2.1 Technical preliminaries

Let the language $\mathcal {L}$ be given by the signature $\sigma (\mathcal {L}) = \{ ` \mathsf {0} \text{'} , ` \mathsf {S} \text{'} , ` + \text{'} , ` \times \text{'} , ` = \text{'} , ` \neg \text{'} , ` \wedge \text{'} , ` \forall \text{'} \}$ . Consider the alphabet A which consists of $\sigma (\mathcal {L})$ together with the symbols $` \mathsf {v} \text{'} $ , $` \prime \text{'} $ , $` ( \text{'} $ and $` ) \text{'} $ . Let $A^\ast $ denote the set of finite strings over A (without the empty string), and let $\ast $ and $\equiv $ denote the concatenation operation and the equality relation on $A^\ast $ respectively. For better readability, we often omit the use of quotation symbols and the concatenaton operation. For instance, for $s,t \in A^\ast $ we write $s \prime $ instead of $s \ast ` \prime \text{'} $ and $(\mathsf {S} t)$ instead of $ ` (\mathsf {S} \text{'} \ast t \ast ` ) \text{'} $ , etc. We take an $\mathcal {L}$ -expression to be any element of $A^\ast $ . The set $\mathit {Var}$ of $\mathcal {L}$ -variables is generated from $\mathsf {v}$ by appending a finite number of strokes. The sets $\mathit {Term}, \mathit {Fml} \subseteq A^\ast $ contain the well-formed $\mathcal {L}$ -terms and $\mathcal {L}$ -formulæ given in infix notation respectively.Footnote ³ Any used symbol not contained in $\mathcal {L}$ is to be understood as the usual abbreviation by symbols of $\mathcal {L}$ . For instance, we define $x \leq y \equiv \exists z \ z + x = y$ , where $x,y,z \in \mathit {Var}$ and $\exists $ is defined by means of $\forall $ and $\neg $ as usual. Since the language $\mathcal {L}$ is fixed, expressions, terms, formulæ and sentences are throughout the paper implicitly assumed to belong to $\mathcal {L}$ . For the sake of brevity, we sometimes use the symbols ${\Rightarrow }$ , , and ${{\exists\kern-0.45em\exists }}$ as abbreviations of the meta-theoretical connectives “implies”, “and”, “for all” and “there is” respectively.

Corresponding to their alphabetical expressions, the following “constructor” operations on strings play an important role throughout this paper:

$$ \begin{align*} \underline{\prime} & \colon \mathit{Var} \to \mathit{Var}, \text{ given by } x \mapsto x \prime ;\\ \underline{\mathsf{S}} & \colon \mathit{Term} \to \mathit{Term}, \text{ given by } t \mapsto (\mathsf{S} t);\\ \underline{+} & \colon \mathit{Term} \times \mathit{Term} \to \mathit{Term}, \text{ given by } (s,t) \mapsto (s + t);\\ \underline{\times} & \colon \mathit{Term} \times \mathit{Term} \to \mathit{Term}, \text{ given by } (s,t) \mapsto (s \times t);\\ \underline{=} & \colon \mathit{Term} \times \mathit{Term} \to \mathit{Fml}, \text{ given by } (s,t) \mapsto (s = t);\\ \underline{\neg} & \colon \mathit{Fml} \to \mathit{Fml}, \text{ given by } \varphi \mapsto (\neg \varphi);\\ \underline{\wedge} & \colon \mathit{Fml} \times \mathit{Fml} \to \mathit{Fml}, \text{ given by } (\varphi,\psi) \mapsto (\varphi \wedge \psi);\\ \underline{\forall} & \colon \mathit{Var} \times \mathit{Fml} \to \mathit{Fml}, \text{ given by } (x,\varphi) \mapsto ( \forall x \varphi ). \end{align*} $$

By slight abuse of notation, both the standard interpretation of $\mathcal {L}$ and its domain is denoted by $\mathbb {N}$ . As usual, $\overline {n}$ denotes the (standard) numeral of the number n. Let $\varphi $ be an expression and let $\alpha $ be a numbering of $A^\ast $ , i.e., $\alpha \colon A^\ast \to \mathbb {N}$ is injective. We write for the numeral of the $\alpha $ -code $\alpha (\varphi )$ of $\varphi $ and we write if no specific numbering is indicated.

We call a function $\alpha \colon B \to \mathbb {N}$ recursive, if $B \subseteq \mathbb {N}$ is decidable and there exists a partial recursive function $\alpha ' \colon \mathbb {N} \to \mathbb {N}$ with $\textrm{{dom}}(\alpha ') = B$ and $\alpha ' \upharpoonright B = \alpha $ . Let $\alpha _1$ and $\alpha _2$ be numberings of given sets $S_1$ and $S_2$ respectively. We call a function $f \colon S_1 \to S_2 $ recursive relative to $\langle \alpha _1, \alpha _2 \rangle $ if $\alpha _2 \circ f \circ \alpha _1^{-1}$ is recursive. If moreover $\alpha _1 = \alpha _2$ , we also say that f is recursive relative to $\alpha _1$ .

In this paper we consider consistent and “sufficiently adequate” $\mathcal {L}$ -theories. We take any “sufficiently adequate” $\mathcal {L}$ -theory to be recursively enumerable and to extend the Tarski–Mostowski–Robinson theory $\mathsf {R}$ , which is given by the following axiom schemata:Footnote ⁴

1. R1. $\overline {m} + \overline {n} = \overline {m + n}$

2. R2. $\overline {m} \times \overline {n} = \overline {m \cdot n}$

3. R3. $\overline {m} \neq \overline {n}, \text { for } m \neq n$

4. R4. $x \leq \overline {n} \rightarrow \bigvee _{k \leq n} x = \overline {k}$

5. R5. $x \leq \overline {n} \vee \overline {n} \leq x$

2.2 Löb’s conditions

Perhaps the most popular approach to abstracting away from specific formalisation choices regarding consistency sentences rests on the so-called Hilbert–Bernays–Löb derivability conditions, or for short: Löb’s conditions, which are defined as follows. A formula $\mathsf {Pr}(x)$ satisfies Löb’s conditions for T, in short: Löb(T), if for all sentences $\varphi $ and $\psi $ :

• Löb1 ;

• Löb2 ;

• Löb3 .

Let $\bot $ be some fixed T-contradiction, such as $\mathsf {0} = \mathsf {S0}$ . This approach results in the following famous generalisation of Gödel’s Second Theorem due to Hilbert & Bernays (Reference Hilbert and Bernays1970) and Löb (Reference Löb1955), which constitutes the starting point of this study:

Indeed, this version of Gödel’s theorem neither resorts to the underlying proof system nor to the specific way T-provability and T’s axiom set are formalised.Footnote ⁵ However, its formulation employs a specific and arbitrarily chosen Gödel numbering. In order to also abstract away from this source of indeterminacy, we first make the numbering used in the formulation of Löb’s conditions explicit, where prima facie any injective function $\alpha \colon A^\ast \to \mathbb {N}$ qualifies as a numbering. We then take a formula $\mathsf {Pr}^\alpha (x)$ to satisfy Löb’s conditions relative to $\alpha $ for T, in short: Löb( $T, \alpha $ ), if for all sentences $\varphi $ and $\psi $ :

• $\boldsymbol{L\ddot{o}b1}^{\boldsymbol{\alpha }}$ ;

• $\boldsymbol{L\ddot{o}b2}^{\boldsymbol{\alpha }}$ ;

• $\boldsymbol{L\ddot{o}b3}^{\boldsymbol{\alpha }}$ .

It is widely believed that Löb’s conditions are sufficient to allow for the “modal reasoning” involved in the proof of Löb’s theorem, and thus in particular, of Theorem 2.1 (for such a proof see e.g., Smith, Reference Smith2007, p. 230). As this “modal reasoning” does not resort to the underlying numbering, one might expect Theorem 2.1 to be invariant in the following sense: whatever numbering $\alpha $ is chosen, for every formula $\mathsf {Pr}^{\alpha }(x)$ satisfying Löb( $T, \alpha $ ).

2.3 Deviant numberings

I now show that this invariance claim is false, as there exist highly artificial and deviant numberings. To start with, for any given set $S \subseteq A^\ast $ of expressions an injective function $\alpha \colon A^\ast \to \mathbb {N}$ can be defined such that the set of $\alpha $ -codes of S is $\Delta ^0_0$ -binumerable.Footnote ⁶ For instance, define $\alpha $ such that

$$ \begin{align*} \alpha(\varphi) \text{ is } \begin{cases} \text{even} & \text{ if } \varphi \in S,\\ \text{odd} & \text{ if } \varphi \not\in S. \end{cases} \end{align*} $$

This can simply be done by enumerating both S and $A^\ast \setminus S$ (without repetitions) and subsequently assigning the number $2 \cdot k$ to the k-th element of the enumeration of S and the number $2 \cdot k +1$ to the k-th element of the enumeration of $A^\ast \setminus S$ . Clearly, these enumerations are possibly noneffective.

Let $S := T^{\vdash }$ be the deductive closure of T and let $\alpha $ be defined as above, such that the set of $\alpha $ -codes of T-theorems are exactly the even numbers. The formula $\mathsf {Pr}^\alpha (x) \equiv \exists y < x \ x = \overline {2} \times y$ then satisfies Löb( $T, \alpha $ ) and we have by the $\Sigma ^0_1$ -completeness of T, in violation of the above invariance claim. The reason for this is that in addition to Löb’s conditions, the fixed point construction is also a crucial ingredient in the “modal reasoning” towards Löb’s theorem. And indeed, the Diagonal Lemma fails with regard to $\alpha $ , i.e., there is no ( $\alpha $ -)fixpoint of $\neg \mathsf {Pr}^\alpha (x)$ . To see this, assume that there exists G such that , which implies $T \not \vdash G$ . But then and therefore $T \vdash G$ , yielding a contradiction.

Similarly, setting $S := \mathsf {Th}(\mathbb {N})$ yields a numbering $\beta $ such that $\exists y < x \ x = \overline {2} \times y$ binumerates $\{ \beta (\varphi ) \mid \mathbb {N} \models \varphi \}$ , thus violating (both the semantic and the syntactic version of) Tarski’s Theorem.

While $\alpha $ and $\beta $ appear to be highly deviant numberings which resist any arithmetisation of the usual syntactic properties, less artificial numberings can be constructed for certain theories $T \supseteq \mathsf {R}$ which still violate Gödel’s Second Theorem and the two versions of Tarski’s Theorem respectively.Footnote ⁷ For instance, based on a more careful construction, both $\alpha $ and $\beta $ can be taken to be monotonic (see $\zeta $ in Appendix 6). That is, the Gödel number of a string is larger than the Gödel numbers of its substrings. There also exists a numbering $\delta $ such that the functions $\underline {\mathsf {S}}$ , $\underline {+}$ , $\underline {\times }$ , $\underline {\prime }$ , $\underline {=}$ , $\underline {\wedge }$ and $\underline {\forall }$ are recursive relative to $\delta $ , yet the set of T-theorems is binumerable by a $\Delta ^0_0(\mathsf {exp})$ -formula $\mathsf {Pr}^{\delta }(x)$ (see Appendix 6).Footnote ⁸ In particular, $\mathsf {Pr}^{\delta }(x)$ once again satisfies Löb( $T, \delta $ ), but .

There exists moreover a numbering $\eta $ such that all the functions $\underline {\mathsf {S}}$ , $\underline {+}$ , $\underline {\times }$ , $\underline {\prime }$ , $\underline {=}$ , $\underline {\neg }$ , $\underline {\wedge }$ and $\underline {\rightarrow }$ are recursive relative to $\eta $ and the set $\{ \eta (\varphi ) \mid \mathcal {N} \models \varphi \}$ of true sentences is binumerable by a $\Delta ^0_0(\mathsf {exp})$ -formula $\mathsf {Pr}^{\eta }(x)$ (see Appendix 6). In particular, the formula $\mathsf {Pr}^{\eta }(x)$ satisfies Löb( $T, \eta $ ) for sound theories $T \supseteq \mathsf {R}$ , but .

Both numberings $\delta $ and $\eta $ thus yield deviant results, even though they allow for the arithmetisation of a large portion of syntactic properties and operations. For instance, the $\mathsf {num}$ -function (mapping numbers to their standard numerals) and the substitution function for terms as well as for atomic formulæ can be binumerated relative to both numberings. However, since $\underline {\neg }$ and $\underline {\forall }$ are not recursive relative to $\delta $ and $\eta $ respectively, the arithmetisation of the substitution function for (complex) formulæ fails for both numberings. This can be seen as the reason why the Diagonal Lemma does not hold when employing these numberings, since its proof crucially relies on the arithmetisation of the substitution function.

In what follows I will argue that these results do not refute Gödel and Tarski’s classical theorems, since the underlying numberings are inadmissible for the contexts they are employed in.Footnote ⁹ Our initial naïve approach to numberings is thus not tenable and has to be replaced by a refined account based on admissible numberings, ruling out such deviant constructions. This is precisely the aim of the next section.

3.1 Computable simulations

The reader is reminded that the set of expressions is given by the set $A^\ast $ of strings over the alphabet A. Let $\alpha \colon A^\ast \to \mathbb {N}$ be any injective function and let $G := \alpha (A^\ast )$ be the set of Gödel numbers, or $\alpha $ -codes, of $A^\ast $ . For any operation $e \colon (A^\ast )^k \to A^\ast $ , we call the function given by

the ( $\alpha $ -)tracking function of e. To illustrate, is defined such that the diagram

(1)

commutes, where $\alpha \times \cdots \times \alpha (a_1, \ldots , a_k) := (\alpha (a_1),\ldots , \alpha (a_k))$ . The function can thus be seen to simulate e on the set of $\alpha $ -codes. For this reason we also sometimes call the e-simulating function (on G).

Intrinsic to the structure of strings is the concatenation operation $\ast $ , which allows to generate any string of $A^\ast $ from elements of A. The operation can then be seen to simulate the concatenation operator on the set G of Gödel numbers. It is easy to check that the associativity of $\ast $ is inherited by . Moreover, as in the case of strings over A, each $\alpha $ -code in G can be finitely generated from the set $\alpha (A)$ of $\alpha $ -codes of A by the $\ast $ -simulating operation . For instance, for any string $s \in A^\ast $ there are alphabetical expressions $s_1, \ldots , s_n \in A$ such that

$$ \begin{align*} s \equiv s_1 \ast \cdots \ast s_n. \end{align*} $$

The $\alpha $ -code of s is then simply the result of “ -concatenating” the $\alpha $ -codes of the alphabetical expressions $s_i$ , i.e.,

This situation can be naturally described in algebraic terms, by conceiving of $A^\ast $ together with $\ast $ as a semigroup. The above considerations then show that also forms a semigroup and $\alpha $ is a (semigroup) isomorphism.

Thus far, no requirements have been imposed on the numbering $\alpha $ . That $\alpha $ computably simulates $A^\ast $ can now be informally expressed by (i) requiring that for each number it is decidable whether or not it $\alpha $ -codes an expression, and by (ii) requiring that Gödel numbers are mechanically generated from the $\alpha $ -codes of alphabetic expressions in A. Since G is a set of natural numbers and is an arithmetical operation which generates G from $\alpha (A)$ , both requirements can be explicated via the Church–Turing Thesis as follows:

This notion of a computable simulation can be traced back to Montague (Reference Montague1957, chap. IV) as well as to the foundational texts Rabin (Reference Rabin1960) and Mal’cev (Reference Mal’cev1961) of the field of computable algebra.Footnote ¹⁰ The reader who questions the need for such a notion is reminded that the explication of a numbering’s computability via the Church-Turing Thesis is not entirely straightforward. Turing machines for instance only operate over strings. A sensible application of Turing-computability to numberings thus requires a representation of their co-domain $\mathbb {N}$ by strings, which in general yields different sets of computable numberings (Montague, Reference Montague1960). The task of singling out admissible or natural representations has proven to be fraught with difficulty, purportedly even resulting in a circular conceptual analysis of (numerical) computability (see Boker & Dershowitz, Reference Boker, Dershowitz, Avron, Dershowitz and Rabinovich2008, Reference Boker, Dershowitz, Blass, Dershowitz and Reisig2010; Quinon, Reference Quinon, Manea, Miller and Nowotka2018; Rescorla, Reference Rescorla2007; Shapiro, Reference Shapiro1982, Reference Shapiro2017).

As will be shown in §4.1, computable simulations do not depend on such arbitrary representational choices, thus providing an adequate explication of computability for our purposes.

3.2 The epistemic role of numberings

While we have seen that computable simulations explicate the notion of computability adequately, the question remains why admissible numberings should be computable in the first place.

Recall the philosophical interpretation of Gödel’s Second Theorem, according to which T does not prove its consistency. This interpretation’s justification rests on the hypothesis that we can formulate the question whether or not T proves a sentence which expresses T’s consistency in a meaningful way. Indeed, the employed Gödel numbering enables us to interpret certain arithmetical closed terms and formulæ as (names of) syntactic objects and properties of syntactical objects respectively. Let us call this the semantic role of Gödel numberings. It allows us to not only reason in T about numbers but also about expressions via their Gödel codes.

The prevalent philosophical reading of Gödel’s theorem as a limitative result rests on yet another crucial feature of numberings. What we are essentially concerned with, is the question of whether or not the consistency sentence is derivable by using only the theory T’s resources. It is therefore pivotal in this context that admissible numberings ensure that reasoning about expressions via their representations in $\mathcal {L}$ requires no resources which lie outside the formal system T. Let us call this the epistemic role of Gödel numberings.

As we have seen in §3.1, every numbering induces a pair , where G is a set of codes and is a $\ast $ -simulating operation on G, which represents the structure of expressions. That this representation does not employ resources exceeding T can be understood in virtue of T “recognising” or “knowing” or “verifying” certain facts and properties concerning . Since T is a formal theory, it can be seen to “recognise,” “know” and “verify” facts and properties by means of proving (formulæ which express) them.

So far, none of these facts and properties have been specified. A minimal approach is to require that T “recognises” the two constituents of the given representation, namely the set G and the operation . Accordingly, T “knows” whether or not ${n \in G}$ , and whether or not , for numbers l, m and n. That is, T binumerates both G and (i.e., its graph). But this is tantamount to G being decidable and being recursive.Footnote ¹¹ Hence by Definition 3.2, the underlying numbering is a computable simulation.

Note that even though the admissibility of a numbering is analysed in terms of T’s resources, the extracted notion of computable simulation is theory-independent. This rests on the fact that theories extending $\mathsf {R}$ binumerate exactly the same sets and functions, namely decidable and recursive ones respectively.

However, one might require T to recognise further properties regarding . For instance, T might reasonably be required to verify the functionality of , i.e., to prove that is total (cf. §4.3). Since different theories extending $\mathsf {R}$ prove the totality of different classes of functions in general, this requirement yields a notion of admissibility which is sensitive to T.

As we will show in §4, even the minimal approach places enough restriction on admissible numberings to establish the invariance of Gödel’s Second Theorem (see Theorem 4.14). It is therefore sufficient for the purpose of this paper to conclude that admissible numberings are computable simulations.Footnote ¹²

In the context of the semantic version of Tarski’s Theorem, we are concerned with the question of whether the set of true sentences is definable in the language $\mathcal {L}$ , rather than with provability in a formal theory T. Accordingly, the epistemic role of numberings in this context is to ensure that reasoning about expressions does not resort to resources exceeding the language $\mathcal {L}$ , rather than the theory T as in the context of Gödel’s theorem. This can be made precise by requiring that $\mathcal {L}$ “captures” or “expresses” relevant facts and properties about admissible representations by means of defining them. Once again resorting to a minimal approach then yields the requirement that both G and (i.e., its graph) are definable in the standard interpretation $\mathbb {N}$ of $\mathcal {L}$ . We call any numbering which gives rise to such a representation an arithmetical simulation (of $A^\ast $ ).

According to the proposed analysis, the notion of admissibility is thus sensitive to the considered metamathematical context. While admissible numberings are computable simulations in the context of Gödel’s Second Theorem and other limitative results which primarily pertain to provability-in-a-theory (such as the syntactic version of Tarski’s Theorem), they are required to be arithmetical simulations in the context of the semantic version of Tarski’s Theorem.

In the remainder of this paper I will consider the admissibility of numberings exclusively in the context of Gödel’s Second Theorem (with Theorem 4.8 and footnote 23 as exceptions):

• Computable Simulativity. Every admissible numbering is a computable simulation.

I now turn briefly to other proposals found in the literature which tie computational considerations to the admissibility of numberings.

3.3 Other routes to admissibility

Let $\gamma $ denote the standard numbering introduced in Smith (Reference Smith2007, sec. 15.1).Footnote ¹³ According to Smith (Reference Smith2007) “the key feature of our Gödelian scheme [ $\gamma $ ] is this: there is a pair of algorithms, one of which takes us from an expression to its code number, the other of which takes us back again from the code number to the original expression” (p. 126). Given any other comparable numbering $\alpha $ , converting the $\alpha $ -code of a certain expression $\varphi $ into its $\gamma $ -code involves running through two algorithms (first mapping $\alpha (\varphi )$ to $\varphi $ and then mapping $\varphi $ to $\gamma (\varphi )$ ), which by assumption do not involve any open-ended searches (ibid.). Conversely, $\gamma $ -codes can be converted into $\alpha $ -codes in a similar fashion. Hence, “there is a p.r. function $tr$ which ‘translates’ code numbers according to [ $\alpha $ ] into code numbers under our official Gödelian scheme [ $\gamma $ ], and another p.r. function $tr^{-1}$ which converts code numbers under our scheme back into code numbers under scheme [ $\alpha $ ]” (ibid.).Footnote ¹⁴ Let us call any numbering $\alpha $ for which such a pair of p.r. translation functions exist, computably translatable to $\gamma $ . The following criterion of admissibility is thus extracted by Smith (Reference Smith2007):

• Computable Translatability. A numbering is admissible iff it is computably translatable to the standard numbering $\gamma $ .

While I agree with Smith’s initial analysis that the key feature of admissibility consists essentially in intuitive computability, the proposed criterion fails to capture this feature in a conceptually satisfactory way. This deficiency can be seen to result from general difficulties in giving a precise account of the computability of numberings (see the remark following Definition 3.2). To circumvent these difficulties, Smith requires admissible numberings to be computably translatable to some designated standard numbering $\gamma $ . The conceptual adequacy of the resulting criterion thus crucially rests on the admissibility of $\gamma $ . For instance, employing one of the deviant numberings of §2.3 as a reference numbering instead of $\gamma $ , would clearly render the criterion inadequate. What justifies the choice of $\gamma $ as the reference numbering is exactly its intuitive computability. Hence, instead of taking the conceptually prior feature of computability as characterising admissibility, a conceptual surrogate is employed.Footnote ¹⁵ Furthermore, Smith’s justification for the existence of p.r. translation functions between numberings which share the same “key feature” (i.e., come with a pair of encoding and decoding algorithms) crucially relies on informal reasoning about computability. A more satisfactory approach would be to first formally characterise the conceptually prior notion of intuitive computability of numberings (cf. Definition 3.2) and then mathematically prove the existence of respective translation functions between them (cf. Theorem 4.6 and Corollary 4.7 below), employing the Church–Turing Thesis only in its “interpretive use” (Smith, Reference Smith2007, p. 275).

A second approach can be found in Smullyan (Reference Smullyan1961), who bases the concept of admissibility of numberings on representability in elementary formal systems (for definitions see Smullyan, Reference Smullyan1961, p. 6). These systems can be seen as devices intended to “explicate the notion of ‘definability by recursion’” (Smullyan, Reference Smullyan1961, p. 2) directly operating on strings, without the usual recourse to numberings. A set S of strings which is definable by recursion in some elementary formal system is called formally representable. If in addition to S also the complement of S is formally representable, S is called solvable. The admissible numberings of S are then taken to be exactly those which preserve these recursion theoretic properties. More precisely, a numbering $\alpha \colon S \to \mathbb {N}$ is called EFS-admissible, if for every subset $U \subseteq S$ the following holds: U is formally representable iff $\alpha (U)$ is recursively enumerable and U is solvable iff $\alpha (U)$ is decidable. This yields:

• EFS-Admissibility. A numbering is admissible iff it is EFS-admissible.

Once again this definition has a computational motivation, since the concept of formal representability is intended to capture the concept of computability, in the sense that formally representable sets characterise sets which are generated by a “computing machine” (Smullyan, Reference Smullyan1961, p. 9).

An account of admissibility which prima facie “strictly contains” the notion of computable simulativity is introduced in Manin (Reference Manin2009). A numbering $\alpha $ is called sequence-admissible, if the image of $\alpha $ is decidable and the length function, the projection function given by $\langle i , \langle a_1, \ldots , a_n \rangle \rangle \mapsto a_i$ as well as the concatenation function are recursive.Footnote ¹⁶ This yields the following criterion:

• Sequence-Admissibility. A numbering is admissible iff it is sequence-admissible.

Despite the different conceptions of computability underlying these proposals, all three notions will be shown in §4 to be extensionally equivalent to computable simulativity (see Corollary 4.7). This endows the criterion of computable simulativity with a desirable amount of robustness.

3.4 Compositionality and monotonicity

I conclude this section by discussing other approaches to the admissibility of numberings based on compositionality and monotonicity rather than computational considerations. As we have seen in §3.2, numberings have a semantic role as they represent strings by natural numbers. It thus seems reasonable to require admissible numberings to satisfy some principle of compositionality. Accordingly, the semantic value of every complex expression can be taken to be determined by its structure and the semantic values of its subexpressions.Footnote ¹⁷ Since for a given numbering $\alpha $ the semantic value of a string is its $\alpha $ -code, the compositionality of $\alpha $ can be understood as $\alpha \colon A^\ast \to G$ being an $\Omega $ -homomorphism, where the signature $\Omega $ contains a binary (concatenation) function symbol . But as we have already seen in §3.1, every numbering $\alpha $ induces such a homomorphism. Hence every numbering is compositional in this sense.

Even resorting to the “mode of presentation” does not yield any proper restriction on numberings. It is a common practice to first assign numbers to the alphabetical symbols and then to define the Gödel number of a string as a function of the Gödel numbers of its entries (e.g., based on prime factorisation in the tradition of Gödel, Reference Gödel1931 or on k-adic notation following Smullyan, Reference Smullyan1961, Reference Smullyan1992). In fact, any numbering $\alpha $ of $A^\ast $ can be reconstructed in this standard “bottom-up” fashion: firstly, map each symbol s of the alphabet A to its $\alpha $ -code. Secondly, consider the $\ast $ -simulating partial function , given in §3.1. These two steps then resemble the usual “bottom-up” definition of numberings and uniquely determine our initially given numbering, since $\alpha $ is the unique function which extends the assignment given in step 1 and is compositional with regard to the partial function given in step 2, i.e., is an $\Omega $ -homomorphism.Footnote ¹⁸ Every numbering irrespective of its presentation can thus be reconstructed in the above manner, which is the prevalent mode of presentation for standard Gödel numbers found in the literature.

In addition to the “bottom-up” fashion of defining numberings, it is ubiquitous in the literature to employ a monotonic partial operation in step 2. This gives rise to the following property of Gödel numberings:

Monotonicity is a prevalent property shared by all standard numberings found in the literature (which are known to the author). One reason for this custom is that monotonicity ensures that the above bottom-up mode of presentation yields an injective function and hence a numbering. Monotonicity thus serves as a technical constraint warranting the intended output of a particular construction principle (see for instance Hilbert & Bernays, Reference Hilbert and Bernays1970, p. 220). Moreover, monotonicity provides a low arithmetical complexity of formulæ representing syntactic properties, allowing the use of convenient tools (such as $\Sigma ^0_1$ -completeness) when proving certain metamathematical theorems.

Apart from these technical considerations, on which conceptual grounds can monotonicity be justified? Can monotonicity be extracted as a necessary condition by analysing the admissibility of numberings as we did in the case of computability?

As we have seen in §3.2, a crucial role of Gödel numberings is to allow reasoning about expressions in an arithmetical theory T. Admissible numberings were thus required to simulate certain structural properties of strings. As we have seen for instance, the operation induced by $\alpha $ simulates the concatenation operation $\ast $ on $A^\ast $ in virtue of the commutativity of diagram 1 (in §3.1). The subexpression-relation ${\preceq }$ on $A^\ast $ can be reasonably taken as yet another important structural feature of strings. Hence for any admissible numbering, the induced semigroup of Gödel numbers should simulate ${\preceq }$ appropriately. This can be made precise by extending the signature $\Omega $ by a binary relation symbol ${\sqsubseteq }$ , yielding , and by construing $(A^\ast ,\ast ,\preceq )$ as an $\Omega ^+$ -structure. A numbering $\alpha $ may then be called $\preceq $ -preserving if $\alpha \colon A^\ast \to G$ is an $\Omega ^+$ -isomorphism. It can be easily seen that in a similar way as in §3.1, for every $\alpha $ a “ $\preceq $ -simulation” on G can be defined, such that is an $\Omega ^+$ -structure and $\alpha $ is an $\Omega ^+$ -isomorphism. Hence, once again, $\preceq $ -preservation does not impose any restriction on numberings.

By additionally drawing on the epistemic role of numberings, the induced structure can be required to not exceed T’s resources, analogously to the analysis in §3.2. Accordingly, it can be required that T “recognises”, and thus binumerates, the relation , which is tantamount to being decidable. Yet, as in the proof of Corollary 4.7 we can show that for every computable simulation $\alpha $ also the “ $\preceq $ -simulating relation” is decidable. This condition therefore does not pose any additional constraint given the requirement of computable simulativity.

As in the case of the $\ast $ -simulating operation , one may require T to verify further principles regarding $\preceq $ . For instance, T might be required to prove that any two strings $s,t$ are subexpressions of $s \ast t$ . This results in a more restrictive notion of admissibility which is sensitive to the considered theory T. Since these principles are formulated in purely string-theoretical terms without access to the arithmetical properties of their codes, they result in limitations, for instance, regarding the growth rate of admissible numberings (cf. §4.3). However, the constraint of monotonicity cannot be obtained using this approach.

As suggested by an anonymous referee, another route to obtain monotonicity may proceed by conceiving of $\leq $ and $\preceq $ as parthood relations on numbers and strings respectively. Taking parthood as an important structural feature of strings, admissible numberings may then be required to preserve structure in virtue of representing the parthood relation $\preceq $ on strings by the parthood relation $\leq $ on numbers. I however do not think that this approach is satisfactory for our purposes, for the following reasons.

Firstly, assuming that both relations ${\preceq }$ and ${\leq }$ are indeed similar in structure by being parthood relations, they still significantly differ with regard to another important structural aspect. While ${\leq }$ is a total order relation, there are strings which are incomparable with respect to $\preceq $ . For this reason, there exists no numbering such that ${\preceq }$ is represented by ${\leq }$ , i.e., that , clearly rendering the above requirement absurd. To require that ${\preceq }$ is represented by ${\leq }$ in virtue of being contained in ${\leq }$ , i.e., that the underlying numbering is monotonic, appears to be an ad-hoc reaction to this structural discrepancy rather than based on conceptual grounds.

Secondly, even if the adequacy of this last step is granted, further problems arise by this conception of structure preservation. This can be illustrated by basing the structure of strings on so-called successor functions $\mathsf {S}_a$ , which append the alphabetical symbol $a \in A$ to any input string (see Corcoran et al.,  Reference Corcoran, Frank and Maloney1974). Instead of employing the concatenation operation $\ast $ , the set of strings can be generated from the empty string by using successor functions $\mathsf {S}_a$ for each $a \in A$ . According to the above approach, admissible numberings can then be required to preserve this structure in virtue of representing the empty string and the successor functions $\mathsf {S}_a$ by their arithmetical counterparts, namely, by $0$ and the successor operation ${n \mapsto n+1}$ . However, even the requirement that (the graph of) the tracking function of some $\mathsf {S}_a$ is contained in (the graph of) the arithmetical successor function cannot be satisfied by any Gödel numbering. As in the case of monotonicity, this requirement stems from a loose and almost metaphorical conception of structure similarity and preservation. In some sense, the empty string and the successor functions $\mathsf {S}_a$ indeed have the structure of $0$ and the arithmetical successor operation, namely in virtue of being generators and constructors respectively. However, they once again differ crucially in other structural respects, resulting in the absurdity of the extracted requirement.

Thirdly, while I agree that admissible numberings should preserve the structure of the subexpression relation $\preceq $ , it is not clear why structure preservation should entail constraints on the arithmetical content of , such as the containment of in $\leq $ , in the first place. Rather, I take an admissible numbering $\alpha $ to preserve the structure of $\preceq $ in virtue of the decidability of whether or not a number $n \alpha $ -codes a subexpression with $\alpha $ -code m for any $m,n \in \mathbb {N}$ , or that the arithmetisations of certain additional string-theoretic properties regarding $\preceq $ are provable in T, as extracted from my initial analysis given above. As we have already seen, monotonicity can not however be derived from this conception of structure preservation.

Finally, the very assumption that both ${\preceq }$ and ${\leq }$ are parthood relations appears to rely on specific and quite arbitrary representational choices of strings and numbers. While $\preceq $ may be understood as a parthood relation on strings via their natural interpretation as geometric objects, it is less clear how $\leq $ can be given such a mereological interpretation. One way is to represent numbers as strings of strokes. Another approach is to represent numbers as Von Neumann ordinals, and to understand the parthood relation on sets as the containment relation $\subseteq $ (see e.g., Lewis, Reference Lewis1991). Then $\leq $ is indeed a parthood relation on numbers. However, this interpretation breaks down for other reasonable representations, such as Zermelo ordinals or equivalence classes of finite sets under the equivalence relation of equinumerosity. Yet another approach is to construe strings and numbers as free algebras and to extract a parthood relation from their underlying term algebras (see Burris & Sankappanavar, Reference Burris and Sankappanavar1981). In the case of numbers, this approach indeed yields $\leq $ as our parthood relation. If we take strings to be generated by the concatenation operation, then $A^\ast $ can be viewed as a free semigroup and we extract $\preceq $ as a parthood relation on strings. However, as we have seen above, we can also take strings to be generated from the empty string by the successor functions $\mathsf {S}_a$ . This results in a different free algebra, according to which the parts of any string are exactly its initial strings. Similarly, parthood holds between strings and their final strings if we conceive of $\mathsf {S}_a$ as prefixing, instead of appending, a to input strings.

In addition to these conceptual reservations, the obtained results also suggest that monotonicity is a property orthogonal to the invariance problems considered in this paper.Footnote ¹⁹ Namely, there exist both monotonic and nonmonotonic numberings which satisfy Gödel’s Second Theorem as well as Tarski’s theorems (take any standard numbering and Visser’s (Reference Visser, Gabbay and Guenthner1989) self-referential numbering respectively), as well as monotonic and nonmonotonic numberings which violate these theorems (e.g., $\zeta $ and $\delta $ respectively).

Consequently, in what follows, admissible numberings are not required to be monotonic. This also permits a more general approach, since monotonicity constitutes a proper constraint on computable simulations.Footnote ²⁰

4.1 Equivalence of numberings

It can be easily checked that $\sim $ is a partial equivalence relation, i.e., $\sim $ is symmetric and transitive. If we require numberings to have a decidable image, then $\sim $ is also reflexive. In particular, $\sim $ is an equivalence relation on the set of computable simulations of S.

A subset $R \subseteq S$ is called decidable, recursively enumerable, arithmetical relative to a numbering $\alpha $ , if the set $\alpha (R)$ has the respective property. The following lemma shows that these properties are invariant with regard to equivalent computable simulations.Footnote ²¹

The next theorem due to Mal’cev (Reference Mal’cev1961) serves as a powerful tool in proving invariance.Footnote ²²

From Lemma 4.5 and Theorem 4.6 we can conclude the extensional equivalence of computable simulativity and the three criteria considered in §3.3.

4.2 Invariance proofs

We start by proving the invariance of the semantic version of Tarski’s Theorem, which is also shown in Manin (Reference Manin2009, p. 240):Footnote ²³

In order to establish the invariance of Gödel’s Second Theorem, we first formalise certain properties of the equivalence of numberings in $\mathsf {R}$ .

Proof. Let $\alpha \colon B \to \mathbb {N}$ be an injective recursive function and let $\alpha '$ be a total recursive function extending $\alpha $ (see §2.1). By Theorem II.6 of Tarski et al. (Reference Tarski, Mostowski and Robinson1953) there is a binumeration $g(x,y)$ of $\alpha '$ and a binumeration $\beta (x)$ of B. Then ${f'(x,y) \equiv \beta (x) \wedge g(x,y)}$ is a binumeration of $\alpha $ . We define

$$ \begin{align*} f(x,y) \equiv f'(x,y) \wedge \forall z \leq x\, (\;f'(z,y) \rightarrow x=z ). \end{align*} $$

In order to show that $f(x,y)$ numerates $\alpha $ , let $n, m \in \mathbb {N}$ be such that $\alpha (n)=m$ . Since $f'(x,y)$ binumerates $\alpha $ , we have $ \mathsf {R} \vdash f'(\overline {n},\overline {m})$ and $\mathsf {R} \vdash \neg f'(\overline {k},\overline {m})$ for all $k < n$ . Thus $\mathsf {R} \vdash f'(\overline {k},\overline {m}) \rightarrow \overline {n} = \overline {k}$ for all $k < n$ , which yields $\mathsf {R} \vdash f'(\overline {k},\overline {m}) \rightarrow \overline {n} = \overline {k}$ for all $k \leq n$ . Using $\mathsf {R4}$ yields ( $\ast $ ) $\mathsf {R} \vdash \forall z \leq \overline {n}\, (\;f'(z,\overline {m}) \rightarrow \overline {n} = z)$ . Thus $\mathsf {R} \vdash f(\overline {n},\overline {m})$ . Since $\mathsf {R} \vdash \neg f'(\overline {n},\overline {m}) \rightarrow \neg f(\overline {n},\overline {m})$ and $\mathsf {R}$ is consistent, $f(x,y)$ also binumerates $\alpha $ .

Let $m \in \alpha (B)$ . Let $n \in \mathbb {N}$ be such that ( $\ast \ast $ ) $\mathsf {R} \vdash f(\overline {n},\overline {m})$ . In order to show that $f(x,y)$ is an inj-binumeration it suffices to show that $\mathsf {R} \vdash \forall x (\;f(x,\overline {m}) \rightarrow x = \overline {n})$ , which we prove in the following derivation in $\mathsf {R}$ :

□

In addition to proving the invariance of Gödel’s Second Theorem, the above method can also be used to simplify proofs of metamathematical theorems. For instance, they provide a proof of the Diagonal Lemma which avoids the usual tedious process of arithmetisation (in particular of the numeral and the substitution functions).

Note that neither the definition of nor the proofs of Theorem 4.6 and Lemma 4.11 employ any arithmetisation of the usual syntactic properties and operations.

Using Lemma 4.12 and the usual schematic “modal reasoning” we can also prove the invariance of Löb’s Theorem as well as its formalised version regarding admissible numberings. In particular, we can prove the desired

The reader might observe that the formulation of Theorem 4.14 already presupposes the employed provability predicates to satisfy Löb’s conditions. Since these conditions allow for the schematic “modal reasoning” sufficient to prove Gödel’s Second Theorem (or more generally, Löb’s Theorem), the problem of the invariance of Gödel’s Second Theorem with regard to numberings was essentially reduced to establishing an invariant version of the Diagonal lemma (Lemma 4.12).

However, no information about the existence of provability predicates satisfying Löb’s conditions with regard to varying numberings is provided by Theorem 4.14. As a result, the reader might wonder how much Theorem 4.14 improves on the classical result by Hilbert, Bernays and Löb (Theorem 2.1). For it could be the case that only the typically employed, standard numberings yield nontrivialFootnote ²⁴ provability predicates satisfying Löb’s conditions, rendering Theorem 4.14 a trivial extension of the standard Theorem 2.1 (at least from an extensional perspective).

The next theorem shows that in fact every admissible numbering $\alpha $ allows for the construction of a nontrivial provability predicate satisfying Löb’s conditions relative to $\alpha $ . This establishes Theorem 4.14 as a proper extension of the classical results based on a specific numbering.

Let $\Omega _1 \equiv \forall x,y\, \exists z\, \mathsf {o}(x,y,z)$ where $\mathsf {o}(x,y,z)$ is a standard binumeration of (the graph of) the polynomially growing function $\omega _1(x,y) = x^{\log y}$ , where $\log x = \max \{y \mid 2^y \leq x \}$ . Let $\mathsf {Exp} \equiv \forall x,y\, \exists z\, \mathsf {e}(x,y,z)$ , where $\mathsf {e}(x,y,z)$ is a standard binumeration of (the graph of) the exponentiation function given by $(x,y) \mapsto x^y$ .Footnote ²⁵

Before proving the theorem we show a useful auxiliary lemma.

Lemma 4.16. Let $\alpha $ and $\beta $ be equivalent numberings. Then there exists a binumeration $f(x,y)$ of $\beta \circ \alpha ^{-1}$ such that for each $\mathsf {Pr}^{\alpha }(x)$ satisfying Löb $(T, \alpha) $ there exists $\mathsf {Pr}^{\beta }(x)$ satisfying Löb $(T, \beta),$ such that for all $n \in \mathbb {N}$ and $m \in \beta (A^\ast )$

$$ \begin{align*} \mathsf{R} \vdash f(\overline{n},\overline{m}) \rightarrow ( \mathsf{Pr}^{\alpha}(\overline{n}) \leftrightarrow \mathsf{Pr}^{\beta}(\overline{m}) ). \end{align*} $$

If moreover T is $\Sigma ^0_2$ -sound and $\mathsf {Pr}^{\alpha }(x)$ is a $\Sigma ^0_1$ -numeration of $\alpha (T^{\vdash } )$ in T, then $\mathsf {Pr}^{\beta }(x)$ numerates $\beta (T^{\vdash } ) $ in T.

Proof. By Lemma 4.10 & 4.11 there exists a binumeration $f(x,y)$ of $\beta \circ \alpha ^{-1}$ and a formula $\mathsf {Pr}^{\beta }(x)$ such that

(2) $$ \begin{align} \mathsf{R} \vdash f(\overline{n},\overline{m}) \rightarrow ( \mathsf{Pr}^{\alpha}(\overline{n}) \leftrightarrow \mathsf{Pr}^{\beta}(\overline{m}) ), \end{align} $$

for all $n \in \mathbb {N}$ and $m \in \beta (A^\ast )$ . We show now that $\mathsf {Pr}^{\beta }(x)$ satisfies Löb( $T, \beta $ ):

1. i. If $T \vdash \varphi $ , then , since $\mathsf {Pr}^{\alpha }(x)$ satisfies Löb1 $^\alpha $ . We have

   
   since $f(x,y)$ binumerates $\beta \circ \alpha ^{-1}$ . It then follows from 2 that
   
   Thus .

2. ii. Let $\varphi , \psi \in \mathit {Fml}$ . Since $f(x,y)$ binumerates $\beta \circ \alpha ^{-1}$ we have

   
   Using 2 we therefore get
   (3)

   
   Similarly, we show
   (4)

   
   and
   (5)

   
   Since $\mathsf {Pr}^{\alpha }(x)$ satisfies Löb2 $^\alpha $ and using 3–5 we conclude that
   

3. iii. Let $\varphi \in \mathit {Fml}$ . As in (ii) we show

   (6)

   
   Since $\mathsf {Pr}^{\alpha }(x)$ satisfies Löb3 $^\alpha $ and using 4 & 6 we get
   (7)

   
   Application of Löb1 $^\beta $ to 4 then yields
   
   Applying Löb2 $^\beta $ then yields
   
   In combination with 7 we thus obtain
   

Assume now that T is $\Sigma ^0_2$ -sound and that $\mathsf {Pr}^{\alpha }(x)$ is a $\Sigma ^0_1$ -numeration of $\alpha (T^{\vdash } )$ in T. Since $\alpha (A^\ast )$ is decidable, inspection of the proof of Lemma 4.10 shows that we can choose $f(x,y) \in \Delta ^0_2$ . Then $\mathsf {Pr}^{\beta }(x)$ numerates $\beta (T^{\vdash } )$ in T by Lemma 4.11. □

The above lemma also provides a different proof of Theorem 4.14.

Alternative Proof of Theorem 4.14

Since $\gamma $ is a computable simulation we have $\alpha \sim \gamma $ , by Theorem 4.6. Applying Lemma 4.16 yields a binumeration $f(x,y)$ of $\gamma \circ \alpha ^{-1}$ and a formula $\mathsf {Pr}^{\gamma }(x)$ satisfying Löb( $T, \gamma $ ), such that

Since $\gamma $ is standard, . Thus .□

4.3 A note on intensionality

Despite its satisfaction of Löb’s conditions as well as its numeration of T-theorems, I believe that even if $\mathsf {Pr}^{\gamma }(x)$ is constructed canonically, the provability predicate $\mathsf {Pr}^{\alpha }(x)$ obtained in Theorem 4.15 hardly qualifies as being intensionally correct. According to the proof of Lemma 4.11, $\mathsf {Pr}^{\alpha }(x)$ is of the form $\exists z (\;f(z,x) \wedge \mathsf {Pr}^{\gamma }(z) )$ , with f being a binumeration of ${\alpha \circ \gamma ^{-1}}$ . The intensional correctness of $\mathsf {Pr}^{\alpha }(x)$ therefore presumably fails, if made precise by a version of the resemblance criterion (see Halbach & Visser, Reference Halbach and Visser2014, p. 676 and footnote 9). If intensional correctness is understood by recourse to meaning postulates (see Halbach & Visser, Reference Halbach and Visser2014, p. 676), it should be noted that computable equivalence is too weak to preserve proof-theoretic properties which are not merely schematic.

For instance, any intensionally correct provability predicate might be taken to satisfy global versions of Löb’s conditions (see for instance D2 $^{\text {G}}$ and D3 $^{\text {G}}$ in Kurahashi, Reference Kurahashi2020, sec. 2.3), as opposed to the local (i.e., schematic) conditions employed in this paper. While the invariance of Gödel’s Second Theorem with regard to these conditions remains unaffected, it can be shown that there is a computable simulation $\alpha $ such that the provability predicate $\mathsf {Pr}^{\alpha }(x)$ obtained in Theorem 4.15 does not satisfy these global conditions.

Furthermore, computability is in general too weak to enforce an intensionally correct arithmetisation of even basic syntactic operations. Before I illustrate this point for $T = \mathsf {PA}$ , the reader is reminded that $A^\ast $ denotes the set of strings over the alphabet A of our language $\mathcal {L}$ . The operation $\underline {\neg }$ maps each well-formed formula $\varphi \in A^\ast $ to its negation $(\neg \varphi )$ . We may reasonably require that every intensionally correct arithmetisation of syntax is based on a numbering such that the tracking functions of $\underline {\neg }$ and the other constructor operations introduced in §2.1 are provably total in $\mathsf {PA}$ (see also Halbach, Reference Halbach2014, p. 33).

We now construct a computable simulation $\beta $ of the set of strings $A^\ast $ , such that the tracking function of $\underline {\neg }$ is recursive but not provably total in $\mathsf {PA}$ . Suppose that the symbols of the alphabet A are ordered. We now effectively enumerate $A^\ast $ using the length-first ordering, according to which $s < t$ , if the length of s is smaller than the length of t, or if s and t have the same length, s is lexicographically smaller than t. Let $\psi _i$ be the i-th string in this enumeration. Let $h \colon \mathbb {N} \to \mathbb {N}$ be a strictly increasing and recursive function which is not provably total in $\mathsf {PA}$ , such that $h(\mathbb {N})$ is decidable, $0 \notin h(\mathbb {N})$ and $| h(\mathbb {N}) | = | \mathbb {N} \setminus h(\mathbb {N}) | = | \mathbb {N} | $ .Footnote ²⁶

We define $\beta \colon A^\ast \to \mathbb {N}$ recursively as follows. Let $\beta (\psi _0) = 0$ . Suppose now that $\beta (\psi _j)$ is defined for all $j < i$ , then

The resulting numbering $\beta $ is bijective, and $\beta \sim \iota $ , where $\iota \colon A^\ast \to \mathbb {N}$ is given by $\psi _i \mapsto i$ . Since $\iota $ is a computable simulation (see Smullyan, Reference Smullyan1961, chap. 9 and Corollary 4.7), also $\beta $ computably simulates $A^\ast $ , by Lemma 4.5. However, the $\beta $ -tracking function of $\underline {\neg }$ coincides with h and is therefore not provably total in $\mathsf {PA}$ . That is, there is no $\Sigma _1$ -numeration $\mathsf {Neg}(x,y)$ of the graph of the function ${\beta (\varphi ) \mapsto \beta ( (\neg \varphi ) )}$ such that $\mathsf {PA} \vdash \forall x\, \exists ! z\, \mathsf {Neg}(x,y)$ . The definition of $\beta $ can be easily extended such that the $\beta $ -tracking functions of all the operations $\underline {=}$ , $\underline {\wedge }$ , $\underline {\rightarrow }$ , $\underline {\forall }$ , etc. are recursive but not provably total in $\mathsf {PA}$ . However, as already mentioned above, this might reasonably be taken as a minimal requirement for an intensionally correct arithmetisation when resorting to the meaning postulate approach.

The Numbering ${\delta }$

For technical convenience we add the constant symbol $\mathsf {1}$ to $\mathcal {L}$ . The reader is reminded that the set of expressions is given by $A^\ast $ (with $\mathsf {1} \in A$ ). Let $T \supseteq \mathsf {PA}^-$ be any fixed consistent theory, where $\mathsf {PA}^-$ is the theory of the non-negative parts of discretely ordered rings (see Kaye, Reference Kaye1991, pp. 16f.). We specify a numbering $\delta $ of $A^\ast $ such that the tracking functions of the operations $\underline {\mathsf {S}}$ , $\underline {+}$ , $\underline {\times }$ , $\underline {\prime }$ , $\underline {=}$ , $\underline {\wedge }$ and $\underline {\forall }$ are recursive relative to $\delta $ and the set of ( $\delta $ -codes of) T-theorems is decidable. For technical convenience, we employ a derivability relation such that for every formula $\varphi $ we have $T \vdash \varphi $ iff $T \vdash \forall \varphi $ , where $\forall \varphi $ denotes the universal closure of $\varphi $ .

The basic idea is to construct $\delta $ such that all ( $\delta $ -codes of) T-theorems can be isolated from the remaining ( $\delta $ -codes of) strings of $A^\ast $ by the decidable property of parity. In order to do so, we first partition the even and odd numbers into infinite decidable sets, each corresponding to a certain syntactic category. Let ${\langle x,y \rangle \equiv \frac {(x+y+1)(x+y+2)}{2}+y}$ be the usual pairing function. We define the following functions:

$$ \begin{align*} & & \Theta^{\mathsf{junk}}(x,y) & = 12 \cdot \langle x,y \rangle +1,\\ & & \Theta^{\mathsf{tm}}(x,y) & = 12 \cdot \langle x,y \rangle +3,\\ \Lambda^{=}(x,y) & = 8 \cdot \langle x,y \rangle, & \Theta^{=}(x,y) & = 12 \cdot \langle x,y \rangle +5,\\ \Lambda^{\neg}(x,y) & = 8 \cdot \langle x,y \rangle +2, & \Theta^{\neg}(x,y) & = 12 \cdot \langle x,y \rangle +7,\\ \Lambda^{\wedge}(x,y) & = 8 \cdot \langle x,y \rangle + 4, & \Theta^{\wedge}(x,y) & = 12 \cdot \langle x,y \rangle +9,\\ \Lambda^{\forall}(x,y) & = 8 \cdot \langle x,y \rangle +6, & \Theta^{\forall}(x,y) & = 12 \cdot \langle x,y \rangle +11. \end{align*} $$

In what follows, we define $\delta $ such that all ( $\delta $ -codes of) terms are elements of $\Theta ^{\mathsf {tm}} (\mathbb {N},\mathbb {N}) = \{ \Theta ^{\mathsf {tm}} (m,n) \mid m,n \in \mathbb {N} \} = 12 \mathbb {N} + 3$ , all ( $\delta $ -codes of) T-theorems of the form $(s = t)$ are elements of $\Lambda ^{=} (\mathbb {N},\mathbb {N}) = 8 \mathbb {N}$ , all ( $\delta $ -codes of) formulæ of the form $(s = t)$ which are not T-theorems are elements of $\Theta ^{=} (\mathbb {N},\mathbb {N}) = 12 \mathbb {N} +5$ , all ( $\delta $ -codes of) of T-theorems of the form $(\varphi \wedge \psi )$ are elements of $\Lambda ^{\wedge } (\mathbb {N},\mathbb {N}) = 8 \mathbb {N} + 2$ , etc. Moreover, all ( $\delta $ -codes of) junk-expressions, i.e., strings which are not well-formed expressions, are elements of $\Theta ^{\mathsf {junk}} (\mathbb {N},\mathbb {N}) = 12 \mathbb {N} + 1$ . Since

$$ \begin{align*} 2 \mathbb{N} & = \Lambda^{=} (\mathbb{N},\mathbb{N}) \cup \Lambda^{\neg} (\mathbb{N},\mathbb{N}) \cup \Lambda^{\wedge} (\mathbb{N},\mathbb{N}) \cup \Lambda^{\forall} (\mathbb{N},\mathbb{N}) \text{ and}\\ 2 \mathbb{N}+1 & = \Theta^{\mathsf{junk}}(\mathbb{N},\mathbb{N}) \cup \Theta^{\mathsf{tm}}(\mathbb{N},\mathbb{N}) \cup \Theta^{=} (\mathbb{N},\mathbb{N}) \cup \Theta^{\neg} (\mathbb{N},\mathbb{N}) \cup \Theta^{\wedge} (\mathbb{N},\mathbb{N}) \cup \Theta^{\forall} (\mathbb{N},\mathbb{N}) \end{align*} $$

all $\delta $ -codes of T-theorems are even, while the $\delta $ -codes of all other expressions are odd. Thus $T \vdash \varphi \text { iff } \delta (\varphi ) \text { is even}$ , for every sentence $\varphi $ . Using $\Sigma ^0_1$ -completeness it is then easy to check that the $\Delta ^0_0$ -formula $\mathsf {Pr}(x) \equiv \exists y < x \ x = \overline {2} \times y$ satisfies Löb( $T, \delta $ ) and . However, $\mathsf {Pr}(x)$ will not define the set of T-theorems, since not every even number will code a T-theorem. We construct a $\Delta ^0_0(\mathsf {exp})$ -formula below which in addition to satisfying Löb’s conditions also binumerates the set of T-theorems.

In order to define the desired numbering $\delta $ , we first construct numberings $\delta _i$ for every stage ${i < \omega }$ . On the term level any standard numbering will do. In order to satisfy the aforementioned desired properties, we set $\delta _0(s) = \Theta ^{\mathsf {tm}}(0, \gamma (s) )$ , for each $s \in \mathit {Term}$ , where $\gamma $ is Smith’s (Reference Smith2007) standard numbering. The arithmetisation of the syntactic properties of and operations on terms proceeds as usual. In particular, the properties of being a ( $\delta _0$ -code of a) variable and being a ( $\delta _0$ -code of a) term are decidable. Moreover, the $\delta _0$ -tracking functions of $\underline {\mathsf {S}}$ , $\underline {+}$ , $\underline {\times }$ , $\underline {\prime }$ are recursive.

In order to extend $\delta _0$ such that also $\underline {=}$ has a recursive tracking function, we first show that for any given expressions $s,t$ , it is decidable whether or not $T \vdash s=t$ (with $s,t$ being possibly open terms).

Since T is $\Pi ^0_1$ -sound and proves the axioms of a commutative semiring, to each term $s(x_1,\ldots ,x_l)$ there corresponds a unique polynomial $p_s \in \mathbb {N}[x_1,\ldots ,x_l]$ such that $p_s = p_t$ iff $T \vdash s = t$ . In order to show the decidability of $T \vdash s = t$ , the rough idea is to define a rewriting system which effectively reduces each term $s(x_1,\ldots ,x_l)$ to its unique normal form $p_s$ . Since commutativity blocks the termination of such systems, we instead define a class-rewriting system which essentially operates on AC-congruence classes of terms, with $AC$ being the equational theory consisting of the associativity and commutativity axioms of both $+$ and $\cdot $ . This gives rise to a system with the following properties:

1. 1. for each term there exists a unique normal form up to permutations under associativity and commutativity,

2. 2. each normal form $s'$ of s is a term and $T \vdash s = s'$ ,

3. 3. normal forms of terms can be effectively computed,

4. 4. for any two $s,t$ and respective normal forms $s',t'$ : $T \vdash s = t$ iff $s' \sim _{AC} t'$ .

We first define the ordinary term rewriting system R, consisting of the following rules:

1. i. $t \times (u + v) \rightarrow t \times u + t \times v$

2. ii. $\mathsf {S}t \rightarrow t + \mathsf {1}$

3. iii. $\mathsf {1} \times t \rightarrow t$

4. iv. $\mathsf {0} \times t \rightarrow \mathsf {0}$

5. v. $t + \mathsf {0} \rightarrow t$

In allowing the rewriting of a term by means of rewriting any $AC$ -equivalent term, R is thus extended to the class-rewriting system $R / AC$ . In order to show that this system terminates, let $A = \{ n \geq 2 \mid n \in \mathbb {N} \}$ and consider the following polynomial weight functions:

1. a. $\mathsf {0}_A = 2$

2. b. $\mathsf {1}_A = 2$

3. c. $\mathsf {S}_A = X + 4$

4. d. $+_A = X + Y +1$

5. e. $\times _A = XY$

All polynomials (a)–(e) satisfy associativity and commutativity. Furthermore the polynomials $F_{l,r}$ (i.e., the result of subtracting the weight of the right hand side from the left hand side of a given rule) of the rules $\text {(i)-(v)}$ are $X-1$ , $1$ , X, $2X-2$ and $3$ respectively, which are all strictly positive (over A). Thus $R / AC$ is terminating (see Terese, Reference Terese2003, chap. 6). Furthermore, R is left-linear. The critical pairs of R are of the form ${(u+v,\mathsf {1} \times u + \mathsf {1} \times v)}$ , $(\mathsf {0},\mathsf {0} \times u + \mathsf {0} \times v)$ and $(t \times u, t \times u + t \times \mathsf {0})$ . It can easily be checked that they converge, for instance, $\mathsf {1} \times u + \mathsf {1} \times v \rightarrow u + \mathsf {1} \times v \rightarrow u + v$ . Hence the class-rewriting system $R / AC$ has the Church-Rosser property modulo $AC$ (see Dershowitz & Jouannaud, Reference Dershowitz, Jouannaud and Leeuwen1990, chap. 7).

We may conclude that for each term, $R / AC$ effectively computes a unique normal form up to permutations under associativity and commutativity. In order to decide whether or not $T \vdash s =t$ for two terms $s,t$ , one proceeds as follows: first effectively rewrite s and t into their respective normal forms $s'$ and $t'$ . Then check whether or not $s' \sim _{AC} t'$ . Since the word problem for $AC$ is decidable and $T \vdash s =t$ iff $s' \sim _{AC} t'$ , the decidability of $T \vdash s =t$ obtains. This decision process can be mimicked on the $\delta _0$ -codes of terms in the usual way, yielding a corresponding decidable arithmetical property, which is binumerated by the formula $\mathsf {PrEqu}(x,y)$ .

In order to extend $\delta _0$ to a numbering of $A^\ast $ satisfying the desired properties, we exploit the fact that for any given expressions $\varphi $ , $\psi $ and x, whether or not $\varphi \wedge \psi $ and $\forall x \varphi $ are T-provable is already fully determined by the T-provability of $\varphi $ and $\psi $ (and by the decidable fact of whether or not x is a variable). Clearly this does not hold for $\neg $ , as neither $T \vdash \neg \varphi $ nor $T \not \vdash \neg \varphi $ are entailed by $T \not \vdash \varphi $ . For this reason, $\underline {\neg }$ will not have a recursive $\delta $ -tracking function.

Let $\mathit {Junk}$ denote all strings of $A^\ast $ which are not well-formed expressions. Let $\{ \chi _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of T-provable formulæ which are of the form $(\neg \varphi )$ and let $\{ \nu _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of non-T-provable formulæ of the form $(\neg \varphi )$ . Moreover, let $\{ \mu _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of $\mathit {Junk}$ .

Set $\delta _0(\chi _n) = \Lambda ^{\neg }(0,n) $ , $\delta _0(\nu _n)= \Theta ^{\neg }(0,n)$ and $\delta ^{\mathsf {junk}}(\mu _n)= \Theta ^{\mathsf {junk}}(0,n)$ . We now successively extend the numbering $\delta _0$ of $\mathit {Term} \cup \{ (\neg \varphi ) \mid \varphi \in \mathit {Fml} \}$ in $\omega $ -many steps to a numbering of $A^\ast $ , by defining a function $\delta _{i+1}$ for each $i < \omega $ as follows:

Finally, we set $\delta = \left ( \bigcup _{i < \omega } \delta _i \right ) \cup \delta ^{\mathsf {junk}}$ , with $\textrm{{dom}}(\delta ) = \left ( \bigcup _{i < \omega } \textrm{{dom}}(\delta _i) \right ) \cup \mathit {Junk}$ . By definition of the functions $\Theta ^{\mathsf {junk}}$ , $\Theta ^{\mathsf {tm}}$ , $\Lambda ^{=}$ , $\Theta ^{=}$ , etc., $\delta $ is an injective function and thus a numbering of $A^\ast $ . In order to show that $\{ \delta (\varphi ) \mid T \vdash \varphi \}$ can be binumerated by a $\Delta ^0_0(\mathsf {exp})$ -formula, we define

$$ \begin{align*} \mathsf{Pr}(x) \equiv & \exists s \ \mathsf{Seq}(s) \wedge x = (s)_{\mathsf{Lh}(s)} \wedge \forall i \leq \mathsf{Lh}(s) \big( \exists n \ (s)_i = \Lambda^{\neg}(0,n)\\ & \vee \exists p,q \ \mathsf{Term}(p) \wedge \mathsf{Term}(q) \wedge \mathsf{PrEqu}(p,q) \wedge (s)_i = \Lambda^{=}(p,q)\\ & \vee \exists j,k < i \ (s)_i = \Lambda^{\wedge}( (s)_j, (s)_k )\\ & \vee \exists j < i, y \leq (s)_i \ \mathsf{Var}(y) \wedge (s)_i = \Lambda^{\forall}(y, (s)_j ) \big). \end{align*} $$

Clearly, $\mathsf {Pr}(x)$ defines $\{ \delta (\varphi ) \mid T \vdash \varphi \}$ . As we have seen above, $\mathsf {PrEqu}(x,y)$ can be taken to be a $\Delta ^0_0(\mathsf {exp})$ -formula, as well as $\mathsf {Seq}(x), \mathsf {Term}(x)$ and $\mathsf {Var}(x)$ . It remains to show that the variable s can be bounded. To do so, construct a $\Delta ^0_0(\mathsf {exp})$ -definable height function $h(x)$ such that $h( \delta (\varphi ) ) = \# \{ \psi \mid \psi \text { is a subformula of } \varphi \}$ . Then the length of the sequence (coded by) s can be bounded by the number of subformulæ of the theorem (coded by) x, i.e., by $h(x)$ . Thus $\mathsf {Pr}(x)$ can be taken to be $\Delta ^0_0(\mathsf {exp})$ .

In a similar way, it can be shown that $\{ \delta (\varphi ) \mid \varphi \in \mathit {Fml} \}$ is $\Delta ^0_0(\mathsf {exp})$ -binumerable, by extending $\mathsf {Pr}(x)$ in each clause with the corresponding set $\Omega ^\neg $ , $\Omega ^{=}$ , $\Omega ^{\wedge }$ or $\Omega ^\forall $ of nonprovable formulæ respectively.

We conclude by showing the operations $\underline {=}$ , $\underline {\wedge }$ and $\underline {\forall }$ are recursive relative to $\delta $ . As above, it can be shown that $\textrm{{dom}}(\delta _i)$ , for all $i < \omega $ as well as $\textrm{{dom}}(\delta )$ is decidable. Moreover, the sets $\mathit {Var}$ , $\mathit {Term}$ , $\mathit {Fml}$ as well as $T^\vdash $ are decidable relative to $\delta $ . To then compute, for instance, the tracking function of $\underline {\wedge }$ given by , one first decides for a given input $(m,n)$ whether or not $\mathsf {Pr}(m)$ and $\mathsf {Pr}(n)$ are true. If both hold, then the output is $\Lambda ^{\wedge }(m,n)$ . If not, the output is $\Theta ^{\wedge }(m,n)$ . Hence, the tracking function $\underline {\wedge }$ is recursive. Showing the recursiveness of the remaining tracking functions proceeds in a similar way.

Remark

The deviant numbering $\delta $ is optimal in the following sense.

In short: invariance is maintained as long as all constructor operations are required to be recursive. The construction of $\delta $ shows that deviance lurks as soon as we drop the requirement of recursiveness for $\underline {\neg }$ . Similarly, deviant results can occur once the recursiveness requirement is given up for $\underline {\forall }$ , as witnessed by the numbering $\eta $ constructed below. However, we cannot expect to find a deviant analogue to $\delta $ and $\eta $ if we relax the requirement of recursiveness for $\underline {\wedge }$ .Footnote ²⁷

We can show similarly that there is no numbering $\delta '$ of $A^\ast $ such that the set of ( $\delta '$ -codes of) true sentences is decidable and the operations $\underline {\mathsf {S}}$ , $\underline {+}$ , $\underline {\times }$ , $\underline {\prime }$ , $\underline {=}$ , $\underline {\neg }$ and $\underline {\forall }$ are recursive relative to $\delta '$ .

The Numbering ${\theta }$

We now construct a numbering $\theta $ of $A^\ast $ such that the tracking function of $\underline {\neg }$ is recursive and the set of ( $\theta $ -codes of) T-theorems is decidable, for any fixed consistent theory $T \supseteq \mathsf {R}$ . As opposed to the above construction, we here employ the standard derivability relation such that only closed sentences are T-theorems. We then use the fact that membership of $\neg \varphi $ in the sets ${\{ \varphi \mid T \vdash \varphi \}}$ , $\{ \varphi \mid T \vdash \neg \varphi \}$ and $\{ \varphi \mid T \not \vdash \varphi , T \not \vdash \neg \varphi \}$ is already fully determined by the respective membership of $\varphi $ . This does however not hold for $\varphi \wedge \psi $ , since for instance ${T \vdash \neg (\varphi \wedge \neg \varphi )}$ but $T \not \vdash \neg (\varphi \wedge \varphi )$ , for any T-independent $\varphi $ . Hence, in what follows, $\theta (\varphi \wedge \psi )$ is not defined along the lines of $\theta (\neg \varphi )$ and thus $\underline {\wedge }$ does not have a recursive $\theta $ -tracking function.

Let $\{ \chi _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of T-theorems which are not of the form $(\neg \varphi )$ , let $\{ \mu _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of T-refutable sentences which are not of the form $(\neg \varphi )$ and let $\{ \nu _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of expressions not of the form $(\neg \varphi )$ which are neither T-provable nor T-refutable (including nonwell-formed expressions).

First, we set $\theta _0(\chi _n) = 3 \cdot \langle 0, n \rangle $ , $\theta _0(\mu _n) = 3 \cdot \langle 0, n \rangle + 1$ and $\theta _0(\nu _n) = 3 \cdot \langle 0, n \rangle +2$ , with $\textrm{{dom}}(\theta _0) = A^\ast \setminus \{ (\neg \varphi ) \mid \varphi \in A^\ast \}$ . We then extend $\theta _0$ to a numbering of $A^\ast $ in $\omega $ -many steps by defining $\theta _{i+1}$ for each $i < \omega $ :

$$ \begin{align*} {\mathrm{dom}}(\theta_{i+1}) & = {\mathrm{dom}}(\theta_i) \cup \{ (\neg \varphi) \mid \varphi \in {\mathrm{dom}}(\theta_i) \}\\ \theta_{i+1}(\varphi) & = \theta_i(\varphi), \text{ if } \varphi \in {\mathrm{dom}}(\theta_i) \\ \theta_{i+1}((\neg \varphi)) & = \begin{cases} 3 \cdot \langle i+1,j \rangle, & \text{ if } T \vdash \neg \varphi \text{ and } \theta_i(\varphi)=3 \cdot \langle i,j \rangle +1\\ 3 \cdot \langle i+1,j \rangle + 1, & \text{ if } T \vdash \varphi \text{ and } \theta_i(\varphi)=3 \cdot \langle i,j \rangle\\ 3 \cdot \langle i+1,j \rangle + 2, & \text{ if } T \not\vdash \varphi, T \not\vdash \neg \varphi \text{ and } \theta_i(\varphi)=3 \cdot \langle i,j \rangle + 2 \end{cases} \end{align*} $$

Finally, we set $\theta = \bigcup _{i < \omega } \theta _i$ , with $\textrm{{dom}}(\theta ) = \bigcup _{i < \omega } \textrm{{dom}}(\theta _i)$ . It is easy to check that $\theta $ is indeed a numbering of $A^\ast $ . We now show that ${\mathsf {Pr}(x) \equiv \exists y < x \ x = \overline {3} \times y}$ defines $\{ \theta (\varphi ) \mid T \vdash \varphi \}$ . First, we define the height function

$$ \begin{align*} h^\neg(\varphi) = \begin{cases} h^\neg(\xi) + 1 & \text{if } \varphi \equiv (\neg \xi), \text{ for some expression } \xi,\\ 0 & \text{otherwise.} \end{cases} \end{align*} $$

It is then easy to show by induction that for every T-theorem $\varphi $ the following holds:

$$ \begin{align*} \theta(\varphi) = 3 \cdot \langle h^\neg(\varphi),n \rangle \text{ and } \varphi \equiv \begin{cases} \underbrace{(\neg \cdots (\neg}_{h^\neg(\varphi)\text{-times}}\chi_n ) \cdots ) & \text{if } h^\neg(\varphi) \text{ is even},\\ \underbrace{(\neg \cdots (\neg}_{h^\neg(\varphi)\text{-times}} \mu_n ) \cdots ) & \text{if } h^\neg(\varphi) \text{ is odd}. \end{cases} \end{align*} $$

Since there is a 1:1-correspondence between T-theorems $\varphi $ and pairs of natural numbers $\langle h^\neg (\varphi ),n \rangle $ , the set of $\theta $ -codes of T-theorems is exactly $3 \mathbb {N}$ . To show that the $\theta $ -tracking function of $\underline {\neg }$ is recursive proceeds as in the case of $\delta $ .

The Numbering ${\eta }$

Let T be given as in the construction of the numbering $\delta $ . We now construct a numbering $\eta $ of $A^\ast $ such that the tracking functions of the operations $\underline {\mathsf {S}}$ , $\underline {+}$ , $\underline {\times }$ , $\underline {\prime }$ , $\underline {=}$ , $\underline {\neg }$ , $\underline {\wedge }$ and $\underline {\rightarrow }$ are recursive relative to $\eta $ and $\{ \eta (\chi ) \mid \mathbb {N} \models \chi \}$ is decidable. As in the case of $\delta $ , we partition the natural numbers into decidable infinite sets corresponding to certain syntactic categories:

$$ \begin{align*} & & & & \Upsilon^{\mathsf{junk}}(x,y) & = 18 \cdot \langle x,y \rangle +2,\\ & & & & \Upsilon^{\mathsf{tm}}(x,y) & = 18 \cdot \langle x,y \rangle +5,\\ \Lambda^{=}(x,y) & = 12 \cdot \langle x,y \rangle, & \Theta^{=}(x,y) & = 12 \cdot \langle x,y \rangle +1, & \Upsilon^{=}(x,y) & = 18 \cdot \langle x,y \rangle +8,\\ \Lambda^{\neg}(x,y) & = 12 \cdot \langle x,y \rangle +3, & \Theta^{\neg}(x,y) & = 12 \cdot \langle x,y \rangle +4, & \Upsilon^{\neg}(x,y) & = 18 \cdot \langle x,y \rangle +11,\\ \Lambda^{\wedge}(x,y) & = 12 \cdot \langle x,y \rangle + 6, & \Theta^{\wedge}(x,y) & = 12 \cdot \langle x,y \rangle +7, & \Upsilon^{\wedge}(x,y) & = 18 \cdot \langle x,y \rangle +14,\\ \Lambda^{\forall}(x,y) & = 12 \cdot \langle x,y \rangle +9, & \Theta^{\forall}(x,y) & = 12 \cdot \langle x,y \rangle +10, & \Upsilon^{\forall}(x,y) & = 18 \cdot \langle x,y \rangle +17. \end{align*} $$

In what follows we define $\eta $ such that all numbers of true sentences of the form $(s = t)$ are elements of $\Lambda ^{=} (\mathbb {N},\mathbb {N}) = \{ \Lambda ^{=} (m,n) \mid m,n \in \mathbb {N} \} = 12 \mathbb {N}$ , all numbers of true sentences of the form $(\varphi \wedge \psi )$ are elements of $\Lambda ^{\wedge } (\mathbb {N},\mathbb {N}) = 12 \mathbb {N} + 3$ , etc. Since

$$ \begin{align*} \Lambda^{=} (\mathbb{N},\mathbb{N}) \cup \Lambda^{\neg} (\mathbb{N},\mathbb{N}) \cup \Lambda^{\wedge} (\mathbb{N},\mathbb{N}) \cup \Lambda^{\forall} (\mathbb{N},\mathbb{N}) = 3 \mathbb{N}, \end{align*} $$

all ( $\eta $ -codes of) true sentences are elements of $3 \mathbb {N} = \{ 3 \cdot n \mid n \in \mathbb {N} \}$ . Similarly, all ( $\eta $ -codes of) false sentences are elements of $3 \mathbb {N} +1$ , and all ( $\eta $ -codes of) expressions which are not sentences are elements of $3 \mathbb {N} +2$ . Thus $\mathsf {Pr}(x) \equiv \exists y < x \ x = \overline {3} \times y$ satisfies Löb( $T, \eta $ ) and , for any sound T. Moreover, the set $\{ \eta (\chi ) \mid \mathbb {N} \models \chi \}$ is definable by a $\Delta ^0_0(\mathsf {exp})$ -formula.

In a similar way to the above definitions of $\delta $ and $\theta $ , we define $\eta $ by exploiting the fact that the membership of $\neg \varphi $ and $\varphi \wedge \psi $ in $\{ \chi \mid \mathbb {N} \models \chi \}$ , ${\{ \chi \mid \mathbb {N} \not \models \chi \}}$ and ${\{ \chi \mid \chi \text { is not a sentence} \}}$ is already fully determined by the respective memberships of $\varphi $ and $\psi $ .Footnote ²⁸ This however does not hold for $\underline {\forall }$ , since there are expressions $\varphi $ , $\psi $ such that both $\mathbb {N} \models \forall x \varphi $ and $\mathbb {N} \models \forall x \psi $ but $\varphi \in \{ \chi \mid \mathbb {N} \models \chi \}$ and ${\psi \in \{ \chi \mid \chi \text { is not a sentence} \}}$ . Hence, in what follows, $\eta (\forall x \varphi )$ is not defined along the lines of $\eta (\neg \varphi )$ and $\eta (\varphi \wedge \psi )$ , resulting in the nonrecursiveness of the $\eta $ -tracking function of $\underline {\forall }$ .

Let $\{ \chi _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of true sentences which are of the form $(\forall x \varphi )$ , let $\{ \nu _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of false sentences which are of the form $(\forall x \varphi )$ , and let $\{ \mu _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of formulæ of the form $(\forall x \varphi )$ which are not sentences (with x and $\varphi $ being expressions). Moreover, let $\{ \xi _n \mid n \in \mathbb {N} \}$ be an enumeration without repetitions of strings which are not well-formed expressions.

We first define a numbering $\eta _0$ of $\mathit {Term} \cup \{ (\forall x \varphi ) \mid x \in \mathit {Var}, \varphi \in \mathit {Fml} \} $ by setting $\eta _0(s) = \Upsilon ^{\mathsf {tm}}(0,\gamma (s))$ for $s \in \mathit {Term}$ , and $\eta _0(\chi _n)= \Lambda ^{\forall }(0,n) $ , $\eta _0(\nu _n)= \Theta ^{\forall }(0,n)$ and $\eta _0(\mu _n)= \Upsilon ^{\forall }(0,n)$ . Moreover, set $\eta ^{\mathsf {junk}}(\xi _n)= \Upsilon ^{\mathsf {junk}}(0,n) $ .

As above we extend $\eta _0$ to a numbering of the well formed expressions, by defining $\eta _{i+1}$ for each $i < \omega $ :

$$ \begin{align*} {\mathrm{dom}}(\eta_{i+1}) & = {\mathrm{dom}}(\eta_i) \cup \{ (s = t) \mid s,t \in {\mathrm{dom}}(\eta_i) \cap \mathit{Term} \} \cup\\ & \quad \cup \{ (\neg \varphi), (\varphi \wedge \psi) \mid \varphi, \psi \in {\mathrm{dom}}(\eta_i) \cap Form\};\\ \eta_{i+1}(\varphi) & = \eta_i(\varphi), \text{ if } \varphi \in {\mathrm{dom}}(\eta_i);\\ \eta_{i+1}((s = t)) & = \begin{cases} \Lambda^{=}(\eta_i(s),\eta_i(t)), & \text{ if } s,t \in {\mathrm{dom}}(\eta_i), s,t \text{ are closed terms, } \mathbb{N} \models s=t,\\ \Theta^{=}(\eta_i(s),\eta_i(t)), & \text{ if } s,t \in {\mathrm{dom}}(\eta_i), s,t \text{ are closed terms, } \mathbb{N} \not\models s = t,\\ \Upsilon^{=}(\eta_i(s),\eta_i(t)), & \text{ if } s,t \in {\mathrm{dom}}(\eta_i) \cap \mathit{Term}, \\ & \;\,\,s \text{ or } t \text{ is not a closed term;} \end{cases}\\ \eta_{i+1}((\neg \varphi)) & = \begin{cases} \Lambda^{\neg}(0,\eta_i(\varphi)), & \text{ if } \varphi \in {\mathrm{dom}}(\eta_i), \mathbb{N} \not\models \varphi,\\ \Theta^{\neg}(0,\eta_i(\varphi)), & \text{ if } \varphi \in {\mathrm{dom}}(\eta_i), \mathbb{N} \models \varphi,\\ \Upsilon^{\neg}(0,\eta_i(\varphi)), & \text{ if } \varphi \in {\mathrm{dom}}(\eta_i) \cap Form, \varphi \text{ is not a sentence;} \end{cases}\\ \eta_{i+1}((\varphi \wedge \psi)) & = \begin{cases} \Lambda^{\wedge}(\eta_i(\varphi),\eta_i(\psi)), & \text{ if } \varphi, \psi \in {\mathrm{dom}}(\eta_i), \mathbb{N} \models \varphi, \psi,\\ \Theta^{\wedge}(\eta_i(\varphi),\eta_i(\psi)), & \text{ if } \varphi, \psi \in {\mathrm{dom}}(\eta_i), \mathbb{N} \not\models \varphi \text{ or } \mathbb{N} \not\models \psi,\\ \Upsilon^{\wedge}(\eta_i(\varphi),\eta_i(\psi)), & \text{ if } \varphi, \psi \in {\mathrm{dom}}(\eta_i) \cap Form, \\ & \,\varphi \text{ or } \psi \text{ is not a sentence.} \end{cases} \end{align*} $$

Finally, we define $\eta = \left ( \bigcup _{i < \omega } \eta _i \right ) \cup \eta ^{\mathsf {junk}}$ , with $\textrm{{dom}}(\eta ) = \left ( \bigcup _{i < \omega } \textrm{{dom}}(\eta _i) \right ) \cup \mathit {Junk}$ . Showing that $\eta $ is a numbering of $A^\ast $ such that $\{ \eta (\chi ) \mid \mathbb {N} \models \chi \}$ can be defined by a $\Delta ^0_0(\mathsf {exp})$ -formula and the $\eta $ -tracking functions of all operations operations $\underline {\mathsf {S}}$ , $\underline {+}$ , $\underline {\times }$ , $\underline {\prime }$ , $\underline {=}$ , $\underline {\neg }$ , $\underline {\wedge }$ and $\underline {\rightarrow }$ are recursive proceeds as in the case of $\delta $ above.

The Numbering ${\zeta }$

We conclude with a construction of a numbering $\zeta $ of $A^\ast $ which is monotonic and such that $\zeta (T^\vdash ) = \{ n \mid {\exists\kern-0.45em\exists } m \geq 10 \ n = m^3 \}$ . Hence the formula ${\mathsf {Pr}(x) \equiv \exists y < x (x = y \times y \times y \wedge y \geq \overline {10})}$ is a $\Delta ^0_0$ -binumeration of ( $\zeta $ -codes) of T-theorems.

Let $\alpha _0 \equiv ( \mathsf {0} = \mathsf {0} )$ and $\alpha _{i+1} \equiv ( \alpha _i \wedge (\mathsf {0} = \mathsf {0}))$ , for $i < \omega $ . Let $(\beta _i)_{i < \omega }$ be an enumeration (without repetitions) of the substrings of elements of $(\alpha _i)_{i < \omega }$ which are themselves not elements of $(\alpha _i)_{i < \omega }$ . We assume that $(\beta _i)_{i < \omega }$ is given in length-first ordering, i.e., $i < j$ iff the length of $\beta _i$ is smaller than the length of $\beta _j$ , or $\beta _i$ and $\beta _j$ have the same length and $\beta _i$ is lexicographically (relative to some total order on A) smaller than $\beta _j$ . Moreover, let $\gamma _0 \equiv ` \prime \text{'} $ and $\gamma _{i+1} \equiv \gamma _i \ast ` \prime \text{'} $ , for $i < \omega $ .

Let $(\chi _i)_{i < \omega }$ be a monotonic enumeration of ${A^\ast \setminus ( (\alpha _i)_{i < \omega } \cup (\beta _i)_{i < \omega } \cup (\gamma _i)_{i < \omega } )}$ without repetitions. We thus have $j \leq i$ if $\chi _j$ is a subexpression of $\chi _i$ . This can for instance be achieved by starting with a surjective and monotonic standard numbering of $A^\ast $ .

We define a bijection $g \colon \mathbb {N} \to A^\ast $ by recursion as follows. Let $\prec $ denote the strict substring relation on $A^\ast $ . Let $n \in \mathbb {N}$ and assume that $g(m)$ is already defined for all $m < n$ . Let $(\mu _i)_{i< \omega }$ be any given enumeration of strings. For the sake of brevity, we also write $\widetilde {\mu }$ instead of $(\mu _i)_{i < \omega }$ . We first set

That is, for a given enumeration $\widetilde {\mu }$ and input n, the function $\mathfrak {m}(\cdot ,\widetilde {\mu }) \colon \mathbb {N} \to \mathbb {N}$ outputs the smallest index k such that $\mu _k$ is not the g-value of any $m < n$ . We now define

Since $\widetilde {\alpha }$ , $\widetilde {\beta }$ , $\widetilde {\gamma }$ and $\widetilde {\chi }$ are pairwise disjoint enumerations without repetitions, g is injective. We now define $\zeta := g^{-1}$ .

We show that $\zeta $ is indeed a function with domain $A^*$ , i.e., we have to show that g is surjective. Informally speaking, this is tantamount to showing that there is no number from which on the values of g only belong to three or less of the four enumerations.

We first show that for each i there exists n such that $g(n) \equiv \alpha _i$ . Assume that this is not the case, and let $i_0 = \max \{i \mid \exists\kern-0.45em\exists n \ g(n) \equiv \alpha _i \}$ . Then there exists an index $j_0$ such that for all $j> j_0$ :

(8)

Let now $i> i_0$ be given such that $(\alpha _{i} \vee \alpha _{i}) \equiv \chi _j$ for some $j> j_0$ (note that such numbers exists since there are infinitely many strings of the form $(\alpha _{i} \vee \alpha _{i})$ ). Then $T \vdash (\alpha _{i} \vee \alpha _{i})$ and $\alpha _{i} \prec (\alpha _{i} \vee \alpha _{i})$ , but since $i> i_0$ there is no k such that $g(\alpha _{i}) \equiv k$ , in contradiction to (8).

Analogously it can be shown that for each i there exists n such that $g(n) \equiv \gamma _i$ (for instance, by considering $` \mathsf {v} \text{'} \ast \gamma _i$ instead of $(\alpha _{i} \vee \alpha _{i})$ ).

We now show that each $\beta _i$ is the g-value of some number, by induction over i. Since $\widetilde {\beta }$ is length-first-ordered, $\beta _0$ has no proper substrings and hence $g(0) \equiv \beta _0$ . Let $i> 0$ , and assume that each $\beta _j$ with $j <i$ is the g-value of some number. If $\psi \prec \beta _i$ , then $\psi \equiv \beta _j$ for some $j < i$ or $\psi \equiv \alpha _l$ for some $l \in \mathbb {N}$ . Since by induction hypothesis each $\beta _j$ with $j < i$ is the g-value of a number, and we have seen above that each $\alpha _l$ is the g-value of a number, there exists a smallest number n such that n is not of the form $m^3$ or $m^3+1$ (for $m \geq 10$ ), and the g-value of each (proper) substring of $\beta _i$ is smaller than n. Then $g(n) \equiv \beta _{\mathfrak {m}(n,\widetilde {\beta })} \equiv \beta _i$ .

We now show that each $\chi _i$ is the g-value of some number, by induction over i. Since $\widetilde {\chi }$ is monotonic, $\chi _0$ is a string of length $1$ . Hence $g(10^3+1) \equiv \chi _0$ .

Let $i> 0$ be given and assume that each $\chi _j$ with $j <i$ is the g-value of some number. If $\psi \prec \chi _i$ , then either $\psi \equiv \chi _j$ with $j < i$ , or $\psi \equiv \alpha _l$ or $\psi \equiv \beta _l$ or $\psi \equiv \gamma _l$ , for some $l \in \mathbb {N}$ . Once again, by induction hypothesis each $\chi _j$ with $j < i$ is the g-value of a number, and we have seen above that each $\alpha _l$ , $\beta _l$ and $\gamma _l$ is the g-value of a number. Hence $g(n) \equiv \chi _{\mathfrak {m}(n,\widetilde {\chi })} \equiv \chi _i$ , where n is the smallest number of the appropriate form, i.e., $m^3$ in case that $T \vdash \chi _i$ , and $m^3+1$ otherwise (for $m \geq 10$ ), such that the g-value of each (proper) substring of $\chi _i$ is smaller than n.

It remains to show that $\zeta $ is monotonic. Let $\varphi \prec \psi $ be given strings.

If $\psi \equiv \gamma _i$ , for some $i \in \mathbb {N}$ . Then $\varphi \equiv \gamma _j$ , for some $j < i$ . It follows immediately from the definition of g that if $\zeta (\gamma _i )=m $ , there exists $k < m$ such that $g(k) \equiv \gamma _j$ . Hence $\zeta (\gamma _j) < \zeta (\gamma _i)$ .

If $\psi \equiv \beta _i$ or $\psi \equiv \chi _i$ , for some $i \in \mathbb {N}$ , the claim immediately follows from the defining clauses of g.

If $\psi \equiv \alpha _i$ , for some $i \in \mathbb {N}$ , then $\varphi \equiv \alpha _j$ , for some $j < i$ , or $\varphi \equiv \beta _l$ for some $l \in \mathbb {N}$ . By definition of g, we have $\zeta (\alpha _j) < \zeta (\alpha _i)$ . It remains to show that for all $i,l \in \mathbb {N}$ :

$$ \begin{align*} (B[i,l]) \quad \text{If } \beta_l \prec \alpha_i \text{ then } g^{-1}(\beta_l) < g^{-1}(\alpha_i). \end{align*} $$

We first show that $A[i]$ implies , for any $i \in \mathbb {N}$ ; where $A[i]$ is the statement:

“ ”: Let $g(m^3) \equiv \alpha _i$ for some m. By $A[i]$ , we find n and l such that $g(n) \equiv \gamma _l$ and $(m-1)^3+1 < n < m^3$ . Since n cannot be of the form $s^3$ or $s^3+1$ , it follows that $g(n)$ is obtained by the second case of the third clause of the definition of g. Hence, there exists $\psi \prec \beta _{\mathfrak {m}(n,\widetilde {\beta })}$ such that ( $\ast $ ) $g(k) \not \equiv \psi $ for all $k < n$ . Clearly, $\psi $ is not contained in $\widetilde {\beta }$ . For otherwise, $\psi \equiv \beta _j$ for some $j \in \mathbb {N}$ . Then $j < \beta _{\mathfrak {m}(n,\widetilde {\beta })}$ by definition of $\widetilde {\beta }$ . But since $\beta _j$ is not the g-value of any number smaller than n, we have $\mathfrak {m}(n,\widetilde {\beta }) \leq j$ , resulting in a contradiction.

Hence, $\psi \equiv \alpha _j$ for some $j \in \mathbb {N}$ . Since $g^{-1}(\alpha _i) \equiv m^3$ , we have $g^{-1}(\alpha _{j'}) \leq (m-1)^3$ for every $j' < i$ . By ( $\ast $ ), $\alpha _j$ is not the g-value of any number smaller than n. Since $(m-1)^3 < n$ we thus conclude that $i \leq j$ . Therefore $\alpha _i \preceq \alpha _j \prec \beta _{\mathfrak {m}(n,\widetilde {\beta })}$ .

We show now that $B[i,l]$ holds, for any given l. If $\beta _l \prec \alpha _i$ , then also $\beta _l \prec \beta _{\mathfrak {m}(n,\widetilde {\beta })}$ . Hence $l < \mathfrak {m}(n,\widetilde {\beta })$ , by definition of $\widetilde {\beta }$ . But then there exists $k < n$ such that $g(k) \equiv \beta _l$ . Since $k < n < m^3$ we conclude that $g^{-1}(\beta _l) < g^{-1}(\alpha _i)$ , as desired.

“ ”: We show now by induction that $A[i]$ holds for all $i \in \mathbb {N}$ .

Since $\alpha _0$ has length $5$ , it has not more than $\frac {5 (5+1)}{2}$ substrings. By definition of g and $\widetilde {\beta }$ , it is easy to check that there is a number $n < 10^3$ such that the g-value of each proper substring of $\alpha _0$ is smaller than n. Hence $\alpha _0 \prec \beta _{\mathfrak {m}(n,\widetilde {\beta })}$ . But since $g^{-1}(\alpha _0) \geq 10^3$ we have $g(n) \equiv \gamma _{\mathfrak {m}(n,\widetilde {\beta })}$ .

Let now $i> 0$ be given, and assume that $A[i-1]$ holds. Let $g^{-1}(\alpha _i) = m^3$ . We assume that $A[i]$ does not hold. Let n be fixed such that $(m-1)^3+1 < n < m^3$ . We then have $g(n) \equiv \beta _{l}$ for some l. Hence,

(9)

If $\beta _{l} \prec \alpha _{i-1}$ , then $g^{-1}(\beta _l) < g^{-1}(\alpha _{i-1}) \leq (m-1)^3$ by $B[i-1,l]$ (using the fact that and the induction hypothesis $A[i-1]$ ). But since $(m-1)^3 < n = g^{-1}(\beta _l)$ , we conclude that $\beta _{l} \prec \alpha _{i-1}$ does not hold. We now show that $\beta _{l} \prec \alpha _{i}$ . Assume this is not the case, then $\alpha _{i} \prec \beta _{l}$ by definition of $\widetilde {\beta }$ . Using (2) we can then derive the contradiction $g^{-1}(\alpha _i) < n < m^3 = g^{-1}(\alpha _i)$ .

We thus have shown that $\beta _{l} \prec \alpha _{i}$ , but $\beta _{l} \not \prec \alpha _{i-1}$ . Since $\beta _{l}$ is the g-value of any given number between $(m-1)^3+1$ and $m^3$ , there are $\Delta $ -many substrings of $\alpha _l$ which are not substrings of $\alpha _{i-1}$ , with $\Delta := (m^3 - ((m-1)^3+1) -1$ .

We derive a contradiction by showing that

(10) $$ \begin{align} \#(\alpha_{i}) - \#(\alpha_{i-1}) < \Delta, \end{align} $$

where $\#(s)$ denotes the number of substrings of s. Let $\mathit {lh}(s)$ denote the length of s. Since $\mathit {lh}(\alpha _{i}) = 5 + 8i$ , it is easy to check that

$$ \begin{align*} \#(\alpha_{i}) - \#(\alpha_{i-1}) \leq 8 \mathit{lh}(\alpha_{i}) = 64i + 40. \end{align*} $$

Moreover $i \leq m - 10$ , since $(i + 10)^3 \leq g^{-1}(\alpha _i) = m^3$ . Hence

$$ \begin{align*} \#(\alpha_{i}) - \#(\alpha_{i-1}) \leq 8 \mathit{lh}(\alpha_{i}) = 64i + 40 \leq 64(m - 10) + 40 = 64m - 600 \end{align*} $$

It remains to check that $64m - 600 < \Delta $ . But this is equivalent to

$$ \begin{align*} 64m - 600 < (m^3 - ((m-1)^3+1) -1 = 3m^2 - 3m -1, \end{align*} $$

which in turn is equivalent to the true equation

$$ \begin{align*} 0 < 3m^2 - 67m +599. \end{align*} $$

This concludes the proof of (3), in contradiction to there being $\Delta $ -many substrings of $\alpha _l$ which are not substrings of $\alpha _{i-1}$ . Thus $A[i]$ obtains.