2.1 Truth theories

Let us introduce the main theories of our interest.

The last clause, called the regularity axiom states that the truth of a sentence does not depend on the exact terms which are used in it, but rather on the values thereof. In the proof, we will actually need a stronger regularity condition which we will discuss in Section 2.4.

One of the fundamental facts about $\text {CT}^-$ is that it does not have any arithmetical content extending $\text {PA}$ .

A number of additional conditions can be imposed on the truth predicate so that the resulting theory remains conservative. One of the most notable is that we can assume that every formula $\phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}$ , considered separately, satisfies induction.

Definition 3. By the internal induction axiom ( $\text {INT}$ ) we mean the following statement:

$$\begin{align*}\forall \phi \in \text{Form}^{\leq 1}_{\mathscr{L}_{\text{PA}}} \Big( T\phi(0) \wedge \forall x \big(T\phi(\underline{x}) \rightarrow T\phi(\underline{S(x)})\big) \rightarrow T \forall x \phi(\underline{x})\Big). \end{align*}$$

By compositional axioms this is equivalent to saying that all induction axioms are true. Perhaps somewhat surprisingly, this also yields a conservative extension of $\text {PA}$ . This result was originally announced in [Reference Kotlarski, Krajewski and Lachlan12]. A proof, by different, model theoretic methods announced in [Reference Enayat, Visser, Achourioti, Galinon, Fernández and Fujimoto6], can be found in the longer, unpublished, and privately circulated manuscript [Reference Enayat and Visser5]. Another argument, now purely proof-theoretic, has been presented in [Reference Leigh14] (where it is proved that one can extend $\text {CT}^-$ with an arbitrary statement of the form “all instances of the axioms scheme $\Gamma $ are true” while still keeping the theory in question conservative. It is easy to check that the compositional axioms allow us to derive our version of the internal induction from the statement that all the induction axioms are true).

2.2 Models of $\text {PA}$

This article will make use of some classical theory of models of $\text {PA}$ . Let us now review some basic facts of this area. The standard references are [Reference Kaye8] (introductory) and [Reference Kossak and Schmerl10] (more advanced), where the proofs of the theorems stated here, and much more, can be found. The first result which we will use repeatedly is the resplendence of recursively saturated models.

The notion of resplendence is very rich in consequences and yet, resplendent models of strong theories are rather easy to find in nature.

End-extensions of models of truth theories play a crucial role in our article. They are also a very classical thread in the theory of the models of $\text {PA}$ .

We will also use a more sophisticated variant of extensions.

Definition 8. Let $M \preceq _e N$ . We say that N is a conservative extension of M iff for any formula $\phi $ , possibly with parametres from N, there exists a formula $\psi $ , possibly with parametres from M, such that for any $a \in M$ ,

$$\begin{align*}N \models \phi(a) \text{ iff } M \models \psi(a). \end{align*}$$

The presence of parametres from N is a crucial requirement in the above definition. Without them, the conclusion would follow trivially by elementarity. One of the key facts from the model theory of $\text {PA}$ states that conservative-extensions exist (in fact, conservative extensions automatically have to be end-extensions).

The statement and the proof of the above result can be found in [Reference Kossak and Schmerl10, Theorem 2.2.8].Footnote ⁵ An important class of models are $\kappa $ -like models, of which we will need a special case.

The proof of the existence of $\omega _1$ -like models is the prototype for the argument presented in this work.

Proof. By repeatedly using Theorem 9, we can construct a sequence of countable models $M_{\alpha }, \alpha <\omega _1$ such that for any $\alpha <\beta $ ,

$$\begin{align*}M_{\alpha} \prec_e M_{\beta}. \end{align*}$$

Let $N = \bigcup _{\alpha < \omega _1} M_{\alpha }$ . Then $N \models \text {PA}$ as a union of an elementary chain, N has cardinality $\aleph _1$ , and for an arbitrary proper initial segment I, there exists an element $a \in N$ , such that $a \notin I$ . Let $\alpha < \omega _1 $ be any ordinal such that $a \in M_{\alpha }$ . Then, since N is an end-extension of $M_{\alpha }$ , $I \subset M_{\alpha }$ and hence it is countable. This shows that N is $\omega _1$ -like.

In our article, we will use a technical condition on cuts of models of $\text {PA}$ .

Cuts in models of arithmetic are a classical subject of intensive study in which a number of their structural properties were isolated. In our argument, we will isolate a new, very weak, regularity condition, whose definition we postpone until Section 4.

2.3 Truth predicates and satisfaction classes

In the literature, there are two competing treatments of the truth-like notions. The first one, more traditional, originates in the theory of models of $\text {PA}$ and speaks of satisfaction classes which are treated primarily as subsets in models of arithmetic. The other stems from discussions in philosophical logic and speaks of truth theories which are treated primarily in the axiomatic manner.

Intuitively, a satisfaction class in a model $M \models \text {PA}$ , should be a set S of pairs $(\phi ,\alpha )$ , where $\phi $ is an arithmetical formula in the sense of the model and $\alpha $ is a $\phi $ -assignment such that, collectively, the pairs in S satisfy Tarski’s compositional conditions. However, we are often interested in such classes where S is only required to work for some subset of formulae in the model. This, however, makes the notion of a satisfaction class somewhat subtle. In particular, the exact definition of a satisfaction class in not quite consistent between different authors. Below, we present the version from [Reference Wcisło20] which, we believe, captures various uses of the notion most smoothly.

Unfortunately, it turns out that in the absence of some form of induction, the connection between truth predicates and satisfaction classes is not as clear-cut as one could hope. A discussion of that phenomenon may be found in [Reference Wcisło20]. However, the distinction between truth and satisfaction classes trivialises if we assume that certain regularity properties holds and the truth value of a formula does not depend on the choice of specific terms, but rather on their values and it does not depend on the choice of the specific bound variables. We will present the exact assumptions on truth and satisfaction classes in question in the next section.

2.4 Syntactic templates and regularity

As we mentioned before, in the main proof we will need a technical condition on the regularity of the constructed truth and satisfaction classes. We will now spell out those, admittedly tedious, technicalities. Similar considerations played the same role in our previous works, such as [Reference Łełyk and Wcisło15] or (in a somewhat different formulation) in [Reference Wcisło20].

In essence, a syntactic template represents the pure syntactic tree of a formula in which all the terms which involve no bound variables were erased and replaced by single variables. Terms which appear under quantifiers are essential for a formula, so we do not modify these.

Example 15. Let $\phi := \neg \left ( S(x + S(0)) = y \times z \right ).$ Footnote ⁷

Then in $\hat {\phi }$ , we replace the term $S(0)$ with a free variable $v$ , and since the term $S(x+ v)$ has only free variables, we replace it with a free variable, say $v_0$ . Similarly, we replace $y \times z$ with a free variable, say $v_1$ . Thus, we obtain:

$$\begin{align*}\hat{\phi} := \neg (v_0 = v_1). \end{align*}$$

Example 16. Let $\phi : = \forall x \exists y \ x = y+y.$ Then, the syntactic template of $\phi $ is $\hat {\phi }$ :

$$\begin{align*}\forall w_0 \exists w_1 \ w_0 = w_1 + w_1, \end{align*}$$

where $w_0$ and $w_1$ are chosen so as to minimise the formula. Notice that we do not simplify any terms, since they all contain only bound variables.

Example 17. Let

$$\begin{align*}\phi = \exists x \forall y \big(x+(y \times 0) = S(0) + (x \times (z \times v))\big). \end{align*}$$

Then the $\hat {\phi }$ should satisfy the following conditions:

• • The constant $0$ and the term $S(0)$ should be replaced with distinct free variables, say $w_0, w_1$ .

• • The term $z \times v$ as a complex term with all variable free should be replaced with a single variable (which we will call $w_2$ ).

• • The term $x \times (z \times v)$ should be replaced with $x \times w_2$ . Note that this is not further replaced with a single variable, since x is not free. Similarly, in $w_1 + x \times (z \times v)$ , the addition should not be simplified any further, since one of terms occurring on the right hand side is not free.

Applying these rules, we obtain:

$$\begin{align*}\widehat{\phi} = \exists v_0 \forall v_1 \big(v_0+(v_1 \times w_0) = w_1 + (v_0 \times w_2)\big), \end{align*}$$

where the variables $v_i, w_i$ are chosen so that the resulting formula is minimal.

One of the reasons to introduce syntactic templates is because, as already mentioned, we would like to work with satisfaction classes and the connection between satisfaction and truth classes is not quite as neat, as one could expect and some regularity assumptions seem to be required to actually ensure that actually the two notions coincide. Below, $\phi [\alpha ]$ is the sentence obtained by substituting the numeral $\underline {\alpha (v)}$ for every instance of the variable v in the formula $\phi $ .

Definition 18. Let $M \models \text {PA}$ . Let $\phi , \psi \in \text {Form}_{\mathscr {L}_{\text {PA}}}(M)$ , let $\alpha \in \text {Asn}(\phi )$ , and let $\beta \in \text {Asn}(\psi )$ . We say that the pairs $(\phi ,\alpha )$ and $(\psi ,\beta )$ are syntactically similar iff $\phi $ is syntactically similar to $\psi $ and there exist sequences of closed terms $\bar {s},\bar {t} \in \text {ClTermSeq}_{\mathscr {L}_{\text {PA}}}(M)$ such that:

• • $\bar {{s}^{\circ }} = \bar {{t}^{\circ }}$ (the terms in both sequences have the same values).

• • $\widehat {\phi }(\bar {s})$ differs from $\phi [\alpha ]$ only by renaming bound variables.

• • $\widehat {\phi }(\bar {t}) (= \widehat {\psi }(\bar {t}))$ differs from $\psi [\beta ]$ only be renaming bound variables.

If the pairs $(\phi ,\alpha ), (\psi ,\beta )$ are syntactically similar, we denote this fact with

$$\begin{align*}(\phi,\alpha) \sim (\psi,\beta). \end{align*}$$

The above notion is actually much simpler than the definition suggests. In essence, we want to say this: Given a formula $\phi $ , we may look at its pure syntactic tree represented by $\bar {\phi }$ . In $\bar {\phi }$ , we will see some free variables. If $\alpha $ is a $\phi $ -assignment, it will actually decide the values of these variables in a unique way. Then the similarity of pairs $(\phi ,\alpha ) \sim (\psi ,\beta )$ means that $\phi $ and $\psi $ have the same pure syntactic tree and $\alpha , \beta $ yield the same assignment of the free variables of in the formula representing that tree.

Example 19. Let

$$ \begin{align*} \phi & : = x = SS(0) \\ \psi & := SSS(0) = y. \end{align*} $$

Let $\alpha $ be a $\phi $ -assignment such that

$$\begin{align*}\alpha(x) = 3. \end{align*}$$

Let $\beta $ be a $\psi $ -assignment such that

$$\begin{align*}\beta(y) = 2. \end{align*}$$

Then $(\psi , \alpha ) \sim (\psi ,\beta )$ . Indeed, we have $\hat {\phi }=\hat {\psi }$ equal to:

$$\begin{align*}v_0 = v_1 \end{align*}$$

and if $\bar {s} = \bar {t}= \langle SSS(0), SS(0) \rangle $ , then:

$$\begin{align*}\hat{\phi}(\bar{s}) = \phi[\alpha] = \psi[\beta] = \hat{\phi}(\bar{t}). \end{align*}$$

Example 20. Let

$$ \begin{align*} \phi & : = \exists x \forall y \ \big( x+(y \times S(S(0))) = z \times (S(0) + S(0)) \big) \\ \psi & : = \exists z \forall v \ \big( z + (v \times (u+w)) = S(S(S(S(0)))) \big). \end{align*} $$

Let $\alpha \in \text {Asn}(\phi )$ be an assignment such that

$$\begin{align*}\alpha(z) = 2. \end{align*}$$

Let $\beta \in \text {Asn}(\psi )$ be an assignment such that

$$\begin{align*}\beta(u) = 1, \beta(w) = 1. \end{align*}$$

Then $(\phi ,\alpha ) \sim (\psi ,\beta )$ . Indeed, this is witnessed by $\widehat {\phi } = \widehat {\psi }$ equal to:

$$\begin{align*}\exists v_0 \forall v_1 \ \big(v_0 + v_1 \times w_0 = w_1\big). \end{align*}$$

and sequences $\bar {s}, \bar {t}$ such that:

$$\begin{align*}\bar{s} = \langle S(S(0)), S(S(0)) + (S(0) + S(0)) \rangle \end{align*}$$

$$\begin{align*}\bar{t} = \langle S(0) + S(0), S(S(S(S(0)))) \rangle.\end{align*}$$

Then

$$ \begin{align*} \widehat{\phi}(\bar{s}) &= \exists v_0 \forall v_1 \ \big(v_0 + (v_1\times S(S(0))) = S(S(0)) + (S(0) + S(0))\big) \\ \phi[\alpha ] &= \exists x \forall y \ \big(x + (y\times S(S(0)) = S(S(0)) + (S(0) + S(0))\big) \end{align*} $$

which differ only by renaming bound variables. Moreover,

$$ \begin{align*} \widehat{\psi}(\bar{t}) = & \exists v_0 \forall v_1 \ \big(v_0 + v_1 \times (S(0) + S(0)) = S(S(S(S(0))))\big) \\ \psi[\beta] = & \exists z \forall v \ \big( z + (v \times (S(0)+S(0))) = S(S(S(S(0)))) \big), \end{align*} $$

which differ only by renaming bound variables.

As we already mentioned, we want to restrict our attention to classes for which good regularity properties hold.

Definition 21. Let $(M,S)$ be a satisfaction class. We say that S is syntactically regular if for any $\phi ,\psi \in \text {Form}_{\mathscr {L}_{\text {PA}}}(M)$ and $\alpha \in \text {Asn}(\phi ), \beta \in \text {Asn}(\psi )$ if $(\phi , \alpha ) \sim (\psi ,\beta )$ , then

$$\begin{align*}(\phi,\alpha) \in S \text{ iff } (\psi,\beta) \in S. \end{align*}$$

Let $(M,T) \models \text {CT}^-$ . We say that T is syntactically regular if for any $\phi , \psi \in \text {Sent}_{\mathscr {L}_{\text {PA}}}(M)$ if $(\phi ,\emptyset ) \sim (\psi , \emptyset )$ (note that $\emptyset $ is the empty function and can be treated as a trivial assignment),

$$\begin{align*}\phi \in T \text{ iff } \psi \in T. \end{align*}$$

Under these regularity assumptions, we can actually make a straightforward connection between truth predicates and satisfaction classes.

Proposition 22. Let $M \models \text {PA}$ and let S be a full syntactically regular satisfaction class on M. Let

$$\begin{align*}T = \lbrace \phi \in \text{Sent}_{\mathscr{L}_{\text{PA}}}(M) \ \mid \ (\phi,\emptyset) \in S \rbrace. \end{align*}$$

Then $(M,T) \models \text {CT}^-$ and, moreover, T is syntactically regular.

Conversely, suppose that $(M,T) \models \text {CT}^-$ and that T is syntactically regular. Let

$$\begin{align*}S = \lbrace (\phi,\alpha) \in M^2 \ \mid \ \phi \in \text{Form}_{\mathscr{L}_{\text{PA}}}(M), \alpha \in \text{Asn}(\phi) \text{, and } \phi[\alpha] \in T \rbrace. \end{align*}$$

Then S is a full syntactically regular satisfaction class.Footnote ⁸

If S and T are interdefinable in the way postulated in the above proposition, we say that S is a satisfaction class corresponding to T and that T is a truth predicate corresponding to S. A slightly different statement, in which we used weaker regularity assumptions appeared as [Reference Wcisło20, Proposition 15]. The definition of syntactic regularity formulated in this work assumes that a syntactically regular satisfaction class is closed under renaming bound variables (equivalently, under $\alpha $ -conversion). This is not needed to obtain the above correspondence, but will be used in the proof of the main theorem. However, some regularity assumptions apparently are needed in order to obtain the above simple correspondence as discussed in the cited article.

Crucially for our article, we have the following.

This fact appeared as [Reference Łełyk and Wcisło15, Theorem 23]. The proof of this fact was strictly speaking omitted, but the proof is a (completely straightforward) modification of an argument which appeared there with a precise comment on what modification is needed.

4.1 Stretching Lemma

In this part, we will discuss the first of the main steps in the end-extension theorem. The key ideas of this subsection appeared already in [Reference Łełyk and Wcisło15]. However, since we will need to extract some additional information from the construction, we will present here the full proof.

As we already mentioned, we will need to impose certain regularity conditions on the cuts arising in our construction. The exact choice of the condition is rather subtle.

Definition 32. Let $I \subset M \models \text {PA}$ be a nonstandard cut. We say that I is locally semiregular in M if for any nonstandard $a \in I$ and any function $f: [0,a] \to M$ coded in M, there exist a nonstandard $a' \leq a $ and $b \in I$ such that the following condition holds:

$$\begin{align*}f[[0,a']] \cap I \subseteq [0,b]. \end{align*}$$

A word of comment is certainly in place. One of the conditions on cuts, classically investigated in the theory of models of $\text {PA}$ , is semiregularity. We say that a cut I is semiregular in M if for any function f coded in M with the domain $[0,a]$ for some $a \in I$ , the set of thhe values $f(i)$ such that $f(i) \in I$ is bounded in I. Local semiregularity wekens this condition, demanding instead that the function f can be restricted to a nonstandard initial segment so that the bounding condition holds.

Admittedly, this is a somewhat technical requirement. However, the choice of this exact condition will be crucial in Section 5, since we were not able to show the main end-extension result from that part which would guarantee any stronger regularity requirements on cuts. On the other hand, any weaker conditions known in the literature do not seem to suffice to perform the copying construction of Theorem 34. However, let us remark that we actually do not need the notion of local semiregularity in the context of the internal induction. In Lemma 33, we can obtain a stronger conclusion that the elementary extension we obtain is conservative and we can assume in Theorem 34 that the cut I is a model of $\text {PA}$ such that M is its conservative extension. However, this would not really simplify the argument in any substantial way, so we decided to prove it in the greater generality. Nevertheless, the reader who is confused by the notion of local semiregularity may completely ignore it in this section and instead think of conservative extensions. Let also us add that we were also not able to find any condition previously known in the literature which would be equivalent to our new notion.

Proof. Fix a model $(M,S)$ as above. For any $\phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}(M)$ , let $S_{\phi }$ be the set:

$$\begin{align*}S_{\phi} = \lbrace \alpha \ \mid \ (\phi,\alpha) \in S \rbrace. \end{align*}$$

Consider the model $(M,S_{\phi })_{\phi \in M}$ . By the INT, this model satisfies full induction. Since M is countable, the signature of the expanded model has only countably many symbols. Thus by MacDowell–Specker Theorem, there exists a proper conservative elementary end-extension

$$\begin{align*}(M,S_{\phi})_{\phi \in M} \prec_e (M',S^{\prime}_{\phi})_{\phi \in M}. \end{align*}$$

Now, let $S' \subset (M')^2$ be defined by the following condition: $(\phi ,\alpha ) \in S'$ iff there exists a pair $(\psi ,\beta )$ such that:

• • $(\psi ,\beta ) \sim (\phi ,\alpha )$ .

• • $\psi \in M$ .

• • $S^{\prime }_{\psi }(\beta )$ holds.

In other words, we take a union of sets $S^{\prime }_{\psi }$ and close it under syntactic similarity. We can then check that by elementarity of the extension $(M',S^{\prime }_{\phi })_{\phi \in M}$ and by the regularity of S, the constructed set $S'$ is a regular satisfaction class satisfying internal induction. The syntactic regularity of $S'$ follows directly by construction.

We have to check that M is a locally semiregular cut of $M'$ . (In fact, we will simply verify that conservativity implies semiregularity). Let

$$\begin{align*}f: [0,a] \to M' \end{align*}$$

be a function coded in $M'$ . We want to check that for any $b \in M$ , there exists $c \in M$ such that the values of $f \upharpoonright M$ are bounded by c. Since f is coded in $M'$ , by conservativity of the extension $(M,S_{\phi })_{\phi \in M} \prec (M',S^{\prime }_{\phi })_{\phi \in M}$ , the set $f \cap M$ is definable in the former structure. However, since this model satisfies full induction in the expanded language, the values of $(f \upharpoonright [0,b]) \cap M$ have to be bounded in M.

We will discuss in Section 5 how internal induction can be eliminated from the above argument.

4.2 Copying Lemma

In the previous section, we have shown how to extend a satisfaction class “upwards” so that it is still defined for all the formulae in the original model. In this section, we will discuss the essentially novel part of our argument: we will show how, under additional model-theoretic assumptions, we can extend a satisfaction class defined on a cut of formulae to the whole model. Our argument crucially uses ideas introduced by Fedor Pakhomov in his construction of a satisfaction class presented in his unpublished note [Reference Pakhomov16].Footnote ¹⁰

Proof. Let, $M,I,S$ be as in the assumptions of the theorem. Let D be the set of all formulae in M whose syntactic depth is in I (i.e., D is the domain of S). We will construct a function $f: \text {Form}_{\mathscr {L}_{\text {PA}}}(M) \to D$ satisfying the following conditions for any $\phi , \psi \in \text {Form}_{\mathscr {L}_{\text {PA}}}$ and any $v \in \text {Var}(M)$ :

• • $f \upharpoonright D = \text {id}$ .

• • For any $\phi , \psi $ if $\phi \sim \psi $ , then $f(\phi ) \sim f(\psi )$ .

• • $f(\neg \phi ) = \neg f(\phi ).$

• • $f(\phi \wedge \psi ) = f(\phi ) \wedge f(\psi ).$

• • $f(\phi \vee \psi ) = f(\phi ) \vee f(\psi ).$

• • $f(\exists v \phi ) = \exists v f(\phi )$ .

• • $f(\forall v \phi ) = \forall v f(\phi )$ .

We will define inductively a sequence of partial functions satisfying the above conditions for a given formula and its subformulae laying at certain depth of the syntactic tree. We will make sure that the functions will be compatible between different formulae and then we will glue them together.

For a formula $\phi \in \text {Form}_{\mathscr {L}_{\text {PA}}}(M)$ and $a \in M$ , let $U(\phi ,a)$ be the set of formulae $\psi $ such that $\psi \sim \psi '$ for some $\psi '$ located at most at the depth a in the syntactic tree of $\phi $ .

Fix an enumeration $(\phi _i)_{i\in \omega }$ of $\text {Form}_{\mathscr {L}_{\text {PA}}}(M)$ . We will define a sequence of nonstandard elements of M, $a_0> a_1>a_2 \ldots $ and functions $f_0, f_1, \ldots $ such that each function $f_i$ satisfies the following conditions:

• • $f_i: U(\phi _i,a_i) \to \text {Form}_{\mathscr {L}_{\text {PA}}}(M)$ .

• • If $i \leq k$ , then $f_i \upharpoonright U(\phi _i,a_k) \cap U(\phi _k,a_k) = f_k \upharpoonright U(\phi _i,a_k) \cap U(\phi _k,a_k)$ .

• • $f_i$ commutes with quantifiers and connectives in the sense postulated for f.

• • $f_i \upharpoonright D = \text {id}$ .

• • If $\phi \sim \psi $ , then $f_i(\phi ) \sim f_i(\psi )$ .

Notice that we do not require that $f_0 \subset f_1 \subset f_2$ or that the functions $f_i$ agree on the whole common domain. Finally, we will set:

$$\begin{align*}f(\phi) = f_i(\phi), \end{align*}$$

where $\phi _i = \widehat {\phi }$ (so i is the index of the template of $\phi $ in our enumeration).

Let us check that f defined in this way indeed satisfies our conditions. It is enough to check that f preserves syntactic operations, since the other conditions follow directly by assumption on the functions $f_i$ . We will check the condition for conjunction, the others being similar or completely analogous. So let us fix formulae $\phi , \psi $ . Suppose that $\phi \sim \phi _k, \psi \sim \phi _l, \phi \wedge \psi \sim \phi _m$ . Let $n = \max (k,l,m).$ Let $U_1 = U(\phi _m,a_n) \cap U(\phi _k,a_n), U_2 = U(\phi _m,a_n) \cap U(\phi _l, a_n)$ . Then

$$\begin{align*}f_k \upharpoonright U_1 = f_m \upharpoonright U_1 \end{align*}$$

and

$$\begin{align*}f_l \upharpoonright U_2 = f_m \upharpoonright U_2. \end{align*}$$

In particular $f(\phi ) = f_m(\phi )$ , $f(\psi ) = f_m(\psi )$ and the claim follows, since $f_m$ preserves the syntactic structure. So it is enough to construct the sequences $(f_i), (a_i)$ as above. We will also construct an auxiliary sequence $(c_i)$ of the elements of M.

Let $a_0$ be an arbitrary element of the cut I. The construction of $f_0$ is very similar to the construction of functions $f_{i+1}$ in the successor steps, so we will go directly to that case indicating the (small and obvious) differences whenever they appear in the proof.

Let $a:=a_{n+1}$ be an arbitrary nonstandard element of I such that:

$$\begin{align*}a_{n+1}2^{a_{n+1}} < \frac{a_n}{2}. \end{align*}$$

Let $\phi := \phi _{n+1}$ . Now, consider the set $U(\phi ,a)$ . Since $a \in I$ , by local semiregularity of this cut we can conclude that (possibly after replacing a with a smaller element $a'$ which we will for simplicity still denote a) there exists $c :=c_{n+1} \in I$ such that all subformulae of $\phi $ which occur at the depth a in its syntactic tree of have themselves syntactic depth either $< c$ or not in I.Footnote ¹¹

Consider the following relation $\unlhd $ :

$$\begin{align*}\xi \unlhd \eta \end{align*}$$

iff there exists a coded sequence of formulae:

$$\begin{align*}\xi_0, \xi_1, \ldots, \xi_p \end{align*}$$

such that $\xi _0 = \xi , \xi _p = \eta $ and for any i, $\xi _i$ is a direct subformula of $\xi _{i+1}$ and $\xi _i \in U(\phi ,a)$ . In other words, $\xi \unlhd \eta $ means that $\xi $ is a subformula of $\eta $ as can be witnessed using only formulae from $U(\phi ,a)$ .

We want to define the function $f:=f_{n+1}: U(\phi ,a) \to \text {Form}_{\mathscr {L}_{\text {PA}}}(M)$ so that it commutes with the syntactic operations. We actually will need a small technical definition. Let us say that a formula $\zeta \in U(\phi ,a)$ is weakly minimal with respect to $\unlhd $ if it has a direct subformula which is not in $U(\phi ,a)$ . Such formulae can be different from $\unlhd $ -minimal formulae in the case their main operator is binary, one of the formulae is in $U(\phi ,a)$ , and the other is not.

It is enough to define the function f on the set weakly minimal formulae from $U(\phi ,a)$ and then extend the definition to the whole $U(\phi ,a)$ by induction (applied internally in the model). In this manner, we will obtain a function $f_{n+1}$ commuting with all connectives and quantifiers, whenever a formula and all its direct subformulae are in the domain.

Now pick any weakly $\unlhd $ -minimal formula $\psi $ . We consider two cases.

4.3 The identity condition

We prove by induction on n that $f_n \upharpoonright D = \text {id}$ . The initial case will be very similar to the induction step, so we only present the latter.

Fix any formula $\psi \in D \cap U(\phi _{n+1},a_{n+1})$ and first assume that $\psi $ is $\unlhd $ -minimal. If $\psi \in \text {dom}(f_k)$ for some $k \leq n$ , then $f_{n+1}(\psi ) = f_l(\psi ),$ where l is the maximal index for which $\psi \in \text {dom}(f_l)$ . Then by induction hypothesis $f_l(\psi ) = \psi $ which proves the claim.

If $\psi \notin \text {dom}(f_k)$ for $k \leq n$ , then $f_{n+1}(\psi )$ is the formula $\psi $ with any subformula at the syntactic level $c_{n+1}$ replaced with a sentence $0=0$ . However, by construction any formula with the syntactic depth from I which belongs to $U(\phi _{n+1},a_{n+1})$ has syntactic depth strictly less than $c_{n+1}$ , so in fact the described substitution is trivial and $f_{n+1}(\psi ) = \psi $ . Then it is enough to observe that on nonminimal formulae $f_{n+1}$ is defined by induction on $\unlhd $ which clearly preserves the identity condition, since the set D is closed under subformulae.

4.4 The congruence condition

We check by induction on n that for any $\psi ,\eta \in \text {dom}(f_n)$ if $\psi \sim \eta $ , then $f_n(\psi ) \sim f_n(\eta )$ .

Suppose that $\psi , \eta $ are weakly minimal in $U(\phi _n,a_n)$ . First suppose that $\psi \in \text {dom}(f_k)$ for some $k < n$ . Notice that the set $U(\phi _k,a_k)$ is closed under $\sim $ , so in such case both $\phi $ and $\psi $ are in $\text {dom}(f_k)$ and by induction hypothesis, $f_n(\psi ) \sim f_n(\eta )$ .

If, on the other hand, $\psi , \eta \notin \text {dom}(f_k)$ for any $k < n$ , then $f_n(\psi )$ and $f_n(\eta )$ are defined by substituting $0=0$ for any subformula at the syntactic depth c in these formulae. Notice that if $\psi \sim \eta $ , then their syntactic trees were equal up to term substitutions and renaming bound variables. In such a case, those trees with $0=0$ substituted for all formulae at the depth c will still be equal.

Now, we can check by induction that $f_n$ is a congruence with respect to $\sim $ on the whole $U(\phi _n,a_n)$ by induction on $\unlhd $ , applied internally.

4.5 The coherence condition

We want to check that if $i \leq k$ , then

$$\begin{align*}f_i \upharpoonright U(\phi_i,a_k) \cap U(\phi_k,a_k) = f_k \upharpoonright U(\phi_i,a_k) \cap U(\phi_k,a_k). \end{align*}$$

We can inductively assume that the desired equality holds for any $j \leq i$ :

$$\begin{align*}f_i \upharpoonright U(\phi_i,a_j) \cap U(\phi_k,a_j) = f_j \upharpoonright U(\phi_i,a_j) \cap U(\phi_j,a_j). \end{align*}$$

Fix any $\phi \in U(\phi _i,a_k) \cap U(\phi _k,a_k).$ Consider any weakly minimal $\psi \in U(\phi _k,a_k)$ which is $\unlhd $ -below $\phi $ in $U(\phi _k,a_k)$ .

Now, observe that any $\unlhd $ -chain in $U(\phi _k,a_k)$ has at most

$$\begin{align*}a_k2^{a_k} \end{align*}$$

elements, since for any $\sim $ -representative $\eta $ of $\phi _k$ , there are at most that many subformulae of $\eta $ in $U(\phi _k,a_k)$ (since they form a tree with at most binary branches of height at most $a_k$ ). Since $a_k2^{a_k} + a_k < a_i$ , all such weakly minimal formulae $\psi $ are in $U(\phi _i,a_i)$ and, in fact in $U(\phi _i,a_j)$ for any $j<k$ . Now, by construction, for any such formula $\psi $ we have:

$$\begin{align*}f_k(\psi) = f_m(\psi), \end{align*}$$

where $m \geq i$ is the greatest index smaller than k such that $\psi \in U(\phi _m,a_m)$ . In particular, by the above remark

$$\begin{align*}\psi \in U(\phi_m,a_m) \cap U(\phi_i,a_m). \end{align*}$$

Hence, by induction hypothesis,

$$\begin{align*}f_m(\psi) = f_i(\psi), \end{align*}$$

and, consequently,

$$\begin{align*}f_k(\psi) = f_m(\psi) = f_i(\psi). \end{align*}$$

Since $f_k(\phi )$ is uniquely determined by the values on weakly minimal formulae which are $\unlhd $ -smaller than $\phi $ in $U(\phi _k,a_k)$ and as we have just argued, all such chains are also contained in $U(\phi _i,a_i)$ , we conclude that

$$\begin{align*}f_k(\phi) = f_i(\phi). \end{align*}$$

This concludes the proof of the coherence clause and therefore of Theorem 34.

4.6 The proof of the main theorem

In this section, we will put together the findings of the two previous parts in order to prove Theorem 28, showing that any countable model of $\text {CT}^- + \text {INT}$ has an end-extension.

Proof of Theorem 28.

As in the formulation of the theorem, let $(M,T) \models \text {CT}^- + \text {INT}$ be a countable model. Let S be a full regular satisfaction class corresponding to T. By Lemma 33, there exists a proper end-extension

$$\begin{align*}(M,S) \subset_e (M',S') \end{align*}$$

such that $S'$ is a regular partial satisfaction class whose domain consists of formulae with depth in M with M being a locally semiregular cut in $M'$ . Morevoer, the obtained satisfaction class satisfies INT.

Now by Theorem 34, there exists $S" \supset S'$ such that $(M',S")$ is a full syntactically regular satisfaction class with the internal induction. Let $T'$ be a truth class corresponding to $S"$ . Then $(M',T') \models \text {CT}^- + \text {INT}$ is a desired end-extension.

Step 1

In order to construct $T^1_{i}$ , consider the sentence $\phi _i$ . If $T_{i} + \phi _i$ is consistent, we let $T^1_{i}$ be that theory. If not, let $T^1_i = T_i + \neg \phi _i$ .

Step 2

In order to construct $T^2_i$ , consider again the sentence $\phi _i$ . If it was not added to the theory constructed in the previous step, let $T^2_i$ be equal to $T^1_i$ . If it was, and it has the form $\exists v \psi (v)$ , we add the sentence $\psi (c_i)$ to our theory. Note that $c_i$ does not appear in $T_i$ by our bookkeeping assumption.

Step 3

In the third step, we check whether the set of sentences of the form

$$\begin{align*}c_i> a, a \in M \end{align*}$$

is consistent with $T^2_i$ . If yes, then we let $T^3_i$ be obtained from $T^2_i$ by adding this set. Otherwise, there exists some $b \in M$ such that

$$\begin{align*}T^2_i \vdash c_i <b. \end{align*}$$

We claim that there exists $d \in M$ such that $T^2_i$ is consistent with the sentence $c_i = d$ .

Suppose otherwise. By assumption $T^2_i$ is $\text {ElDiag}(M)$ together with finitely many types of the form $c_j>a, a \in M$ together with finitely many additional formulae. By considering the minimum of the elements $c_j$ and coding the whole tuple as a single element, we can assume without loss of generality that $T^2_i$ extends $\text {ElDiag}(M)$ by a single formula $\psi (t,c_i)$ and a single theory $t>a: a \in M$ . (The constant $c_i$ could be of course also eliminated, but we keep it for the clarity of the rest of the argument.) By assumption $T^2_i \vdash c_i < b$ for some $b \in M.$

Now, suppose that a theory $t> a: a \in M + \text {ElDiag}(M) + \psi (t,c_i)$ proves for any $d \in M$ that $c_i \neq d$ . Notice, however, that this implies that for any $d < b \in M$ , there exists $a \in M$ such that:

$$\begin{align*}M \models t>a \rightarrow \neg \psi(t,d). \end{align*}$$

Then, by Compression Lemma, there would exists a single element $r \in M$ for which:

$$\begin{align*}M \models \forall x> r \forall y < b \ \neg \psi (x,y). \end{align*}$$

However this contradicts our assumption on $\psi $ . Therefore, there exists in M an element $d <b$ such that for any a, we can find $t \in M$ with:

$$\begin{align*}M \models \psi(t,d) \wedge t>a. \end{align*}$$

Let us pick any such d and set $T^3_i : = T^2_i + c_i = d.$

Step 4

Finally, we want to ensure local semiregularity. Suppose that $\phi _i$ is a sentence expressing that some $c_k, k \leq i$ codes a function with the domain $[0,a]$ for some $a \in M$ . Specifically,

$$\begin{align*}\phi_i = \forall x \leq a \exists! y \Big(\langle x,y \rangle \in c_k \Big). \end{align*}$$

In the rest of the argument, let us denote $c_k$ with f for cleaner presentation. We will also replace $\langle x,y \rangle \in f$ with the more usual notation $f(x) = y$ and treat $f(x)$ as if it were an independent term.

Claim

There exists $a' < a$ and $b \in M$ such that the following theory is consistent (which clearly ensures local semiregularity):

$$\begin{align*}T^3_i + \forall x <a' \Big( \big(f(x) < b \vee f(x)> d_i\big) \Big) + d_i >p: p \in M. \end{align*}$$

We begin with a bit of notation. Recall that $T^3_i$ extends $\text {ElDiag}(M)$ with a sentence $\eta (c_0,\ldots ,c_i,d_0, \ldots , d_{i-1} )$ and finitely many sets of sentences of the form $c_j, d_j>q, q\in M$ . By considering the minimum of all the parameters and via coding of tuples, we may assume that $T^3_i$ extends $\text {ElDiag}(M)$ simply with a theory of the form:

$$\begin{align*}\eta(f,t) + t> q: q \in M, \end{align*}$$

where $\eta $ is a single formula. Notice that by assumption $\eta (f,t)$ implies that f codes a function with the domain $[0,a]$ in the sense explained above.

Now, let us say that an interval $[0,r]$ eventually visits an interval $[b,p]$ if the following holds:

$$\begin{align*}\exists N \forall q> N \forall t,f \exists x <r \Big( \eta(t,f) \wedge t>q \rightarrow f(x) \in [b,p] \Big). \end{align*}$$

We denote it with $\xi (r,b,p)$ .

Subclaim

There exists a nonstandard $a_0 \in M$ such that for some B, $[0,a_0]$ does not eventually visit any interval $[b,p]$ with $b> B$ .

To prove the Subclaim, notice first that for any $n \in \omega $ , there exists B for which $\xi (n,b,p)$ does not hold for any $b>B$ . Indeed, if no such B exists, then by successively taking intervals which are located higher up in the model, there exists a family of intervals

$$\begin{align*}[b_0,p_0] < [b_1,p_1] < \ldots < [b_n,p_n] < [b_{n+1},p_{n+1}] \end{align*}$$

such that $[0,n]$ eventually visits all these intervals. However, this is impossible by the pigeonhole principle, since the image of $[0,n]$ under f cannot intersect $n+2$ disjoint sets.

Now, we have just shown that the following set of formulae is a type over M (with the free variable v):

$$\begin{align*}\exists B \forall b,p>B \ \neg \xi(v,b,p) \wedge v>\underline{n}: n \in \omega. \end{align*}$$

This type actually uses only finitely many symbols from the signature, so by local recursive saturation, it is realised in M. By fixing $a'$ as an element realising this type, we prove the Subclaim.

We are now ready to prove the claim. Let $a'$ be any element satisfying the subclaim with a bound B. Fix any $p, q \in M$ . We claim that there exist $t,f,d_i \in M$ such that:

$$\begin{align*}M \models \eta(t,f) \wedge d_i>p \wedge t> q \wedge \forall x <a' \Big( \big(f(x) < B +1 \vee f(x)> d_i\big) \Big) \end{align*}$$

Indeed, by assumption there is no interval I above b such that $[0,a']$ eventually visits I. In particular, there exist arbitrarily large t such that $\eta (t,f)$ holds and

$$\begin{align*}\forall x <a' \ f(x) \notin [B+1,p+1]. \end{align*}$$

By fixing any such f and $t>q$ , and setting $d_i = p+1$ , we find our desired interpretation of these three constants in M such that our fixed finite portion of the theory is satisfied. Let $T_i^4 (=: T_i)$ , be the theory:

$$\begin{align*}T^3_i + \forall x <a' \Big( \big(f(x) < b' \vee f(x)>d_i\big) \Big) + d_i >p: p \in M, \end{align*}$$

with $a'$ chosen as in the proof and $b' := B+1$ . (If $\phi _i$ was not of the specific form considered, of course we set $T^4_i = T^3_i$ .)

The rest of the argument is completely routine. Let $T_{\omega } = \bigcup _{i \in \omega } T_i$ . Consider the Henkin model N given by $T_{\omega }$ . In the step for the theories $T^3_i$ , we have ensured that every element of N is either greater than all the elements of M or one of these elements. In the fourth step, we have ensured that any function on a domain bounded in M which is definable in N can be restricted to one which has all values either smaller than a fixed element of M or greater than a fixed element of N which ensures local semiregularity.

In the above proof, we have used the specific facts that our models admit some degree of saturation which also involved some coding of functions. Admittedly, all these assumptions seem somewhat tangential to the main idea of the stated result. In fact, one could imagine another property which makes sense in a more general context. Let M be any model over the signature with a binary symbol $<$ interpreted as a linear order. Let I be an initial segment of M. We say that I has a gap property in M if for any formula $\phi (x,y)$ and any $a \in I$ , if

$$\begin{align*}M \models \forall x < a \exists y \ \phi(x,y), \end{align*}$$

then there exist $b \in I, c \in M$ such that

$$\begin{align*}M \models \forall x < a \exists y \ \phi(x,y) \wedge (y<b \vee y>c). \end{align*}$$

It is an interesting question whether any countable model in a signature with a linear order satisfying full collection has an elementary end-extension with a gap property. However, we were unable to obtain our result in this generality.