We write:

• • $K:U \lhd V$ for: K is an interpretation of U in V.

• • $U \lhd V$ for: there is a K such that $K:U \lhd V$ . We also write $V\rhd U$ for: $U \lhd V$ .

• • $U \lhd _{\mathsf {loc}} V$ for: for every finitely axiomatisable sub-theory $U_0$ of U, we have $U_0 \lhd V$ .

• • $U \lhd _{\mathsf {mod}} V$ for: for every V-model $\mathcal {M}$ , there is a translation $\tau $ from the U-language in the V-language, such that $\tau $ defines an internal U-model $\mathcal {N} = \widetilde \tau (\mathcal {M})$ of U in $\mathcal {M}$ .

In Appendix B, we will have a brief look at effective versions of local interpretability, essentially concluding that all such versions collapse either to ordinary local interpretability or, somewhat surprisingly, to global interpretability.

In [Reference Visser28], the relation of essential tolerance was studied since it has backwards preservation of essential hereditary undecidability. In Appendix C, we briefly consider effective essential tolerance. This relation has backwards preservation of effective essential hereditary undecidability.

2.4 The non-effective notions

In this subsection, we introduce the non-effective notions. We will then discuss what the appropriate corresponding effective versions should be in the next subsection.

Our first building blocks are decidability, completeness, and separability.

• • A theory U is decidable if there is an algorithm that decides provability in U. In other words, U is decidable iff $U_{{\mathfrak {p}}}$ is computable. A theory is undecidable if it is not decidable.

• • A theory U is complete if, for every U-sentence $\varphi $ , we have $U\vdash \varphi $ or $U \vdash \neg \,\varphi $ . In other words, U is complete iff $U_{{\mathfrak {p}}} \cup U_{{\mathfrak {r}}}= \mathsf {Sent}_U$ . A theory is incomplete if it is not complete.

Suppose $\mathcal P$ is a property of theories. We say that U is essentially $\mathcal P$ if all consistent c.e. extensions (in the same language) of U are $\mathcal P$ . We say that U is hereditarily $\mathcal P$ if all consistent c.e. sub-theories of U (in the same language) are $\mathcal P$ . We say that U is potentially $\mathcal P$ if some consistent c.e. extension (in the same language) of U is $\mathcal P$ .

We defined essential and potential and hereditary with respect to $\subseteq ^{{\mathfrak {s}}}$ . In this paper we also will consider these notions with respect to $\subseteq ^{{\mathfrak {s}}}_{{\mathfrak {f}}}$ .

If $\mathcal R$ is a relation between theories the use of essential and hereditary and potential always concerns the first component aka the subject. Thus, e.g., we say that U essentially tolerates V meaning that U essentially has the property of tolerating V. Tolerance itself is defined as potential interpretation. So U essentially tolerates V if U essentially potentially interprets V. We will have a closer look at essential tolerance in Appendix C.

An important recursion theoretic notion is computable (in)separability. Two c.e. sets $\mathcal {X}$ and $\mathcal {Y}$ are computably separable iff, there is a computable $\mathcal {Z}$ such that $\mathcal {X}\subseteq \mathcal {Z}$ and $\mathcal {Y} \subseteq \mathcal {Z}^{\mathsf {c}}$ . Two sets are computably inseparable iff they are not computably separable. We want to apply computable (in)separability to theories and pairs of theories by designating certain sets of sentences associated with the theories as candidates for computable (in)separability.

Let us say that a pair of theories $(U,V)$ is acceptable iff U and V have the same signature and are jointly consistent. Let $(U,V)$ be acceptable. We define:

• • $(U,V)$ is computably (in)separable iff $U_{{\mathfrak {p}}}$ and $V_{{\mathfrak {r}}}$ are computably (in)separable.

• • U is computably (in)separable iff $(U,U)$ is computably (in)separable.

• • U is strongly computably (in)separable iff $(U, 0_U)$ is computably (in)separable.

Here, $0_U$ denotes the pure predicate calculus in the language of U.

We define: $(\mathcal {X},\mathcal {Y}) \leq _1 (\mathcal {Z},\mathcal {W})$ iff there is an injective computable function f, such that $n\in \mathcal {X}$ iff $f(n) \in \mathcal {Z}$ , and $n\in \mathcal {Y}$ iff $f(n) \in \mathcal {W}$ . Our definition generalises [Reference Soare20, Definition 2.4.9, p. 40], which coincides with our definition when we restrict ourselves to disjoint pairs of sets. We have $(\mathcal {X},\mathcal {X}) \leq _1 (\mathcal {Y},\mathcal {Y})$ iff $\mathcal {X}\leq _1\mathcal {Y}$ . Clearly, if $(\mathcal {X},\mathcal {Y})$ and $(\mathcal {Z},\mathcal {W})$ are disjoint pairs and if $(\mathcal {X},\mathcal {Y})$ is computably inseparable and $(\mathcal {X},\mathcal {Y}) \leq _1(\mathcal {Z},\mathcal {W})$ , then $(\mathcal {Z},\,\mathcal {W})$ is computably inseparable.

We have the following simple insights:

Proof The function $\psi \mapsto (\varphi \wedge \psi )$ witnesses that

$$\begin{align*}((U+\varphi)_{{\mathfrak {p}}},(V+\varphi)_{{\mathfrak {r}}})\leq_1 ((U+\varphi)_{{\mathfrak {p}}},V_{{\mathfrak {r}}}).\\[-37pt]\end{align*}$$

The computable inseparability of a theory is closely related to the undecidability and the incompleteness of the theory. Indeed, the computable inseparability of a theory U implies the essential undecidability of U. For example, the computable inseparability of the theory $\mathsf {R}$ of weak arithmetic follows from the work by Smullyan [Reference Smullyan14], and then the essential undecidability of $\mathsf {R}$ that was first established by Tarski, Mostowski, and Robinson [Reference Tarski, Mostowski and Robinson21] immediately follows. It is well-known that, for any consistent c.e. theory, essential incompleteness and essential undecidability are equivalent, and so the computable inseparability of $\mathsf {R}$ also yields the essential incompleteness of $\mathsf {R}$ . Here, the essential incompleteness of $\mathsf {R}$ is also strengthened. Mostowski [Reference Mostowski11] proved that $\mathsf {R}$ is uniformly essentially incomplete, that is, for any computable sequence $(T_i)_{i \in \omega }$ of consistent c.e. extensions of $\mathsf {R}$ , there exists a sentence simultaneously independent of all theories in the sequence. Interestingly, Ehrenfeucht [Reference Ehrenfeucht4] proved that Mostowski’s theorem is equivalent to the computable inseparability of $\mathsf {R}$ , namely, he proved that, for any consistent c.e. theory U, U is computably inseparable if and only if U is uniformly essentially incomplete.

Finitely axiomatisable theories sometimes behave well. Tarski, Mostowski, and Robinson [Reference Tarski, Mostowski and Robinson21] showed that for a finitely axiomatisable theory, essential undecidability is equivalent to essential hereditary undecidability. Also, it follows from the Subtraction Theorem (Theorem 2.4) that for a finitely axiomatised theory, computable inseparability is equivalent to strong computable inseparability. Note that strong effective inseparability implies essential hereditary undecidability. So, the finitely axiomatised theory $\mathsf {Q}$ which is an extension of $\mathsf {R}$ is strongly computably inseparable and essentially hereditarily undecidable. Here, since $\mathsf {R}$ is not finitely axiomatisable, the essential hereditary undecidability of $\mathsf {R}$ is non-trivial. This was proved by Cobham, but his proof of the result was not published. Vaught [Reference Vaught, Nagel, Suppes and Tarski22] gave a proof of Cobham’s theorem by proving the strong computable inseparability of $\mathsf {R}$ . For a detailed study of the notion of essential hereditary undecidability, see [Reference Visser28]. See [Reference Visser26] and [Reference Kurahashi and Visser9] for new proofs of Cobham and Vaught’s theorems.

Relating to these notions, we also introduce the following two notions:

• • U is f-essentially incomplete iff, for any U-sentence $\varphi $ , if $U \cup \{\varphi \}$ is consistent, then $U \cup \{\varphi \}$ is incomplete.

• • U is f-uniformly essentially incomplete iff, for any $k \in \omega $ , whenever $U_0$ , …, $U_{k-1}$ are consistent c.e. extensions of U in the same language, then there is a sentence independent of each of the $U_0, \ldots , U_{k-1}$ .

It is easy to see that a theory U is f-essentially incomplete iff the Lindenbaum algebra of U is atomless. For f-uniform essential incompleteness, we have the following:

The relationships between these non-effective notions are visualised in Figure 1. In [Reference Murwanashyaka, Pakhomov and Visser12], the existence of a decidable f-essentially incomplete theory was proved. Also, essential hereditary undecidability and computable inseparability are incomparable in general (cf. [Reference Visser28, Example 6]). Therefore, none of the implications in Figure 1 are reversible. Related to this figure, one could consider the notions such as f-essential undecidability and f-essential hereditary undecidability etc., but we will not deal with these notions, as they are beside the main subject of this paper.

Figure 1

Implications between non-effective notions.

In what follows, we explore the effectivisations of these notions of incompleteness, undecidability, and inseparability.

2.5 What is effective?

Notions like essential and hereditary operate extensionally on the notions they modify. The situation is not so simple for adding effective. Adding “effective” in front of an expression operates intensionally on the definition of the concept.

Suppose the definition of $\mathcal {P}$ has the form $\forall \vec {x}\,\exists \vec {y}\, \varphi (\vec {x},\vec {y})$ , where $\varphi $ does not start with an existential quantifier. We propose to say that effectively $\mathcal {P}$ means that there are computable functions $\vec {\varPhi }$ such that $\forall \vec {x}\, \varphi (\vec {x},\vec {\varPhi }(\vec {x}))$ . If the given definition of $\mathcal {P}$ has the form $\forall \vec {x}\,(\psi (\vec {x}) \to \exists \vec {y}\, \varphi (\vec {x},\vec {y}))$ , where $\varphi $ does not start with an existential quantifier, then, effectively $\mathcal {P}$ means that there are partial computable functions $\vec {\varPhi }$ such that

$$\begin{align*}\forall \vec{x}\, (\psi(\vec{x}) \to \exists \vec{z}\, (\vec{\varPhi}(\vec{x})\simeq \vec{z} \wedge \varphi(\vec{x},\vec{z}))).\end{align*}$$

2.5.1 Effective undecidability

A c.e. set $\mathcal {X}$ is decidable iff

$$\begin{align*}\exists i \, ((\mathcal {X} \cap \mathsf {W}_i) = \emptyset \wedge (\mathcal {X} \cup \mathsf {W}_i) = \omega ).\end{align*}$$

So, $\mathcal {X}$ is undecidable means:

$$\begin{align*}\forall i\, (\exists y\, (y \in \mathcal {X} \wedge y\in \mathsf{W}_i) \vee \exists x\, (x\not\in \mathcal {X} \wedge x \not\in \mathsf{W}_i)).\end{align*}$$

Equivalently,

$$\begin{align*}\forall i\, \exists x\, ((x \in \mathcal {X} \wedge x\in \mathsf{W}_i) \vee (x\not\in \mathcal {X} \wedge x \not\in \mathsf{W}_i)).\end{align*}$$

The constructivisation of this is: there is a computable $\varPhi $ such that

$$\begin{align*}\forall i\, ((\varPhi(i) \in \mathcal {X} \wedge \varPhi(i)\in \mathsf{W}_i) \vee (\varPhi(i)\not\in \mathcal {X} \wedge \varPhi(i) \not\in \mathsf{W}_i)).\end{align*}$$

So this is the notion of being constructively non-computable (see [Reference Rogers13, p. 162]).

Alternatively, $\mathcal {X}$ is undecidable also means:

$$\begin{align*}\forall i\, \exists y\, ((\mathcal {X} \cap \mathsf{W}_i) = \emptyset \Rightarrow (y \not \in \mathcal {X} \wedge y \not \in \mathsf{W}_i)).\end{align*}$$

So, the effectivisation of this is: there is a computable $\varPhi $ such that

$$\begin{align*}\forall i\, ((\mathcal {X} \cap \mathsf{W}_i) = \emptyset \Rightarrow (\varPhi(i) \not \in \mathcal {X} \wedge \varPhi(i) \not \in \mathsf{W}_i)).\end{align*}$$

This is exactly the notion of being creative.

Every constructively non-computable set is exactly a c.e. set whose complement is completely productive, which is a notion introduced by Dekker [Reference Dekker3]. It is proved in [Reference Rogers13, p. 183, Theorem VI] that productivity and complete productivity coincide, and hence creativity and constructive non-computability also coincide. So, these notions serve stable effectivisation of the notion of undecidability.

2.5.2 Effective essential undecidability

Let us assume the definition of the essential undecidability of U is

$$ \begin{align*} \forall i\,\forall j \, \Big(\big(\mathsf{W}_{i} \vdash U\text{ and } &\mathsf{con}(\mathsf{W}_i)\big)\; \Rightarrow \\ & \exists x\, \big((x\not\in \mathsf{W}_{i{\mathfrak {p}}} \wedge x \not\in \mathsf{W}_j) \vee (x \in \mathsf{W}_{i{\mathfrak {p}}} \wedge x\in \mathsf{W}_j)\big)\Big). \end{align*} $$

Here $\mathsf {con}(\mathsf {W}_i)$ is an abbreviation of the statement ‘ $\mathsf {W}_i$ is consistent’. So our recipe gives: there is a partial computable $\varPsi $ such that

$$ \begin{align*} & \forall i\,\forall j \, \Big(\big(\mathsf{W}_{i} \vdash U\text{ and } \mathsf{con}(\mathsf{W}_i)\big) \Rightarrow \\ & \big(\varPsi(i,j) {\downarrow} \wedge \big((\varPsi(i,j)\not\in \mathsf{W}_{i{\mathfrak {p}}} \wedge \varPsi(i,j) \not\in \mathsf{W}_j) \vee (\varPsi(i,j) \in \mathsf{W}_{i{\mathfrak {p}}} \wedge \varPsi(i,j)\in \mathsf{W}_j)\big)\big)\Big). \end{align*} $$

By the usual argument, we can always choose $\varPsi $ total. Moreover, our definition is equivalent to: there is a total computable $\varTheta $ such that

$$\begin{align*}\forall i \, \big((\mathsf{W}_{i} \vdash U\text{ and } \mathsf{con}(\mathsf{W}_i)\big) \Rightarrow \big( \lambda j.\varTheta(i,j) \text{ witnesses that}\ \mathsf{W}_{i{\mathfrak {p}}}\ \text{is creative)}\big). \end{align*}$$

So this gives us that effective essential undecidability is the same thing as effective essential creativity. We will work with the last notion. This notion was suggested in Feferman’s paper [Reference Feferman5, Footnote 11] and investigated by Smullyan [Reference Smullyan15].

In the rest of the paper, we will simply stipulate the effective versions of the relevant notions. The reader may amuse herself by deriving the definitions following our recipe. We briefly discuss why there is not separate notion of effective local interpretability in Appendix B.

2.6 Effective inseparability

Two disjoint c.e. sets $\mathcal {X}$ and $\mathcal {Y}$ are said to be effectively inseparable iff, there exists a partial computable function $\varPhi $ such that for any c.e. sets $\mathsf {W}_i$ and $\mathsf {W}_j$ , if $\mathsf {W}_i$ and $\mathsf {W}_j$ weakly bi-separate $\mathcal {X}$ and $\mathcal {Y}$ , then $\varPhi (i, j)$ converges and $\varPhi (i, j) \notin \mathsf {W}_i \cup \mathsf {W}_j$ . Let $(U,V)$ be acceptable pair of theories. We define:

• • $(U,V)$ is effectively inseparable iff $U_{{\mathfrak {p}}}$ and $V_{{\mathfrak {r}}}$ are effectively inseparable.

• • U is effectively inseparable iff $(U,U)$ is effectively inseparable.

• • U is strongly effectively inseparable iff $(U, 0_U)$ is effectively inseparable.

We define witness comparison notation. For every c.e. relation $R(\vec {x})$ , we can effectively find a primitive computable relation $R^\star (\vec {x}, y)$ such that $R(\vec {x})$ iff $\exists y\, R^\star (\vec {x}, y)$ . For all pairs of c.e. relations $R_0(\vec {x})$ and $R_1(\vec {x})$ , we define:

• • $R_0(\vec {x}) \leq R_1(\vec {x}) :\leftrightarrow \exists y\, (R_0^\star (\vec {x}, y) \wedge \forall z < y\, \neg \,R_1^\star (\vec {x}, z))$ ,

• • $R_0(\vec {x}) < R_1(\vec {x}) :\leftrightarrow \exists y\, (R_0^\star (\vec {x}, y) \wedge \forall z \leq y\, \neg \,R_1^\star (\vec {x}, z))$ .

We note that the witness comparison notation is intensional. The procedure to find $R^\star $ given R operates on a presentation of R and gives a presentation of $R^\star $ as output.

Smullyan analyzed the notion of effective inseparability extensively, and in particular various notions related to effective inseparability are discussed in his [Reference Smullyan17, Reference Smullyan18]. For more information on this topic, see the recent paper by Yong Cheng [Reference Cheng2].

The effective inseparability of the theory $\mathsf {R}$ follows from a result established by Smullyan [Reference Smullyan15]. Smullyan mentioned that effective inseparability implies effective essential creativity. Also, for any consistent c.e. theory, effective essential creativity clearly implies effective essential incompleteness. Ehrenfeucht [Reference Ehrenfeucht4] provided an essentially incomplete theory that is not computably inseparable, but, in the effective case, such an example cannot exist. Namely, for any consistent c.e. theory, effective essential incompleteness implies effective inseparability. This result was established by Pour-El. Actually, Pour-El proved more. We say that a theory U is effectively if-essentially $\mathcal {F}$ -incomplete iff there is a partial computable function $\varPhi $ such that for any i, if $\mathsf {W}_i$ is a consistent finite extension of U, then $\varPhi (i)$ converges, $\varPhi (i) \in \mathcal {F}(\mathsf {W}_i)$ , and $\varPhi (i)$ is independent of $\mathsf {W}_i$ . Here, ‘if’ stands for ‘intensional finite extensions’. The notion of effective ef-essential $\mathcal {F}$ -incompleteness is studied in Section 4, where ‘ef’ stands for ‘extensional finite extensions’. Pour-El called effective essential incompleteness effective extensibility. She called effective if-essential incompleteness weak effective extensibility. Pour-El’s theorem is stated as follows:

In the next section, we prove a slightly strengthened version of Pour-El’s theorem (Theorem 3.1). Furthermore, our proof is slightly simpler than the original one. For completeness, we describe Pour-El’s original argument in Appendix A.

We say that a theory U is effectively uniformly essentially $\mathcal {X}$ -incomplete iff, there is a partial computable function $\varPhi $ such that for any j, if j is the index of a computable sequence of consistent c.e. extensions $(T_i)_{i \in \omega }$ of U, then we have that $\varPhi (j)$ converges, $\varPhi (j)\in \mathcal {X}$ , and $\varPhi (j)$ is independent of $T_i$ for all $i \in \omega $ . For any consistent c.e. theory, effective inseparability easily implies effective uniform essential incompleteness. So, one could say that, in the effective case, Ehrenfeucht’s result on the equivalence between computable inseparability and uniform essential incompleteness is superseded by Pour-El’s work. However, we do think it is instructive to include the effective analogues of Ehrenfeucht’s theorem here, also since these results have entirely self-reference free proofs. We also present the effective version of Ehrenfeucht’s results from the mono-perspective.

Proof “(a) to (b)”. Consider a computable sequence $(U^{\prime }_i)_{i\in \omega }$ of consistent c.e. extensions of U. Say our sequence has index j. Let $V:= \bigcup _{i\in \omega } U^{\prime }_{i{\mathfrak {p}}}$ and let $W := \bigcup _{i\in \omega } U^{\prime }_{i{\mathfrak {r}}}$ . We can find indices k and $\ell $ for V and W effectively from j. Clearly, V and W weakly bi-separate $U_{{\mathfrak {p}}}$ and $U_{{\mathfrak {r}}}$ . We take $ \varPsi _0(j) := \varPhi (k,\ell )$ . It follows that $ \varPsi _0(j)$ is in $\mathcal {X}$ and is independent of each $U^{\prime }_i$ .

“(b) to (c)”. We can effectively transform an index j of the sequence $(\nu _i)_{i\in \omega }$ into an index $j'$ of the sequence of theories $(U+\nu _i)_{i\in \omega }$ . We set $\varPsi _1(j) := \varPsi _0(j')$ .

“(c) to (d)”. We can take $\varPsi _2:= \varPsi _1$ .

“(d) to (e)”. Suppose $\mathsf {W}_j \Supset U$ and $\mathsf {W}_j$ is mono-consistent. We can effectively find an index k of an enumeration $(\chi _s)_{s\in \omega }$ of the elements of $\mathsf {W}_j$ from j. Since $\mathsf {W}_j$ and $U_{{\mathfrak {r}}}$ are disjoint, we have that $\chi _s$ is consistent with U for each $s \in \omega $ . We obtain that $\varPsi _2(k)$ converges, $\varPsi _2(k) \in \mathcal {X}$ , and for all s, $0_U \nvdash \chi _s \to \varPsi _2(k)$ and $0_U \nvdash \chi _s \to \neg \, \varPsi _2(k)$ . We then have $\varPsi _2(k) \notin \mathsf {W}_{j {\mathfrak {m}}} \cup \mathsf {W}_{j {\mathfrak {n}}}$ . We take $\varPsi _3(j) := \varPsi _2(k)$ .

“(e) to (f)”. Consider any c.e. $\mathsf {W}_j$ such that $W:=\widehat U \Cup \mathsf {W}_j$ is mono-consistent. It is easy to see that $W \Supset U$ . We can easily find an index k of W from j. We take $\varPsi _4(j) := \varPsi _3(k)$ .

“(f) to (a)”. Suppose $\mathsf {W}_i$ and $\mathsf {W}_j$ weakly bi-separate $U_{{\mathfrak {p}}}$ and $U_{{\mathfrak {r}}}$ . Let

$$\begin{align*}V := \mathsf{W}_i \cup \{ \chi \mid (\neg\,\chi) \in \mathsf{W}_j \}.\end{align*}$$

If $\widehat U \Cup V$ were mono-inconsistent, there would be a $\nu \in V$ and $U \vdash \neg \, \nu $ . The second conjunct tells us that $ \nu \not \in \mathsf {W}_i$ and $(\neg \,\nu )\not \in \mathsf {W}_j$ . A contradiction. So, $\widehat U \Cup V$ is mono-consistent. We clearly can find an index k of V effectively from i and j. We take $\varPhi (i,j) := \varPsi _4(k)$ .

We prove the equivalence between (a) and (a $'$ ). The (a $'$ )-to-(a) direction is trivial. We assume (a). Consider any pair of indices $i,j$ . We define $\varPhi ^\ast $ . We can effectively find indices $i',j'$ of $U_{{\mathfrak {p}}} \cup \mathsf {W}_i$ and $U_{{\mathfrak {r}}} \cup \mathsf {W}_j$ . We compute $\varPhi (i',j')$ and, simultaneously, we seek a counterexample to the claim that $\mathsf {W}_{i'},\mathsf {W}_{j'}$ weakly biseparate $\mathsf {U}_{{\mathfrak {p}}}$ and $U_{{\mathfrak {r}}}$ . If we find a value of $\varPhi (i',j')$ first then we give that as output of $\varPhi ^\ast (i,j)$ . If we find a counterexample first, we give that as output (or, alternatively, we give some fixed chosen sentence as output). It is easy to see that $\varPhi ^\ast $ is total and satisfies our specification.

The equivalences of (b) and (b $'$ ), (c) and (c $'$ ), (d) and (d $'$ ), (e) and (e $'$ ), and (f) and (f $'$ ) are proved in an analogous way.

3.1 The Theorem

In this subsection, we give our version of Pour-El’s result.

Proof Let U be effectively if-essentially $\mathcal {F}$ -incomplete with partial computable witness $\varPhi $ . Suppose $\mathsf {W}_i$ and $\mathsf {W}_j$ weakly biseparate $U_{{\mathfrak {p}}}$ and $U_{{\mathfrak {r}}}$ .

Using the Recursion Theorem with parameters (cf. [Reference Soare19, Theorem 3.5]), we effectively find a $k^\ast $ (depending of i and j) such that

$$ \begin{align*} & \mathsf{W}_{k^\ast} = U \cup \{ \varphi \mid \varPhi(k^\ast)\simeq \varphi \text{ and }\varphi \in \mathsf{W}_i \}\; \cup \\ & \{ \psi \mid \exists \varphi \,(\varPhi(k^\ast)\simeq \varphi \text{ and }\varphi \in \mathsf{W}_j \text{ and } \psi = \neg\,\varphi) \}. \end{align*} $$

We note that we have

$$\begin{align*}\mathsf{W}_{k^\ast} = \begin{cases} U \cup \{\varPhi(k^\ast)\}, & \ \text{if}\ \varPhi(k^\ast)\ \text{converges and}\ \varPhi(k^\ast) \in \mathsf{W}_i,\\ U \cup \{\neg\, \varPhi(k^\ast)\}, & \ \ \text{if}\ \varPhi(k^\ast)\ \text{converges and}\ \varPhi(k^\ast) \in \mathsf{W}_j, \\ U, & \ \text{otherwise}. \end{cases} \end{align*}$$

Let $U^\ast :=\mathsf {W}_{k^\ast }$ .

• • Suppose $\varPhi (k^\ast )$ converges to, say, $\varphi ^\ast $ and $\varphi ^\ast \in \mathsf {W}_i$ . It follows that $U^\ast = U\cup \{ \varphi ^\ast \}$ . Since $\varphi ^\ast \not \in U_{{\mathfrak {r}}}$ , we have that $U^\ast $ is consistent and, hence, $\varphi ^\ast $ is independent of $U^\ast $ . A contradiction.

• • Suppose $\varPhi (k^\ast )$ converges to, say, $\varphi ^\ast $ and $\varphi ^\ast \in \mathsf {W}_j$ . It follows that $U^\ast = U\cup \{ \neg \,\varphi ^\ast \}$ . Since $\varphi ^\ast \not \in U_{{\mathfrak {p}}}$ , we have that $U^\ast $ is consistent, and, hence, $\varphi ^\ast $ is independent of $U^\ast $ . A contradiction.

We may conclude that $U^\ast = U$ . Hence, $\varPhi (k^\ast )$ converges, say to $\varphi ^\ast $ . We have $\varphi ^\ast \not \in \mathsf {W}_i\cup \mathsf {W}_j$ .

Clearly, our argument delivers a total computable $\varPsi $ , such that, whenever $\mathsf {W}_i$ and $\mathsf {W}_j$ weakly biseparate $U_{{\mathfrak {p}}}$ and $U_{{\mathfrak {r}}}$ , we have $\varPsi (i,j) \not \in \mathsf {W}_i\cup \mathsf {W}_j$ .

Finally, we note that the $\mathsf {W}_{k^\ast }$ are all equal to U and, thus $\varPsi (i,j)= \varPhi (k^\ast ) \in \mathcal {F}(U)$ .

As a consequence of Theorem 3.1, we obtain the following corollary showing the equivalence of several relating notions. For each formula class $\mathcal {X}$ and theory U, let $[\mathcal {X}]_U$ be the closure of $\mathcal {X}$ under U-provable equivalence.

3.2 A variant: double generativity

In Smullyan’s book [Reference Smullyan18], many notions that are equivalent to effective inseparability were introduced (see also [Reference Cheng2]). As a sample of variants of the argument in Section 3.1, we focus on double generativity among them as the double analogue of constructive non-computability, and discuss its witness constraining version.

We say that a theory U is doubly $\mathcal {X}$ -generative iff there exists a total computable function $\varPhi $ such that for any $i, j \in \omega $ , if $\mathsf {W}_i \cap \mathsf {W}_j = \emptyset $ , then:

• • $\varPhi (i, j) \in U_{{\mathfrak {p}}}$ iff $\varPhi (i, j) \in \mathsf {W}_j$ ,

• • $\varPhi (i, j) \in U_{{\mathfrak {r}}}$ iff $\varPhi (i, j) \in \mathsf {W}_i$ ,

• • if $\varPhi (i, j) \notin \mathsf {W}_i \cup \mathsf {W}_j$ , then $\varPhi (i, j) \in \mathcal {X}$ .

We obtain a prima facie less constrictive notion if we demand that the $\mathsf {W}_i$ , $\mathsf {W}_j$ are the $V_{{\mathfrak {p}}}$ , $V_{{\mathfrak {r}}}$ of some theory V. A consistent c.e. theory U is $\mathcal {X}$ -theory-generative iff, there is a total computable function $\varPsi $ , such for every consistent theory V in the U-language with index i, we have:

• • $\varPsi (i) \in U_{{\mathfrak {p}}}$ iff $\varPsi (i) \in V_{{\mathfrak {r}}}$ ,

• • $\varPsi (i) \in U_{{\mathfrak {r}}}$ iff $\varPsi (i) \in V_{{\mathfrak {p}}}$ ,

• • if $\varPsi (i) \notin V_{{\mathfrak {p}}} \cup V_{{\mathfrak {r}}}$ , then $\varPsi (i) \in \mathcal {X}$ .

Proof “(a) to (b)”. Suppose that a total computable function $\varPhi $ witnesses the double $\mathcal {X}$ -generativity of U. Let V be any consistent c.e. theory with index i. We can effectively find $k_0$ and $k_1$ from i such that $\mathsf {W}_{k_0} =V_{{\mathfrak {p}}}$ and $\mathsf {W}_{k_1} = V_{{\mathfrak {r}}}$ . Let $\varPsi (i) := \varPhi (k_0,k_1)$ . It is immediate that $\varPhi $ witnesses that U is $\mathcal {X}$ -theory-generative.

“(b) to (c)”. Suppose $\varPhi $ witnesses that U is $\mathcal {X}$ -theory-generative and let V be a consistent c.e. theory that if-extends U with index i. Since the first two cases of $\mathcal {X}$ -theory-generativity cannot be active, it follows that $\varPhi (i)\not \in V_{{\mathfrak {p}}} \cup V_{{\mathfrak {r}}}$ and $\varPhi (i) \in \mathcal {X}$ .

“(c) to (a)”. Suppose that U is effectively if-essentially $\mathcal {X}$ -incomplete. Let $\varPhi $ be a partial computable function that witnesses the effective if-essential $\mathcal {X}$ -incompleteness of U. We may assume that $\varPhi (k)$ converges if $\mathsf {W}_k$ is an if-extension of U, whether $\mathsf {W}_k$ is consistent or not. By the Recursion Theorem with parameters, there exists a computable function $\varPsi (x, y)$ such that, setting $\varPhi ^*(x, y) : = \varPhi (\varPsi (x, y))$ , we have

$$\begin{align*}\mathsf{W}_{\varPsi(x, y)} = \begin{cases} U \cup \{\varPhi^*(x, y)\}, & \ \text{if}\ \big(\varPhi^*(x, y) \in (U_{{\mathfrak {p}}} \cup \mathsf{W}_{x}) \big) \leq \big(\varPhi^*(x, y) \in (U_{{\mathfrak {r}}} \cup \mathsf{W}_{y}) \big),\\ U \cup \{\neg \,\varPhi^*(x, y)\}, & \ \text{if}\ \big(\varPhi^*(x, y) \in (U_{{\mathfrak {r}}} \cup \mathsf{W}_{y}) \big) < \big(\varPhi^*(x, y) \in (U_{{\mathfrak {p}}} \cup \mathsf{W}_{x}) \big), \\ U, & \ \text{otherwise}. \end{cases} \end{align*}$$

Since $\mathsf {W}_{\varPsi (i, j)}$ is an if-essential extension of U for all i and j, we find that $\varPhi ^*(i, j)$ is a total computable function.

We show that $\varPhi ^*$ witnesses the double $\mathcal {X}$ -generativity of U. Let i and j be such that $\mathsf {W}_i \cap \mathsf {W}_j = \emptyset $ and let $\varphi ^\ast := \varPhi ^\ast (i,j).$

• • Suppose $\big (\varphi ^* \in (U_{{\mathfrak {p}}} \cup \mathsf {W}_{i}) \big ) \leq \big (\varphi ^* \in (U_{{\mathfrak {r}}} \cup \mathsf {W}_{j}) \big )$ holds. Then, $\mathsf {W}_{\varPsi (i, j)} = U \cup \{\varphi ^*\}$ . Since $\mathsf {W}_{\varPsi (i, j)} \vdash \varPhi (\varPsi (i, j))$ , we have that $\mathsf {W}_{\varPsi (i, j)} = U \cup \{\varphi ^*\}$ is inconsistent. So, $\varphi ^* \in U_{{\mathfrak {r}}}$ . Since $U_{{\mathfrak {p}}} \cap U_{{\mathfrak {r}}} = \emptyset $ and $\varphi ^* \in U_{{\mathfrak {p}}} \cup \mathsf {W}_i$ , we obtain $\varphi ^* \in U_{{\mathfrak {r}}} \cap \mathsf {W}_i$ .

• • Suppose $\big (\varphi ^* \in (U_{{\mathfrak {r}}} \cup \mathsf {W}_{j}) \big ) < \big (\varphi ^* \in (U_{{\mathfrak {p}}} \cup \mathsf {W}_{i}) \big )$ holds. As above, it is shown that $\varphi ^* \in U_{{\mathfrak {p}}} \cap \mathsf {W}_j$ .

• • Otherwise, $\varphi ^* \notin U_{{\mathfrak {p}}} \cup \mathsf {W}_j$ and $\varphi ^* \notin U_{{\mathfrak {r}}} \cup \mathsf {W}_i$ . Since $\mathsf {W}_{\varPsi (i, j)} = U$ is a consistent if-essential extension of U, we have $\varphi ^* = \varPhi (\varPsi (i, j)) \in \mathcal {X}$ .

A simple exercise in propositional logic now shows that $\varPhi ^*$ witnesses the double $\mathcal {X}$ -generativity of U.

3.3 Orey-sentences of extensions of Peano Arithmetic

In this subsection we treat Orey-sentences for extensions of PA. The results of Section 3.4 will extend these results, but it is nice to see the simple case first.

Let U be any consistent c.e. extension of Peano Arithmetic. An Orey-sentence of U is a sentence O, such that $U \rhd (U+O)$ and $U \rhd (U+ \neg \, O)$ . Clearly, an Orey sentence O of U is independent of U. We also note that the Orey property is extensional in the sense that it only depends on the theorems of the given theory.

Here is one way to construct an Orey-sentence for U. Let $\mathsf {W}_i$ be an enumeration of the axioms of U. We define $\mathsf {W}_{i,n}$ as the theory axiomatised by the first n axioms enumerated in $\mathsf {W}_i$ . We define

Here stands for provability in $\mathsf {W}_{i,x}$ and stands for , so, arithmetizes the consistency of $\mathsf {W}_{i,n}$ . We write ${\triangledown }$ for $\neg {\vartriangle }\neg $ . We find that ${\vartriangle }_i$ satisfies the modal laws of K. Moreover, we have the seriality axiom D, i.e., ${\triangledown }_i\top $ . Finally, we can prove ${\triangledown }_i\varphi \rhd _U \varphi $ (see [Reference Feferman6] or [Reference Visser27]).

We find $\gamma _i$ with $\mathsf {PA} \vdash \gamma _i \leftrightarrow \neg \, {\vartriangle }_i \gamma _i$ . We claim that $\gamma _i$ is an Orey-sentence of U. We have, temporarily omitting the subscripts i and U:

$$ \begin{align*} \gamma & \vdash \neg\,{\vartriangle} \gamma \\ & \vdash {\triangledown} \neg\, \gamma \\ & \rhd \neg\, \gamma \\ \neg \, \gamma & \vdash {\vartriangle} \gamma \\ & \vdash {\triangledown} \gamma \\ & \rhd \gamma. \end{align*} $$

Since, we have $\gamma \rhd \neg \, \gamma $ and $\neg \,\gamma \rhd \neg \, \gamma $ , we find, using a disjunctive interpretation, $\top \rhd \neg \, \gamma $ . Similarly, we find $\top \rhd \gamma $ . Thus, $\gamma _i$ is an Orey-sentence of U as promised.

Let $\mathbb O$ be the function that assigns to sets of sentences $\mathcal {A}$ in the signature of arithmetic the Orey-sentences of the theory axiomatised by $\mathcal {A}$ . Note that it is possible that there are no such Orey-sentences, so the empty set will be in the range of this function. We have shown the following:

Applying Corollary 3.3, we find the following:

Theorem 3.7 illustrates the important insight that independence,per se, has nothing to do with strength. Of course, we are familiar with this point, e.g., from the well-known results on, e.g., the continuum hypothesis which is an Orey-sentence of ZF that is independent of many extensions that have to do with strength. However, Theorem 3.7 has a somewhat different flavor in that it presents a systematic construction of such sentences for a wide range of theories.

3.4 Consistency and conservativity

We say a formula $\alpha (x)$ is a binumeration of a theory U iff for any U-sentence $\varphi $ , if $\varphi \in U$ , then $U \vdash \alpha (\ulcorner {\varphi }\urcorner )$ ; and if $\varphi \notin U$ , then $U \vdash \neg \, \alpha (\ulcorner {\varphi }\urcorner )$ . For each binumeration $\alpha (x)$ of U, we can naturally construct a formula $\mathsf {Prf}_{\alpha }(x, y)$ saying that y is the code of a proof of a formula with the code x in the theory defined by $\alpha $ (cf. [Reference Feferman6]). Let $\mathsf {Con}_\alpha $ be the consistency statement $\neg \, \exists y\, \mathsf {Prf}_\alpha (\ulcorner {\bot }\urcorner , y)$ of U, where $\bot $ is some U-refutable sentence. Let $\mathbb C$ be the function that assigns to sets of sentences $\mathcal {A}$ in the signature of arithmetic the set of all sentences of the form $\mathsf {Con}_\alpha $ , where $\alpha $ is a binumeration of some computable axiomatisation of an extension A of $\mathsf {PA}$ axiomatised by $\mathcal {A}$ . It is known that $\mathsf {PA}$ is effectively essentially $\mathbb {C}$ -incomplete (cf. Lindström [Reference Lindström10, Theorem 2.8]). Thus, $\mathsf {PA}$ is effectively $\mathbb {C}(\mathsf {PA})$ -inseparable and effectively essentially $\mathbb {C}(\mathsf {PA})$ -incomplete.

For each $n \geq 1$ , let $\mathbb D_n$ be the function that assigns to sets of sentences $\mathcal {A}$ in the signature of arithmetic the set of all sentences $\varphi $ such that $\varphi $ and $\neg \, \varphi $ are both $\Pi _n$ -conservative over the theory axiomatised by $\mathcal {A}$ . It is well-known that for any consistent c.e. extension U of $\mathsf {PA}$ , $\mathbb D_1(U) = \mathbb {O}(U)$ (cf. [Reference Lindström10, Theorem 6.6]). For every pair of functions $\mathcal {F}$ and $\mathcal {G}$ from sets of sentences to sets of sentences, let $\mathcal {F} \cap \mathcal {G}$ be the function defined by $(\mathcal {F} \cap \mathcal {G})(\mathcal {A}) = \mathcal {F}(\mathcal {A}) \cap \mathcal {G}(\mathcal {A})$ .

For each binumeration $\alpha $ of an extension U of $\mathsf {PA}$ , the formula $\mathsf {Prf}_{\alpha }^{\Sigma _n}(x, y)$ is defined as

$$\begin{align*}\exists u \leq y\, (\Sigma_n(u) \land \mathsf{true}_{\Sigma_n}(u) \land \mathsf{Prf}_\alpha(u \dot{\to} x, y)). \end{align*}$$

Then, we have the following:

Proposition 3.8 (The small reflection principle (cf. [Reference Lindström10, Lemma 5.1(ii)])).

Let U be any computable extension of $\mathsf {PA}$ and $\alpha $ be a binumeration of U. Then, for any sentence $\varphi $ and natural number m, we have

$$\begin{align*}U \vdash \exists y < \underline{m}\, \mathsf{Prf}_{\alpha}^{\Sigma_n}(\ulcorner{\varphi}\urcorner, y) \to \varphi.\end{align*}$$

Proof Suppose that $\mathsf {W}_i$ is a consistent c.e. extension of $\mathsf {PA}$ . By Craig’s trick, we can effectively find a k from i such that $\mathsf {W}_k$ is a primitive computable axiomatisation of $\mathsf {W}_i$ . Let $\beta (x)$ be an effectively found primitive computable binumeration of $\mathsf {W}_k$ . We can effectively find a formula $\alpha (x)$ such that

$$\begin{align*}\mathsf{PA} \vdash \alpha(x) \leftrightarrow \Big( \big(\beta(x) \lor \exists y < x\, \mathsf{Prf}_{\beta}^{\Sigma_n}(\ulcorner{\mathsf{Con}_\alpha}\urcorner, y) \big) \land \forall z < x\, \neg\, \mathsf{Prf}_{\beta}^{\Sigma_n}(\ulcorner{\neg\, \mathsf{Con}_\alpha}\urcorner, z) \Big). \end{align*}$$

We show that $\mathsf {Con}_\alpha $ is $\Pi _n$ -conservative over $\mathsf {W}_k$ . Let $\pi $ be any $\Pi _n$ sentence such that $\mathsf {W}_k \cup \{\mathsf {Con}_\alpha \} \vdash \pi $ . Then, $\mathsf {W}_k \cup \{\neg \, \pi \} \vdash \neg \, \mathsf {Con}_\alpha $ . There exists a number q such that $\mathsf {PA} \cup \{\neg \, \pi \} \vdash \mathsf {Prf}_{\beta }^{\Sigma _n}(\ulcorner {\neg \, \mathsf {Con}_\alpha }\urcorner , \underline {q})$ . Thus, $\mathsf {PA} \cup \{\neg \, \pi \} \vdash \alpha (x) \to x \leq \underline {q}$ . We have

$$ \begin{align*} \mathsf{PA} \cup \{\neg\, \pi\} \vdash \forall y < \underline{q}\, \neg\, \mathsf{Prf}_{\beta}^{\Sigma_n}(\ulcorner{\mathsf{Con}_\alpha}\urcorner, y) & \to \Big (\alpha(x) \to \big( x\leq \underline{q} \;\wedge \\ & \hspace{1cm} \forall y < x\, \neg\, \mathsf{Prf}_{\beta}^{\Sigma_n}(\ulcorner{\mathsf{Con}_\alpha}\urcorner, y) \big)\kern-1pt\Big) \\ & \to \big(\alpha(x) \to (\beta(x) \land x \leq \underline{q})\big) \\ & \to (\mathsf{Con}_{\beta \leq \underline{q}} \to \mathsf{Con}_\alpha). \\ & \to \mathsf{Con}_\alpha. \end{align*} $$

The last step uses the fact that $\beta (x)$ a is binumeration of $\mathsf {W}_k$ in combination with the essential reflexiveness of $\mathsf {PA}$ . By combining this with the small reflection principle, we obtain $\mathsf {W}_k \cup \{\neg \, \pi \} \vdash \mathsf {Con}_\alpha $ . Since $\mathsf {W}_k \cup \{\mathsf {Con}_\alpha \} \vdash \pi $ , we conclude $\mathsf {W}_k \vdash \pi $ .

We show that $\neg \, \mathsf {Con}_\alpha $ is also $\Pi _n$ -conservative over $\mathsf {W}_k$ . Let $\pi $ be any $\Pi _n$ sentence such that $\mathsf {W}_k \cup \{\neg \, \mathsf {Con}_\alpha \} \vdash \pi $ . Then, $\mathsf {W}_k \cup \{\neg \, \pi \} \vdash \mathsf {Con}_\alpha $ . There exists a number p such that $\mathsf {PA} \cup \{\neg \, \pi \} \vdash \mathsf {Prf}_{\beta }^{\Sigma _n}(\ulcorner {\mathsf {Con}_\alpha }\urcorner , \underline {p})$ . For each $q> p$ , we have $\mathsf {PA} \cup \{\neg \, \pi \} \vdash \exists y < \underline {q}\, \mathsf {Prf}_{\beta }^{\Sigma _n}(\ulcorner {\mathsf {Con}_\alpha }\urcorner , y)$ . We obtain

$$\begin{align*}\mathsf{PA} \cup \{\neg\, \pi\} \vdash \forall z < \underline{q}\, \neg\, \mathsf{Prf}_{\beta}^{\Sigma_n}(\ulcorner{\neg\, \mathsf{Con}_\alpha}\urcorner, z) \to \alpha(\underline{q}). \end{align*}$$

So, for a sufficiently large $q> p$ , we have

$$\begin{align*}\mathsf{PA} \cup \{\neg\, \pi\} \vdash \forall z < \underline{q}\, \neg\, \mathsf{Prf}_{\beta}^{\Sigma_n}(\ulcorner{\neg\, \mathsf{Con}_\alpha}\urcorner, z) \to \neg\, \mathsf{Con}_\alpha. \end{align*}$$

By combining this with the small reflection principle, we have $\mathsf {W}_k \cup \{\neg \, \pi \} \vdash \neg \, \mathsf {Con}_\alpha $ . Since $\mathsf {W}_k \cup \{\neg \, \mathsf {Con}_\alpha \} \vdash \pi $ , we conclude $\mathsf {W}_k \vdash \pi $ .

We have proved that $\mathsf {Con}_\alpha \in \mathbb {D}_n(\mathsf {W}_i)$ . Consequently, we have that $\mathsf {Con}_{\alpha }$ is independent of $\mathsf {W}_i$ .

By $\Pi _n$ -conservativity, we obtain that for each $m \in \omega $ ,

• • $\mathsf {W}_k \vdash \forall y < \underline {m}\, \neg \, \mathsf {Prf}_{\beta }^{\Sigma _n}(\ulcorner {\mathsf {Con}_\alpha }\urcorner , y)$ and

• • $\mathsf {W}_k \vdash \forall z < \underline {m}\, \neg \, \mathsf {Prf}_{\beta }^{\Sigma _n}(\ulcorner {\neg \, \mathsf {Con}_\alpha }\urcorner , z)$ .

Hence, we have $\mathsf {W}_k \vdash \alpha (\underline {m}) \leftrightarrow \beta (\underline {m})$ . This means that $\alpha (x)$ is also a binumeration of $\mathsf {W}_k$ , and, thus, $\mathsf {Con}_{\alpha } \in \mathbb C(\mathsf {W}_i)$ . We take $\varPhi (i) : = \mathsf {Con}_\alpha $ . Then, $\varPhi $ witnesses the effective essential $\mathbb {C} \cap \mathbb {D}_n$ -incompleteness of $\mathsf {PA}$ .