Compositional truth with propositional tautologies and quantifier-free correctness

Abstract

In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as $\Delta _0$-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.

1 Introduction

It is a very widespread phenomenon in logic that if a theory $S_1$ can formulate a truth predicate for a theory $S_2$, then $S_1$ is stronger than $S_2$, a claim which can be made precise in many different ways.

This phenomenon, stripped down to its essence, is investigated in the area of truth theory. Truth theories are axiomatic theories which arise by adding a fresh predicate T(x) to a base theory B which handles syntactic notions (Peano arithmetic, $\text {PA}$, is an example of such a theory). The intended interpretation of T is the set of (codes of) true sentences of the base theory. By considering various possible axioms governing the behaviour of T, we investigate the impact of various notions of truth on the properties on the obtained theory.

One line of research in this area asks what precise properties of the truth predicate make a theory with a truth predicate non-conservative over the base theory. (A theory $S_1$ is conservative over its subtheory $S_2$ if it does not prove any theorems in the language of $S_2$ which are not already provable in that subtheory.)

It is rather straightforward to see that if we add to $\text {PA}$ a unary truth predicate which satisfies compositional axioms and the full induction scheme in the arithmetical language extended with the truth predicate, then by induction on lengths of proofs we can show that all theorems of $\text {PA}$ are true and hence arithmetic is consistent. On the other hand, by a nontrivial result of Kotlarski, Krajewski, and Lachlan from Kotlarski et al. [1], the theory of pure compositional truth predicate with no induction is conservative over $\text {PA}$. Recent research brought much better understanding of which exact principles weaker than full induction yield a nonconservative extension of $\text {PA}$.¹

One of the persistent open questions in this line of research asks whether the compositional truth theory over $\text {PA}$ with an additional axiom expressing that all propositional tautologies are true is conservative over arithmetic. We know that related principles such as “truth is closed under propositional logic” or “valid sentences of first-order logic are true” are not conservative and indeed are all equivalent to $\Delta _0$-induction for the truth predicate.²

In this article, we provide a partial answer to Cieśliński’s question. We show that $\text {CT}^-$ extended with the principle expressing that propositional tautologies are true becomes nonconservative upon adding quantifier-free correctness principle $\text {QFC}$ which states that T predicate agrees with partial arithmetical truth predicates on quantifier-free sentences. The principle $\text {QFC}$ can itself be easily seen to be conservative over $\text {PA}$ (we include a proof in the Appendix B; it is routine).

Our result can therefore be seen as a certain no-go theorem. Our methods for showing conservativity of truth theories behave very well when we demand that several such properties are satisfied at once. Therefore our theorem seems to impose certain restriction on what methods can be used to attack the problem of propositional tautologies.

The argument presented in this article is the original proof of nonconservativity of the compositional truth with the principles “propositional tautologies are true” and the quantifier-free correctness. However, subsequently, another proof, based on different ideas, has been found by Cieśliński and, after placing in a more general framework, published in Cieśliński et al. [5], Proposition 15. We believe however, that the argument presented in the present article might still be of interest, as it is based on a significantly different technique, disjunction with stopping conditions, introduced implicitly in Smith [6] and discussed more systematically in Kossak and Wcisło [7], which we think might find still further applications in the study of models and proof-theoretic properties of truth predicates.

2 Preliminaries

2.1 Arithmetic

In this paper, we consider truth theories over Peano Arithmetic ($\text {PA}$) formulated in the language $\{+,\times , S,0\}$. It is well known that $\text {PA}$, as well as its much weaker subsystems, are capable of formalising syntax. This topic is standard and the reader can find its discussion e.g. in Kaye [8] or Hájek and Pudlák[9]. Below, we list some formulae defining formalised syntactic notions which we will use throughout the paper.

Definition 1

 

• $\text {Var}(x)$ defines the set of (codes of) first-order variables.

• $\text {Term}_{\mathscr {L}_{\text {PA}}}(x)$ defines the set of (codes of) terms of the arithmetical language.

• $\text {ClTerm}_{\mathscr {L}_{\text {PA}}}(x)$ defines the set of (codes of) closed terms of the arithmetical language.

• $\text {Num}(x,y)$ means that y is (the code of) the canonical numeral denoting x. We will use the expression $y = \underline{x}$ interchangeably.

• ${t}^{\circ }=x$ means that t is (a code of) a closed arithmetical term and its formally computed value is x.

• $\text {Form}_{\mathscr {L}_{\text {PA}}}(x)$ defines the set of (codes of) arithmetical formulae.

• $\text {Form}_{\mathscr {L}_{\text {PA}}}^{\le 1}(x)$ defines the set of (codes of) arithmetical formulae with at most one free variable.

• $\text {Sent}_{\mathscr {L}_{\text {PA}}}(x)$ defines the set of (codes of) arithmetical sentences.

• $\text {SentSeq}_{\mathscr {L}_{\text {PA}}}(x)$ defines the set of (codes of) sequences of arithmetical sentences.

• $\text {qfSent}_{\mathscr {L}_{\text {PA}}}(x)$ defines the set of (codes of) quantifier-free arithmetical sentences.

• $\text {Pr}_{\text {PA}}(d,\phi )$ means that d is (a Gödel code of) a proof of $\phi$ in $\text {PA}$. $\text {Pr}_{\text {PA}}(\phi )$ means that $\phi$ is provable in $\text {PA}$.

• $\text {FV}(x,y)$ means that y is (a code of) an arithmetical formula and x is amongst its free variables.

• $\text {Asn}(\alpha ,x)$ means that x is (a code of) an arithmetical term or formula and $\alpha$ is an assignment for x, i.e., a function whose domain contains its free variables.

• If $t \in \text {Term}_{\mathscr {L}_{\text {PA}}}$ and $\alpha$ is an assignment for t, then by $t^{\alpha }=x$, we mean that x is the formally computed value of the term t under the assignment $\alpha$.

In the paper, we will make an extensive use of a number of conventions.

Convention 2

 

• We will use formulae defining syntactic objects as if they were denoting the defined sets. For instance, we will write $x \in \text {Sent}_{\mathscr {L}_{\text {PA}}}$ interchangeably with $\text {Sent}_{\mathscr {L}_{\text {PA}}}(x)$.

• We will often omit expressions defining syntactic operations and simply write the results of these operations in their stead. For example, we will write $T(\phi \wedge \psi )$ meaning “$\eta$ is the conjunction of (the codes of) the sentences $\phi , \psi ,$ and $T(\eta )$.”

• We will use formulae defining functions as if they actually were function symbols, e.g. writing $\underline{x}$ or ${t}^{\circ }$ like stand-alone expressions.

• We will in general omit Quine corners and conflate formulae with their Gödel codes. This should not lead to any confusion.

• We will use expressions $x \in \text {FV}(\phi )$ and $\alpha \in \text {Asn}(\phi )$ interchangeably with $\text {FV}(x,\phi )$ and $\text {Asn}(\alpha ,\phi )$. Moreover, we will use the expressions $\text {FV}(\phi ), \text {Asn}(\phi )$ as if they had a stand-alone meaning, denoting sets of free variables and of $\phi$-assignments respectively.

In this paper, we analyse the compositional truth theory. Let us define the theory in question.

Definition 3

By $\text {CT}^-$ we mean a theory formulated in the arithmetical language extended with a fresh unary predicate T(x) obtained by adding to $\text {PA}$ the following axioms:

• 1. $\forall s,t \in \text {ClTerm}_{\mathscr {L}_{\text {PA}}} \ \Big (T(s=t) \equiv {s}^{\circ } = {t}^{\circ }\Big ).$

• 2. $\forall \phi \in \text {Sent}_{\mathscr {L}_{\text {PA}}} \ \Big (T \lnot \phi \equiv \lnot T \phi \Big ).$

• 3. $\forall \phi , \psi \in \text {Sent}_{\mathscr {L}_{\text {PA}}} \ \Big (T (\phi \vee \psi ) \equiv T\phi \vee T \psi \Big ).$

• 4. $\forall \phi \in \text {Form}^{\le 1}_{\mathscr {L}_{\text {PA}}} \forall v \in \text {FV}(\phi ) \ \Big (T\exists v \phi \equiv \exists x T\phi (\underline{x})\Big ).$

• 5. $\forall {\bar{s}}, {\bar{t}} \in \text {ClTermSeq}_{\mathscr {L}_{\text {PA}}} \forall \phi \in \text {Form}_{\mathscr {L}_{\text {PA}}} \ \Big ( \phi ({\bar{s}}), \phi ({\bar{t}}) \in \text {Sent}_{\mathscr {L}_{\text {PA}}} \wedge \bar{{s}^{\circ }} = \bar{{t}^{\circ }} \rightarrow T\phi ({\bar{s}}) \equiv T \phi ({\bar{t}})\Big ).$

Notice that in the axioms of $\text {CT}^-$ we do not assume any induction for the formulae containing the compositional truth predicate.

Definition 4

By $\text {CT}$ we mean the theory obtained by adding to $\text {CT}^-$ the full induction scheme for formulae in the full language (i.e., arithmetical language extended with the unary truth predicate).

By $\text {CT}_n$ we mean $\text {CT}^-$ with $\Sigma _n$-induction in the extended language, for $n \ge 0$.

It is very well known that $\text {PA}$ (and,in fact, its much weaker fragments) can define partial truth predicates, i.e., formulae which satisfy axioms of $\text {CT}^-$ for sentences of some specific syntactic shape.³ In this paper, we will only need a very special case of this fact.

Proposition 5

There exists an arithmetical formula $\text {Tr}_0(x)$ which satisfies axioms 1–3 of $\text {CT}^-$ restricted to $\phi , \psi \in \text {qfSent}_{\mathscr {L}_{\text {PA}}}$, provably in $\text {PA}$.

2.2 The Tarski boundary

Recall that a theory $S_1$ is conservative over $S_2$ if $S_1 \supseteq S_2$ and whenever $\phi$ is a sentence from the language of $S_2$ and $S_1 \vdash \phi$, then $S_2 \vdash \phi$. It is a persistent phenomenon in logic that the presence of a truth predicate adds substantial strength to theories in question, as witnessed by the following classical theorem⁴:

Theorem 6

$\text {CT}$ is not conservative over $\text {PA}$.

The compositional truth predicate can be employed to prove by induction on the size of proofs that whatever is provable in $\text {PA}$ is true. This allows us to derive the consistency statement for $\text {PA}$ which is unprovable in Peano Arithmetic itself by Gödel’s Second Theorem. The straightforward argument mentioned above uses $\Pi _1$-induction for the compositional truth predicate, but as shown in Łełyk and Wcisło [11], one can do better:

Theorem 7

$\text {CT}_0$ is not conservative over $\text {PA}$.

As a matter of fact, as shown in Łełyk [12], $\Delta _0$-induction is equivalent over $\text {CT}^-$ to the following Global Reflection Principle ($\text {GRP}$):

$$\begin{aligned} \forall \phi \in \text {Sent}_{\mathscr {L}_{\text {PA}}} \ \Big (\text {Pr}_{\text {PA}}(\phi ) \rightarrow T \phi \Big ). \end{aligned}$$

Note that $\text {GRP}$ is, in a way, the exact reason why $\text {CT}$ is not conservative over $\text {PA}$. On the other hand, one of the most important features of $\text {CT}^-$ is that it cannot prove any new arithmetical theorems.

Theorem 8

(Essentially Kotlarski–Krajewski–Lachlan) $\text {CT}^-$ is conservative over $\text {PA}$.

Now, as we can see, compositional truth by itself can be deemed “weak,” but it becomes strong upon adding some induction. One of the main goals of our research is to understand what principles can be added to $\text {CT}^-$ in order to make it nonconservative. It turns out that $\text {CT}_0$ plays a crucial role in this research. A number of apparently very distinct principles turn out to be exactly equivalent with $\Delta _0$-induction for the truth predicate. Let us present the one which largely motivates the research in this paper.

Definition 9

By Propositional Closure Principle $(\text {PC})$ we mean the following axiom:

$$\begin{aligned} \forall \phi \in \text {Sent}_{\mathscr {L}_{\text {PA}}} \Big (\text {Pr}^{\text {Prop}}_{T}(\phi ) \rightarrow T\phi \Big ). \end{aligned}$$

The formula $\text {Pr}^{\text {Prop}}_T(x)$ means that x is provable from true premises in propositional logic. By “true premises,” we mean the set of arithmetical sentences $\phi$ such that $T(\phi )$.

It was proved in Cieśliński [3] that $\text {PC}$ is actually equivalent over $\text {CT}^-$ to $\text {CT}_0$. This is a very surprising result: the mere closure of truth under propositional logic is actually enough to show that consequences of $\text {PA}$ are true.

We can form principles similar to $\text {PC}$ which employ stronger closure conditions:

• “Truth is closed under provability in first-order logic.”

• “Truth is closed under provability in $\text {PA}$.”

We can also weaken these principles so that they only express soundness of discussed systems, not closure properties.

• “Any sentences provable in first-order logic is true.”

• “Any sentence provable in $\text {PA}$ is true.”

It turns out that all the principles listed above are equivalent to each other over $\text {CT}^-$.⁵ One axiom which is noticeably absent from the list is the soundness counterpart of $\text {PC}$. This is not an accident. Whether this principle is conservative over $\text {PA}$ is still an open problem. Let us state our official definition.

Definition 10

By propositional soundness principle ($\text {PS}$), we mean the following axiom:

$$\begin{aligned} \forall \phi \in \text {Sent}_{\mathscr {L}_{\text {PA}}} \Big (\text {Pr}^{\text {Prop}}_{\emptyset }(\phi ) \rightarrow T\phi \Big ). \end{aligned}$$

The formula $\text {Pr}^{\text {Prop}}_{\emptyset }(\phi )$ expresses that $\phi$ is provable in propositional logic from the empty set of premises. In other words, $\text {PS}$ states that any propositional tautology is true.

Enayat and Pakhomov [4] proved that actually a very modest fragment of propositional closure, $\text {PC}$, is already enough to yield a non-conservative theory.

Definition 11

By Disjunctive Correcntess ($\text {DC}$), we mean the following principle:

$$\begin{aligned} \forall (\phi _i)_{i \le c} \in \text {SentSeq}_{\mathscr {L}_{\text {PA}}}\Big (T \bigvee _{i \le c} \phi _i \equiv \exists i \le c \ T \phi _i \Big ). \end{aligned}$$

In other words, $\text {DC}$ expresses that any finite disjunction is true iff one of its disjuncts is. Here “finite” is understood in the formalised sense, so that it may refer to nonstandard objects. We treat the symbol $T\bigvee _{i \le c} \phi _i$ as denoting disjunctions with parentheses grouped to the left for definiteness.

Theorem 12

(Enayat–Visser) $\text {CT}^- + \text {DC}$ is equivalent to $\text {CT}_0$. Consequently, $\text {CT}^-+ \text {DC}$ is not conservative over $\text {PA}$.

This theorem is really striking. Admittedly, $\text {DC}$ can be viewed as a natural extension of compositional axioms. We simply want to allow that the truth predicate behaves compositionally with respect not just to binary (or standard) disjunctions, but to arbitrary finite ones.

2.3 Disjunctions with stopping conditions

The main technical tool which we are going to use in this article are disjunctions with stopping conditions, a tool implicitly introduced (but not officially defined), in Smith [6]. This is a particular propositional construction which is a very useful tool in the analysis of $\text {CT}^-$. The motivation and proofs of the cited facts concerning disjunctions with stopping conditions can be found in Kossak and Wcisło [7].

Definition 13

Let $(\alpha _i)_{i \le c}, (\beta _i)_{i \le c}$ be sequences of sentences. We define the disjunction of $\beta _i$ with the stopping condition $\alpha$ for $i \in [j,c]$ by backwards induction on j:

$$\begin{aligned} \bigvee _{i = c}^{\alpha ,c} \beta _i= & {} \alpha _c \wedge \beta _c \\ \bigvee _{i = j}^{\alpha ,c} \beta _i= & {} (\alpha _j \wedge \beta _j) \vee (\lnot \alpha _j \wedge \bigvee _{i = j+1}^{\alpha ,c} \beta _i ). \end{aligned}$$

The key feature of disjunctions with stopping conditions is that they allow us to use disjunctive correctness in some very limited range of cases which suffice for certain applications without actually committing to the full strength of this axiom.

Theorem 14

Let $(M,T) \models \text {CT}^-$. Let $(\alpha _i)_{i \le c}, (\beta _i)_{i \le c} \in \text {SentSeq}_{\mathscr {L}_{\text {PA}}}(M)$ be sequences of sentences. Suppose that $k_0 \in \omega$ is the least number j such that $(M,T) \models T \alpha _{j}$ holds. Then

$$\begin{aligned} (M,T) \models T \bigvee _{i = 0}^{\alpha ,c} \beta _i \equiv T \beta _{k_0}. \end{aligned}$$

Notice that above we assume that $k_0 \in \omega$, i.e., it is in the standard part of M. In other words: if we are guaranteed that some $\alpha _k$ holds for a standard k, we can make an infinite case distinction of the form: “either $\alpha _0$ holds and then $\beta _0$ or $\alpha _1$ holds and then $\beta _1$... or $\alpha _c$ holds and then $\beta _c$” so that it actually works correctly in the presence of compositional axioms alone without assuming any induction whatsoever. The proof of Theorem 14 (together with applications) may be found in Kossak and Wcisło [7].

The following proposition explains why disjunctions with stopping conditions are so named.

Proposition 15

Suppose that $\alpha _i \beta _i, i \le c$ are sentences of propositional logic. Then every boolean valuation which makes exactly one of $\alpha _i$ satisfied makes the following equivalence satisfied:

$$\begin{aligned} \bigvee _{i=0}^c \alpha _i \wedge \beta _i \equiv \bigvee _{i=0}^{\alpha ,c} \beta _i. \end{aligned}$$

Moreover, this is provable in $\text {PA}$.

Proof

We work in $\text {PA}$. Fix any valuation which makes exactly one of the sentences $\alpha _i$ true, say, $i=k$. It is clear that the disjunction $\bigvee _{i = k}^c \alpha _i \wedge \beta _i$ is equivalent to $\beta _k$. We will show by backwards induction on j that all formulae $\bigvee _{i = j}^{\alpha _i,c} \beta _i$ are equivalent to $\beta _k$.

Suppose that $j=k$. Since $\alpha _k$ holds, we immediately have the following equivalence:

$$\begin{aligned} \bigvee _{i=k}^{\alpha ,c} \beta _i = (\alpha _k \wedge \beta _k) \vee (\lnot \alpha _k \wedge \bigvee _{i=k+1}^{\alpha ,c } \beta _i ) \equiv \beta _k. \end{aligned}$$

Suppose that the claim holds for $j+1 \le k$. Since $j<k$, by assumption $\alpha _j$ is not true. Hence, again by elementary manipulations, the following equivalence holds:

$$\begin{aligned} \bigvee _{i=j}^{\alpha ,c} \beta _i = (\alpha _j \wedge \beta _j) \vee (\lnot \alpha _j \wedge \bigvee _{i=j+1}^{\alpha ,c } \beta _i ) \equiv \bigvee _{i=j+1}^{\alpha ,c} \beta _i. \end{aligned}$$

By induction hypothesis, the last formula is equivalent to $\beta _k$. This proves our claim. $\square$

Theorem 14 can be proved by following the above argument, starting with $k_0$ instead of k and noticing that in this case, we only need to perform standardly many steps in of induction, so it can be carried out externally. Let us also remark, that Proposition 15 can be clearly proved in much weaker subsystems of $\text {PA}$ such as $\text {I}\Delta _0 +\exp$.

Most importantly for this article, the behaviour of disjunctions with stopping conditions can be partly encoded as a propositional tautology.

We will use the following notation: if $(\alpha _i)_{i \le c}$ is a sequence of sentences, then by $\bigcirc _{i \le c} \alpha _i$, we mean the following sentence:

$$\begin{aligned} \bigvee _{i \le c} \Big (\alpha _i \wedge \bigwedge _{j \ne i} \lnot \alpha _j \Big ). \end{aligned}$$

It expresses that exactly one of $\alpha _i$s is true.

Corollary 16

For any sentences $\alpha _i,\beta _i, i \le c$, the following is a propositional tautology:

$$\begin{aligned} \bigcirc _{i \le c} \alpha _i \rightarrow \Big (\bigvee _{i = 0}^{c, \alpha } \beta _i \equiv \bigvee _{i=0}^c \alpha _i \wedge \beta _i \Big ). \end{aligned}$$

Morevoer, this is provable in $\text {PA}$.

3 The main result

In this section, we prove the main result of our paper. We will show that the propositional soundness principle added to $\text {CT}^-$ becomes non-conservative (and actually equivalent to $\text {CT}_0$) upon adding an innocuous principle which can be easily shown to be conservative by itself.

Definition 17

By the quantifier-free correctness principle $(\text {QFC})$, we mean the following axiom:

$$\begin{aligned} \forall \phi \in \text {qfSent}_{\mathscr {L}_{\text {PA}}} \Big (T\phi \equiv \text {Tr}_0\phi \Big ). \end{aligned}$$

In other words, on quantifier-free sentences arithmetical partial truth and truth in the sense of the T predicate agree. Notice that this allows us to use full induction when reasoning about the truth predicate applied to quantifier-free sentences, since the truth predicate restricted to such sentences is equivalent to an arithmetical formula. It turns out that this innocuous principle is enough to yield propositional soundness nonconservative.

Theorem 18

The theory $\text {CT}^- + \text {QFC}+ \text {PS}$ is not conservative over $\text {PA}$. In fact, it is exactly equivalent to $\text {CT}_0$.

Crucially, $\text {CT}^- + \text {QFC}$ is by itself conservative over $\text {PA}$.

Theorem 19

The theory $\text {CT}^- + \text {QFC}$ is conservative over $\text {PA}$.

The proof of this fact is a routine application of Enayat–Visser proof of conservativeness of $\text {CT}^-$. For completeness, we present it in Appendix B.

Now, we can present the last crucial ingredient of our proof. As we have already mentioned, disjunctive correctness was proved to be equivalent to $\text {CT}_0$ (over $\text {CT}^-$) in Enayat and Pakhomov [4]. However, by inspection of the proof, it can be seen that actually somewhat weaker assumption is employed, as the disjunctive correctness is used only with respect to one rather specific kind of formulae.

Definition 20

By Atomic Case Distinction Correctness ($\text {ACDC}$) we mean the following axiom: For any sequence of arithmetical sentences $(\phi _i)_{i \le c} \in \text {SentSeq}_{\mathscr {L}_{\text {PA}}}$ and any closed term $t \in \text {ClTerm}_{\mathscr {L}_{\text {PA}}}$, the following equivalence holds:

$$\begin{aligned} T \left( \bigvee _{i \le c} \big ( t= \underline{i} \wedge \phi _i \big ) \right) \equiv \exists a \le c \left( {t}^{\circ } = a \wedge T\phi _a \right) . \end{aligned}$$

Theorem 21

(Essentially Enayat–Pakhomov) $\text {CT}^- + \text {ACDC}$ is equivalent to $\text {CT}_0$. In particular it is not conservative over $\text {PA}$.

As we already mentioned, this theorem is proved by a straightforward inspection of the earlier argument by Enayat and Pakhomov. For the convenience of the reader, we will discuss it in Appendix A. Now, we are ready to present the proof of our main result, Theorem 18.

Proof of Theorem 18

Fix any model $(M,T) \models \text {CT}^- + \text {QFC}+ \text {PS}.$ We will show that

$$\begin{aligned} (M,T) \models \text {CT}^- + \text {ACDC}, \end{aligned}$$

which shows by Theorem 21 that $(M,T) \models \text {CT}_0$.

Fix any $c\in M$, a closed term $t \in \text {ClTerm}_{\mathscr {L}_{\text {PA}}}(M)$, and an arbitrary sequence of sentences $(\phi _i)_{i\le c} \in \text {SentSeq}_{\mathscr {L}_{\text {PA}}}(M)$.

First, suppose that there exists $a \le c$ such that ${t}^{\circ } = a$ and $T\phi _a$ holds. Observe that:

$$\begin{aligned} (t = \underline{a} \wedge \phi _a) \rightarrow \bigvee _{i \le c} \left( t= \underline{i} \wedge \phi _i \right) \end{aligned}$$

is recognised in M as a propositional tautology. Hence, by $\text {CT}^- + \text {PS}$, we obtain:

$$\begin{aligned} (M,T) \models T \left( \bigvee _{i \le c} \big (t= \underline{i} \wedge \phi _i \big )\right) . \end{aligned}$$

This proves one direction of $\text {ACDC}$. For the harder direction, assume that

$$\begin{aligned} (M,T) \models T \left( \bigvee _{i \le c} \big ( t= \underline{i} \wedge \phi _i \big )\right) . \end{aligned}$$

We first show that indeed $M \models {t}^{\circ } \le c$. Suppose otherwise. Then, this fact is recognised by the partial arithmetical truth predicate as follows:

$$\begin{aligned} M \models \text {Tr}_0 \bigwedge _{i \le c} \lnot ( t = \underline{i}). \end{aligned}$$

By $\text {QFC}$, the same holds for the truth predicate T rather than $\text {Tr}_0$. Moreover, notice that the following sentence is a propositional tautology:

$$\begin{aligned} \bigwedge _{i \le c} \lnot (t = \underline{i}) \rightarrow \lnot \bigvee _{i \le c} \big (t= \underline{i} \wedge \phi _i \big ). \end{aligned}$$

Hence, by propositional soundness $\text {PS}$ and our assumption that $T \bigvee _{i \le c} (t= \underline{i} \wedge \phi _i)$ holds, the value of t, as computed in M, is below c.

Now, fix $a \le c$ such that ${t}^{\circ } = a$. Fix any permutation $\sigma : \{0, \ldots , c\} \rightarrow \{0, \ldots , c\}$ such that $\sigma (a) = 0$ (so after permuting, the only disjunct which can be true is placed as the first one). Since disjunctions are associative and commutative provably in $\text {PA}$ (and in much weaker systems), by propositional soundness $\text {PS}$, the following holds:

$$\begin{aligned} (M,T) \models T \left( \bigvee _{i \le c} \big ( t= \underline{i} \wedge \phi _i \big ) \right) \equiv T \left( \bigvee _{i \le c} \big ( t= \underline{\sigma (i)} \wedge \phi _{\sigma (i)} \big ) \right) . \end{aligned}$$

Now, notice that exactly one of the formulae $t = \underline{i}$ is true, and this can be expressed as follows:

$$\begin{aligned} M \models \text {Tr}_0 \bigvee _{i \le c} \Bigl ( t= \underline{i} \wedge \bigwedge _{j \ne i} \lnot t = \underline{j} \Bigr ). \end{aligned}$$

By $\text {QFC}$, using our notation from previous section, this is equivalent to:

$$\begin{aligned} M \models T \bigcirc _{i \le c} t = \underline{i}. \end{aligned}$$

The same argument applies, if we consider sentences $t = \underline{\sigma (i)}$ rather than $t = \underline{i}$. By Corollary 16, the following is a propositional tautology, hence true in the sense of the predicate T by $\text {PS}$:

$$\begin{aligned} \bigcirc _{i \le c} t = \underline{i} \rightarrow \left( \left( \bigvee _{i \le c} \big ( t= \underline{i} \wedge \phi _i \big ) \right) \equiv \bigvee _{i = 0}^{t = \underline{i}, c} \phi _i \right) . \end{aligned}$$

Again, this holds if we consider sequences $t=\underline{\sigma (i)}$ and $\phi _{\sigma (i)}$ instead. Putting it all together, we know that the following formulae are true:

$$\begin{aligned} (M,T) \models T \bigcirc _{i \le c} t = \sigma (i) \wedge T \bigvee _{i = 0}^c \left( t= \underline{\sigma (i)} \wedge \phi _{\sigma (i)} \right) . \end{aligned}$$

Therefore,

$$\begin{aligned} (M,T) \models T \bigvee _{i = 0}^{t= \underline{\sigma (i)}, c} \phi _{\sigma (i)}. \end{aligned}$$

By Theorem 14 on disjunctions with stopping conditions, as the above disjunction stops at $i = 0$, we obtain:

$$\begin{aligned} (M,T) \models T \phi _{\sigma (0)}. \end{aligned}$$

Since $\sigma (0) = a = {t}^{\circ }$, this concludes our argument. $\square$

Appendices

Appendix A: The strength of $\text {ACDC}$

In the main part, we crucially used the observation that Atomic Case Distinction Correctness, $\text {ACDC}$ is equivalent to $\text {CT}_0$. As we already mentioned, this result is really due to Enayat and Pakhomov, as this is what their arguments in [4] actually show. However, since verifying this claim would be admittedly cumbersome, we will rather repeat their argument below.

Following closely the presentation in the original paper, we split our argument into two parts. We first show that $\text {ACDC}$ together with internal induction yields $\Delta _0$-induction for the truth predicate. Subsequently, we show that $\text {ACDC}$ implies internal induction. Before any of this happens let us define what internal induction actually is.

Definition 22

By Internal Induction ($\text {INT}$), we mean the following axiom:

$$\begin{aligned} \forall \phi \in \text {Form}^{\le 1}_{\mathscr {L}_{\text {PA}}} \Big (\forall x \bigl ( T \phi (\underline{x}) \rightarrow T \phi (\underline{x+1}) \bigr ) \rightarrow \forall x \bigl ( T\phi (\underline{0}) \rightarrow \forall x \ T \phi (\underline{x}) \bigr ) \Big ). \end{aligned}$$

In other words, internal induction expresses that any arithmetical formula satisfies induction under the truth predicate.

Theorem 23

$\text {CT}^- + \text {ACDC}+ \text {INT}$ is equivalent to $\text {CT}_0$.

Proof

It can be directly verified that $\text {CT}_0$ implies $\text {INT}$ and full $\text {DC}$. Therefore, we will focus on the harder direction, showing that $\text {CT}^- + \text {ACDC}+ \text {INT}$ implies $\text {CT}_0$.

Fix any model $(M,T) \models \text {CT}^- + \text {ACDC}+ \text {INT}$. We want to show that $(M,T) \models \text {CT}_0$. It is enough to demonstrate that for any $c \in M$, the set $T \cap [0,c]$ is coded, i.e., there exists $s \in M$ such that $a \in T \cap [0,c]$ iff the a-th bit of s in the binary expansion is equal to 1.

Fix the unique sequence $(\phi _i)_{i \le c}$ of sentences such that $\ulcorner \phi _i \urcorner = i$ if i happens to be an arithmetical sentence (that is, $i \in \text {Sent}_{\mathscr {L}_{\text {PA}}}(M)$) and $\phi _i = \ulcorner 0 \ne 0 \urcorner$ otherwise. Consider the following formula $\Theta (a,x)$:

$$\begin{aligned} \Theta _c(x):= \bigvee _{i \le c} (x = \underline{i} \wedge \phi _i). \end{aligned}$$

By $\text {ACDC}$, for $\phi \in \text {Sent}_{\mathscr {L}_{\text {PA}}}(M) \cap [0,c]$,

$$\begin{aligned} (M,T) \models T \Theta _c(\underline{\phi }) \equiv T \phi . \end{aligned}$$

On the other hand, by $\text {INT}$, the formula $T \Theta _c(\underline{x})$ satisfies full induction. In particular, the set of elements smaller than c satisfying this formula is coded. $\square$

Now, we can move to the second ingredient of the proof:

Theorem 24

$\text {CT}^- + \text {ACDC}$ implies $\text {INT}$.

In the paper [4] which we closely follow in this presentation, the analogue of Theorem 24 is proved by an extremely elegant detour via a theory of iterated truth predicates.

Definition 25

By $\text {ITB}$ (Iterated Truth Biconditionals), we mean a theory with two sorts: a number sort and index sort, over the language with the following symbols:

• The function symbols of $\mathscr {L}_{\text {PA}}$, whose arguments come from the number sort.

• A fresh predicate $T(\alpha , x)$, where $\alpha$ comes from the index sort and x from the number sort. We will also denote it with $T_{\alpha }(x)$.

• A fresh predicate $\alpha \prec \beta$, whose arguments come from the index sort.

Its axioms consist of $\text {PA}$, axioms saying that $\prec$ is a linear ordering of the index sort and the following scheme:

$$\begin{aligned} \forall \alpha \Big (T_{\alpha } \phi \equiv \phi ^{\prec \alpha } \Big ), \end{aligned}$$

where $\phi$ comes from the full language and $\phi ^{\prec \alpha }$ is $\phi$ with the index-sort quantifiers $\forall \beta , \exists \beta$ replaced with $\forall \beta \prec \alpha , \exists \beta \prec \alpha$.

$\text {ITB}$ axiomatises a hierarchy of truth predicates over a linear order. The key point is that this order cannot have infinite descending chains. The theorem below was proved in [4], based on the main result in [13].

Theorem 26

The theory $\text {ITB}$ together with the axioms $\forall \alpha \exists \beta \ \beta \prec \alpha$ and the sentence expressing nonemptiness of the index sort is inconsistent.

By the above theorem, there exists a finite fragment $\Gamma$ of $\text {ITB}$ which proves that $\prec$ has the least element. This theory contains finitely many biconditionals of the form:

$$\begin{aligned} \forall \alpha \Big (T_{\alpha } \phi \equiv \phi ^{\prec \alpha } \Big ). \end{aligned}$$

Let $\phi _1,\ldots ,\phi _n$ be the enumeration of sentences which occur in the biconditionals from $\Gamma$. Let us denote the biconditional involving $\phi _i$ with $B(\phi _i)$.

Proof of Theorem 24

Let $(M,T) \models \text {CT}^- + \text {ACDC}.$ Fix any $\phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}^{\le 1}(M)$ such that for some $c_0 \in M$, $(M,T) \models T \phi (\underline{c_0})$. We will show that there exists the least $c \in M$ such that $(M,T) \models T \phi (\underline{c})$. Since $\phi$ is arbitrary, and by compositionality of T, this implies that internal induction holds in (M, T).

By induction we will construct in M a sequence of interpretations $\iota _a, a \in M$ of $\Gamma \subset \text {ITB}$, i.e., a sequence of tuples of formulae: the definitions of domains for number and index sorts, the interpretations of the arithmetical symbols, and the interpretations for the predicates $\prec$, $T(\alpha ,x)$.

• For all a, $\iota _a$ interprets arithmetical symbols by identity and the domain of number quantifiers is the whole M (i.e., the domain is defined by the formula $x=x$).

• The a-th domain of index quantifiers is given by $d_a(x):= x \le a \wedge \phi (x)$.

• The index inequality $\prec$ is interpreted by the usual inequality <.

• The predicate $T(\alpha ,x)$ is defined recursively as follows:

  $$\begin{aligned} \bigvee _{i \le n} \Bigl ( x = \underline{\phi _i} \wedge \bigvee _{j<a} \big ( \alpha = \underline{j} \wedge \phi (\underline{j}) \wedge \iota _j(\phi _i) \big ) \Bigr ). \end{aligned}$$

We will show that for all a, if $(M,T) \models T\phi (\underline{a})$, then $\iota _a$ is indeed an interpretation of $\Gamma$ under the truth predicate. This means that for all sentences $\psi \in \Gamma$,

$$\begin{aligned} (M,T) \models T \iota _a(\psi ). \end{aligned}$$

This is immediate for arithmetical axioms and the ordering axioms for $\prec$. Thus it is enough to check that the claim is satisfied for the truth biconditionals. Fix $k \le n$ and $a \in M$. We want to check that:

$$\begin{aligned} (M,T) \models T \iota _a \forall \alpha \Big (T_{\alpha } \phi _k \equiv \phi _k^{\prec \alpha } \Big ). \end{aligned}$$

If there are no $a'<a$ such that $(M,T) \models T\phi (\underline{a'})$, then the interpretation of the universal quantifier $\forall \alpha$ makes the sentence trivially true. So suppose otherwise and fix any $\alpha < a$ such that $Td_a(\underline{\alpha })$ holds. We want to check that the following holds:

$$\begin{aligned} T\iota _a T(\alpha ,\phi _k) \equiv T\iota _a \phi _k^{\prec \alpha }. \end{aligned}$$

Expanding the definition of $\iota _a$ on the left-hand side of the equivalence yields:

$$\begin{aligned} T \biggl (\bigvee _{i \le n} \Bigl ( \phi _k = \underline{\phi _i} \wedge \bigvee _{j<a} \alpha = \underline{j} \wedge \phi (\underline{j}) \wedge \iota _j(\phi _i) \Bigr ) \biggr ). \end{aligned}$$

The first disjunction has standardly many disjuncts, of which only one, namely k, is true, so by the compositional axioms, this is equivalent to:

$$\begin{aligned} T \bigvee _{j<a} \big ( \alpha = \underline{j} \wedge \phi (\underline{j}) \wedge \iota _j(\phi _k) \big ). \end{aligned}$$

By $\text {ACDC}$, this is equivalent to:

$$\begin{aligned} T \phi (\underline{\alpha }) \wedge T \iota _\alpha (\phi _k). \end{aligned}$$

By assumption on $\alpha$, this is equivalent to

$$\begin{aligned} T \iota _\alpha (\phi _k). \end{aligned}$$

Now, it is enough to check that the following equivalence holds:

$$\begin{aligned} T \iota _\alpha (\phi _k) \equiv T \iota _a \phi _k^{\prec \alpha }. \end{aligned}$$

We essentially check by induction on complexity of subformulae $\psi$ of $\phi _k$ that this equivalence holds for all $\psi$. To make this more precise, we introduce the following definition. We say that a tuple $t_1, \ldots , t_m \in \text {ClTerm}_{\mathscr {L}_{\text {PA}}}(M)$ is suitable for a formula $\psi$ if $\psi$ has m free variables and for every term t corresponding to an index variable $\beta$, $(M,T) \models T d_{\alpha }(t)$.

Now, by induction on the complexity of formulae, we will show that for any subformula $\psi$ of $\phi _k$, and any suitable tuple ${\bar{t}}$ of closed terms in the sense of M, the following equivalence holds:

$$\begin{aligned} T \iota _{\alpha } (\psi )({\bar{t}}) \equiv T \iota _a(\psi )^{\prec \alpha }({\bar{t}}). \end{aligned}$$

The induction steps for connectives and number quantifiers, as well as the initial step for the arithmetical atomic formulae and the atomic formula $\beta \prec \gamma$ are immediate. Let us now focus on the initial case for the formula $T(\beta ,x)$. Fix any suitable pair of closed terms $t_1, t_2$. In particular this means that the value of $t_2$ is no greater than $\alpha < a$. $T \iota _\alpha T(t_1,t_2)$ is the following sentence:

$$\begin{aligned} T \biggl (\bigvee _{i \le n} \Bigl ( t_1 = \underline{\phi _i} \wedge \bigvee _{j< \alpha } t_2 = \underline{j} \wedge \phi (\underline{j}) \wedge \iota _j(\phi _i) \Bigr ) \biggr ). \end{aligned}$$

By $\text {ACDC}$ and the fact that ${t_2}^{\circ } \le \alpha < a$, this is equivalent to:

$$\begin{aligned} T \biggl (\bigvee _{i \le n} \Bigl ( t_1 = \underline{\phi _i} \wedge \bigvee _{j<a} t_2 = \underline{j} \wedge \phi (\underline{j}) \wedge \iota _j(\phi _i) \Bigr ) \biggr ). \end{aligned}$$

(The two formulae differ by the range of the second disjunction.) Since the second formula is equal to $T \iota _a T(t_1,t_2) = T \iota _aT(t_1,t_2)^{\prec \alpha }$, the atomic case is proved.

What remains to be proved is the induction step for the index quantifier. Suppose that our claim holds for $\psi$ and consider the formula $\forall \beta \psi (\beta ).$ Notice that the following equalities hold:

$$\begin{aligned} \iota _\alpha \forall \beta \psi (\beta )= & {} \forall x \Big (d_{\alpha }(x) \rightarrow \iota _{\alpha }\psi (x) \Big ) \\ \iota _a \Big ( (\forall \beta \psi (\beta ))^{\prec \alpha } \Big )= & {} \forall x \Big (d_{\alpha }(x) \rightarrow \iota _a\psi ^{\prec \alpha }(x) \Big ). \end{aligned}$$

By induction hypothesis and the compositional axioms if we substitute suitable terms in the formulae on the right-hand side, then the first one is true if and only if the second one is. This concludes the induction argument and the whole proof. $\square$

Appendix B: Conservativeness of $\text {CT}^- + \text {QFC}$

In the main part, we claimed that the quantifier-free correctness can be added to $\text {CT}^-$ still yielding a conservative theory. As we already noted, this is a very simple application of the Enayat–Visser construction, but we could not find this exact statement in the literature.⁶ Therefore, we decided to include a proof of this claim. However, the reader should feel entirely free to skip it.

Definition 27

Let $M \models \text {PA}$. We say that a set $T_0 \subset \text {Sent}_{\mathscr {L}_{\text {PA}}}(M)$ is a partial compositional truth predicate if there exists a set of formulae D closed under taking direct subformulae such that:

• $T_0$ satisfies the axioms of $\text {CT}^-$ for all sentences resulting from substituting closed terms into formulae from D;

• If $T_0(\phi )$ holds, then any formula resulting from replacing terms in $\phi$ with other terms (possibly with free variables) is in D.

We will derive Theorem 19 from the following, more general fact.

Theorem 28

Let $M_0 \models \text {PA}$ and let $T_0 \subset M_0$ be a partial compositional truth predicate. Then there exists an elementary extension $(M_0,T_0) \preceq (M',T)$ and $T' \supseteq T$ such that $(M',T') \models \text {CT}^-$.

Proof of Theorem 19 from Theorem 28

Let $M \models \text {PA}$ and let $T_0 \subset M_0$ be defined as the set of sentences $\phi$ such that $M \models \text {Tr}_0(\phi ).$ We apply Theorem 28 to $(M_0,T_0)$ obtaining an elementary extension $(M',T) \succeq (M_0,T_0)$ and $T' \supseteq T$ such that $(M',T') \models \text {CT}^-$.

Now, observe that actually $(M',T') \models \text {CT}^- + \text {QFC}$. Indeed, by elementarity T is exactly the set of $\phi \in \text {Sent}_{\mathscr {L}_{\text {PA}}}(M')$ such that $M' \models \text {Tr}_0(\phi )$. $\square$

Now we turn to the proof of Theorem 28. Since we are dealing with truth predicates for a language with terms and we include extensionality in our axioms, we have to take care of certain additional technicalities. Before we proceed to the proof, we will introduce some definitions and notation.

Definition 29

Let $M \models \text {PA}$ and let $\phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}(M)$. By a trivialisation of $\phi$, we mean a formula ${\widehat{\phi }}$ such that:

• There exists a sequence of terms ${\bar{t}} \in \text {TermSeq}_{\mathscr {L}_{\text {PA}}}(M)$ such that $\phi = {\widehat{\phi }}({\bar{t}})$.

• No variable occurs in ${\widehat{\phi }}$ both free and bound.

• No free variable occurs in ${\widehat{\phi }}$ more than once.

• No closed term occurs in ${\widehat{\phi }}$.

• No complex term whose all variables are free occurs in ${\widehat{\phi }}$.

• ${\widehat{\phi }}$ is the least formula with the above properties. (In order to guarantee the uniqueness.)

For instance, if $\phi = \exists x \forall y \Big ( x +( z \times S0 + 0 \times u ) = x \times y + 0 \Big )$, then

$$\begin{aligned} {\widehat{\phi }} = \exists x \forall y \Big ( x + v_1 = x \times y + v_2 \Big ), \end{aligned}$$

where $v_1, v_2$ are chosen so as to minimise the formula ${\widehat{\phi }}$.

• We say that two formulae $\phi _1, \phi _2$ are syntactically similar if $\widehat{\phi _1} = \widehat{\phi _2}.$ We denote it with $\phi _1 \sim \phi _2$.

• If $\phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}$ and $\alpha \in \text {Asn}(\phi )$, then by $\phi [\alpha ]$ we mean the sentence resulting by substituting the numeral $\underline{\alpha (v)}$ for each variable v.

• If $\phi _1, \phi _2 \in \text {Form}_{\mathscr {L}_{\text {PA}}}, \alpha _1 \in \text {Asn}(\phi _1), \alpha _2 \in \text {Asn}(\phi _2)$, then we say that $(\phi _1,\alpha _1)$ is extensionally equivalent to $(\phi _2, \alpha _2)$ if $\phi _1 \sim \phi _2$ and there exist two sequences of closed terms $\bar{t_1}, \bar{t_2} \in \text {ClTermSeq}_{\mathscr {L}_{\text {PA}}}$ such that $\overline{{t_1}^{\circ }} = \overline{{t_2}^{\circ }}$ (the values of terms in $\bar{t_1}, \bar{t_2}$ are pointwise equal), $\phi _1 = \phi (\bar{t_1}), \phi _2 = \phi (\bar{t_2})$, where $\phi = \widehat{\phi _1}= \widehat{\phi _2}$. We denote this relation with $(\phi _1, \alpha _1) \sim (\phi _2, \alpha _2)$.

Notice that the syntactic similarity and extensional equivalence are both equivalence relations.

Proof

Let $M_0$ be any model of $\text {PA}$ and let $T_0 \subset M_0$ be a partial truth predicate. We will construct a chain of models $(M_i,T_i,S_i), i \in \omega$. The chain of models $(M_i,T_i)$ will be elementary and the binary predicates $S_i$ will be partial satisfaction predicates extending one another and extending $T_i$.

We perform the construction in the following way: once we have constructed the model $(M_i,T_i,S_i)$, we let $(M_{i+1},T_{i+1}, S_{i+1})$ be any model of the theory $\Theta _{i+1}$ consisting of the following axioms in the arithmetical language with additional predicates $S_{i+1}, T_{i+1}$:

• $\text {ElDiag}(M_i,T_i)$. (The elementary diagram of $(M_i, T_i)$, formulated with $T_{i+1}$ replacing $T_i$.)

• $\text {Comp}(\phi ), \phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}(M_i)$. (The compositionality scheme, to be defined later.)

• $\forall \phi ,\phi ' \in \text {Form}_{\mathscr {L}_{\text {PA}}} \forall \alpha \in \text {Asn}(\phi ), \alpha ' \in \text {Asn}(\phi ') \ \Big ((\phi ,\alpha ) \sim (\phi ', \alpha ') \rightarrow S_{i+1}(\phi ,\alpha ) \equiv S_{i+1}(\phi ',\alpha ') \Big )$. (The extensionality axiom)

• $\forall x \Big (T_{i+1}(x) \rightarrow S_{i+1}(x,\emptyset )\Big )$. (The satisfaction predicate $S_{i+1}$ agrees with $T_{i+1}$.)

• $S_{i+1}(\phi ,\alpha )$, where $\phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}(M_{i-1})$, $\alpha \in \text {Asn}(\phi )$ and $(\phi ,\alpha ) \in S_i$. (The preservation scheme.)

An instance of the compositionality scheme $\text {Comp}(\phi )$ is defined as the disjunction of the following clauses:

• 1. $\exists s, t \in \text {Term}_{\mathscr {L}_{\text {PA}}} \Big ( \phi = (s=t) \wedge \forall \alpha \in \text {Asn}(\phi ) \ S(\phi , \alpha ) \equiv s^{\alpha } = t^{\alpha }\Big ).$

• 2. $\exists \psi \in \text {Form}_{\mathscr {L}_{\text {PA}}} \Big ( \phi = (\lnot \psi ) \wedge \forall \alpha \in \text {Asn}(\phi ) \ S(\phi , \alpha ) \equiv \lnot S(\psi ,\alpha ) \Big ).$

• 3. $\exists \psi , \eta \in \text {Form}_{\mathscr {L}_{\text {PA}}} \Big (\phi = (\psi \vee \eta ) \wedge \forall \alpha \in \text {Asn}(\phi ) \ S(\phi ,\alpha ) \equiv S(\psi , \alpha ) \vee S(\eta , \alpha ) \Big ).$

• 4. $\exists \psi \in \text {Form}_{\mathscr {L}_{\text {PA}}}, v \in \text {Var}\Big (\phi = (\exists v \psi ) \wedge \forall \alpha \in \text {Asn}(\phi ) \Big (S(\phi ,\alpha ) \equiv \exists \beta \sim _{v} \alpha \ S(\psi ,\beta )\Big )\Big ).$

For the time being, suppose that all theories $\Theta _n$ are consistent. We will finish the proof under this assumption and return to it afterwards.

Let $M' = \bigcup _{n \in \omega } M_n$, $T = \bigcup T_n$. Let

$$\begin{aligned} T' = \left\{ \phi \in \text {Sent}_{\mathscr {L}_{\text {PA}}}(M) \ \mid \ \exists n \in \omega \ \phi \in M_n \wedge (\phi , \emptyset ) \in S_{n+1} \right\} . \end{aligned}$$

It can be directly verified that the predicate $T'$ defined in such a way satisfies axioms of $\text {CT}^-$ thanks to the assumption that the predicates $S_{n}$ satisfy the compositionality scheme together with preservation and extensionality axioms. Similarly, we check that $T' \supset T$, because each of the predicates $S_n$ extends $T_n$. The details are rather straightforward. The reader can consult the Appendices in [7] or [15], where a very similar construction is presented.

We have yet to check by induction that all theories $\Theta _n$ are consistent. So assume that this is true for a given $\Theta _n$ and let $M_n \models \Theta _n$. In order to make the proof work uniformly for the successor and the initial steps of induction, we set by convention $M_{-1} = T_{-1} = S_{-1} = \emptyset$.

We will prove consistency of $\Theta _{n+1}$ in the following way. Consider any finite subtheory $\Gamma \subset \Theta _n$. In the model $M_n$, we will find a binary relation S which satisfies $\Gamma$.

Since $\Gamma$ is finite, there are only finitely many formulae $\phi _1, \ldots , \phi _k$ which occur in the compositionality scheme.

Consider the equivalence classes $[\phi _i]$ of the formulae $\phi _i$ under the syntactic similarity relation $\sim$. Let $\unlhd$ be the transitive closure of the following relation on classes: $[\phi ] \unlhd [\psi ]$ if there exist $\phi ' \in [\phi ], \psi ' \in [\psi ]$ such that $\phi '$ is a direct subformula of $\psi '$. This is indeed an ordering: transitivity and reflexivity is clear, so it is enough to check weak antisymmetry. However, this is clear, since if $[\phi ] \unlhd [\psi ]$, then the total number of connectives and quantifiers in $\phi$ is no greater than in $\psi$.

We define the extension of S as follows. We define the set $S^0$ by the following conditions. A pair $(\phi ,\alpha )$ belongs to $S^0$ if one of the following conditions is satisfied:

• $[\phi ] \cap M_{n -1} \ne \emptyset$ and $(\phi ',\alpha ') \sim (\phi ,\alpha )$ for some $\phi ' \in M_{n-1}$ and $\alpha ' \in M_n$ such that $(\phi ',\alpha ') \in S_n$.

• There exists $\phi ' \in M_n$ such that $(\phi ',\emptyset ) \sim (\phi ,\alpha )$ and $\phi ' \in T_n$.

• $\phi$ is an atomic formula of the form $t=s$ for some terms $t=s$ and $t^{\alpha } = s^{\alpha }.$

In the above list, we do not explicitly include the case when $[\phi ]$ is minimal among $[\phi _i]$ with respect to the relation $\unlhd$ and $[\phi ] \cap M_{n-1} = \emptyset$, but we also implicitly treat this case as covered. Such formulae are simply not satisfied under any assignment. Hence, they effectively define the empty set under the satisfaction predicate.

Then we inductively construct a series of predicates $S^j$. We define $S^{j+1}$ as the union of $S^j$ with the set of $(\phi ,\alpha )$ such that $[\phi ] = [\phi _i]$ for some $i \le k$, $[\phi _i]$ is not minimal with respect to the relation $\unlhd$, and $\phi$ satisfies one of the following conditions:

• There exists $\psi \in M_n$ such that $\phi = \lnot \psi$ and $(\psi ,\alpha ) \notin S^j$.

• There exist $\psi , \eta \in M_n$ such that $\phi = \psi \vee \eta$ and $(\psi ,\alpha ) \in S^j$ or $(\eta ,\alpha ) \in S^j$.

• There exists $\psi ,v \in M_n$ such that $\phi = \exists v \psi$, and $\beta \sim _v \alpha$ such that $(\psi ,\beta ) \in S^j$.

Since we considered only finitely many classes $[\phi _i]$, the construction terminates at some point. Let S be the predicate obtained as the final one in this construction. We claim that $(M_n,T_n,S)$ satisfies the finite theory $\Gamma$. The elementary diagram of $(M_n,T_n)$ is obviously satisfied in the obtained model. Our construction and the fact that $S_n$ and $T_n$ were compositional and extensional immediately guarantee that the constructed predicate S agrees with $T_n$, preserves $S_n$ for formulae from $M_{n-1}$ and satisfies the instances of the compositional scheme from $\Gamma$. Finally, we check by induction on j that each $S^j$ satisfies the extensionality axiom. This concludes the proof of consistency of $\Gamma$, the proof of consistency of $\Theta _{n+1}$ and consequently, the proof of Theorem 28. $\square$