Claim 1. there is a pair of effectively inseparable c.e.sets \(Z^0\) and \(Z^1\) and a set \(Y\) such that
\(Y\) separates \(Z^0\) and \(Z^1\), and
\(Y\) is computable in \(\mathsf{V}(T)\).
June 10, 2025
In this paper, we study the collection of first-order logical schemata all of whose instances are theorems of a given theory \(T\); we call these the validities of \(T\) (\(\mathsf{V}(T)\)). It is easy to see (Corollary 1 below) that if \(T\) is a decidable theory, then \(\mathsf{V}(T)\) is distinct from the set of valid formulas of first-order logic as customarily understood. We provide a complete model-theoretic characterization of the complexity, in the sense of Turing degree, of \(\mathsf{V}(T)\) for decidable theories \(T\); answer a question posed by [1] concerning the complexity of the collection of validities common to all decidable theories; and begin the study of the case of undecidable theories.
We fix a first-order language \(S\) with effective syntax, without identity, without constant or function symbols, and with countably many relation symbols \(X^n_i\) of arity \(n\), for each \(n\geq 0\). We think of \(S\) as a schematic language for formulating validities. We use \(L\) to denote an effective first-order language with vocabulary disjoint from \(S\) that may contain identity, function symbols, and constants.3
Definition 1. Let \(\varphi(X^1_1, \ldots,X^1_k,\ldots,X^n_1, \ldots,X^n_k)\) be a formula of the first-order language \(S\) all of whose predicate symbols are among those exhibited, and let \[\psi^1_1(y_1), \ldots,\psi^1_k(y_1),\ldots,\psi^n_1(y_1,\ldots,y_n), \ldots,\psi^n_k(y_1,\ldots,y_n)\] be formulas of a first-order language \(L\) in which the displayed variables \(y_i\) all appear free. Then \[\varphi(\psi^1_1(y_1), \ldots,\psi^1_k(y_1),\ldots,\psi^n_1(y_1,\ldots,y_n), \ldots,\psi^n_k(y_1,\ldots,y_n))\] is the formula of \(L\) obtained by replacing \(X^j_i(t_{i1}\cdots t_{ij})\) with \(\psi^j_i(t_{i1},\ldots,t_{ij})\) in \(\varphi\), for \(1\leq i \leq k\) and \(1\leq j \leq n\).
We write \(\mathsf{V}\) for the set of valid formulas of \(S\). The following well-known theorem summarizes two of the fundamental results of mathematical logic, the Gödel Completeness Theorem and the Church-Turing Undecidability Theorem.
Theorem 1. \(\mathsf{V}\) is a \(\Sigma_1\)-complete set.
The next definition introduces the notion of the “set of validities of a theory,” the central concept of this paper.
Definition 2. A set of sentences \(T\) of a first-order language \(L\) is an \(L\)-theory* if and only if \(T\) is closed under logical consequence. We assume all theories we consider are consistent.*
Let \(T\) be an \(L\)-theory and let \(\varphi(X^1_1, \ldots,X^1_k,\ldots,X^n_1, \ldots,X^n_k)\) be a sentence of the first-order language \(S\) all of whose predicate symbols are among those exhibited. The sentence \(\varphi\) is a validity of \(T\) if and only if for all \(L\)-formulas \[\psi^1_1(y_1), \ldots,\psi^1_k(y_1),\ldots,\psi^n_1(y_1,\ldots,y_n), \ldots,\psi^n_k(y_1,\ldots,y_n)\] in which the displayed variables \(y_i\) all appear free, \[\varphi(\psi^1_1(y_1), \ldots,\psi^1_k(y_1),\ldots,\psi^n_1(y_1,\ldots,y_n), \ldots,\psi^n_k(y_1,\ldots,y_n))\] is a member of \(T\).
We write \(\mathsf{V}(T)\) for the set of valid sentences of \(T\).
It is evident that for every theory \(T\), \(\mathsf{V}\subseteq\mathsf{V}(T).\) It is well-known that identity holds for some computably axiomatizable theories. The following result is a corollary to the “Hilbert-Bernays Completeness Theorem.”4
Theorem 2. \(\mathsf{V}(\mathbf{PA})=\mathsf{V}\).
In this section we focus on the complexity of the collection of validities of a decidable theory.
Theorem 3. If \(T\) is a decidable* theory, then \(\mathsf{V}(T)\) is \(\Pi_1\).*
Proof. Let \(\varphi(X_1,\ldots,X_k)\) be an \(S\)-sentence and suppose that \(T\) is a decidable \(L\)-theory. We can effectively enumerate all possible choices of \(L\)-formulas \(\psi_1,\ldots,\psi_k\). By Definition 2 \[\varphi(X_1,\ldots,X_k)\in\mathsf{V}(T)\] if and only if for all \(L\)-formulas \(\psi_1,\ldots,\psi_k\), \[\varphi(\psi_1,\ldots,\psi_k)\in T.\] ◻
In contrast to Theorem 2, we have the following result concerning decidable theories.
Corollary 1. If \(T\) is decidable, \(\mathsf{V}\subsetneq\mathsf{V}(T)\).
Example 1. Consider \(T_{dlo}=\mathsf{Th}(\mathbb{Q},<)\), the theory of dense linear orders without endpoints. Let \(\varphi(X)\) be the formula \[\forall v,w(Xv\leftrightarrow Xw).\] Then \(\varphi(X)\in\mathsf{V}(T_{dlo})\), because for any formula \(\psi(y)\) with a single free-variable in the language \(\{<,=\}\), either \(\forall y\,\psi\in T_{dlo}\) or \(\forall y\,\neg\psi\in T_{dlo}\).
In the remainder of this section, we explore the interaction between the complexity of \(\mathsf{V}(T)\) and \(\aleph_0\)-categoricity. In particular, in Section 3.1 we show, as a corollary to a result of [4], that for every c.e.degree \(\mathbf{a}\in[\mathbf{0}, \mathbf{0^\prime}]\), there is a decidable \(\aleph_0\)-categorical theory \(T\) with \(\mathsf{deg}(\mathsf{V}(T))=\mathbf{a}\). In Section 3.2, we establish that if a decidable theory \(T\) is not \(\aleph_0\)-categorical, then \(\mathsf{V}(T)\) is \(\Pi_1\)-complete.
We begin by showing that the set of valid sentences of the theory of dense linear orders without endpoints is decidable.
Theorem 4. \(\mathsf{V}(T_{dlo})\) is decidable.
Proof. \(T_{dlo}\) has quantifier elimination, so every formula on \(n\) variables is equivalent to a disjunction of a conjunction of clauses \(x_i<x_j\) or \(x_i=x_j\). There are at most \(2^{3^{n^2}}\) such formulas. For instance, the only formulas on \(2\) variables are the \(8\) Boolean combinations of \[x_1<x_2,\;x_2<x_1,\;\text{and}\;x_1=x_2.\] So to test the validity of \(\varphi(X_1,\ldots,X_k)\) we can iterate through the finitely many choices for each \(X_i\). ◻
Example 2. The proof of Theorem 4 indicates further validities in \(\mathsf{V}(T_{dlo})\setminus \mathsf{V}\) along the lines of Example 1. For example, the following sentence is in \(\mathsf{V}(T_{dlo})\setminus \mathsf{V}\). \[\bigvee_{1\leq i<j\leq 9}(\forall y_1)(\forall y_2)(X_i(y_1,y_2)\leftrightarrow X_j(y_1,y_2)).\]
Theorem 4 is a special case of a general result characterizing the complexity of \(\mathsf{V}(T)\) for all decidable \(\aleph_0\)-categorical theories \(T\) in terms of an important model-theoretic invariant associated to \(T\).
Theorem 5 (Engeler/Ryll-Nardzewski/Svenonius). A complete theory \(T\) with infinite models is \(\aleph_0\)-categorical if and only if for each \(n\) there are finitely many \(n\)-types consistent with \(T\).
Definition 3. The Ryll-Nardzewski function* of a complete \(\aleph_0\)-categorical theory \(T\) is the function \(\mathop{\mathrm{RN}}_T:\mathbb{N}\rightarrow\mathbb{N}\) such that, for each \(n\), \(\mathop{\mathrm{RN}}_T(n)\) is the number of \(n\)-types consistent with \(T\).*
Theorem 6. If \(T\) is a complete, decidable, \(\aleph_0\)-categorical theory, then \(\mathsf{V}(T)\equiv_\mathrm{T} \mathop{\mathrm{RN}}_T\).
Proof. Let \(X^n_i\), \(1\leq i\leq 2^k\) be predicate letters of arity \(n\) and let \(\overline{y} = (y_1,\ldots,y_{n})\). Let \(\varphi(X_1,\ldots,X_{2^k})\) be the formula \[\bigvee_{1\leq i<j\leq 2^k}(\forall \overline{y})(X_i(\overline{y})\leftrightarrow X_j(\overline{y})).\] Then, \(\varphi\in \mathsf{V}(t)\) if and only if \(\mathop{\mathrm{RN}}_T(n)<k\). Therefore \(\mathop{\mathrm{RN}}_T\leq_\mathrm{T} \mathsf{V}(T)\).
Given an oracle for \(\mathop{\mathrm{RN}}_T\) and a decidable theory \(T\), consider a formula \(\varphi(X_1,\ldots,X_k)\). For each \(X_i\) with arity \(n_i\) and \(d_i=\mathop{\mathrm{RN}}_T(n_i)\), enumerate lists \((\psi_1,\ldots,\psi_{d_i})\) of formulas of arity \(n_i\) until we find a list of formulas, each consistent with \(T\), and pairwise nonequivalent modulo \(T\). Such a list must be an enumeration of the \(n_i\)-types of \(T\). Then all formulas of arity \(n_i\) are equivalent to disjunctions of the \(\psi_i\). We can test \(\varphi(X_1,\ldots,X_k)\) with each of the finitely many possible combinations of these disjunctions. Therefore \(\mathsf{V}(T)\leq_\mathrm{T} \mathop{\mathrm{RN}}_T\). ◻
It follows at once that If \(T\) is complete, decidable, and \(\aleph_0\)-categorical, and \(\mathop{\mathrm{RN}}_T\) is computable, then \(\mathsf{V}(T)\) is computable.5
The following result of [4] now yields a complete characterization of the possible Turing degrees of \(\mathsf{V}(T)\) for decidable \(\aleph_0\)-categorical theories \(T\).
Theorem 7 (Schmerl). For any c.e.degree \(\mathbf{a}\in[0,0']\), there are complete, decidable, \(\aleph_0\)-categorical theories \(T\) such that \(\mathop{\mathrm{RN}}_T\) is in \(\mathbf{a}\).
Corollary 2. For any c.e.degree \(\mathbf{a}\in[0,0']\), there are complete, decidable, \(\aleph_0\)-categorical theories \(T\) such that \(\mathsf{V}(T)\) is in \(\mathbf{a}\).
Note that Corollary 2 is optimal, since \(\mathsf{V}(T)\) must have c.e.degree, by Theorem 3.
As a corollary to the foregoing proof that \(\mathop{\mathrm{RN}}_T\leq_\mathrm{T} \mathsf{V}(T)\), we have
Corollary 3. If \(T\) and \(U\) are complete \(\aleph_0\)-categorical theories, then \[\text{if }\mathsf{V}(T)=\mathsf{V}(U),\text{then } \mathop{\mathrm{RN}}_T=\mathop{\mathrm{RN}}_U.\]
The following example, due to Alex Kruckman, shows that the converse to Corollary 3 fails.6
Example 3. There are complete \(\aleph_0\)-categorical theories \(T\) and \(U\) such that \[\mathop{\mathrm{RN}}_T=\mathop{\mathrm{RN}}_U,\text{ but }\mathrm{V}(T)\neq\mathrm{V}(U).\] For example, let \(T\) be the theory of the countable random graph, and let \(U\) be the theory of the countable random tournament. Since both theories admit elimination of quantifiers, \(k\)-types are determined, in the case of \(T\), by the presence or absence of edges, and in the case of \(U\), by the orientation of edges. Hence, \(\mathop{\mathrm{RN}}_T=\mathop{\mathrm{RN}}_U\). On the other hand, for all distinct \(a,b,c,d\in |U|\), \[\mathsf{type}_U(a,b)=\mathsf{type}_U(c,d)\text{ or } \mathsf{type}_U(a,b)=\mathsf{type}_U(d,c),\] which generates a formula in \(\mathrm{V}(U)\) that is absent from \(\mathrm{V}(T)\).
Theorem 8. If \(T\) is a complete, decidable, non-\(\aleph_0\)-categorical theory with infinite models, then \(\mathsf{V}(T)\) is \(\Pi_1\)-complete.
Proof. Let \(T\) be a complete, decidable, non-\(\aleph_0\)-categorical theory with infinite models. We will show that
Claim 1. there is a pair of effectively inseparable c.e.sets \(Z^0\) and \(Z^1\) and a set \(Y\) such that
\(Y\) separates \(Z^0\) and \(Z^1\), and
\(Y\) is computable in \(\mathsf{V}(T)\).
It follows from Claim (1) that \(\mathsf{V}(T)\) has PA degree, and from Theorem 3 that it has c.e.degree. Since the only c.e.PA degree is \(0^\prime\), we conclude that \(\mathsf{V}(T)\) is \(\Pi_1\)-complete.7 We proceed to establish Claim (1).
We consider Turing machines with the unary tape alphabet \(\{1, B\}\). We think of such a machine as computing 0-1-valued partial functions on \(\omega\) via the convention that input \(n\) is represented by a tape initialized with 1’s on its \(n+1\) left-most squares and otherwise blank, and with output 0, if it halts on a blank square, and with output 1, if it halts on a square inscribed with a 1. Let \(M\) be such a Turing machine with the property that the sets \[\begin{align} Z^0 &= \{n\mid M outputs 0 with input n\},and \\ Z^1 &= \{n\mid M outputs 1 with input n\}, \end{align}\] are effectively inseparable. In order to establish Claim 1, it suffices to show that there is a total 0-1-valued function \(f\) computable in \(\mathsf{V}(T)\) that extends the partial function computed by \(M\). Toward this end, we specify formulas \(\varphi^l_n\), \(n\in \omega\), \(l\in\{0,1\}\) that describe the behavior of \(M\) with input \(n\). We suppose that \(M\) has \(m\) states labeled \(1,\dots,m\).
Since \(T\) is not \(\aleph_0\)-categorical, we may choose a \(k\) so that there are infinitely many \(k\)-types. For all \(n\in \omega\), \(l\in\{0,1\}\), the vocabulary of \(\varphi^l_n\) consists of the following predicates:
\(X_{\mathrm{step}}, X_{\mathrm{start}},X_{\mathrm{end}},X_{\mathrm{tape}}, X_{\mathrm{left}},\) and \(X_{\mathrm{state}=i}, i\in\{1,\ldots,m\}\) are \(k\)-ary predicates, and
\(X_\leq, X_{\preceq},X_0,X_1, X_{\mathrm{head}}\) are all \(2k\)-ary predicates.
The sentences \(\varphi_n^l\) will state that the predicates encode the execution of the Turing machine \(M\) with input \(n\) and output \(l\), in accord with the foregoing input-output conventions. \(X_{\mathrm{step}}\) will be a predicate satisfied by those elements that encode a time step in the running of \(M\). \(X_{\leq}\) will be the subset of \(X_{\mathrm{step}}^2\) encoding when one step in time happens before another. In general we cannot expect that a time step is represented by a single element, so we will only require that \(X_{\leq}\) be a preorder on \(X_{\mathrm{step}}\). \(X_{\mathrm{start}}\) and \(X_{\mathrm{end}}\) encode the first and last time steps, respectively. Similarly, \(X_{\mathrm{tape}}\) will be a predicate satisfied by those elements that encode a position on the tape. \(X_{\preceq}\) will encode a preorder on \(X_{\mathrm{tape}}\) and \(X_{\mathrm{left}}\) will be the leftmost position.
\(X_0\) and \(X_1\) partition \(X_{\mathrm{step}}\times X_{\mathrm{tape}}\) encoding which positions on the tape are blank (\(X_0\)) or hold a \(1\) (\(X_1\)) at a given time step, and \(X_{\mathrm{head}}\) will be a subset of \(X_{\mathrm{step}}\times X_{\mathrm{tape}}\) encoding where the head is at a given step.
Finally, the sets \(X_{\mathrm{state}=1},\ldots,X_{\mathrm{state}=m}\) will be subsets of \(X_{\mathrm{step}}\) indicating what state the machine is in at a given step.
Formally, \(\varphi^l_n\) is a conjunction of clauses stating:
\(X_\leq\) is a discrete preorder on \(X_{\mathrm{step}}\) with \(X_{\mathrm{start}}\) and \(X_{\mathrm{end}}\) as the first and last equivalence classes, respectively;
\(X_\preceq\) is a discrete preorder on \(X_{\mathrm{tape}}\) with \(X_{\mathrm{left}}\) as the first equivalence class;
\(X_0\) and \(X_1\) partition \(X_{\mathrm{step}}\times X_{\mathrm{tape}}\) respecting \(X_\leq\) and \(X_\preceq\) equivalence classes;
a clause stating that in the initial configuration the leftmost \(n+1\) tape squares hold a \(1\) and the remainder of the tape is blank;
\(X_{\mathrm{head}}\) is a subset of \(X_{\mathrm{step}}\times X_{\mathrm{tape}}\) that respects \(X_\leq\) and \(X_\preceq\) equivalence classes. \(X_{\mathrm{head}}\) represents the graph of a function from \(X_{\mathrm{step}}\) equivalence classes to \(X_{\mathrm{tape}}\) equivalence classes, and contains \(X_{\mathrm{start}}\times X_{\mathrm{left}}\);
\(X_{\mathrm{state}=1}\) through \(X_{\mathrm{state}=m}\) form a partition of \(X_{\mathrm{step}}\) respecting \(X_\leq\) equivalence classes;
letting \(U\) be the set of terminal states, \(X_{\mathrm{end}}=\bigcup_{i\in U}X_{\mathrm{state}=i}\);
the \(X_{\mathrm{head}}\) image of \(X_{\mathrm{end}}\) is contained in \(X_l\);
for each rule of the Turing machine saying “when in state \(i\) with \(a\) under the head, change the head to \(b\), change to state \(j\), and move in a particular direction”, the implication corresponding to this rule.
We proceed to show that there is a total 0-1-valued function \(f\) computable in \(\mathsf{V}(T)\) that extends the partial function computed by \(M\). First, suppose that \(M\) halts on input \(n\) with output \(l\in\{0,1\}\). Then, for some \(d>n\), \(M\) halts with input \(n\) in at most \(d\) many steps with output \(l\). We need at most \(2d\) many elements to encode this computation as above: \(d\) of them to represent the steps of the computation, and \(d\) of them to represent the tape positions the head might occupy during the computation. By our choice of \(k\), we can specify a list \(\psi_1,\ldots,\psi_{2d}\) of formulas on \(k\)-tuples, that are consistent with \(T\) and pairwise incompatible over \(T\). We can then construct a substitution instance \(\Phi^l_n\) of the formula \(\varphi^l_n\) by substituting all the predicate variables with suitable boolean combinations of these formulas so that \(T\cup\{\Phi^l_n\}\) is satisfiable. It follows at once that for \(l\in\{0,1\}\) and \(n\in\omega\), \[\label{eq1} \text{if \neg\varphi^l_n\in \mathsf{V}(T), then M does not halt on input n with output l}.\tag{1}\] Moreover, for \(l\in\{0,1\}\) and \(n\in\omega\), \[\label{eq2} \text{if \neg\varphi^l_n\not\in \mathsf{V}(T), then M does not halt on input n with output 1-l}.\tag{2}\] For suppose \(\neg\varphi^l_n\not\in \mathsf{V}(T)\). It follows that \(\varphi^l_n\) is satisfiable, and hence that there is a computation \(C\) on input \(n\) in which \(M\) enters an end state scanning \(l\).8 But then if \(M\) halted on input \(n\) with output \(1-l\), \(C\) would contain a finite initial segment in which \(M\) enters an end state scanning \(1-l\). Thus, it follows that \(M\) does not halt on input \(n\) with output \(1-l\).
It is evident from (1 ) and (2 ) that the following instructions specify a total 0-1-valued function \(f\) computable in \(\mathsf{V}(T)\) that extends the partial function computed by \(M\). Given input \(n\), output \(l\) for the least \(l\in\{0,1\}\) such that \(\neg\varphi^l_n\not\in \mathsf{V}(T)\); if there is no such \(l\), output \(0\). ◻
In this section we follow [1] and study the validities common to all decidable theories. We begin with the following definition from [1].9
Definition 4. \(\mathsf{V_{dec}}=\bigcap_T\mathsf{V}(T)\) where \(T\) ranges over all decidable theories.
The next two results concerning the complexity of \(\mathsf{V_{dec}}\) are due to [6] and [7], respectively.
Theorem 9 (Cobham). \(\mathsf{V_{dec}}\) is \(\Pi_3\).
Theorem 10 (Hanf). \(\mathsf{V_{dec}}\) is not \(\Pi_1\).
[1] framed the problem of classifying \(\mathsf{V_{dec}}\) in the arithmetical hierarchy; indeed, Theorem 10 was the outcome of Hanf’s successful effort to resolve Vaught’s specific question whether \(\mathsf{V_{dec}}\) is co-c.e.
The next theorem resolves Vaught’s problem.
Theorem 11. \(\mathsf{V_{dec}}\) is \(\Pi_3\)-complete.
The proof of Theorem 11 makes use of the following results due to [7], Han75? and [8],10 and to [9], respectively. We write \(T_e\) for the \(\Pi^0_1\) subclass of \(2^\omega\) with index \(e\), and \(\theta_e\) for an effective enumeration of the sentences of the language \(L\) of graphs, which contains a single binary predicate \(E\) and the identity predicate.
Theorem 12 (Hanf/Peretyat’kin). There is a computable function \(\sigma\) such that, for each \(e\), the degrees of paths in the \(\Pi^0_1\)-class \(T_e\) is the same as the degrees of complete \(L\)-extensions of the theory axiomatized by the sentence \(\theta_{\sigma(e)}\); in particular, \(T_e\) has a computable path if and only if \(\theta_{\sigma(e)}\) has a decidable complete \(L\)-extension.
Theorem 13 (Gasarch-Martin). The set of \(e\) such that \(T_e\) has no computable path is \(\Pi_3\)-complete.
Since our schematic language \(S\) for validities does not contain identity, we need to specify a translation of \(L\)-sentences \(\theta\) into \(S\)-sentences \(\theta^*\) in order to derive Theorem 11 from Theorems 12 and 13. Let \(R\) and \(\approx\) be binary predicate letters of \(S\) and let \(\chi(R,\approx)\) be the \(S\)-sentence that expresses that \(\approx\) is a congruence with respect to \(R\). If \(\theta\) is an \(L\)-sentence, \(\theta^*\) is the \(S\)-sentence \[\chi(R,\approx)\wedge \theta[E/R,{=}/{\approx}].\] Theorem 11 is an immediate corollary to Theorems 12 and 13, and the next lemma.
Lemma 1. An \(L\)-sentence \(\theta\) has a complete decidable \(L\)-extension if and only if \(\neg\theta^*\not\in\mathsf{V_{dec}}\).
Proof. The left to right direction is immediate: if \(T\) is a decidable complete \(L\)-extension of \(\theta\), then \(\neg\theta^*\not\in\mathsf{V}(T)\), hence \(\neg\theta^*\not\in\mathsf{V_{dec}}\).
For the right to left direction, suppose that \(\neg\theta^*\not\in\mathsf{V_{dec}}\), and let \(T\) be a decidable theory with \(\neg\theta^*\not\in\mathsf{V}(T)\). Thus, there are formulas \(\varphi\) and \(\psi\) in the language of \(T\) such that \[\label{lem-eq-1} \neg\theta^*[R/\varphi,{\approx}/\psi]\not\in T.\tag{3}\] It follows from (3 ) that we may extend \(T\) to a decidable complete theory \(T^*\) with \[\label{lem-eq-2} \{\theta^*[R/\varphi,{\approx}/\psi],(\forall x)(\forall y)(Exy\leftrightarrow\varphi), (\forall x)(\forall y)(x\approx y\leftrightarrow\psi)\}\subseteq T^*.\tag{4}\] But now it follows from (4 ) that there is a complete decidable \(L\)-extension of \(\theta\). ◻
Proof. (of Theorem 11) It follows at once from Theorem 12 and Lemma 1 that the map from \(e\) to \(\neg\theta_{\sigma(e)}^*\) is a computable reduction from the collection of \(\Pi^0_1\) classes \(T_e\) with no computable path to \(\mathsf{V_{dec}}\). Hence, by Theorems 9 and 13, \(\mathsf{V_{dec}}\) is \(\Pi_3\)-complete. ◻
Note that the above proof still works if we restrict \(T\) to decidable complete theories in the definition of \(\mathsf{V_{dec}}\).
Let \(\mathsf{V_{fa}}=\bigcap_T\mathsf{V}(T)\) where \(T\) ranges over all finitely axiomatizable complete theories.11 Vaught notes that \(\mathsf{V_{fa}}\) is \(\Pi_3\), and suggests the problem of determining its exact arithmetical classification.12 So far as we are aware, this problem remains open.
It is also interesting to consider the complexity of \(\mathsf{V}(T)\) when \(T\) is undecidable. In the proof of Theorem 8, the decidability of \(T\) is not used in showing that \(\mathsf{V}(T)\) has PA degree, so we have the following result.
Theorem 14. If \(T\) is a non-\(\aleph_0\)-categorical complete theory with infinite models, then \(\mathsf{V}(T)\) has PA degree.
Combining this fact with the extension of Arslanov’s Completeness Criterion by [11], we have the following extension of Theorem 8.
Corollary 4. Let \(\mathbf{d}\) be an \(n\)-CEA degree for some \(n\). If \(T\) is a \(\mathbf{d}\)-computable, non-\(\aleph_0\)-categorical, complete theory with infinite models, then \(\mathbf{d} \oplus \mathsf{V}(T)\) computes \(\emptyset'\).
Proof. By Theorem 14, the degree of \(\mathsf{V}(T)\) is PA, and by the relativized form of Theorem 3, it is also \(\mathbf{d}\)-c.e. Thus the degree of \(\mathbf{d} \oplus \mathsf{V}(T)\) is PA and \((n+1)\)-CEA. By the aforementioned work of Jockusch, Lerman, Soare, and Solovay, this degree computes \(\emptyset'\). ◻
Model-theoretic properties can also have an effect on the complexity of \(\mathsf{V}(T)\). For instance, as noted in Footnote 6, in the proof of Theorem 8, the computation \(C\) might be non-standard. But in this case, still using the notation in that proof, the formula substituted in for \(X_\leq\) in witnessing the satisfaction of \(\varphi^l_n\) defines a preorder with an infinite strict chain, which is one way to say that \(T\) has the strict order property. Thus, if T is NSOP, i.e., if it does not have the strict order property, then \(\neg\varphi^l_n \notin \mathsf{V}(T)\) actually implies that \(M\) halts on input \(n\) with output \(l\). This fact allows us to compute the sets \(Z^0\) and \(Z^1\), rather than merely separate them, using \(\mathsf{V}(T)\). Choosing M to be a universal Turing machine now yields the following.
Theorem 15. If \(T\) is a non-\(\aleph_0\)-categorical complete NSOP theory with infinite models, then \(\mathsf{V}(T)\) computes \(\emptyset'\).
The following result stands in contrast to the previous two results.
Theorem 16. Let \(\mathbf{d}\) be a PA degree. Then there is a \(\mathbf{d}\)-computable, complete, non-\(\aleph_0\)-categorical theory \(T\) with infinite models, such that \(\mathsf{V}(T)\) is also \(\mathbf{d}\)-computable.
Proof. We will use the fact that if \(\Gamma\) is a consistent set of sentences with no finite models in a language \(\mathcal{L}_0\), and \(\sigma\) is a consistent, equality-free sentence in a language \(\mathcal{L}_1\) disjoint from \(\mathcal{L}_0\), then \(\Gamma \cup \{\sigma\}\) is a consistent set of \((\mathcal{L}_0 \cup \mathcal{L}_1)\)-sentences.
Let \(\mathcal{L}\) be a language with infinitely many predicate symbols of each arity. (The proof will work the same whether or not \(\mathcal{L}\) includes equality.) Let \(P_0,P_1,\ldots\) form an infinite, coinfinite subset of the unary predicates of \(\mathcal{L}\), and let \(\Gamma\) consist of \(\exists x P_i(x)\) for each \(i\), and \(\forall x \neg (P_i(x) \wedge P_j(x))\) for each \(i \neq j\). Then \(\Gamma\) is consistent and has no finite models, and any consistent extension of \(\Gamma\) has infinitely many \(1\)-types, and hence cannot be \(\aleph_0\)-categorical.
For each binary string \(\alpha\), we will define a (not necessarily consistent) set of \(\mathcal{L}\)-sentences \(\Sigma_\alpha\), and a set of \(\mathcal{S}\)-sentences \(V_\alpha\). Let \(\sigma_0,\sigma_1,\ldots\) be an effective list of all \(\mathcal{L}\)-sentences, and let \(\varphi_0,\varphi_1,\ldots\) be an effective list of all \(\mathcal{S}\)-sentences (both without repetitions). For the empty string \(\lambda\), let \(\Sigma_\lambda=\Gamma\) and let \(V_\lambda=\emptyset\). Given \(\Sigma_\alpha\) and \(V_\alpha\), proceed as follows.
If \(|\alpha|=2n\) for some \(n\) then let \(\Sigma_{\alpha 0}=\Sigma_\alpha \cup \{\neg \sigma_n\}\), let \(\Sigma_{\alpha 1}=\Sigma_\alpha \cup \{\sigma_n\}\), and let \(V_{\alpha 0} = V_{\alpha 1} = V_\alpha\).
If \(|\alpha|=2n+1\) for some \(n\) then let \(V_{\alpha 0} = V_\alpha\), let \(V_{\alpha 1} = V_{\alpha} \cup \{\varphi_n\}\), let \(\Sigma_{\alpha 1} = \Sigma_\alpha\), and define \(\Sigma_{\alpha 0}\) as follows. Write \(\varphi_n\) as \(\varphi_n(X^1_1,\ldots,X^1_k,\ldots,X_1^m,\ldots,X_k^m)\) (i.e., the predicate symbols are among the ones exhibited). Let \(P^1_1,\ldots,P^1_k,\ldots,P_1^m,\ldots,P_k^m\) be pairwise distinct predicate symbols of \(\mathcal{L}\) such that each \(P^j_i\) has arity \(j\), and no \(P^j_i\) appears in \(\Sigma_\alpha\). Let \(\Sigma_{\alpha 0} = \Sigma_\alpha \cup \{\neg\varphi_n(P^1_1,\ldots,P^1_k,\ldots,P_1^m,\ldots,P_k^m)\}\). Notice that if \(\Sigma_\alpha\) is consistent and \(\varphi_n\) is not a validity, then \(\Sigma_{\alpha 0}\) is consistent, by the fact mentioned in the beginning of the proof.
Now for an infinite binary sequence Z, let \(T_Z = \bigcup_{\alpha \prec Z} \Sigma_{\alpha}\), and let \(V_Z = \bigcup_{\alpha \prec Z} V_\alpha\). It is easy to check that \(T_Z\) and \(V_Z\) are both \(Z\)-computable. Now suppose that \(T_Z\) is consistent. It is easy to check that \(T_Z\) is a complete, non-\(\aleph_0\)-categorical theory. Furthermore, if \(\varphi_n \notin V_Z\), then \(Z(2n+1)=0\), so \(T_Z\) contains a sentence witnessing that \(\varphi_n \notin \mathsf{V}(T_Z)\). Thus \(V_Z = \mathsf{V}(T_Z)\) iff \(V_Z \subseteq \mathsf{V}(T_Z)\), i.e., iff \(\{\varphi_n : Z(2n+1)=1\} \subseteq \mathsf{V}(T_Z)\).
Let \(\mathcal{P}\) be the class of all \(Z\) such that \(T_Z\) is consistent and \(\{\varphi_n : Z(2n+1)=1\} \subseteq \mathsf{V}(T_Z)\). Both of these conditions are \(\Pi^0_1\), so \(\mathcal{P}\) is a \(\Pi^0_1\) class. We now show that it is nonempty. Define a binary sequence \(Y\) as follows. Given \(\alpha = Y \upharpoonright 2n\), let \(Y(2n)=1\) if \(\Sigma_{\alpha} \cup \{\sigma_n\}\) is consistent, and let \(Y(2n)=0\) otherwise. Let \(Y(2n+1)=1\) if \(\varphi_n \in \mathsf{V}\) (i.e., \(\varphi_n\) is a validity), and let \(Y(2n+1)=0\) otherwise. It is easy to show by induction that \(\Sigma_\alpha\) is consistent for all \(\alpha \prec Y\), so that \(T_Y\) is consistent, and clearly \(\{\varphi_n : Y(2n+1)=1\} = \mathsf{V} \subseteq \mathsf{V}(T_Y)\).
Thus \(\mathcal{P}\) is a nonempty \(\Pi^0_1\) class, and hence, by the Scott Basis Theorem, has a \(\mathbf{d}\)-computable member \(Z\). Then \(T_Z\) is a \(\mathbf{d}\)-computable, complete, non-\(\aleph_0\)-categorical theory with infinite models, and \(\mathsf{V}(T_Z)=V_Z\) is also \(\mathbf{d}\)-computable. ◻
Notice that the sequence \(Y\) in the above proof is \(\emptyset'\)-computable, and \(T_Y\) is thus a \(\emptyset'\)-computable complete theory such that \(\mathsf{V}(T_Y)=\mathsf{V}\). Conversely, if \(\mathsf{V}(T) = \mathsf{V}\) then \(\mathsf{V}\) is both \(\Sigma_1\) and \(\Pi_1\) relative to \(T\), so \(T\) computes \(\mathsf{V}\), and hence computes \(\emptyset'\).
Notice also that, in the above proof, for any \(\varphi_n \notin \mathsf{V}\), the restriction of \(\mathcal{P}\) to elements \(Z\) such that \(Z(2n+1)=0\) is still a \(\Pi^0_1\) class containing \(Y\), and hence has a \(\mathbf{d}\)-computable member. Thus, for any PA degree \(\mathbf{d}\), the set of sentences that are validities of every \(\mathbf{d}\)-computable complete theory is just \(\mathsf{V}\), and hence is \(\Sigma^0_1\), in contrast to Theorem 11. It would be interesting to study this set for degrees \(\mathbf{d}\) that are neither computable nor PA.
There is also likely more that can be said about the possible complexity of \(\mathsf{V}(T)\) for undecidable \(T\), including how this complexity is affected by model-theoretic properties of \(T\) beyond \(\aleph_0\)-categoricity and NSOP.
Partially supported by NSF grant DMS-1854279.↩︎
Supported by NSF grant DMS-2054379.↩︎
In somewhat dated terms, \(S\) corresponds to the language of the pure predicate calculus without identity, and the languages \(L\) to applied predicate calculi with identity. See, for example, [2].↩︎
Cf. Theorem 36 of [2]. The theory \(\mathbf{PA}\) is first-order Peano Arithmetic. A perspicuous, new proof of this result is presented by [3].↩︎
Notice that the same argument shows that if \(T\) is a complete decidable theory with a finite model, then \(\mathsf{V}(T)\) is computable.↩︎
Alex Kruckman, private communication.↩︎
The reader is advised to consult [5] for background on PA degrees.↩︎
Note that \(C\) may be a “non-standard computation,” that is, a linearly ordered sequence of steps which behaves like a computation, but need not have finite length.↩︎
[1] remarks that \(\mathsf{V_{dec}}\) was considered by Tarski, though he gives no citation, and by [6].↩︎
Specifically, Theorem 12 is a corollary to Peretyat’kin’s result that from every computably axiomatizable theory \(A\), a finitely axiomatizable theory \(F\) can be constructed such that \(F\) is computably isomorphic to \(A\); moreover the construction of \(F\) is effective in a c.e.index for \(A\). This result was conjectured in [7] with a proof sketched in Han75?. A full proof is presented in [8].↩︎