2 Technical Preliminaries

In this section, we introduce the necessary basic notions concerning syntax, theories and numberings.

2.3 Gödel numberings

Let S be a domain which permits a robust notion of effectiveness for functions $\xi \colon S \to \omega $ .Footnote ⁵ We say that a function $\xi \colon S \to \omega $ is a Gödel numbering or coding of S, if $\xi $ is injective and effective.Footnote ⁶ We also call $\xi (A)$ the ( $\xi $ -)code of A (for terms and formulæ A). In this paper we consider numberings of languages $\mathcal {L}$ given in infix or Polish notation. We note that choice of infix language versus Polish language is, in a sense, immaterial, since the two languages/representations of the language are connected by a standard bijection.

We occasionally also consider a language as embedded in a set of strings. Then, a numbering of the strings over the given finite alphabet will induce a numbering of the language. For instance, the infix expressions of ${\mathcal {L}^0}$ can be conceived of as strings over an alphabet containing $17$ symbols, while its Polish notations can be formulated as strings over an alphabet with $15$ symbols. (So, when we consider the language as embedded in strings the difference between infix and Polish suddenly does have some role.)

We note that, e.g., if we would think of formulæ as labeled directed acyclic graphs (dags), the numbering’s domain S could be the totality of all finite labeled dags with labels in a fixed alphabet. Such choices often reflect a syntax theory. If we view syntax as sui generis, S will usually be the set of expressions itself. If we view the syntactic objects as specima of a wider variety X, we will usually take $S :=X$ .

Remark 2.1. We could allow one extra degree of freedom for Gödel numberings: we could drop functionality. For example, suppose we want to an existing coding of syntax in the finite sets to define our Gödel numbering via coding the finite sets as numbers. There are various ways to code the finite sets. Suppose we do it by interpreting sequences first and then ignoring the order of the components. This gives us a non-functional Gödel numbering. Note that it would still be injective. For the purposes of the present paper we will not need the extra flexibility of non-functionality.

Remark 2.2. The Gödel numbering employed in Reference Feferman[4] is a nice example of a Gödel numbering that looks directly at the language-as-algebra without considering it as embedded in strings. For Feferman’s coding, the choice between infix and Polish is irrelevant.

An important example of a numbering of strings is the length-first ordering. Let ${\mathcal A}^{\ast }$ be a finite alphabet and suppose some ordering of ${\mathcal A}$ is given. We order the strings of ${\mathcal A}$ using the length-first ordering $(\alpha _n)_{n\in \omega }$ in which we enumerate the strings according to increasing length, where the strings of same length are ordered alphabetically. We set ${\mathfrak g}(\alpha ) = n$ if $\alpha = \alpha _n$ .Footnote ⁷ We write $|\alpha |$ for the length of $\alpha $ .

Throughout this paper, we will use the following basic fact about the length-first ordering.

Lemma 2.3. Suppose the alphabet ${\mathcal A}$ has $N \geq 2$ letters. We then have:

$$ \begin{align*} \frac{N^{|\alpha|}-1}{N-1} \leq {\mathfrak g} (\alpha) < \frac{N^{|\alpha|+1}-1}{N-1}. \\[-15pt]\end{align*} $$

It follows that $|\alpha | \leq {\mathfrak g}(\alpha )$ . Moreover, whenever $|\alpha | < |\beta |$ , then ${\mathfrak g}(\alpha ) < {\mathfrak g}(\beta )$ .

Proof. Clearly, for any string $\alpha $ of ${\mathcal A}$ we have

$$ \begin{align*} \overbrace{1\cdots 1}^{|\alpha| \times} \leq {\mathfrak g}(\alpha) \leq \overbrace{1\cdots 1}^{|\alpha|+1\; \times}, \\[-15pt]\end{align*} $$

where, for any m, $\overbrace {1\cdots 1}^{m \times }$ is considered as an N-adic notation. The claim follows immediately from a well-known property of geometric series:

$$ \begin{align*} \overbrace{1\cdots 1}^{m \times} = \sum_{i=0}^{m - 1} N^i = \frac{N^m-1}{N-1}. \\[-42pt]\end{align*} $$

□

2.4 Naming devices

Let $\mathsf {ClTerm}_{\mathcal {L}}$ denote the set of closed ${\mathcal {L}}$ -terms. We call a function ${\nu \colon \omega \to \mathsf {ClTerm}_{\mathcal {L}}}$ a numeral function for $\mathcal {L}$ , if $\nu $ is injective and effective and the closed term $\nu (n)$ has value n, for each $n \in \omega $ . We call $\nu (n)$ the $\nu $ -numeral of n. Standard numerals form a canonical choice of a numeral function and are defined as follows.

• • $\alpha ::= {\sf 0} \mid {\sf S}\alpha $ .

We write $\underline {n}$ for the ordinary, or standard, numeral with value n, and also write $\underline {\cdot }$ for the standard numeral function. These numerals provide unique normal forms for the natural numbers. These normal forms reflect the fact that we naturally read the axioms for addition and multiplication as directed. We have $(x+{\sf 0}) \rightsquigarrow x$ , $(x+{\sf S}y) \rightsquigarrow {\sf S}(x+y)$ , $(x\times {\sf 0}) \rightsquigarrow {\sf 0}$ , $(x \times {\sf S}y) \rightsquigarrow ((x\times y)+x)$ . The numerals are the normal forms for the closed terms in this reduction system.

In the literature on weak systems, however, another kind of numeral is used that corresponds to binary or dyadic notations. The main reason for this is simply that the arithmetisation of the numeral function for standard numerals and the usual Gödel numberings exhibits exponential growth. This holds for example for the Gödel numbering ${\mathfrak g}$ based on the length-first ordering. The numeral function for the numerals in the other style is more efficient. Thus, these alternative numerals are also called efficient numerals. We define efficient numerals for dyadic representations as follows.

• • $\alpha ::= {\sf 0} \mid {\sf S}({\sf SS}{\sf 0} \times \alpha ) \mid {\sf SS}({\sf SS}{\sf 0} \times \alpha )$ .

We write $\overline {n}$ for the efficient numeral with value n and write $\overline {\cdot }$ for the efficient numeral function. We note that efficient numerals also correspond more naturally to reduction systems in a different signature, e.g., a theory of strings formulated with two successors.

Remark 2.4. Consider the Gödel numbering ${\mathfrak g}$ for the language $\mathcal {L}^0$ based on the length-first ordering, where we assume we used infix notations. We note that the cardinality of the alphabet is 17. Consider any number n.

Clearly, we have $|\underline {n}|= n+1$ . So, $\frac {17^{n+1}-1}{16} \leq {\mathfrak g}(\underline {n})$ . So ${\mathfrak g}(\underline {n})$ has an exponential lower bound in n.

We give an upper bound for ${\mathfrak g}(\overline {n})$ . Let the length of the dyadic notation of n be k. We have: $ 2^k-1 \leq n < 2^{k+1}-1$ . We can estimate the length of the dyadic numeral $\overline {n}$ by considering the worst case where the dyadic notation of n is a string of 2’s. We find that $|\overline {n}| \leq 8k+1$ . Thus, using that $17<2^5$ , we see that:

$$ \begin{align*} {\mathfrak g}(\overline{n}) < \frac{17^{8k+2} -1}{16} \leq 64\cdot 2^{40k} \leq 64\cdot (n+1)^{40}. \end{align*} $$

So, we see that $ {\mathfrak g}(\overline {n})$ has a polynomial upper bound in n.

3 Formalisations of m-Self-Reference

The notion of self-reference considered here is defined by means of a reference relation on sentences. According to Leitgeb Reference Leitgeb[17], “a singular sentence might […] be defined to refer to all the referents of all of its singular terms, and only to them” (p. 4). A sentence is then said to be self-referential if and only if it refers to itself. By employing a Gödel numbering, this notion of self-reference can be formalised in an arithmetical framework according to the following definition.Footnote ⁸

This notion of self-reference is for example captured by the Kreisel–Henkin Criterion for self-reference (see [Reference Halbach and Visser10, p. 684]). According to this criterion, given a formula $A(x)$ expressing a property P, a sentence $A(t)$ says of itself that it has property P iff $A(t)$ is m-self-referential with respect to $A(x)$ (in the sense of the above definition).

In order to make this notion of m-self-reference mathematically precise, Definition 3.1 has to be specified with respect to the formalisation choices (i)–(iii). Accordingly, we call a triple $\langle \mathcal {L}, \xi , \nu \rangle $ a formalisation choice, if

• • $\mathcal {L}$ is a language as specified in Section 2.1;

• • $\xi $ is a Gödel numbering of $\mathcal {L}$ ; and

• • $\nu $ is a numeral function for $\mathcal {L}$ .

It is easy to see that the attainability of m-self-reference is invariant regarding the choice of the numeral function. That is, m-self-reference is attainable for $\langle \mathcal {L}, \xi , \nu _1 \rangle $ iff m-self-reference is attainable for $\langle \mathcal {L}, \xi , \nu _2 \rangle $ , for any language $\mathcal {L}$ , numbering $\xi $ and numeral functions $\nu _1$ and $\nu _2$ for $\mathcal {L}$ . Hence, instead of requiring the provability of $t = \nu (\xi (A(t)))$ in Definition 3.2, one may equivalently require that ${\mathsf {R}_{\mathcal {L}} \vdash t = \underline {\xi (A(t))}}$ . The reason for considering arbitrary numeral functions as parameters in formalisation choices is that certain adequacy conditions on numberings will be shown to be sensitive to this aspect of formalisation.

Moreover, instead of using provability in $\mathsf {R}_{\mathcal {L}}$ , one can equivalently require in Definition 3.2 that $\mathbb {N} \models t = \underline {\xi (A(t))}$ , since $\mathsf {R}_{\mathcal {L}}$ proves every true equation of closed $\mathcal {L}$ -terms.

3.2 Monotonicity and self-referentiality

The notion of self-referential numberings can be generalised to arbitrary numeral functions.

Definition 3.3. A numbering $\xi $ is called self-referential for a numeral function $\nu $ if, for any formula $A(x)$ , there exists $n \in \omega $ such that $n=\xi ( A(\nu (n)) )$ .

As opposed to the notion of m-self-reference for sentences (see Definition 3.1), self-reference for numberings is a mere technical concept and not intended to capture a pre-theoretical or philosophical notion. However, as we have seen, self-referential numberings immediately yield m-self-referential sentences.

In the following sections we will investigate whether adequate numberings can be self-referential, i.e., whether self-referential numberings are indeed as contrived as typically believed. We will first examine whether monotonicity serves as an adequacy condition on numberings which is sufficiently restrictive to exclude self-referential numberings and will then turn to additional more restrictive adequacy constraints for numberings.

We start by showing that a monotonic numbering cannot be self-referential for a numeral function $\nu $ , if the number of subterms of $\nu (n)$ grows sufficiently fast. In order to make this precise, let $\mathsf {st} \colon \mathsf {ClTerm}_{\mathcal {L}} \to \omega $ be the function assigning to each closed $\mathcal {L}$ -term the number of its proper subterms-qua-type. We then get the following incompatibility result.

Lemma 3.4. Let $\nu $ be a numeral function and suppose that there is a constant $c \in \omega $ such that ${\sf st}(\nu(n))< c$ , for all $n \in \omega $ . Then, there is no monotonic numbering which is self-referential for $\nu $ .

Proof. Assume that $\xi $ is monotonic and self-referential for $\nu $ . Let $B(x)$ be an $\mathcal {L}$ -formula in which x does occur freely at least once. Set

$$ \begin{align*} A(x) := \underbrace{B(x) \vee \cdots \vee B(x)}_{(c+1) \times}. \end{align*} $$

Then there exists $n \in \omega $ such that $\xi (A(\nu (n))) = n$ . By monotonicity we then get

$$ \begin{align*} n & = \xi(A(\nu(n)))\\ & = \xi(\underbrace{B(\nu(n)) \vee \cdots \vee B(\nu(n))}_{(c+1) \times})\\ & \geq \xi(\underbrace{B(\nu(n)) \vee \cdots \vee B(\nu(n))}_{c \times}) +1 \geq \cdots\\ & \geq \xi(B(\nu(n))) + c\\ & \geq \xi(\nu(n)) +c\\ & \geq \mathsf{st}(\nu(n)) + c, \end{align*} $$

in contradiction to the assumption that $n - \mathsf {st}(\nu (n)) < c$ .□

Since $n - \mathsf {st}(\underline {n}) < 1$ for all $n \in \omega $ , we conclude from the above lemma:

Corollary 3.5. There is no self-referential numbering for standard numerals which is monotonic.

After the second author’s lecture on Cogwheels of Self-reference at the workshop Ouroboros, Formal Criteria of Self-Reference in Mathematics and Philosophy on February 17, 2018, Joel Hamkins asked whether the result on the non-monotonicity of self-referential Gödel numberings also holds when we consider efficient numerals.

We first note that Lemma 3.4 does not apply to efficient numerals. We define:

• • $\widetilde 0 := {\sf 0}$ and

• • $\widetilde {n+1} := {\sf S}({\sf SS}{\sf 0} \times \widetilde {n})$ .

Let ev be the evaluation function for arithmetical closed terms and let $|\alpha |$ denote the length of the string $\alpha $ . We find:

Lemma 3.6. For every $n \in \omega $

• • ${\sf ev}(\widetilde {n} ) = 2^n -1$ ;

• • $|\widetilde {n}| = 7n+1$ ; and

• • the number of subterms-qua-type of $\widetilde {n}$ is $\leq 2n+3$ .

From this lemma we conclude that the function $\lambda n.2^n -1$ grows exponentially while the number of subterms of $\overline {2^n -1}$ is not larger than $2n+3$ and thus only grows linearly. Hence, the assumption of Lemma 3.4 cannot be satisfied for efficient numerals, since there is no constant c such that $n - {\sf st}(\overline {n}) < c$ for all $n \in \omega $ .

Indeed, in what follows we show that the answer to Hamkins’ question is no, by constructing a monotonic numbering which is self-referential for efficient numerals.

4 A Monotonic Self-Referential Numbering

Let ${\mathcal L}$ be an arithmetical language, as introduced in Section 2.1, and let ${\mathcal L}(\mathsf {c})$ be ${\mathcal L}$ extended with a fresh constant $\mathsf {c}$ . Let $A_0,A_1,\dots $ be an effective enumeration of all expressions of ${\mathcal L}(\mathsf {c})$ . We assume that if $C \preceq A_n$ , then, for some $k\leq n$ , we have $C=A_k$ .

We define ${\lceil A \rceil }$ as the number of sub-expressions-qua-type of A, in other words, ${\lceil A \rceil } $ is the cardinality of $\{ B\mid B\preceq A \}$ . The following trivial observation is very useful:

Let $A [{\sf c} := t]$ denote the result of substituting t for all occurrences of ${\sf c}$ in A. Here is our construction.

The construction in stage k of the list $\Lambda _k$ can be illustrated as follows:

It follows from the construction that if $A^{\ast }_k$ is not in $\Lambda _{k-1}$ , then

(1) $$ \begin{align} B_{{\mathfrak n}(k)-1} = A_k^{\ast} = A_k[\mathsf{c} := \mathsf{S}(\widetilde{k+4})]. \end{align} $$

We will show in Lemma 4.6 that (1) holds for each $A_k$ which contains $\mathsf {c}$ .

To see that our construction is well-defined it is sufficient that ${\mathfrak n}(k)-\ell \geq {\mathfrak n}(k-1)$ or, equivalently, $\ell \leq {\mathfrak n}(k)-{\mathfrak n}(k-1)$ . We note that:

$$ \begin{align*} \ell \leq {\lceil A_k^{\ast} \rceil} \leq {\lceil A_k \rceil}+{\lceil {\sf S}\widetilde {(k+4)} \rceil}-1 \leq k +1 + 2(k+4)+3 = 3 k + 12. \end{align*} $$

(The $-1$ in the right-hand-side of the second inequality can be seen as follows. In case c occurs in $A_k$ it is subtracted in the substitution. If c does not occur, we have $A_k = A_k^{\ast }$ and, from this, the inequality follows immediately.)

If $k=0$ , we have $3\cdot 0 +12 = 12 < 17 = 2^{0+4}+1-0 = {\mathfrak n}(0) - {\mathfrak n}(-1)$ . Let $k>0$ . We find:

$$ \begin{align*} 3k+12 < 2^{k+3} = 2^{k+4}+1 - 2^{k+3}-1 = {\mathfrak n}(k)-{\mathfrak n}(k-1). \end{align*} $$

Setting ${\sf gn}_0(A)$ to be the unique n such that $A = B_n$ , we have found a monotonic self-referential Gödel numbering ${\sf gn}_0$ , where monotonicity is defined with respect to the sub-expression relation.

5 Strings

Let ${\mathcal L}$ be an arithmetical language, as introduced in Section 2.1, and let ${\mathcal A}$ be the alphabet of ${\mathcal L}$ . Let $\mathsf {c}$ be a fresh letter. We define:

• • ${\mathcal A}^{\ast }$ is the set of all strings of ${\mathcal A}$ , including the empty string $\varepsilon $ .

• • ${\mathcal A}_{\sf c}$ is ${\mathcal A}$ extended with c and ${\mathcal A}_{\sf c}^{\ast }$ is the set of all strings of ${\mathcal A}_{\sf c}$ . (In ${\mathcal L}({\sf c})$ viewed as a subset of ${\mathcal A}_{\sf c}^{\ast }$ , we treat c as a constant-symbol.)

• • $\alpha \sqsubseteq \beta $ iff $\alpha $ is a sub-string of $\beta $ .

• • $|\alpha |$ is the length of $\alpha $ .

• • ${\lfloor \alpha \rfloor }$ is the number of sub-strings-qua-type of $\alpha $ , in other words, ${\lfloor \alpha \rfloor } $ is the cardinality of $\{ \beta \mid \beta \sqsubseteq \alpha \}$ .

We have the following well-known fact.

Since we are not striving for maximal efficiency, we will estimate ${\lfloor \alpha \rfloor }$ by $|\alpha |^2+1$ . Let $\mathfrak a := {\sf S}({\sf SS}{\sf 0}\times {}$ . Then, ${\sf S}\widetilde {n} = {\sf S}\mathfrak a^n{\sf 0}{})^n$ . So, $| {\sf S}\widetilde {n} | = 1+6n +1+n = 7n+2$ and, thus, ${\lfloor {\sf S}\widetilde {n} \rfloor } \leq (7n+2)^2+1 = 49 n^2+ 28 n +5$ .

We will use the following insights.

Let $\alpha _0,\alpha _1,\dots $ be an effective enumeration of the strings in ${\mathcal A}^{\ast }_{\sf c}$ . We assume that if $\gamma \sqsubseteq \alpha _n$ , then, for some $k\leq n$ , we have $\gamma = \alpha _k$ . The following trivial observation is very useful:

Here is our construction.

To see that our construction is well-defined it is sufficient that ${\mathfrak n}(k)-\ell \geq {\mathfrak n}(k-1)$ or, equivalently, $\ell \leq {\mathfrak n}(k)-{\mathfrak n}(k-1)$ . We note that, if $k=0$ , we have $\alpha _{0} = \varepsilon $ , and hence $\ell =1$ . Moreover, ${\mathfrak n}(0)-{\mathfrak n}(0-1)= 2^{15}+1$ . So, we are done. Suppose $k>0$ . Clearly, ${\mathfrak n}(k)-{\mathfrak n}(k-1)= 2^{k+14}$ . In case c does not occur in $\alpha _k$ , we have $\ell \leq {\lfloor \alpha _k^{\ast } \rfloor }= {\lfloor \alpha _k \rfloor } \leq k+1 \leq 2^{k+14}$ . In case c does occur $\alpha _k$ , we have:

$$ \begin{align*} \ell \leq {\lfloor \alpha_k^{\ast} \rfloor} \leq {\lfloor \alpha_k \rfloor}|{\sf S}\widetilde {k+15}|^2 + 1 \leq (k+1)(7(k+15)+2)^2+1 \leq 2^{k+14} \end{align*} $$

We leave the verification of the last inequality to the diligent reader.

Setting ${\sf gn}_1(\alpha )$ to be the unique n such that $\alpha = \beta _n$ , we have found a monotonic self-referential Gödel numbering ${\sf gn}_1$ . Here monotonicity is defined with respect to the sub-string relation.

6 Gödel Numerals as Quotations

In the philosophical literature, Gödel numerals are typically conceived of as canonical names for their coded expressions. One of the most canonical naming devices is quotation, assigning to each expression A its name “ $A.$ ” Accordingly, the Tarski biconditional $\mathsf {T}(\underline {\xi (A)}) \leftrightarrow A$ for instance is often taken as an arithmetical proxy for the sentence “‘A’ is true iff $A.$ ”

The standard numeral $\underline {\xi (A)}$ is prima facie not the only candidate for an arithmetical proxy for quotation. As there are different adequate ways to quote an expression A, such as “A”, ‘A’, , etc., different numerals may be reasonably taken as adequate arithmetical proxies of A’s quotation, such as $\underline {\xi (A)}$ , $\overline {\xi (A)}$ , etc.Footnote ¹¹

Once $\nu $ -numerals are taken as adequate arithmetical proxies for quotations, another criterion for numberings may be extracted from the basic idea underlying monotonicity, which has been suggested by Halbach Reference Halbach[9]. Recall that monotonic numberings have the property that the code of an expression A is larger than the code of any expression (strictly) contained in A. Since each quotation properly contains its quoted expression, it may be argued, along the lines of Section 3.1, that adequate numberings should preserve this structural feature by requiring that the code of the Gödel $\nu $ -numeral of an expression A is larger than the code of A itself:

• M4 $(\nu )$ . $\ \xi (A) < \xi ( \nu (\xi (A)) )$ for all $A \in \mathcal {L}$ ,

We call a numbering $\xi $ of $\mathcal {L}$ strongly monotonic for $\nu $ -numerals, if $\xi $ satisfies M1–M3 as well as M4( $\nu $ ).

We note that M4( $\nu $ ) rules out that $\xi ({\sf 0}) = 0$ and $\nu (0) = {\sf 0}$ , which holds for certain length-first orderings and standard numerals. These standard numberings are not excluded by its weakened version M4 $^{\ast }$ ( $\nu $ ), which is obtained by replacing $<$ by $\leq $ in M4( $\nu $ ). In what follows, we will always employ the version of M4( $\nu $ ) which yields the stronger result.

Since the standard numeral $\underline {n}$ has n-many proper subterms, any numbering satisfying M1 also satisfies M4( $\underline {\cdot }$ ). Strong monotonicity is however not entailed by monotonicity when efficient numerals are employed. For instance, as will follow from the next lemma, the numbering ${\sf gn}_0$ is monotonic but not strongly monotonic for efficient numerals. In fact, strong monotonicity is incompatible with the self-referentiality of numberings.

Proof. Suppose $\xi $ is a self-referential numbering of $\mathcal {L}$ . Consider any formula $A(x)$ with free variable x and $n \in \omega $ such that $\xi (A(\nu (n))) = n$ . Since $\nu (n) \prec A(\nu (n))$ , using M3 and M4 $^{\ast }$ ( $\nu $ ) yields the contradiction

$$ \begin{align*} n = \xi(A(\nu(n))) \leq \xi(\nu(\xi(A(\nu(n))))) = \xi(\nu(n)) < \xi(A(\nu(n))) = n. \\[-35pt]\end{align*} $$

□

Thus, if we strengthen the conception of monotonicity in the above manner, then, by Halbach’s standards, there are no reasonable numberings which are self-referential.

6.1 A critical remark

We are somewhat sceptical whether strong monotonicity really is a necessary constraint for reasonable numberings (see also footnote 11). For instance, we now show that the length-first numbering is not strongly monotonic for a fairly reasonable numeral function based on prime decomposition. Any philosophically stable view which excludes self-referential numberings as adequate formalisation choices along the above lines, hence appears to be committed to render the introduced numeral function below an inadequate naming device.

Example 6.2. Let $\mathcal {L}^{\sf Q} := \mathcal {L}^0 \cup \{ {\sf Q} \}$ , where ${\sf Q}$ is a unary function symbol for taking the square. Let ${\mathfrak g}$ be a numbering of $\mathcal {L}^{\sf Q}$ based on the length-first ordering.

Are there plausible numerals for which strong monotonicity fails w.r.t. ${\mathfrak g}$ ? This is indeed the case. We give an example of such numerals that seems reasonably natural.

We consider the following numeral function $\mathsf {pd}$ . (‘ $\mathsf {pd}$ ’ stands for prime decomposition.)

1. i. ${\sf pd}(0) := {\sf 0}$ and ${\sf pd}(1) := {\sf S}{\sf 0}$ .

2. ii. Suppose $n> 1$ , and n is not a power of a prime. Let p be the smallest prime that divides n. Say $n = p^i\cdot m$ , where p does not divide m and $i>0$ . Then, ${\sf pd}(n) := \times {\sf pd}(p^i){\sf pd}(m)$ .

3. iii. Suppose $n = p^i$ , where p is prime and $i>0$ . We take ${\sf pd}(p) := \underline p$ . In case $i = 2k$ , for $k>0$ , we set ${\sf pd}(n) := {\sf Q}\,{\sf pd}(p^k)$ . In case $i = 2k+1$ , for $k>0$ , we set ${\sf pd}(n) := \times \underline p\, {\sf Q}\,{\sf pd}(p^k)$ .

We find that, e.g.,

$$ \begin{align*} {\sf pd}(2^{2^m}) = \overbrace{{\sf Q} \cdots {\sf Q}}^{m \times}\underline 2. \end{align*} $$

It follows that $|{\sf pd}(2^{2^m})| = m + 3$ . Hence, by Lemma 2.3,

$$ \begin{align*} {\mathfrak g}({\sf pd}(2^{2^m})) \leq \frac{16^{m+4}-1}{15}. \end{align*} $$

Since $2^{2^m}$ has double exponential growth, the failure of strong monotonicity is immediate.

We note that ${\sf pd}$ has good properties for multiplication but does not seem to be very friendly towards successor and addition.

One can imagine many variants of ${\sf pd}$ . For example, we can replace $\underline p$ by $\overline p$ in the definition. Alternatively, we can make $\mathfrak {p}$ where we replace clause (iii) by:

• • Suppose $n = p^i$ , where p is prime and $i>0$ . We take ${\mathfrak p}(p) := {\sf S}{\mathfrak p}(p-1)$ . In case $i = 2k$ , for $k>0$ , we set $\mathfrak p(n) := {\sf Q}\,\mathfrak p(p^k)$ . In case $i = 2k+1$ , for $k>0$ , we set $\mathfrak p(n) := \times {\sf S}\mathfrak p(p-1)\,{\sf Q}\,\mathfrak p(p^k)$ .

We leave an in-depth discussion of this constraint’s adequacy for another occasion and, from now on, simply assume that there are good reasons to consider strong monotonicity to be a necessary constraint for reasonable numberings. In the next section we further examine the implications of this constraint on the attainability of self-reference.

6.2 A strongly monotonic numbering with strong diagonalisation

While strong monotonicity excludes self-referential numberings (Lemma 6.1), we now show that this constraint is not sufficiently restrictive to rule out m-self-reference in $\mathcal {L}^0$ . In particular, we construct a strongly monotonic numbering ${\sf gn}_2$ for efficient numerals which satisfies the Strong Diagonal Lemma for $\mathcal {L} \supseteq \mathcal {L}^0$ :

Lemma 6.3 (Strong Diagonal Lemma for ${\mathcal {L}}$ )

Let C be an $\mathcal {L}$ -expression with x as a free variable. Then there exists a closed term t in $\mathcal {L}$ such that ${\mathsf {R}_{\mathcal {L}} \vdash t = \underline {{\sf gn}_2(C[x := t])}}$ .

We note that we escape Lemma 6.1 here by allowing t to be different from the efficient numeral of the value of t.

Let ${\mathcal L}$ be an arithmetical language, as introduced in Section 2.1, and let ${\mathcal L}(\mathsf {c})$ be ${\mathcal L}$ extended with a fresh constant $\mathsf {c}$ . Let $A_0,A_1,\dots $ be an effective enumeration of all expressions of ${\mathcal L}(\mathsf {c})$ . We assume that if $C \preceq A_n$ , then, for some $k\leq n$ , we have $C=A_k$ .

We define ${\lceil A \rceil }$ as the number of subexpressions-qua-type of A, in other words, ${\lceil A \rceil } $ is the cardinality of $\{ B\mid B\preceq A \}$ . The following trivial observation is very useful:

Lemma 6.4. ${\lceil A_k \rceil } \leq k+1$ .

Let the function $\widehat {\cdot } \colon \omega \to \mathcal {L}$ be given by setting $\widehat {0} := \mathsf {v}$ and $\widehat {n+1} := \widehat {n} '$ . As in the case of standard numerals, expressions of the form $\widehat m$ are downwards closed with regard to the sub-expression relation and we have that ${\lceil \widehat m \rceil } = m+1$ . In the following construction, we use expressions of the form $\widehat m$ instead of standard numerals as fillers. This ensures that the constructed enumeration does not start with $\overline 0$ and thus that the resulting numbering is strongly monotonic.

Here is our construction.

We construct a list $\Lambda :=(B_n)_{n\in \omega }$ in stages k. Let ${\mathfrak n}(k) := 2^{2k+4}+1$ . Each stage k will result in a list $\Lambda _k = B_0,\dots , B_{{\mathfrak n}(k)-1}$ . To simplify the presentation, we make $\Lambda _{-1}$ the empty list and ${\mathfrak n}(-1) := 0$ .

In stage k, we act as follows. Let $A_k^{\ast } := A_k [{\sf c} := {\sf S}\widetilde {(k+2)}]$ . Let the sub-expressions of $A_k^{\ast }$ that do not occur in $\Lambda _{k-1}$ and are not of the form $\widehat m$ be $A_{i_0},\dots , A_{i_{\ell -1}}$ , where the sequence $i_j$ is strictly increasing. We note that $\ell $ could be 0. We define $B_{{\mathfrak n}(k)-\ell +j} := A_{i_j}$ , for $j <\ell $ . Let s be the smallest number such that $\widehat s$ is not in $\Lambda _{k-1}$ . We set $B_{{\mathfrak n}(k-1)+p} := \widehat {s+p}$ , for $p< {\mathfrak n}(k)-{\mathfrak n}(k-1)-\ell $ .

To see that our construction is well-defined it is sufficient that ${\mathfrak n}(k)-\ell \geq {\mathfrak n}(k-1)$ or, equivalently, $\ell \leq {\mathfrak n}(k)-{\mathfrak n}(k-1)$ . We note that:

(*) $$ \begin{align} \ell \leq {\lceil A_k^{\ast} \rceil} \leq {\lceil A_k \rceil}+{\lceil {\sf S}\widetilde {(k+2)} \rceil}-1 \leq k +1 + 2(k+2)+3 = 3 k + 8. \end{align} $$

(The $-1$ in the right-hand-side of the second inequality can be seen as follows. In case c occurs in $A_k$ it is subtracted in the substitution. If c does not occur, we have $A_k = A_k^{\ast }$ and, from this, the inequality follows immediately.)

If $k=0$ , we have $3\cdot 0 +8 = 8 < 17 = 2^{2(0+2)}+1-0 = {\mathfrak n}(0) - {\mathfrak n}(-1)$ . Let $k>0$ . We find:

$$ \begin{align*} 3k+8 < 3 \cdot 2^{2k+2} = 2^{2k+4}+1 - 2^{2k+2}-1 = {\mathfrak n}(k)-{\mathfrak n}(k-1). \end{align*} $$

Lemma 6.5. Suppose $A_k \in {\mathcal L}$ . Then, $A_k$ is in $\Lambda _k$ and, hence, in $\Lambda $ .

Proof. Consider stage k. We note that $A_k^{\ast } = A_k$ . In case $A_k$ occurs in $\Lambda _{k-1}$ , we are done. In case $A_k$ is not in $\Lambda _{k-1}$ and not of the form $\widehat m$ , clearly, $A_k$ will be added, so we are again done. Suppose $A_k = \widehat m$ , for some m. It follows that $m+1={\lceil \widehat m \rceil } \leq k+1$ . We note that all sub-expressions of $A_k$ are of the form $\widehat {n}$ , for $n\leq m$ . So $\ell = 0$ . Clearly, $m+1 \leq k+1 < {\mathfrak n}(k) - {\mathfrak n}(k-1)$ . So, all sub-expressions of $A_k$ are either in $\Lambda _{k-1}$ or will be added.□

Lemma 6.6. Suppose $C \preceq B_n$ . Then, for some $j\leq n$ , we have $C = B_j$ .

Proof. Let $C \preceq B_n$ . Suppose $B_n$ is added in stage k. Then, $B_{n}$ is either of the form $\widehat m$ or a sub-expression of $A^{\ast }_k$ not of the form $\widehat m$ .

Suppose $B_n = \widehat m$ . Let s be the smallest number such that $\widehat s$ is not in $\Lambda _{k-1}$ . For all $u< s$ , we have $\widehat u\in \Lambda _{k-1}$ . By our construction, all $\widehat v$ , for $s\leq v \leq m$ are added at stage k in ascending order. So, for all $w \leq m$ , we find that $\widehat w$ precedes $\widehat m$ in $ \Lambda _k$ . Finally, each sub-expression C of $\widehat m$ is of the form $\widehat w$ , for some $w \leq m$ .

Suppose $B_n$ is added as a sub-expression of $A^{\ast }_k$ not of the form $\widehat m$ . First suppose $C= \widehat p$ . We note that $p+1 = {\lceil \widehat p \rceil } < {\lceil A_k \rceil } \leq k+1$ . This tells us, by Lemma 6.5, that $\widehat p$ is in $\Lambda _{k-1}$ . Hence, it is added to $\Lambda $ before $B_n$ . Now suppose that C is not of the form $\widehat p$ . Then, either C is in $\Lambda _{k-1}$ or added to $\Lambda _k$ before $B_n$ .□

Lemma 6.7. The enumeration $\Lambda $ is without repetitions.

Proof. Consider any $C \in {\mathcal L}$ . If C is not of the form $\widehat m$ by our construction it will be only added once.

Suppose $ C = \widehat m$ . We note that, by Lemma 6.6, the n such that $\widehat n \in \Lambda _{k-1}$ are downwards closed. This means that, in our construction, there is no $m\geq s$ , such that $\widehat m \in \Lambda _{k-1}$ . So all the $\widehat m$ that are added in any stage k are new.□

Lemma 6.8. Suppose ${\sf S}\widetilde {n}$ occurs in $\Lambda _k$ , then $n \leq k+2$ .

Proof. We prove this by induction on k. In stage 0, we easily verify that the largest term of the form $\mathsf {S}\widetilde {n}$ can be at most $\mathsf {S} \widetilde 2$ .

Suppose, we have our desired estimate for $k-1$ , where $k>0$ . We prove our estimate for k. Clearly, the $\mathsf {S}\widetilde {n}$ occurring in $\Lambda _{k-1}$ do satisfy our estimate. Moreover, all the $\widehat p$ added in stage k do not provide new elements of the form $\mathsf {S}\widetilde {n}$ . So, the only interesting case is the case where $A_k$ is not of the form $\widehat p$ . Suppose it is not. The ${\sf S}\widetilde {n}$ that are added are all sub-expressions of $ A^{\ast }_{k}= A_k[{\sf c} := {\sf S}\widetilde {(k+2)}]$ . So, we should focus on the largest such subterm. Suppose ${\sf S}\widetilde {n} \preceq A^{\ast }_k$ . There are two possibilities:

1. i. ${\sf S}\widetilde {n}\preceq A_k$ . In this case, we have $n < {\lceil \mathsf {S}\widetilde {n} \rceil } \leq {\lceil A_k \rceil } \leq k+1 \leq k +2$ .

2. ii. $n= k+2$ , since the subterm results from the substitution for c.

In each of the possible cases, we are done.□

Setting ${\sf gn}_2(A)$ to be the unique n such that $A = B_n$ , we have found a monotonic Gödel numbering ${\sf gn}_2$ .

In order to show that ${\sf gn}_2$ satisfies M4( $\overline {\cdot }$ ), i.e., that ${\sf gn}_2$ is strongly monotonic for efficient numerals, we prove the following auxiliary lemma.

Lemma 6.9. Suppose $\overline {p} \preceq A_k[\mathsf {c} := \overline r]$ , for some $p,r \in \omega $ . Then, $p < 2^{k}(r+2)$ .

Proof. Let $p,r \in \omega $ be given such that $\overline {p} \preceq A_k[\mathsf {c} := \overline r]$ . Then (i) $\overline {p} \preceq A_k$ or (ii) $\overline {p}\preceq \overline r$ or (iii) there exists $t \preceq A_k$ containing $\mathsf {c}$ such that $\overline p = t[\mathsf {c} := \overline r]$ . In case (i), we have, $p < 2^{{\lceil \overline p \rceil }} \leq 2^{{\lceil A_k \rceil }} \leq 2^{k+1} \leq 2^{k}(r+2)$ . In case (ii), we have $p\leq r$ . So both in Case (i) and (ii), we find $p< 2^{k}(r+2)$ .

Suppose we are are in case (iii). Since $\overline {r}$ and $t[\mathsf {c} := \overline r]$ are efficient numerals, there exist $k,j$ such that $\overline {r}$ and t are of the following forms:

• • $\overline {r} = \mathsf {S}_{a_0}( \cdots \mathsf {S}_{a_k}({\sf 0}) \cdots )$ ;

• • $t = \mathsf {S}_{c_0}( \cdots \mathsf {S}_{c_j}(\mathsf {c}) \cdots )$ ;

where $a_p, c_q \in \{ 1, 2 \}$ for each $p \leq k$ and $q \leq j$ , and $\mathsf {S}_1(x) := \mathsf {S} (\mathsf {S}\mathsf {S} \mathsf {0} \times x)$ and $\mathsf {S}_2(x) := \mathsf {S} \mathsf {S} (\mathsf {S}\mathsf {S} \mathsf {0} \times x)$ . Moreover, $\mathsf {ev}(\overline {r}) = \sum ^k_{i = 0} a_i b^i$ . Since

$$ \begin{align*} t(\overline{r}) = \mathsf{S}_{c_0}( \cdots \mathsf{S}_{c_j}(\mathsf{S}_{a_0}( \cdots \mathsf{S}_{a_k}(\mathsf{0}) \cdots )) \cdots ), \end{align*} $$

we have

$$ \begin{align*} p & = \mathsf{ev}(t(\overline{r}))\\ & = \sum^j_{i = 0} c_i 2^i + \sum^k_{i = 0} a_i 2^{i + j +1}\\ & = \sum^j_{i = 0} c_i 2^i + 2^{j +1} \sum^k_{i = 0} a_i 2^{i}\\ & < 2^{j+2} + 2^{j +1} r\\ & = 2^{j+1}(r+2). \end{align*} $$

For each $i \leq j$ , the expression $\mathsf {S}_{c_i}( \cdots \mathsf {S}_{c_{j}}(\mathsf {c}) \cdots )$ is a subterm of $A_k$ . Since also $\mathsf {c}$ is a subterm of $A_k$ , we have $j+2 \leq {\lceil A_k \rceil } \leq k+1$ . We thus conclude that $p < 2^{k}(r+2)$ .□

We now show that ${\sf gn}_2$ satisfies M4( $\overline {\cdot }$ ).

Lemma 6.10. $m < {\sf gn}_2( \overline {m} )$ for all $m \in \omega $ .

Proof. Let $\overline m$ be added in stage k, i.e., $\overline m \in \Lambda _k \setminus \Lambda _{k-1}$ . We then have $B_{{\mathfrak n}(k) - \ell + j} = \overline {m}$ , for some $j < \ell $ . (We use here the fact that no filler is of the form $\overline {m}$ ). We have $\ell \leq 3k + 8$ , as noted in (*) subsequent to presenting our construction. Hence,

$$ \begin{align*} 2^{2k+4} - 3k - 7 = {\mathfrak n}(k) - (3k + 8) \leq {\mathfrak n}(k) - \ell \leq {\sf gn}_2( \overline{m} ). \end{align*} $$

We moreover have $\overline {m} \preceq A_k [{\sf c} := {\sf S}\widetilde {(k+2)}]$ . Since ${\sf S}\widetilde {(k+2)} = \overline {2^{k+2}}$ we get

$$ \begin{align*} m < 2^{k} (2^{k+2}+2) = 3 \cdot 2^{k+1}, \end{align*} $$

by Lemma 6.9. It is easy to check that $3 \cdot 2^{k+1} < 2^{2k+4} - 3k - 7$ . Hence, $m < {\sf gn}_2( \overline {m} )$ .□

Even though ${\sf gn}_2$ cannot be self-referential for efficient numerals by Lemma 6.1, the following modification holds.

Lemma 6.11. Consider any A such that c occurs in A. Then there exists n such that ${\sf gn}_2(A[{\sf c}:= \overline {n}]) = n^2$ .

Proof. Suppose c occurs in A. Let $A := A_k$ . The only thing we have to show is that $A^{\ast }_k$ is not in $\Lambda _{k-1}$ . This is certainly true for $k=0$ . Suppose $k>0$ . We note that ${\sf S}\widetilde {(k+2)}$ occurs in $A^{\ast }_k$ . So, it is sufficient to show that ${\sf S}\widetilde {(k+2)}$ does not occur in $\Lambda _{k-1}$ . By Lemma 6.8, whenever $\mathsf {S}\widetilde {n}$ occurs in $\Lambda _{k-1}$ , we have $n \leq k+1$ . We may conclude that $B_{{\mathfrak n}(k)-1} = A_k[\mathsf {c} := \mathsf {S}(\widetilde {k+2})]$ . Let $n := 2^{k+2}$ . Since $n^2 = {\mathfrak n}(k)-1$ and $\mathsf {S}(\widetilde {k+2}) = \overline {n}$ , we are done.□

We now can immediately derive the Strong Diagonal Lemma.

Proof of Lemma 6.3

Let $C(x)$ be given and set $A := C[x := \mathsf {c} \times \mathsf {c}]$ . By Lemma 6.11 there exists $n \in \omega $ such that ${\sf gn}_2(A[\mathsf {c} := \overline {n}]) = n^2$ . Set $t := \overline {n} \times \overline {n}$ . We then have $A[\mathsf {c} := \overline {n}] = C[x := t]$ . Moreover $\mathsf {R}_{\mathcal {L}} \vdash t = \underline {n^2}$ . Hence ${\mathsf {R}_{\mathcal {L}} \vdash t = \underline {{\sf gn}_2(C[x := t])}}$ .□

By suitably adapting the above construction, we obtain a numbering which is monotonic with respect to the sub-string relation, satisfies M4( $\overline {\cdot }$ ) and provides the existence of m-self-referential sentences for $\mathcal {L}$ (i.e., satisfies Lemma 6.3). We leave the details to the diligent reader.

7 Computational Constraints

In addition to requiring monotonicity, other adequacy constraints for Gödel numberings can be extracted from the literature. Let $\mathfrak {Po}$ , $\mathfrak {El}$ and $\mathfrak {Pr}$ denote the classes of p-time, (Kalmár) elementary and primitive recursive functions respectively. As usual, these classes can be extended to relations and partial functions on $\omega $ as follows.

Definition 7.1. Let $\mathcal {C} \in \{\mathfrak {Po},\mathfrak {El},\mathfrak {Pr}\}$ . We say that a subset of $\omega ^k$ or a relation on $\omega $ is in $\mathcal {C}$ , if its characteristic function is in $\mathcal {C}$ .

Let $R \subseteq \omega ^k$ be given. We say that a function $f \colon R \to \omega $ is in $\mathcal {C}$ , if the total function $f' \colon \omega ^k \to \omega $ given by

$$ \begin{align*} f'(m_1, \ldots, m_k) := \begin{cases} f(m_1, \ldots, m_k) +1 & \text{if } m_1, \ldots, m_k \in R;\\ 0 & \text{otherwise.} \end{cases} \end{align*} $$

is in $\mathcal {C}$ .

Reasonable numberings are commonly required to represent certain syntactic relations and operations by primitive recursive relations and operations on $\omega $ (see [Reference Halbach8, sec. 5.1]). The usual definitions of the syntactic relations (i.e., their extensions) $\mathsf {Var}_{\mathcal {L}}$ (“is a variable”), $\mathsf {Ter}_{\mathcal {L}}$ (“is a term”), $\mathsf {AtFml}_{\mathcal {L}}$ (“is an atomic formula”), $\mathsf {Fml}_{\mathcal {L}}$ (“is an a formula”), $\mathsf {Sent}_{\mathcal {L}}$ (“is a sentence”), “is a free variable in,” “is a $\mathsf {PA}$ -derivation of” etc., are either explicit or of simple recursive structure. Adequate numberings may be required to preserve this simple algorithmic nature, in virtue of representing these relations by primitive recursive relations on $\omega $ . More precisely, for any adequate numbering $\xi $ , the characteristic functions of the sets of $\xi $ -codes of $\mathsf {Var}_{\mathcal {L}}$ , $\mathsf {Ter}_{\mathcal {L}}$ , etc. are required to be primitive recursive. A similar constraint can be extracted for the syntactic functions $\dot {=}$ , $\dot \neg $ , $\dot \wedge $ , $\dot \vee $ , $\dot \forall $ , $\dot \exists $ , $\dot {\mathsf {S}}$ , $\dot {+}$ , $\dot \times $ , $\mathsf {Sub}$ etc., where for instance, $\dot {\wedge } \colon \mathsf {Fml}_{\mathcal {L}}^2 \to \mathsf {Fml}_{\mathcal {L}}$ maps a pair $\langle A,B \rangle $ of formulæ to the conjunction $A \wedge B$ , $\dot {\forall } \colon \mathsf {Fml}_{\mathcal {L}} \times \mathsf {Var}_{\mathcal {L}} \to \mathsf {Fml}_{\mathcal {L}}$ maps a pair $\langle A,x \rangle $ to the universal formula $\forall x A$ , etc., and where $\mathsf {Sub}(A,x,t)$ is the result of substituting the term t for the variable x in the formula A. Once again, adequate numberings may be required to represent these syntactic functions by primitive recursive functions on $\omega $ . Finally, the standard and the efficient numeral function may be required to be represented primitive recursively.

The arithmetisation of syntax in weaker theories, like Buss’ ${\sf S}^1_2$ and $\mathrm{I}\Delta _0+\Omega _1$ , requires representation of the syntactic relations and functions in weaker theories.Footnote ¹² A closer inspection of what is going on shows that, in the presence of a reasonable Gödel numbering like the one based on the length-first ordering, the syntactic relations and functions can be made p-time without extra effort. See [Reference Buss2, Reference Hájek and Pudlák7]. The Gödel numberings we construct here fit most naturally in the intermediate function class $\mathfrak {El}$ .

The definition of ${\mathcal C}$ -adequate is somewhat awkward since, for a good numbering like the one based on the length-first ordering, the function $\xi \circ (\underline \cdot )$ will be exponential. This is the reason for the use of efficient numerals in the context of weak theories. It would be interesting, and conceivably genuinely useful, to explore whether there are $\mathfrak {Po}$ -adequate numberings $\xi $ for which $\xi \circ (\underline \cdot )$ is p-time.

We now introduce a useful proof-theoretical characterisation of the classes $\mathfrak {Po}$ , $\mathfrak {El}$ and $\mathfrak {Pr}$ . This will result in a specification of the theories in which ${\mathcal C}$ -adequate numberings permit the arithmetisation of syntax. To this end, we first introduce so-called provably recursive functions.

The classes $\mathfrak {El}$ and $\mathfrak {Pr}$ can now be characterised as exactly the functions which are provably recursive in a certain theory. The situation for the p-time computable functions is a bit more delicate.

Hence, for example, for any $\mathfrak {Pr}$ -adequate numbering $\xi $ there exists a $\Sigma _1$ -formula $\mathsf {Conj}(x,y,z) \in \mathcal {L}$ , such that

• • $\mathbb {N} \models \mathsf {Conj}(\underline {\xi (A)},\underline {\xi (B)},\underline {\xi (C)}) $ iff $C = A \wedge B$ , for all formulæ $A,B,C \in \mathcal {L}$ ;

• • $\mathsf {I \Sigma _1} \vdash \forall x,y \exists ! z \, \mathsf {Conj}(x,y,z) $ ;

i.e., the $\xi $ -tracking function of $\dot \wedge $ is provably recursive in $\mathsf {I \Sigma _1}$ . Indeed, the representation of $\dot \wedge $ by a provably recursive function (in $\mathsf {PA}$ ) is required as a necessary condition for reasonable numberings in [Reference Halbach8, p. 33]. As a result of the above theorem, $\mathfrak {Pr}$ -adequate numberings allow a large portion of syntax to be formalised in the theory $\mathsf {I \Sigma _1}$ and, similarly, for $\mathfrak {El}$ and EA and for $\mathfrak {Po}$ and ${\sf S}^1_2$ .Footnote ¹³

Recall that the constructions of the numberings ${\sf gn}_0$ , ${\sf gn}_1$ and ${\sf gn}_2$ rely on an enumeration $(A_n)_{n \in \omega }$ of $\mathcal {L}$ -expressions. Thus far, assuming $(A_n)_{n \in \omega }$ to be effective has sufficed to ensure the effectiveness of the resulting numberings ${\sf gn}_0$ , ${\sf gn}_1$ and ${\sf gn}_2$ . Since no further computational constraint has been imposed on $(A_n)_{n \in \omega }$ , there is however no reason to expect these numberings to be even $\mathfrak {Pr}$ -adequate. In the remainder of this paper, we will assume that the enumeration $(A_n)_{n \in \omega }$ (without repetitions) employed in the constructions of the numberings ${\sf gn}_0$ , ${\sf gn}_1$ and ${\sf gn}_2$ is obtained from a standard numbering which is $\mathfrak {El}$ -adequate, such that $(A_n)_{n \in \omega }$ also satisfies the remaining assumptions imposed in the constructions of ${\sf gn}_0$ , ${\sf gn}_1$ and ${\sf gn}_2$ respectively. That is, $(A_n)_{n \in \omega } = ( \mathsf {gn}_{\ast }^{-1}(n))_{n \in \omega }$ , for some suitable standard numbering $\mathsf {gn}_{\ast }$ .

Under the assumption that $(A_n)_{n \in \omega }$ is $\mathfrak {El}$ -adequate, we now show that the numberings ${\sf gn}_0$ , ${\sf gn}_1$ and ${\sf gn}_2$ are $\mathfrak {El}$ -adequate, and, thus, also $\mathfrak {Pr}$ -adequate. This will follow from the fact that the standard numbering $\mathsf {gn}_{\ast }$ is $\mathfrak {El}$ -adequate and that $\mathsf {gn}_{\ast }$ , ${\sf gn}_0$ , ${\sf gn}_1$ and ${\sf gn}_2$ are elementarily intertranslatable. Let $\mathsf {tr}_{i,j} \colon \mathsf {gn}_j(\mathcal {L}) \to \omega $ be the translation function, given by $\mathsf {tr}_{i,j} := \mathsf {gn}_i \circ \mathsf {gn}_j^{-1}$ , for each $i, j \in \{ \ast ,0,1,2 \}$ . We then have:

Proof. We treat the case of the intertranslation of $\ast $ and 2. To show that $\mathsf {tr}_{\ast ,i}$ and $\mathsf {tr}_{i,\ast }$ are elementary for $i = 0,1$ proceeds similarly. From these cases we can conclude that $\mathsf {tr}_{i,j}$ is elementary for $i,j \leq 2$ .

We use the fact that the coding machinery for finite sequences can be developed in $\mathfrak {El}$ (see, for example, [Reference Schwichtenberg and Wainer26, chap. 2]). Let $\# \colon \omega ^{< \omega } \to \omega $ be an elementary coding function of finite sequences and let ${\sf lh}$ be the length function. Let $g \colon \omega \to \omega $ serve as a parameter. We define the function $\sigma _g \colon \omega \to \omega $ , by setting $\sigma _g(k) := \# \langle i_0, \ldots , i_{\ell -1} \rangle $ , where $A_{i_0}, \ldots , A_{i_{\ell -1}}$ are exactly the sub-expressions of $A_k [{\sf c} := {\sf S}\widetilde {(k+2)}]$ which are not of the form $\widehat {m}$ such that $g(n) \neq i_j$ for all $n < {\mathfrak n}(k-1)$ and $j < {\sf lh}(\sigma _g(k)) = \ell $ . Since ${\mathfrak n} \in \mathfrak {El}$ and ${\sf gn}_{\ast }$ is $\mathfrak {El}$ -adequate, $g \in \mathfrak {El}$ implies $\sigma \in \mathfrak {El}$ , by the usual closure properties of elementary functions. In case that $g := \mathsf {tr}_{2,\ast }$ , the last clause is equivalent to $A_{i_j} \notin \Lambda _{k-1}$ for each $j < \ell $ .

Set $\sigma := \sigma _{\mathsf {tr}_{2,\ast }}$ . For any $n \in \omega $ , we compute $\mathsf {tr}_{2,\ast }(n)$ by course-of-values recursion as follows. As usual, we will take the empty sum to be 0.

Compute the smallest k ( ${\leq } n$ ) such that $n\leq 2^{2k+4}$ . Is $p := 2^{2k+4}-n < {\sf lh}(\sigma (k))$ ?

If yes, set $\mathsf {tr}_{2,\ast }(n) := [\sigma (k)]_{{\sf lh}(\sigma (k)) -1 -p}$ .

If no, take

$$ \begin{align*} m := n - \sum_{i=0}^{k-1}{\sf lh}(\sigma(i)). \end{align*} $$

Set $\mathsf {tr}_{2,\ast }(n) := {\sf gn}_{\ast }(\widehat {m})$ .

It is left to the reader to verify that this computation really defines the function $\mathsf {tr}_{2,\ast }$ . Note that when computing $\mathsf {tr}_{2,\ast }(n)$ , we only resort to values of $\mathsf {tr}_{2,\ast }$ for arguments smaller than n, since ${\mathfrak n}(k-1) \leq n$ . Moreover, let $\mathsf {B} \colon \omega \to \omega $ the elementary function given by $\mathsf {B}(k) := \mathsf {gn}_{\ast }(A_k [{\sf c} := {\sf S}\widetilde {(k+2)}])$ . Note that $\mathsf {tr}_{2,\ast }(n)$ codes a filler expression of the form $\widehat {m}$ , or a sub-expression of $A_k [{\sf c} := {\sf S}\widetilde {(k+2)}]$ . Since in each stage k, there are less than ${\mathfrak n}(k)$ -many fillers added, we obtain the estimate $\mathsf {tr}_{2,\ast }(n) \leq \max ({\sf gn}_{\ast }(\widehat {(k+1) \cdot {\mathfrak n}(k)},\mathsf {B}(k))$ . Hence, the employed recursion is limited. Since elementary functions are closed under limited course-of-values recursion, we conclude that $\mathsf {tr}_{2,\ast } \in \mathfrak {El}$ .

In order to show that $\mathsf {tr}_{\ast ,2}$ is elementary, we use the fact that a function is elementary if (1) its graph is elementary and (2) it can be dominated by an elementary function. Since $\mathsf {tr}_{2,\ast }$ is elementary, it is easy to check that the graph of $\mathsf {tr}_{\ast ,2}$ is elementary. Hence, it is sufficient to show (2). Let $A_k \in \mathcal {L}$ . By Lemma 6.5, $A_k \in \Lambda _k$ . Hence, $\mathsf {tr}_{\ast ,2}(k) < {\mathfrak n}(k)$ and we are done.□

It is easy to show that the elementary (or primitive recursive) relations and functions are invariant regarding numberings which are elementarily (or primitive recursively) intertranslatable.

We have seen in this section that $\mathfrak {Pr}$ -adequacy can be extracted as a necessary condition for reasonable numberings from the literature. Since $\mathsf {gn}_{\ast }$ is $\mathfrak {El}$ -adequate we conclude from Lemmas 7.7 and 7.8 that the numberings ${\sf gn}_0$ , ${\sf gn}_1$ and ${\sf gn}_2$ are $\mathfrak {El}$ -adequate and, in particular, $\mathfrak {Pr}$ -adequate. Thus, the computational constraint of $\mathfrak {Pr}$ -adequacy (or $\mathfrak {El}$ -adequacy) is not sufficiently restrictive to deny these numberings the status of adequate formalisation choices.

8 Domination and Regularity

We have seen in Section 6 that strong monotonicity is not sufficient to rule out m-self-referential sentences formulated in $\mathcal {L}$ . This is due to the fact that the “fixed point” term t used in the proof of Lemma 6.3 is not an efficient numeral. Hence, conditions M3 and M4( $\overline {\cdot }$ ) do not force the value of t to be smaller than the code of the formula containing t.

For the remainder of this section, let $\mathcal {L}$ , $\mathcal {K}$ be fixed languages with $\mathcal {L}^0 \subseteq \mathcal {K} \subseteq \mathcal {L}$ , as specified in Section 2.1. We introduce two related constraints on numberings of $\mathcal {L}$ which prohibit the code of a closed $\mathcal {K}$ -term to be smaller than its value. We will see that together with requiring monotonicity, these constraints, applied to $\mathcal {L}$ , rule out m-self-referential sentences formulated in $\mathcal {L}$ .