Abstract
There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific choice of the Gödel numbering and the notation system. A similar observation applies to Gödel’s and Church’s theorems, which are commonly taken to impose severe limitations on what can be proved and computed using the resources of certain formalisms. The philosophical force of these limitative results heavily relies on the belief that these theorems do not depend on contingencies regarding the underlying formalisation choices. The main aim of this paper is to provide metamathematical facts which support this belief. While employing a fixed notation system, I showed in previous work (Review of Symbolic Logic, 2021, 14(1):51–84) how to abstract away from the choice of the Gödel numbering. In the present paper, I extend this work by establishing versions of Tarski’s, Gödel’s and Church’s theorems which are invariant regarding both the notation system and the numbering. This paper thus provides a further step towards absolute versions of metamathematical results which do not rely on contingent formalisation choices.
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Acknowledgements
I am indebted to Arnon Avron, Volker Halbach, Léon Probst, Gil Sagi, Albert Visser and two anonymous referees for their detailed and helpful comments. I would also like to thank Nachum Dershowitz, Rea Golan, Michael Goldboim, David Kashtan, Carlo Nicolai, Lavinia Picollo, Carl Posy, Johannes Stern, Dan Waxman and Lingyuan Ye for helpful conversations on topics presented in this paper. Finally, I am grateful to the audiences of my talks on this material for their valuable feedback.
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Appendices
Appendix A. Many-Sorted Algebras
Let \(S = \{ s_0, \ldots , s_k \}\) be a set (of sorts), for some \(k \in \omega \). The family \(A := \langle A^s \rangle _{s \in S}\) is called an S-sorted set. We sometimes write \(\langle A_0, \ldots , A_k \rangle \) for the family A. For technical convenience, I will assume throughout that for any S-sorted set \(\langle A^s \rangle _{s \in S}\), we have \(A^s \cap A^t = \varnothing \) for all \(s \ne t \in S\). We sometimes want to consider the elements of a family A, without distinguishing their sorts. In that case we take the union \(\bigcup _{s \in S} A^s\) of all sets \(A^s\), which we also write as \(\bigcup A\).
The basic set-theoretic notions can be defined for S-sorted sets in the usual sortwise fashion. More specifically, let A and B be S-sorted sets. We say that A is finite, if \(A_s\) is finite for each \(s \in S\). We write \(A \subseteq B\), and call A an S-sorted subset of B, if \(A^s \subseteq B^s\) for every \(s \in S\). We moreover set \(A \cap B := \langle A^s \cap B^s \rangle _{s \in S}\) and define general S-sorted intersections in a similar way. Moreover, we call the S-sorted set \(\langle f^s :A^s \rightarrow B^s \rangle _{s \in S}\) an S-sorted function, which we denote by \(f :A \rightarrow B\). We call f injective (or surjective), if \(f^s\) is injective (or surjective) for each \(s \in S\). Finally, we write \(\varnothing \) for the S-sorted set \(\langle \varnothing \rangle _{s \in S}\).
Let \(S^\star \) denote the set of finite sequences over S, including the empty sequence \(\lambda \). In the context of signatures, we exclusively write \(s_0 \times \cdots \times s_{k-1}\) for the sequence \((s_0 , \ldots , s_{k-1})\). An S-sorted algebraic signature \(\Omega \) is a finite \(S^\star \times S\)-sorted set \(\langle \Omega ^{w,s} \rangle _{w \in S^{\star },s \in S}\) of pairwise disjoint, non-empty finite sets. We say that \(\sigma \) is a symbol of \(\Omega \) and write \(\sigma \in \Omega \), if \(\sigma \in \bigcup _{w \in S^{\star },s \in S} \Omega ^{w,s}\). The elements of \(\Omega ^{w,s}\) are called function symbols of type \(w \rightarrow s\) and of sort s. If a function symbol has type \(\lambda \rightarrow s\), we also call it a constant symbol of sort s.
Many-sorted algebras consist of a many-sorted domain, as well as designated elements and fundamental operations defined on that set. The following definition makes this precise.
Definition 2.1
An \(\Omega \)-algebra \({\textbf {A}}\) is an ordered pair \(\langle \langle |{\textbf {A}}|^s \rangle _{s \in S}, \langle \sigma _{{\textbf {A}}} \rangle _{\sigma \in \Omega }\rangle \), such that
-
\(|{\textbf {A}}|^s \) are non-empty sets, for all \(s \in S\);
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\(\sigma _{{\textbf {A}}} \in |{\textbf {A}}|^s\) if \(\sigma \) is a constant symbol of sort s;
-
\(\sigma _{{\textbf {A}}} :|{\textbf {A}}|^{w(1)} \times \cdots \times |{\textbf {A}}|^{w(\textrm{lh}(w))} \rightarrow |{\textbf {A}}|^s \) if \(\sigma \) is a function symbol of type \(w \rightarrow s\), where \(\textrm{lh}(w)\) denotes the length of w.
We call \(\langle |{\textbf {A}}|^s \rangle _{s \in S}\) the domain of A, which we sometimes write as \(|{\textbf {A}}|\). If we do not want to distinguish between different sorts, we consider the union \(\bigcup _{s \in S} |{\textbf {A}}|^s\), which we sometimes write as \(\bigcup |{\textbf {A}}|\). Finally, we call the functions \(\sigma _{{\textbf {A}}}\), with \(\sigma \in \Omega \), the fundamental operations of A.
Let \({\textbf {A}}\), \({\textbf {B}}\) be \(\Omega \)-algebras. An \(\Omega \)-homomorphism from A to B is an S-sorted function \(\alpha :{\textbf {A}} \rightarrow {\textbf {B}}\), such that
-
\(\alpha ^s(\sigma _{{{\textbf {A}}}}) = \sigma _{{\textbf {B}}}\), for all constant symbols \(\sigma \in \Omega \); and
-
\(\alpha ^s( \sigma _{{{\textbf {A}}}} (a_1, \ldots , a_{n})) = \sigma _{{\textbf {B}}}( \alpha ^{s_1}(a_1), \ldots , \alpha ^{s_n}(a_n) )\), for all \(a_i \in |{\textbf {A}}|^{s_i} \) (with \(1\le i \le n\)) and every function symbol \(\sigma \in \Omega \) of type \(s_1 \times \cdots \times s_n \rightarrow s\).
We say that an S-sorted subset B of the domain \(\vert {\textbf {A}} \vert \) is closed under the fundamental operations of \({\textbf {A}}\), if for every function symbol \(\sigma \in \Omega ^{w,s}\) and \(b_1 \in B^{w(1)}, \ldots , b_{\textrm{lh}(w)} \in B^{w(\textrm{lh}(w))}\) we have \(\sigma _{{\textbf {A}}}(b_1, \ldots ,b_{\textrm{lh}(w)}) \in B^s\). Let now X be an S-sorted set with \(X \subseteq ~\vert A \vert \). We call
the algebraic closure of X. We say that a set U is generated by the fundamental operations of A from X, if \(U = \text {Cl}_{{\textbf {A}}}(X)\). We call A minimal, if \(\text {Cl}_{{\textbf {A}}}(\varnothing ) = \vert {\textbf {A}} \vert \). We call A finitely generated, if there is a finite S-sorted set X such that \(\text {Cl}_{{\textbf {A}}}(X) = \vert {\textbf {A}} \vert \).
We now provide an example of a finitely generated algebra that is not single-sorted.
Example A.2
Let S contain the two sorts \(\textrm{label}\) and \(\textrm{tree}\). A labelled binary tree is a tree where each node has at most two subtrees, called the left and right subtree respectively. Moreover, each node of the tree is labelled by exactly one element of some non-empty set L of labels. For a graphical representation of a labelled binary tree see Fig. 4.
Let \(\textrm{B}_L\) be the set of all labelled binary trees with labels from L. Let [a] be the tree in \(\textrm{B}_L\) which only consists of a root with label \(a \in L\). Let \(\Omega \) be a S-sorted signature containing the following function symbols:
-
\(\textsf{leaf} :\textrm{label} \rightarrow \textrm{tree}\)
-
\(\textsf{right} :\textrm{label} \times \textrm{tree} \rightarrow \textrm{tree}\)
-
\(\textsf{left} :\textrm{tree} \times \textrm{label} \rightarrow \textrm{tree}\)
-
\(\textsf{both} :\textrm{tree} \times \textrm{label} \times \textrm{tree} \rightarrow \textrm{tree}\)
We now turn \(\textrm{B}_L\) into an \(\Omega \)-algebra \({\textbf {B}}_L\) by setting
-
\(\vert {\textbf {B}}_L \vert ^{\textrm{label}} = L\) and \(\vert {\textbf {B}}_L \vert ^{\textrm{tree}} = \textrm{B}_L\);
-
\(\textsf{leaf}_{{\textbf {B}}_L}(a) = [a]\)
-
\(\textsf{right}_{{\textbf {B}}_L}(a,T)\) adds to [a] the right subtree T.
-
\(\textsf{left}_{{\textbf {B}}_L}(T,a)\) adds to [a] the left subtree T.
-
\(\textsf{both}_{{\textbf {B}}_L}(T,a,U)\) adds to [a] the left subtree T and the right subtree U.
The algebra \({\textbf {B}}_L\) is finitely generated, as long as the set L of labels is finite.
We now introduce the notion of initiality, which is of central importance in abstract algebra.
Definition 2.2
Let \(\mathfrak {A}\) be a class of \(\Omega \)-algebras and let \({\textbf {D}} \in \mathfrak {A}\). If for every \({\textbf {E}} \in \mathfrak {A}\) there is a unique \(\Omega \)-homomorphism \(\alpha ^*:{\textbf {D}} \rightarrow {\textbf {E}}\), then we say that \({\textbf {D}}\) has the universal property for \(\mathfrak {A}\) and we also call \({\textbf {D}}\) initial in \(\mathfrak {A}\).
If \({\textbf {D}}\) is initial in the class \(\textrm{Mod}(\Omega )\) of all \(\Omega \)-algebras, we also call \({\textbf {D}}\) an absolutely free \(\Omega \)-algebra.
An important example of an absolutely free algebra is the term algebra \({\textbf {T}}_{\Omega }\), defined as follows.
Definition 2.3
For each \(s \in S\), the set \(T^s_{\Omega }\) of \(\Omega \)-terms of sort s is the smallest set such that
-
(1)
\(\Omega ^{\lambda ,s} \subseteq T^s_{\Omega }\);
-
(2)
If \(\sigma \in \Omega \) is of type \(s_1 \times \cdots \times s_n \rightarrow s\) and \(t_1 \in T^{s_1}_{\Omega }, \ldots , t_n \in T^{s_n}_{\Omega }\), then the string \(\sigma (t_1, \ldots , t_n)\) is in \(T^s_{\Omega }\).
The S-sorted set \(T_\Omega \) is called the Herbrand universe of \(\Omega \). Each element of \(\bigcup T_\Omega \) is called an \(\Omega \)-term.
We can naturally transform the Herbrand universe \(T_\Omega \) of \(\Omega \) into an \(\Omega \)-algebra \({\textbf {T}}_\Omega \) as follows:
Definition 2.4
Let \(T_\Omega \) be the Herbrand universe of \(\Omega \). We turn \(T_\Omega \) into an \(\Omega \)-algebra \({\textbf {T}}_\Omega \) by setting
-
\(\sigma _{{\textbf {T}}_\Omega } := \sigma \), for each constant symbol \(\sigma \in \Omega \);
-
\(\sigma _{{\textbf {T}}_\Omega }(t_1, \ldots , t_n) := \sigma (t_1, \ldots , t_n)\), for each function symbol \(\sigma \in \Omega \) of type \(s_1 \times \cdots \times s_n \rightarrow s\) and terms \(t_1 \in T_\Omega ^{s_1}, \ldots , t_n \in T_\Omega ^{s_n}\).
We call the resulting \(\Omega \)-algebra \({\textbf {T}}_\Omega \) the term algebra or Herbrand algebra of \(\Omega \).
Appendix B. Quine-Bourbaki notation
In what follows, I show how the Quine-Bourbaki notation system (see Section 3.4) can be accommodated in the framework developed in Section 4.
Let A be the finite alphabet which contains the symbols of \(\mathcal {L}_0\) together with the auxiliary symbols ‘\(\textsf{v}\)’, ‘\(\prime \)’, \(`\square \)’, ‘(’ and ‘)’. Let \(\mathbb{H}\mathbb{F}_A\) be the set of hereditarily finite sets with the elements of A as urelements. \(\mathbb{H}\mathbb{F}_A\) together with the adjunction operation \(\mathcal A\) given by \(\mathcal A(x, y) := x \cup \{ y \}\) forms a finitely generated algebra \({\textbf {HF}}_A\). Assume that finite sequences of elements of \(\mathbb{H}\mathbb{F}_A\) and the concatenation operation \(*\) on finite sequences are defined in \(\mathbb{H}\mathbb{F}_A\) in a fixed and standard way.
It will be convenient to conceive of A-strings as sequences \(s_0 *\cdots *s_n\), where each sequence entry \(s_i\) is either a symbol of A or the result of appending finitely many strokes \(\prime \) to \(\mathsf v\) such that \(\textsf{v} *\prime \) is not a subsequence of \(s_0 *\cdots *s_n\). Call such a sequence simplified. In simplified sequences, we treat A-strings of the form \(\textsf{v} \prime \ldots \prime \) as “alphabetical symbols” of length 1.
Consider now an ordered pair \(\langle s_0 *\cdots *s_n, C \rangle \), such that the sequence \(s_0 *\cdots *s_n\) is simplified and such that C is a (possibly empty) set of unordered pairs \(\{ i , j \}\) with \(i \ne j \le n\). In what follows, the ordered pair \(\langle s_0 *\cdots *s_n, C \rangle \) will represent the result of adding curved lines to the A-string \(s_0 *\cdots *s_n\) as follows: there is a curved line between \(s_i\) and \(s_j\) iff \(\{ i , j \} \in C\).
We now define an implementation \(\iota \) of p-expressions of \(\mathcal {L}_0\) into \({\textbf {HF}}_A\). We write the projection to the i-th component as \(\langle \cdot , \cdot \rangle _i\), for \(i=1,2\). For example, \(\langle s, C \rangle _1 = s\). Moreover, for any unordered pair \(\{ i, j \}\) of numbers, we write \(\{ i, j \} + n\) for \(\{ i+n, j +n\}\), for \(n \in \omega \).
We first define the implementation \(\iota ^{\textrm{var}}\) of p-variables into \({\textbf {HF}}_A\) recursively as follows.
We now define the implementation \(\iota ^{\textrm{ter}}\) of p-terms recursively as follows:
We finally define the implementation \(\iota ^{\textrm{fml}}\) of p-formulas:
where \(\iota ^{\textrm{fml}}(\varphi )_1[\square / \iota ^{\texttt {var}}(x)]\) is the result of substituting each occurrence of \(\iota ^{\texttt {var}}(x)\) in \(\iota ^{\textrm{fml}}(\varphi )_1\) by \(\square \). We observe that \(\iota ^{\textrm{fml}}(\varphi )_1\) and \(\iota ^{\textrm{fml}}(\varphi )_1[\square / \iota ^{\texttt {var}}(x)]\) are simplified sequences of the same length.
For any (distinct) p-variables x, y, z, the p-expression
is implemented by \(\iota \) as the same object, viz.
which corresponds to the following Quine-Bourbaki notation (with the omission of some parentheses):
More generally, \(\iota \)-equivalence coincides with \(\alpha \)-equivalence, i.e., syntactic identity up to renaming of bounded variables.
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Grabmayr, B. A Step Towards Absolute Versions of Metamathematical Results. J Philos Logic 53, 247–291 (2024). https://doi.org/10.1007/s10992-023-09731-6
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DOI: https://doi.org/10.1007/s10992-023-09731-6