2.1 Coding conventions

We assume a standard coding of syntax in $\mathrm {PA}$ (defined as in [Reference Hájek and Pudlák9]): primitive symbols of the language are assigned numbers in a recursive way, and then terms, formulae, sentences, etc. are treated as well-formed sequences of such numbers. The notion of sequence is based on the definable Ackermanian membership predicate $\in $ . $\mathrm {Term}(x)$ , $\mathrm {ClTerm}(x)$ , $\mathrm {Var}(x)$ , $\mathrm {Form}(x)$ , and $\mathrm {Sent}(x)$ denote the arithmetical formulae expressing that x is an arithmetical term, a closed term, a variable, an arithmetical formula, and an arithmetical sentence, respectively. $x \in \mathrm {Subf}(y)$ expresses that x is a subformula of y. We define $x\in \mathrm {Term}(y)$ and $x\in \mathrm {ClTerm}(y)$ analogously (y is required to be either a formula or a term). $x\in \mathrm {FV}(y)$ expresses that x is a variable which has a free occurrence in (a formula) y.

The choice of coding apparatus is irrelevant as long as the coding is $\mathrm {PA}$ provably monotone, i.e., the following is provable in $\mathrm {PA}$ :

$$\begin{align*}\forall \phi,\psi \ \ \big( \phi\in \mathrm{Subf}(\psi) \rightarrow \phi \leq \psi\big).\end{align*}$$

We require a similar condition for (the given formalisation) of $x\in \mathrm {Term}(y)$ . Various codings which violate this condition are studied in [Reference Grabmayr8, Reference Halbach and Visser11].

Throughout the paper we distinguish between variables of the metalanguage, for which we reserve the symbols $x,y,z, x_0, x_1,\ldots , y_0, y_1,\ldots ,$ and variables of the arithmetized language, which are denoted $v, v_0, v_1,\ldots $ . We assume a fixed correspondence between the first and the second ones. $\bar {x}$ , $\bar {v}$ , $\ldots $ denote sequences of variables. For a formula $\phi $ , $\ulcorner \phi \urcorner $ denotes its Gödel code.

2.2 Some model theory of $\mathrm {PA}$

All the definitions and conventions regarding models of $\mathrm {PA}$ are as in [Reference Kaye12]. By default $\mathcal {M}$ , $\mathcal {N}$ , $\mathcal {K}$ (possibly with indices) range over nonstandard models of $\mathrm {PA}$ and M, N, K denote their respective universes. If $\mathcal {M}$ is any model (possibly for $\mathcal {L}_S$ ) and $\phi (\bar {x})$ a formula (possibly with parameters from $\mathcal {M}$ ; $\bar {x}$ denotes a sequence of variables), then $\phi ^{\mathcal {M}}$ denotes the set definable by $\phi $ in $\mathcal {M}$ , i.e., $\{\bar {a}\in M \ \ | \ \ \mathcal {M}\models \phi (\bar {a}) \}$ . If $\mathcal {M}\models \mathrm {PA}$ and $X\subseteq M^n$ , then $X{\upharpoonright _{<b}}$ denotes the restriction of X to all elements smaller than b. In the case when $\mathcal {N}\subseteq \mathcal {M}$ , $X{\upharpoonright _{\mathcal {N}}}$ denotes $\bigcup _{b\in N} X{\upharpoonright _{<b}}$ (the restriction of a relation to the submodel).

Let $I\subseteq M$ . We write $d> I$ if d is greater than all the elements of I. I is called an initial segment of $\mathcal {M}$ if I is closed downwards with respect to $\leq $ . We say that I is a cut if I is an initial segment which is closed under successor, i.e.,

$$\begin{align*}\forall x\ \ x\in I \rightarrow x+1 \in I.\end{align*}$$

If I is a cut of $\mathcal {M}$ , then we call $\mathcal {M}$ an end-extension of I and write $I\subseteq _e \mathcal {M}$ (note that I need not be a submodel of $\mathcal {M}$ ). Any element $c\in M$ such that $c>\omega $ is called nonstandard.

The following is one of the most basic consequences of induction in models of $\mathrm {PA}^*$ :

In particular, if $\emptyset \subsetneq I\subsetneq _e M$ is a cut and $\phi (x)$ is any formula such that

$$\begin{align*}\forall a\in I\ \ \mathcal{M}\models \phi(a),\end{align*}$$

then there exists a $d>I$ such that $\mathcal {M}\models \phi (d)$ .

One last notion which is very tightly linked to the topic of satisfaction classes is recursive saturation:

Definition 2.2. Fix $\mathcal {M}$ and $\bar {a}\in M$ . Let $p(x)$ be a set of formulae with at most one variable x and parameters $\bar {a}$ . We say that $p(x)$ is recursive (or computable) if so is the set

$$ \begin{align*}\{\ulcorner \phi(x,\bar{y}) \urcorner \ \ | \ \ \phi(x,\bar{a})\in p(x) \}.\end{align*} $$

We say that $p(x)$ is a type over $\mathcal {M}$ if every finite subset of $p(x)$ is satisfied in $\mathcal {M}$ . We say that $\mathcal {M}$ is realized if there is a $b\in M$ such that for every $\phi (x)\in p(x)$ , $\mathcal {M}\models \phi (b)$ . We say that $\mathcal {M}$ is recursively saturated (or computably saturated) if every recursive type over $\mathcal {M}$ is realized in $\mathcal {M}$ . $\triangle $

2.3 Some model theory in $\mathrm {PA}$

Models of theories extending Robinson’s arithmetic are infinite objects; thus inside arithmetic they become essentially second-order objects. In what follows a set means a second-order object and we distinguish it from a coded set (a first-order object). The notion of a $\Delta _n$ set is explained below. $x\in X$ should be understood as a membership relation between a first- and a second-order object, whereas $x\in y$ denotes the Ackermanian membership (mentioned earlier) between first-order objects.

We recall the notion of a $\Delta _n$ -set (see [Reference Hájek and Pudlák9]): $\mathrm {Sat}_{\Sigma _n}(x,y) (\mathrm {Sat}_{\Pi _n}(x,y)$ ) denotes the arithmetically definable partial satisfaction predicate for $\Sigma _n$ ( $\Pi _n$ respectively) formulae (as in [Reference Kaye12] or [Reference Hájek and Pudlák9]) and $\mathrm {Tr}_{\Sigma _n}(x) (\mathrm {Tr}_{\Pi _n}(x))$ abbreviates $\mathrm {Sat}_{\Sigma _n}(x,\varepsilon )$ ( $\mathrm {Sat}_{\Pi _n}(x,\varepsilon )$ ). We stress that the construction of $\mathrm {Sat}_{\Sigma _n} (\mathrm {Sat}_{\Pi _n})$ is elementary in n, so it gives rise to an $\mathrm {EA}$ -provably total $\Delta _1$ map sending n to (the formula) $\mathrm {Sat}_{\Sigma _n}$ .

In $\mathrm {PA}$ , a $\Sigma _n$ set ( $\Pi _n$ set) is any $\Sigma _n (\Pi _n)$ formula $\phi (v)$ with precisely one free variable. We define a $\Delta _n$ set to be a pair of formulae $(\phi ,\psi )$ such that $\phi $ is $\Sigma _n$ , $\psi $ is $\Pi _n$ , and

$$\begin{align*}\forall x\ \ \big(\mathrm{Sat}_{\Sigma_n}(\phi,x)\equiv \mathrm{Sat}_{\Pi_n}(\psi,x)\big).\end{align*}$$

The notions of a $\Sigma _n (\Pi _n, \Delta _n)$ relation is defined analogously. If X is a $\Delta _k$ set given by the $\Sigma _k$ formula $\phi (v)$ and a $\Pi _k$ formula $\psi (v)$ , then $x\in _k X$ abbreviates $\mathrm {Sat}_{\Sigma _k}(\phi , x)$ . Note that $x\in _k X$ is $\Delta _k$ .

Observe that a set $A\subseteq M$ is definable from parameters in a model $\mathcal {M}$ if and only if, for some $k\in \omega $ , there exists a $\Delta _k$ set X such that

$$\begin{align*}A:=\{x\in M \ \ | \ \ x\in_k X \}.\end{align*}$$

In the paper, except for side remarks, in which case the definitions below can clearly be adapted, we will only need to talk about models for very specific signatures consisting of two binary functions $+$ , $\times $ , one binary relational symbol $S(x,y)$ , reserved for a satisfaction class and two constants, $0$ and $1$ .

We note that since in $\mathrm {PA}$ models for theories extending some basic arithmetic (which we are uniquely interested in) are class-size objects, we do not always have a satisfaction relation for them. Models without the satisfaction relation will called partial to contrast them with the full ones for which the truth of an arbitrary sentence can be decided.

Definition 2.4. If $\mathcal {N}$ is any model of $\mathrm {PA}$ then we say that $\mathbf {M}$ is a partial $\mathcal {N}$ -definable model if for some $k\in \omega $ ,

$$\begin{align*}\mathcal{N}\models “\mathbf{M}\ \mathrm{ is\ a\ partial}\ \Delta_k\ \mathrm{model}.” \\[-3pc]\end{align*}$$

$\triangle $

Note that, according to our convention, “ $\mathcal {N}$ -definable $” $ means $“\mathcal {N}$ -definable with parameters $.”$

Remark 2.6. If $\mathcal {N}\models \mathrm {PA}$ and $\mathbf {M} = (U_{\mathbf {M}},+_{\mathbf {M}},\times _{\mathbf {M}}, S_{\mathbf {M}}, 0_{\mathbf {M}}, 1_{\mathbf {M}})$ is a partial $\mathcal {N}$ -definable model, then, outside of $\mathcal {N}$ , it gives rise to a model for the signature $\{+,\times , S, 0,1\}$ . Indeed, we may define model $\mathcal {M}$ by putting

$$\begin{align*}\mathcal{M}:= ((U_{\mathbf{M}})^{\mathcal{N}},(+_{\mathbf{M}})^{\mathcal{N}},(\times_{\mathbf{M}})^{\mathcal{N}}, (S_{\mathbf{M}})^{\mathcal{N}}, 0_{\mathbf{M}}, 1_{\mathbf{M}}).\end{align*}$$

Such a model will be denoted by $(\mathbf {M})^{\mathcal {N}}$ . $\triangle $

Definition 2.9 ( $\mathrm {ACA}_0$ )

Let $\mathbf {M}$ be a partial model. An $\mathbf {M}$ -evaluation of terms is a partial function f of type

$$\begin{align*}\mathrm{Term}\times \mathrm{Asn}(\mathbf{M})\rightarrow \mathbf{M}\end{align*}$$

such that for every term t, $\{t\}\times \mathrm {Asn}(t,\mathbf {M})\subseteq \mathrm {dom}(f)$ and for every terms $s,t$ and every $\mathbf {M}$ -assignment $\alpha $ :

1. 1. $\langle t, \alpha \rangle \in \mathrm {dom}(f)\leftrightarrow \mathrm {FV}(t)\subseteq \mathrm {dom}(\alpha )$ ,

2. 2. $\alpha \subseteq \beta \wedge \langle t, \alpha \rangle \in \mathrm {dom}(f)\rightarrow f(t,\alpha )=f(t,\beta )$ ,

3. 3. $f(0,\alpha ) = 0^{\mathbf {M}}$ , $f(1,\alpha ) = 1^{\mathbf {M}}$ ,

4. 4. $f(v,\alpha ) = \left \{\begin {array}{l} \alpha (v), \mathrm { if }\ v\in \mathrm {dom}(\alpha ),\\ \mathrm {undefined,}\ \mathrm { otherwise},\end {array} \right.$

5. 5. $f((s+t),\alpha ) = f(s,\alpha )+^{\mathbf {M}}f(t,\alpha )$ , $f((s\cdot t),\alpha ) = f(s,\alpha )\cdot ^{\mathbf {M}} f(t,\alpha )$ . $\triangle $

We stress that $(\mathbf {M}, \mathrm {Sat}_{\mathbf {M}})$ being c-full presupposes that $\mathbf {M}$ and $\mathrm {Sat}_{\mathbf {M}}$ satisfy full induction (treated as additional predicates).

Observe that if for every $n\in \omega $ , $(\mathbf {M}, \mathrm {Sat}_{\mathbf {M}})$ is an $\mathcal {N}$ -definable n-full model, then we have two satisfaction classes for $\mathbf {M}$ at our disposal: the metatheoretical one and $\mathrm {Sat}_{\mathbf {M}}$ . The two relations agree in the following sense: for every $\phi (x_1,\ldots , x_n)\in \mathcal {L}_S$ and for all $a_1,\ldots , a_n \in (U_{\mathbf {M}})^{\mathcal {N}}$ ,

$$\begin{align*}\mathcal{M}\models \phi[a_1/x_1,\ldots, a_n/x_n] \iff \mathcal{N}\models \mathrm{Sat}_{\mathbf{M}}(\ulcorner \phi(x_0,\ldots,x_n) \urcorner, [a_1,\ldots,a_n]),\end{align*}$$

where $[a_1,\ldots , a_n]$ denotes the assignment $x_i\mapsto a_i$ , $i\leq n$ .

By the Tarski’s undefinability of truth theorem one obtains that if $\mathcal {M}$ is any model of $\mathrm {PA}$ , then there is no formula $\mathrm {Sat}_{\mathbf {V}}$ with parameters from $\mathcal {M}$ such that for every n, $(\mathbf {V},\mathrm {Sat}_{\mathbf {V}})$ is an n-full model. However, relativizing the standard partial truth predicates (see [Reference Hájek and Pudlák9]) one obtains the following observation.

Proof. We reason in $\mathrm {ACA}_0$ and assume the contrary. Then there is a sequent-calculus proof of the sequent

$$\begin{align*}\Gamma\Rightarrow 0=1\end{align*}$$

in the pure first-order logic, where $\Gamma \subseteq Y$ is a finite set. By cut-elimination we may assume that this proof has a subformula property, so every formula occurring in it is a subformula of a formula from $\Gamma \cup \{0=1\}$ . By induction on the length of the proof we can show that for every sequent $\Theta \Rightarrow \Delta $ it holds that

$$\begin{align*}\forall \phi\in\Theta \big(\mathbf{M}\models_{\mathrm{Sat}_{\mathbf{M}}}\phi\big) \rightarrow \exists \psi\in\Delta \big(\mathbf{M}\models_{\mathrm{Sat}_{\mathbf{M}}}\psi\big).\end{align*}$$

This contradicts that the proof ends with $0=1$ .□

The above notions of partial and full model lead to the definition of two interpretability relations between structures:

Definition 2.17 (Interpretable models; see [Reference Kaye12])

Let $\mathcal {M}$ and $\mathcal {N}$ be two models of an extension of $\mathrm {PA}$ (not necessarily satisfying $\mathrm {PA}^*$ ). We say that $\mathcal {M}$ interprets $\mathcal {N}$ if there exists a partial $\mathcal {M}$ -definable model $\mathbf {N}$ such that

$$\begin{align*}\mathcal{N} = (\mathbf{N})^{\mathcal{M}}.\end{align*}$$

We say that $\mathcal {M}$ strongly interprets $\mathcal {N}$ iff there exists $\mathbf {K}$ witnessing that $\mathcal {M}$ interprets $\mathcal {N}$ and there exists an $\mathcal {M}$ -definable satisfaction predicate $\mathrm {Sat}_{\mathbf {K}}$ making $\mathbf {K}$ a full model. Interpretability and strong interpretability will be denoted by $\lhd $ and $\lhd _S$ , respectively. $\triangle $

Observe that, as defined neither interpretability nor strong interpretability is preserved under isomorphism, in the sense that from $\mathcal {M}\lhd \mathcal {N}$ and $\mathcal {N}\simeq \mathcal {K}$ we cannot conclude that $\mathcal {M}\lhd \mathcal {K}$ . The next two propositions uncover the important properties of $\lhd $ . The following routine notion will come in handy:

Definition 2.18 ( $\mathrm {ACA}_0$ ; relativization)

Suppose that $\mathbf {M}$ is a partial model. For every formula $\phi $ we define its relativization $\phi ^{\mathbf {M}}$ by induction on the complexity of $\phi $ :

$$ \begin{align*} (s(\bar{v_s}) = t(\bar{v_t}))^{\mathbf{M}} &:= \mathrm{val}_{\mathbf{M}}(s, [\bar{v_s}]) = \mathrm{val}_{\mathbf{M}}(t,[\bar{v_t}]),\\ (S(t(\bar{v})))^{\mathbf{M}} &:= \mathrm{val}_{\mathbf{M}}(t,[\bar{v}]) \in S_{\mathbf{M}},\\ (\phi\vee \psi)^{\mathbf{M}} &:= (\phi)^{\mathbf{M}}\vee (\psi)^{\mathbf{M}},\\ (\neg \phi)^{\mathbf{M}} &:= \neg (\phi)^{\mathbf{M}},\\ (\exists x \phi)^{\mathbf{M}} &:= \exists x \in U_{\mathbf{M}}\ \ (\phi)^{\mathbf{M}}. \end{align*} $$

Above, $\mathrm {val}_{\mathbf {M}}(t, [\bar {v_s}])=y$ abbreviates the formula

$$\begin{align*}\exists \alpha \big(\alpha\in\mathrm{Asn}(t,\mathbf{M}) \wedge \bigwedge_{i} \alpha(v_i)= x_i \wedge y = \mathrm{val}_{\mathbf{M}}(t,\alpha)\big). \\[-3.8pc]\end{align*}$$

$\triangle $

Proof. Suppose that $\mathcal {N} = (\mathbf {N})^{\mathcal {M}}$ and $\mathcal {K} = (\mathbf {K})^{\mathcal {N}}$ . Suppose further that $\mathbf {N}$ is partial $\Delta _k$ model in $\mathcal {M}$ . Hence using partial satisfaction predicate for $\Sigma _k$ formulae, we can see that the $\mathbf {K}^{\mathbf {N}}$ (see Definition 2.18) makes sense in $\mathcal {M}$ , and, in $\mathcal {M}$ , $\mathbf {K}^{\mathbf {N}}$ is a partial $\Delta _k$ model. Moreover it is easy to observe that

$$\begin{align*}\left(\mathbf{K}^{\mathbf{N}}\right)^{\mathcal{M}} = \mathcal{K},\end{align*}$$

which ends the proof.□

The following proposition will play a crucial role in some of our arguments. Its proof consists in internalizing the argument from Remark 2.6 and makes use of the arithmetization of the relativization function introduced above.

Proof. By Proposition 2.19 we have $\mathcal {M}\lhd \mathcal {K}$ and $(\mathbf {K})^{\mathbf {N}}$ is a partial $\mathcal {M}$ -definable model witnessing the interpretability. We define the satisfaction relation for $(\mathbf {K})^{\mathbf {N}}$ via the formula

$$\begin{align*}\mathrm{Sat}_{\mathbf{K}^{\mathbf{N}}}(x,y):= \mathrm{Form}_{\mathcal{L}_S}(x) \wedge y\in\mathrm{Asn}(x,\mathbf{K}^{\mathbf{N}}) \wedge \mathrm{Sat}_{\mathbf{N}}(x^{\mathbf{K}}, y). \\[-34pt] \end{align*}$$

□

In $\mathrm {PA}$ we can prove that every consistent theory admits a full model. Since in most cases both the theory and the model are infinite objects, this is in fact a parametrized family of theorems:

Theorem 2.21 (Arithmetized Completeness Theorem)

For every $n\in \omega $ , $\mathrm {PA}^*$ proves the sentence

$$ \begin{align*} \mathit{Every}\ \Delta_n\ \mathit{consistent\ theory\ has\ a}\ \Delta_{n+1}\ \mathit{full\ model}.\\[-3pc] \end{align*} $$

$\boxplus $

Since the proof of this fact (apart from axioms for arithmetical operations) depends only on the presence of induction, it can be proved also in every extension of $\mathrm {PA}$ which includes full induction scheme (for the extended language). This will be crucial in the second part of the paper. Let us complete this introductory part with two classical observations which give us some information about the structure of interpretable models. The first one shows that in fact interpretability can be seen as refined end-extendibility.

Proof. Suppose $\mathbf {N}:=\langle U_{\mathbf {N}},+_{\mathbf {N}},\times _{\mathbf {N}}, S_{\mathbf {N}}, 0_{\mathbf {N}}, 1_{\mathbf {N}} \rangle $ is an $\mathcal {M}$ definable partial model witnessing the interpretability of $\mathcal {N}$ in $\mathcal {M}$ . Let $\mathrm {val}_{\mathbf {N}}$ be a valuation function for $\mathbf {N}$ . We define the embedding $\iota :\mathcal {M}\rightarrow \mathcal {N}$ via the formula

$$\begin{align*}\iota(x) := \mathrm{val}_{\mathbf{N}}(\underline{x}, \varepsilon),\end{align*}$$

where $\underline {x}$ is a canonical numeral (in the sense of $\mathcal {M}$ ) naming x. By the earlier remarks there exists a satisfaction predicate $\mathrm {Sat}_{\mathbf {N}}$ making $\mathbf {N}$ a $1$ -full model. The fact that $\iota $ is an initial embedding follows since for every x we can build a quantifier-free sentence (in the sense of $\mathcal {M}$ )

$$\begin{align*}\forall v\ \ \bigg(v<\underline{x} \rightarrow \bigvee_{z<x} v = \underline{z}\bigg),\end{align*}$$

and by induction on x show that every such sentence is true in $\mathbf {N}$ according $\mathrm {Sat}_{\mathbf {N}}$ . But this is equivalent to $\iota $ being an initial embedding. Now, if $\iota '$ is any other $\mathcal {M}$ definable isomorphism between $\mathcal {M}$ and an initial segment of $\mathcal {N}$ , then it follows that

$$\begin{align*}\mathcal{M}\models “\iota'(0) = 0_{\mathbf{N}} \wedge \forall x\big(\iota'(x+1) = \iota'(x)+_{\mathbf{N}} 1_{\mathbf{N}}\big).”\end{align*}$$

Then, by induction it follows that $\mathcal {M}\models \forall x\big (\iota (x) = \iota '(x)\big )$ .□

If we strengthen the assumption to strong interpretability, then we can conclude that the interpreted model is always $“$ longer. $”$

Proof. Let $\mathcal {M}$ , $\mathcal {N}$ , $\iota $ be as above and let $\mathbf {N}$ be a partial $\mathcal {M}$ -definable model such that $\mathcal {N} = (\mathbf {N})^{\mathcal {M}}$ . That $\iota $ is not elementary follows from Tarski undefinability of truth theorem. Indeed, since $\iota $ and $\mathrm {Sat}_{\mathbf {N}}$ are $\mathcal {M}$ definable, we can define in $\mathcal {M}$ a predicate $S(x,y)$ by putting

$$\begin{align*}S(\phi, \alpha) \equiv \mathrm{Sat}_{\mathbf{N}}(\phi, \iota\circ \alpha).\end{align*}$$

With such a definition for every formula $\phi (x_1,\ldots , x_n)$ and all $a_1,\ldots , a_n\in M$ we have

$$\begin{align*}\mathcal{M}\models S(\ulcorner \phi(x_1,\ldots, x_n) \urcorner, [a_1,\ldots, a_n])\iff \mathcal{N}\models \phi(\iota(a_1),\ldots, \iota(a_n)).\end{align*}$$

Then, if $\iota $ were elementary, then the condition on the right-hand side would be equivalent to $\mathcal {M}\models \phi (a_1,\ldots , a_n)$ which contradicts Tarski’s theorem. The last part follows easily from the above and Proposition 2.23.□

Let us note one immediate corollary. Recall that $\mathcal {M}$ is $\kappa $ -like if $|M| = \kappa $ but every proper initial segment of $\mathcal {M}$ has cardinality strictly smaller than $\kappa $ .

Moreover, models strongly interpretable in a nonstandard model of $\mathrm {PA}$ have to be recursively saturated. This is a corollary to the proposition below (see also [Reference Kossak and Schmerl14]):

Proof. Suppose that $\mathcal {M} = (\mathbf {M},\mathrm {Sat}_{\mathbf {M}})$ is an $\mathcal {N}$ definable depth-d-full model, $\mathcal {N}$ and d being as above. Fix an arbitrary recursive type $p(x) = \{\phi _i(x,a) \ \ | \ \ i\in \omega \}$ with (without loss of generality) a single parameter a and let $\sigma (x,y)$ be the $\Delta _1$ formula representing its recursive enumeration, i.e., for every $i\in \omega $ ,

$$\begin{align*}\mathrm{PA}\vdash \forall w\ \ \big( \sigma(i,w)\leftrightarrow w = \underline{\ulcorner \phi_i(x,z) \urcorner}\big).\end{align*}$$

Now, since the depth of every $\phi _i$ is less than d and $p(x)$ is a type we have for every $n\in \omega $

$$\begin{align*}\mathcal{N}\models \exists z \forall i\leq n\forall w\ \ \big(\sigma(i,w)\rightarrow \mathrm{Sat}_{\mathbf{M}}(w,[x\mapsto z, y\mapsto a])\big)\end{align*}$$

( $[x\mapsto z, y\mapsto a]$ denotes the unique assignment sending the variable x to z and the variable y to a). Hence, by overspill for some nonstandard c we have

$$\begin{align*}\mathcal{N}\models \exists z \forall i\leq c\forall w\ \ \big(\sigma(i,w)\rightarrow \mathrm{Sat}_{\mathbf{M}}(w,[x\mapsto z, y\mapsto a])\big),\end{align*}$$

which shows that $p(x)$ is realised in $\mathcal {M}$ .□

2.4 Satisfaction classes

Satisfaction classes provide truth conditions for $\mathbf {V}$ .Footnote ⁶ Usually they are studied in the context of nonstandard models of $\mathrm {PA}$ . Let $\mathcal {M}$ be such a model.

Definition 2.27. We say that $S\subseteq M^2$ is a partial satisfaction class on $\mathcal {M}$ if there is a nonstandard c such that $(\mathcal {M},S)\models \mathrm {CS}^-(\mathrm {dp}(c+1),\mathbf {V}, S)$ . Equivalently the following holds in $(\mathcal {M},S)$ :

1. 1. $ \forall x,y \big (S(x,y)\rightarrow \mathrm {Form}(x)\wedge x\in \mathrm {dp}(c) \wedge y\in \mathrm {Asn}(x)\big )$ .

2. 2. $\forall s,t\forall \alpha \in \mathrm {Asn}(s,t)\ \ \left (S(s=t,\alpha )\equiv s^{\alpha }= t^{\alpha }\right )$ .

3. 3. $\forall \phi \in \mathrm {dp}(c)\forall \alpha \in \mathrm {Asn}(\phi )\ \ \left (S(\neg \phi ,\alpha )\equiv \neg S(\phi ,\alpha )\right )$ .

4. 4. $\forall \phi ,\psi \in \mathrm {dp}(c)\forall \alpha \in \mathrm {Asn}(\phi ,\psi )\ \ \left (S(\phi \vee \psi ,\alpha )\equiv S(\phi ,\alpha {\upharpoonright _{\phi }})\vee S(\psi ,\alpha {\upharpoonright _{\psi }})\right )$ .

5. 5. $\forall \phi \in \mathrm {dp}(c) \forall v \forall \alpha \in \mathrm {Asn}(\exists v\phi )\ \left (S(\exists v\phi ,\alpha )\equiv \exists \beta \geq _v\alpha \big (S(\phi ,\beta {\upharpoonright _{\phi }})\right )$ .

Henceforth, the conjunction of $1$ – $5$ will be denoted by $\mathrm {CS}^-(c)$ . If additionally $(\mathcal {M},S)\models \mathrm {PA}^*$ , then S is called a partial inductive satisfaction class. If $(\mathcal {M},S)\models \forall x\ \ \mathrm {CS}^-(x)$ , then S is called a full satisfaction class. Further define:

$$ \begin{align*} \mathrm{UTB}^-&:= \{\mathrm{CS}^-(\underline{n}) \ \ | \ \ n\in \omega \},\\ \mathrm{UTB}_n&:= \mathrm{UTB}^- + I\Sigma_n(S),\\ \mathrm{UTB} &:= \bigcup_{n\in\omega} \mathrm{UTB}_n. \end{align*} $$

Now we define an analogue of arithmetical hierarchy over the theory of a full satisfaction class.

$$ \begin{align*} \mathrm{CS}^- &:=\forall x\ \ \mathrm{CS}^-(x),\\ \mathrm{CS}_n &:= \mathrm{CS}^- + I\Sigma_n(S),\\ \mathrm{CS} &:= \bigcup_{n\in\omega} \mathrm{CS}_n. \end{align*} $$

If S is a partial satisfaction class on $\mathcal {M}$ and $b\in M$ , then we put

$$\begin{align*}S_b:= \{\langle \phi, \alpha\rangle\in S \ \ | \ \ \phi\in\Sigma^*_b \}.\end{align*}$$

Note that $S_b$ is $\Delta _0$ definable from S, and hence for every n, if S is $\Sigma _n$ inductive, then so is $S_b$ . $\triangle $

Note that if S is a full satisfaction class, then $(\mathcal {M},S)$ strongly interprets $\mathcal {M}$ and $\mathbf {V}$ is a $\Delta _0$ partial model witnessing the interpretation. However, $(\mathbf {V},S)$ need not be full, as we need not have any induction for S. In particular, it does not follow that

$$\begin{align*}(\mathcal{M},S) \models \forall \phi \in \mathrm{Ax}_{\mathrm{PA}} \ \ S(\phi,\varepsilon),\end{align*}$$

which, arguably, would mean that $\mathcal {M}$ knows that $\mathbf {V}$ is a model of $\mathrm {PA}$ . Let us call such a satisfaction class $\mathrm {PA}$ -correct. It can be shown that for a countable $\mathcal {M}$ the following conditions are equivalent:

1. 1. There exists a full satisfaction class on $\mathcal {M}$ .

2. 2. There exists a $\mathrm {PA}$ -correct full satisfaction class on $\mathcal {M}$ .

3. 3. $\mathcal {M}$ is recursively saturated.

The implication $3.\Rightarrow 2.$ has been shown for the first time in [Reference Kotlarski, Krajewski and Lachlan16]. References [Reference Enayat, Visser, Achourioti, Galinon, Fernández and Fujimoto7, Reference Leigh17] contain different proofs. The implication $1.\Rightarrow 3.$ is a consequence of Lachlan’s theorem (see Theorem 2.31).

The name $\mathrm {UTB}^-$ stands for Uniform Tarski Biconditionals Footnote ⁷ and is normally used for the theory having as axioms all sentences of the form

$$\begin{align*}\forall \alpha \in \mathrm{Asn}(\phi)\ \ S(\underline{\ulcorner \phi \urcorner},\alpha)\equiv \phi((\alpha(x_1),\ldots, (\alpha(x_n)),\end{align*}$$

for every $\phi (x_1,\ldots ,x_n)\in \mathrm {Form}_{\mathcal {L}}$ .Footnote ⁸ One can show that, over $\mathrm {EA}$ , this set of sentences is equivalent to the one we’ve officially taken as a definition of $\mathrm {UTB}^-$ . By Observation 2.15 each finite portion of $\mathrm {UTB}^-$ is definable in $\mathrm {PA}$ and consequently we obtain the following proposition (which formalizes in $\mathrm {EA}$ ).

Furthermore, observe that $(\mathcal {M}, S)\models \mathrm {CS}^-$ iff S is a full satisfaction class on $\mathcal {M}$ and $(\mathcal {M},S)\models \mathrm {CS}$ iff S is a full inductive satisfaction class in the sense of [Reference Kossak and Schmerl14]. For further usage let us observe that the relation of $\mathrm {CS}$ to $\mathrm {CS}_n$ is similar to that between $\mathrm {PA}$ and $I\Sigma _n$ . In particular there are definable partial $\mathrm {Sat}_{\Sigma _n}$ satisfaction predicates for $\Sigma _n \mathcal {L}_{S}$ formulae. Each $\mathrm {Sat}_{\Sigma _n}$ is a $\Sigma _n \mathcal {L}_S$ formula. As a consequence we obtain:

We note that if S is a partial satisfaction class on a model $\mathcal {M}$ , then for an arbitrary standard formula $\phi (x_0,\ldots , x_n)\in \mathcal {L}$ ,

$$\begin{align*}(\mathcal{M}, S)\models \forall \alpha \ \ \left(S(\ulcorner \phi \urcorner,\alpha) \equiv \phi(\alpha(x_1),\ldots,\alpha(x_n))\right).\end{align*}$$

In particular it follows from Tarski’s theorem that S is never definable in $\mathcal {M}$ (even if we allow parameters).

Nonstandard satisfaction classes provide a very useful tool for investigating nonstandard models of $\mathrm {PA}$ . The first point of interest is that their existence implies recursive saturation. For starters we cite a proposition which directly follows from Proposition 2.26:

Interestingly, with a much more complicated proof one can strengthen the above proposition lifting the assumption that the chosen satisfaction class is inductive.

The converse to Lachlan’s theorem fails, as was shown by Smith.

The condition that $(\mathcal {M},S')\models I\Delta _0(S)$ implies that $S'$ is piecewise coded (in the sense of [Reference Hájek and Pudlák9]) or, using set-theoretical notions, a class on $\mathcal {M}$ . Since $(\mathcal {M},S')\models \mathrm {CS}^-(c)$ it follows that $S'$ is not definable (even allowing parameters) in $\mathcal {M}$ (or, is a proper class). Since there are recursively saturated models of $\mathrm {PA}$ in which every class is definable (with parameters; see [Reference Kossak and Schmerl14]), Smith’s result shows that there are recursively saturated models which do not carry a full satisfaction class.

A common strengthening of theorems of Smith’s and Lachlan’s was obtained by Wcisło in [Reference Wcisło and Łełyk25]:

An interesting open problem in the model theory of $\mathrm {PA}$ is whether the converse to the above theorem is true, i.e., whether every $\mathcal {M}\models \mathrm {PA}$ which admits a partial inductive satisfaction class admits a full satisfaction class. If one allows to prolong the given model, then there is a positive answer to this question.

The proof of this theorem is given in [Reference Łełyk and Wcisło20, Theorem 43]. In Section 5 we shall give an analogous result for $\mathcal {M}\models \mathrm {REF}^{\omega }(\mathrm {PA})$ and $\mathrm {CS}_0$ instead of $\mathrm {CS}^-$ .

Remark 2.35. The theory $\mathrm {CS}^-$ is a cousin of a compositional truth theory $\mathrm {CT}^-$ which admits a unary predicate T. All compositional axioms of $\mathrm {CS}^-$ can be easily adapted to this new setting; however in the case of the universal quantifier we have two natural ways to go. The first candidate is the $“$ numeral $”$ version, i.e.,

$$\begin{align*}\forall v\forall \phi(v)\ \ \big(T(\forall v \phi)\equiv \forall x T(\phi[x/v])\big).\end{align*}$$

We stress that $\phi [x/v]$ denotes the result of substituting the numeral naming x for every free occurrence of the variable v. If such an axiom is adopted, then the resulting theory, denote it $n\mathrm {CT}^-$ , can define the satisfaction predicate satisfying $\mathrm {CS}^-$ via the formula

$$\begin{align*}S(\phi,\alpha):= \mathrm{Form}_{\mathcal{L}}(\phi) \wedge \alpha \in \mathrm{Asn}(\phi) \wedge T(\phi[\alpha]).\end{align*}$$

Let us stress that the above formula is $\Delta _0(\exp )$ . The second option is the $“$ term $”$ version of $\mathrm {CT}^-$ , denote it $t\mathrm {CT}^-$ , where the axiom for $\forall $ is the following:

$$\begin{align*}\forall v\forall \phi(v)\ \ \big(T(\forall v \phi)\equiv \forall t\in\mathrm{Term} \ \ T(\phi[t/v])\big).\end{align*}$$

Using Enayat–Visser methods from [Reference Enayat, Visser, Achourioti, Galinon, Fernández and Fujimoto7] it can be shown that $n\mathrm {CT}^-$ and $t\mathrm {CT}^-$ are independent of each other, i.e., neither of them implies the other one (over the remaining axioms of $\mathrm {CS}^-$ ). Moreover, it is an open problem whether $t\mathrm {CT}^-$ can define the predicate of $\mathrm {CS}^-$ . In this paper $\mathrm {CT}^-$ will be introduced in Section 3 and will denote the numeral version, i.e., $n\mathrm {CT}^-$ . $\triangle $

2.5 Reflection principles

Reflection principles are various (families of) statements expressing the soundness of a given theory $\mathrm {Th}$ in a way which is transparent for $\mathrm {Th}$ . In other words, their aim is to capture the meaning of the metatheoretical assertion:

Every theorem of $\mathrm {Th}$ is true.

In order to avoid the problem of choosing the presentation for (an abstractly given theory $\mathrm {Th}$ ), we will assume that $\mathrm {Th}$ is an elementary formula, which, provably in $\mathrm {EA}$ , defines a set of sentences. Such a formula will be called a Gödelized theory. We use $\mathrm {PA}$ to abbreviate the canonical elementary formula saying “x is an axiom of $\mathrm {PA}^-$ or an axiom of induction.” Having a satisfaction predicate $S(x,y)$ at our disposal we can express the above in the form of the Global Reflection Principle

(GR(Th)) $$ \begin{align} \forall \phi\ \ \mathrm{Pr}_{\mathrm{Th}}(\phi)\rightarrow S(\phi,\varepsilon). \end{align} $$

If S satisfies $\mathrm {UTB}^-$ , then from this one can derive instantiation of the uniform reflection

(REF(Th)) $$ \begin{align} \forall x_1\ldots\forall x_n\big( \mathrm{Pr}_{\mathrm{Th}}(\ulcorner \phi \urcorner[\underline{x_1},\ldots,\underline{x_n}])\rightarrow \phi(x_1,\ldots,x_n)\big). \end{align} $$

Hence $\mathrm {REF}(\mathrm {Th})$ contains all the formulae of the above form for the language of $\mathrm {Th}$ . If $\Gamma $ is a set of formulae of the language of $\mathrm {Th}$ , then denote the restriction of $\mathrm {REF}(\mathrm {Th})$ to formulae from class $\Gamma $ . Below we will need also its iterated versions:

In the successor step we tacitly fix the canonical representation of

.

The last definition which is relevant to formalizing soundness claims introduces the oracle provability predicates.

The oracle provability predicate defined above enables us to (uniformly) define a closure conditions on various satisfaction classes. For example we shall often encounter the assertion

$$\begin{align*}\forall \phi\big(\mathrm{Pr}_{\emptyset}^{S}(\phi)\rightarrow S(\phi,\varepsilon)\big),\end{align*}$$

which should be read as “Every first-order consequence of true sentences is true,” where “true” abbreviates that $S(\phi ,\varepsilon )$ holds. In the above assertion we simply substitute the definable class $\{x \ \ | \ \ S(x,\varepsilon ) \}$ for the free second-order variable X. Let us also observe that formally $\mathrm {Pr}_{\mathrm {Th}}^X$ is the same as $\mathrm {Pr}_{\mathrm {Th}\cup X}$ ; however, for heuristic reasons we prefer to keep the lower index for absolute definitions and the upper one for arbitrary sets of formulae.

2.5.2 Reflection and satisfaction classes

Full satisfaction classes in non-standard models embody the conception of a satisfaction relation for $\mathbf {V}$ . However, as we have already remarked, not every satisfaction class provides us with a reasonable truth predicate for $\mathbf {V}$ . One property that one would require from such a truth predicate is the closure under the internal provability relation. In particular the satisfaction relation for $\mathbf {V}$ should make true all the (internal) theorems of first-order logic. This corresponds to the sentence

(GR(∅)) $$ \begin{align} \forall \phi\ \ \big(\mathrm{Pr}_{\emptyset}(\phi)\rightarrow S(\phi,\varepsilon)\big) \end{align} $$

being true in a model $(\mathcal {M}, S)$ . However, as shown for the first time in [Reference Cieśliński4], over $\mathrm {CS}^-$ the above sentence implies GR( $\mathrm {PA}$ ). In particular, in a countable recursively saturated model $\mathcal {M}$ in which there is a proof of inconsistency of $\mathrm {PA}$ there is no such a reasonable class for $\mathbf {V}$ (although there are many unreasonable ones). A characterization of models admitting a well-behaved satisfaction class was essentially first given by Kotlarski (in [Reference Kotlarski15]):Footnote ⁹

Theorem 2.39 (Kotlarski)

Suppose $\mathcal {M}$ is a countable recursively saturated model of $\mathrm {PA}$ . Then, there exists a full satisfaction class S such that $(\mathcal {M},S)\models \mathrm {GR}(\mathrm {PA})$ if and only if $\mathcal {M}\models \mathrm {REF}^{\omega }(\mathrm {PA})$ . $\boxplus $

A different natural question is how much induction is required to prove the global reflection for $\mathrm {PA}$ . Here a partial answer was given by Wcisło in [Reference Wcisło and Łełyk25], who showed that the satisfaction predicate satisfying $\mathrm {CS}^- + \mathrm {GR}(\mathrm {PA})$ is definable in $\mathrm {CS}_0$ :

Theorem 2.40 (Wcisło)

There exists an $\mathcal {L}_S$ formula $S'(x,y)$ such that

$$\begin{align*}\mathrm{EA} + \mathrm{CS}_0\vdash [\mathrm{CS}^- + \mathrm{GR}(\mathrm{PA})][S'/S].\\[-3pc] \end{align*}$$

$\boxplus $

In the above $\phi [S'/S]$ denotes the result of a uniform substitution of a formula $S'(s,t)$ for each occurrence of a formula $S(s,t)$ (renaming variables if necessary). In the next section we improve this result and show that GR $(\mathrm {PA})$ is provable in $\mathrm {EA} + \mathrm {CS}_0$ .

3.1 Kotlarski’s proof

Kotlarski’s proof of GR( $\mathrm {PA})$ in $\mathrm {CS}_0$ starts by observing that each $\Delta _0$ inductive satisfaction class makes all (in the sense of the ground model) the axioms of induction true. The argument runs as follows: working in $\mathrm {CS}_0$ fix a formula $\phi (v)$ with a free variable v. Then, $S(\phi (v), [x])$ is a $\Delta _0(\exp )$ formula with a free variable x, where $[x]$ denotes the assignment $\{\langle v, x\rangle \}$ . Hence, the following is an axiom of $\mathrm {CS}_0$ :

$$\begin{align*}S(\phi(v), [0])\wedge \forall x\big(S(\phi(v),[x])\rightarrow S(\phi(v),[x+1])\big)\longrightarrow \forall x S(\phi(v), [x]).\end{align*}$$

Here Kotlarski’s proof of $\mathrm {PA}$ -correctness ends. However, it is not obvious whether the above is equivalent to $S(\mathrm {Ind}(\phi (v)), \varepsilon )$ , where $\mathrm {Ind}(\psi )$ denotes the axiom of induction for a formula $\psi $ . Repeated applications of the compositional clauses yield the equivalence of $S(\mathrm {Ind}(\phi (v)), \varepsilon )$ with

$$\begin{align*}S(\phi[0/v], \varepsilon)\wedge \forall x\big(S(\phi(v),[x])\rightarrow S(\phi[v+1/v],[x])\big)\longrightarrow \forall x S(\phi(v), [x]).\end{align*}$$

Firstly, on the grounds of $\mathrm {CS}^-$ alone $S(\phi (v),[{0}])$ neither implies $S(\phi [0/v],\varepsilon )$ nor is implied by it. Similarly with $S(\phi (v), [x+1])$ and $S(\phi [v+1/v], [x])$ . To see this one should think of a nonstandard $\phi (v)$ in which v occurs nonstandardly deep in $\phi (v)$ (i.e., at a nonstandard level of $\phi (v)'$ s syntactical tree). For example one can take $\phi (v)$ to be

$$\begin{align*}\underbrace{0=0 \vee (0=0 \vee (0=0\vee\cdots}_{a\ \mathrm{ times }\ 0=0} \vee v=v)\cdots),\end{align*}$$

for a nonstandard element a. This is the first problem.

The second problem lies in showing that a $\Delta _0$ –inductive satisfaction class is closed under provability, i.e., proving the sentence

$$\begin{align*}\forall \phi \ \ \big(\mathrm{Pr}_{\emptyset}^{S}(\phi) \rightarrow S(\phi,\varepsilon)\big).\end{align*}$$

In the above $\mathrm {Pr}_{\emptyset }^S$ derives from the oracle provability predicate defined in Definition 2.36. Kotlarski’s idea was to work (internally in $\mathrm {PA}$ ) with a Hilbert-style proof calculus with Modus Ponens as the only rule of reasoning and then using $\Delta _0(\exp )$ induction for an $\mathcal {L}_S$ formula

$$\begin{align*}\theta(x,p):= “\mathrm{If }\ \phi\ \mathrm{ is\ the}\ x\mathrm{-th\ sentence\ in}\ p,\ \mathrm{then\ } S(\phi,\varepsilon).”\end{align*}$$

In $\theta (x,p)$ all quantifiers can be bounded by p,Footnote ¹⁰ that can be taken as a parameter. So it is indeed a $\Delta _0(\exp )$ -formula. In the base step $\phi $ is either a logical axiom or a true sentence, so it’s truth is either trivial (the latter case) or seems to follow from the compositional axioms. In the inductive step, we have to check that if $\phi $ and $\phi \rightarrow \psi $ are true, then so is $\psi $ . This is indeed guaranteed by the compositional axioms.

However, problems arise while verifying the base step. For example (working in a nonstandard model) we might encounter the following logical axiom:

$$\begin{align*}\psi:=\forall v \phi(v)\rightarrow \phi(t),\end{align*}$$

for some nonstandard formula $\phi $ and a term t. Then $S(\psi ,\varepsilon )$ is equivalent to

$$\begin{align*}\forall y S(\phi(v), [y])\rightarrow S(\phi(t),\varepsilon),\end{align*}$$

so we encounter problems similar to the ones discussed while dealing with the truth of induction axioms. Moreover, there are more generic problems: if one does not want to incorporate the rule of universal generalisation, then one has to accept universal generalisations of all instances of propositional tautologies as axioms. In particular (working in a nonstandard model $(\mathcal {M},S)$ ) in the base step one might encounter an axiom of the form

$$\begin{align*}\xi:=\forall v_1\ldots \forall v_a \big( \phi\vee \neg\phi\big).\end{align*}$$

If it is true that that for any full satisfaction class S on $\mathcal {M}$ , $(\mathcal {M},S)\models \forall \alpha \in \mathrm {Asn}(\phi ) S(\phi \vee \neg \phi ,\alpha )$ , inferring that $(\mathcal {M},S)\models S(\xi , \varepsilon )$ requires some argument which cannot be carried out in $\mathrm {CS}^-$ alone.

3.2 The idea

To fill in the gaps in Kotlarski’s reasoning it is sufficient to establish within $\mathrm {CS}_0$ a kind of induction on the buildup of formulae. Indeed, what we missed were (inter alia) the following properties:

$$ \begin{align*} &\forall\phi\in\mathrm{Form}_{\mathcal{L}}\forall \alpha \in \mathrm{Asn}(\phi)\big(S(\phi,\alpha)\equiv S(\phi[\alpha],\varepsilon)\big).\\ &\forall\phi\in\mathrm{Form}^{\leq 1}_{\mathcal{L}}\forall s,t\forall \alpha\in\mathrm{Asn}(s)\forall \beta\in\mathrm{Asn}(t)\big(s^{\alpha} = t^{\beta}\rightarrow S(\phi[s/v],\alpha)\equiv S(\phi[t/v],\beta)\big). \end{align*} $$

The above hold (provably in $\mathrm {CS}^-$ ) if $\phi $ is an atomic formula and clearly are preserved by taking disjunctions, negations, and applying existential quantification. However, in order to secure the step for $\exists $ we need a $\Pi _1$ assumption, saying that the equivalence

$$\begin{align*}S(\psi,\alpha)\equiv S(\psi[\alpha],\varepsilon)\end{align*}$$

holds under an arbitrary assignment $\alpha $ . It turns out, however, that such assumptions can be expressed with a $\Delta _0$ formula. Firstly, the above is clearly equivalent to

(1) $$ \begin{align} \forall \alpha \in \mathrm{Asn}(\psi)S(\psi\equiv \psi[\alpha],\alpha). \end{align} $$

Secondly, if S commuted with the blocks of universal quantifiers, the above could have been further reduced to

(2) $$ \begin{align} S(\mathrm{ucl}(\psi\equiv \psi[\alpha]),\varepsilon), \end{align} $$

where $\mathrm {ucl}(\cdot )$ is a (definable) function which given a formula returns its universal closure. Our problem thus reduces to showing the equivalence between conditions of types (1) and (2). The standard strategy is to use induction on the length of the quantifier prefix. However, the proof of this once again uses $\Pi _1$ induction. To bypass this problem, we shall first establish commutation with blocks of uniformly bounded universal quantifiers, i.e., the principle

(B∀C) $$ \begin{align} \forall \phi \forall x \big(S(\mathrm{bucl}(\phi,x), \varepsilon)\equiv \forall \alpha \prec [x]_{\phi}\big(\alpha\in\mathrm{Asn}(\phi)\rightarrow S(\phi,\alpha)\big), \end{align} $$

where $\mathrm {bucl}(\phi ,x)$ denotes the universal closure of $\phi $ , in which every quantifier in the prefix is bounded by (the term) x and $\alpha \prec [x]_{\phi }$ says that $\mathrm {dom}(\alpha )=\mathrm {FV}(\phi )$ and each value of $\alpha $ is less than x. Having this, we will express $\forall \alpha S(\phi ,\alpha )$ in a $\Delta _0$ way via

$$\begin{align*}S(\forall v\mathrm{bucl}(\phi,v),\varepsilon),\end{align*}$$

where v is a variable which do not occur in $\phi $ . We now proceed to the details.

3.3 The proof

One more preparation step will be helpful. We shall expand the language $\mathcal {L}_S$ with symbols for all primitive recursive functions and extend $\mathrm {CS}_0$ with the defining equations of them. Let $\mathcal {L}_S^+$ denote the expanded language and $\mathrm {CS}_0^+$ the extended theory. Since, trivially, $\mathrm {CS}_0^+$ is $\mathcal {L}_S$ -conservative over $\mathrm {CS}_0$ , it is sufficient to prove (GR(Th)) in $\mathrm {CS}_0^+$ . Let us observe that the latter theory proves $\Delta _0$ induction for the language with new function symbols.

Proof. Fix a model $\mathcal {M}\models \mathrm {CS}_0^+$ and a $\Delta _0(\mathcal {L}_S^+)$ formula with parameters $\phi (x)$ . Without loss of generality all the terms occurring in $\phi (x)$ are of the form $f(\underline {n},y)$ where $f(x,y)$ is a two place p.r. function. Let $\psi _f(x,y,z)$ be the $\Delta _0$ formula of $\mathcal {L}$ defining the graph of f. Fix an arbitrary $a\in M$ , assume that $\phi (a)$ holds, and let b be greater than a and all parameters from $\phi (x)$ . Without loss of generality assume that b is nonstandard. Let c be such that

$$\begin{align*}\mathcal{M}\models \forall x<b\forall y<b \psi_f(x,y)< c.\end{align*}$$

Now let $\phi '(x)$ result from $\phi (x)$ by recursively changing all subformulae of $\phi (x)$ of the form

$$\begin{align*}R(f(\underline{k},x),f(\underline{n},y)) \end{align*}$$

into

$$\begin{align*}\exists z\exists w \big(z< c \wedge w< c \wedge \psi_f(\underline{n},y,w) \wedge \psi_f(\underline{k},x,z) \wedge R(w,z)\big),\end{align*}$$

where z, w are fresh variables. $\phi '(x)$ is a $\Delta _0(\mathcal {L}_S)$ formula and clearly we have

$$\begin{align*}\mathcal{M}\models \forall x< b\ \ \big(\phi(x)\equiv \phi'(x)\big).\end{align*}$$

Hence, since $a<b$ and $\phi (a)$ holds, $\phi '(a)$ holds as well. Since for $\phi '(x)$ we have an induction axiom there is the least $d<a$ such that $\phi '(d)$ holds. Hence d is the least element satisfying $\phi (x)$ .□

In the proof below we shall use the following primitive recursive functions (we identify p.r. relations with their characteristic functions):

• • $\prec $ is a primitive recursive product ordering on functions. That is, if $\alpha $ and $\beta $ are two functions, then $\alpha \prec \beta $ holds if $\alpha $ is smaller than $\beta $ in the product ordering, i.e.,

  $$\begin{align*}\mathrm{dom}(\alpha)=\mathrm{dom}(\beta) \wedge \forall x\in\mathrm{dom}(\alpha)\ \ \alpha(x)<\beta(x).\end{align*}$$
  $\preccurlyeq $ is the partial ordering based on $\prec $ .

• • $\mathrm {bucl}(\phi ,x)$ denotes the universal closure of $\phi $ , in which every quantifier in the prefix is bounded by (the term) x.

• • $\mathrm {bucl}(\phi , x, c)$ returns the formula

  $$\begin{align*}\forall v_{i_1}<x\ldots \forall v_{i_y}<x\ \ \phi,\end{align*}$$
  where $x_{i_1},\ldots , x_{i_y}$ are all the elements of the set c listed in the order of decreasing indices (i.e., $i_y<i_{y-1}<\cdots < i_1$ ). In particular x has to be a term and c a set of variables. Officially $\forall x< t \psi $ abbreviates $\forall x\big (x<t \rightarrow \psi )$ ; hence the above formula is slightly more complicated than it seems to be. Moreover c need not contain uniquely variables which are free in $\phi $ ; hence some of the quantifiers in the prefix of $\mathrm {bucl}(\phi , x, c)$ might be dummy.

• • $[x]_{c}$ returns the constant function assigning value x to every variable from the set c.

• • For a coded set of variables c and a number a, $c{\uparrow _a}$ denotes the set consisting of first a elements of c and $c{\downarrow ^{a}}$ the set consisting of last a elements of c. For simplicity we assume that the ordering of variables is given by their indices.

• • Syntactical relations mentioned at the beginning of Section 2.

Lemma 3.2 ( $\mathrm {CS}_0^+$ )

For every formula $\phi $ , every a, every set of variables c, and every assignment $\alpha $ for $\mathrm {bucl}(\phi , \underline {a}, c)$ ,

$$\begin{align*}S(\mathrm{bucl}(\phi, \underline{a}, c),\alpha)\equiv \forall \beta\prec [a]_{c} S(\phi,\alpha \cup \beta{{\upharpoonright_{\cdot}}}).\end{align*}$$

Proof. Fix $\phi ,a,c$ and let b be the cardinality of c. We formalize the standard argument on the length of the quantifier prefix in $\mathrm {bucl}(\phi , \underline {a}, c)$ : starting from the assumption that $\forall \beta \prec [a]_{c} S(\phi ,\alpha \cup \beta {{\upharpoonright _{\cdot }}})$ we check that we can prefix $\phi $ with b quantifiers and arrive at $S(\mathrm {bucl}(\phi , \underline {a}, c),\alpha )$ . Formally, we use induction on y up to b in the $\Delta _0(\mathcal {L}_S^+)$ -formula

$$\begin{align*}\forall \beta\prec [a]_{c{\downarrow^{b-y}}}S(\mathrm{bucl}(\phi, \underline{a}, c{\uparrow_{y}}), \alpha\cup \beta{{\upharpoonright_{\cdot}}})\equiv \forall \beta \prec [a]_{c} S(\phi,\alpha\cup\beta{{\upharpoonright_{\cdot}}}).\end{align*}$$

Observe that $c{\uparrow _{b}} = c{\downarrow ^{b}} = c$ , $c{\downarrow ^{0}} = \emptyset $ . Hence $\mathrm {bucl}(\phi , \underline {a}, c{\uparrow _{0}}) = \phi $ , $[a]_{c{\downarrow ^{b}}} = [a]_{c}$ . Moreover, $\varepsilon $ is the unique assignment $\gamma $ satisfying $\gamma \prec [a]_{\emptyset }$ . Consequently, for $y=b$ the left-hand side is equivalent to $S(\mathrm {bucl}(\phi , \underline {a}, c), \alpha )$ and for $y=0$ both sides of the equivalence are the same.

Assume that the inductive hypothesis holds for d and let ${i_d}$ be the index of the $(d+1)$ -st variable in c. Observe that

$$\begin{align*}\mathrm{bucl}(\phi, \underline{a}, c{\uparrow_{d+1}}) = \forall v_{i_d}{<\underline{a}}^\frown\mathrm{bucl}(\phi, \underline{a}, c{\uparrow_{d}}).\end{align*}$$

Hence by compositional axioms, $\forall \beta \prec [a]_{c{\downarrow ^{b-(d+1)}}}S(\mathrm {bucl}(\phi , \underline {a}, c{\uparrow _{d+1}}), \alpha \cup \beta {{\upharpoonright _{\cdot }}})$ is equivalent to

(*) $$ \begin{align} \forall \beta\prec [a]_{c{\downarrow^{b-(d+1)}}}\forall \gamma{\geq_{v_{i_d}}}\alpha\cup \beta{{\upharpoonright_{\mathrm{bucl}(\phi, \underline{a}, c{\uparrow_{d+1}})}}}\big(\gamma(v_{i_d})<a\rightarrow S(\mathrm{bucl}(\phi, \underline{a}, c{\uparrow_{d}}), \gamma{{\upharpoonright_{\cdot}}})\big). \end{align} $$

Observe that the above is equivalent to

$$\begin{align*}\forall \beta\big(\beta \prec [a]_{c{\downarrow^{b-d}}} \rightarrow S(\mathrm{bucl}(\phi, \underline{a}, c{\uparrow_{d}}),\alpha\cup\beta{{\upharpoonright_{\cdot}}})\big),\end{align*}$$

which, by induction hypothesis, is equivalent to $\forall \beta \prec [a]_{c} S(\phi ,\alpha \cup \beta {{\upharpoonright _{\cdot }}})$ .□

The above lemma motivates the following abbreviation: we define

$$\begin{align*}\mathrm{cl}(\phi,c):= \forall v^\frown \mathrm{bucl}(\phi, v, c).\end{align*}$$

In the above v is a variable with the least index among those which do not occur in $\phi $ . In particular $\mathrm {cl}(x,y)$ is a (partial) primitive recursive function, so we have a symbol for it in $\mathrm {CS}_0^+$ . $\mathrm {cl}(\phi )$ abbreviates $\mathrm {cl}(\phi ,\mathrm {FV}(\phi ))$ .

Now, the following corollary clearly follows from Lemma 3.2.

Now, we demonstrate how to use the above corollary for establishing, within $\mathrm {CS}_0^+$ , the induction on the buildup of formulae. It will be convenient to isolate a few more definitions.

Example 3.5. Let $\psi := \big (x_0 = x_1 \wedge \exists x_2( x_2 = ((0+1)+1)\cdot x_0)\big )$ and $\phi := (\forall x_0 \psi ) \wedge x_0 = x_0$ . Then

$$\begin{align*}\mathrm{Var}(\phi/\psi) = \{x_0\}.\end{align*}$$

Consequently $\mathrm {cl}_{\phi }(\psi ) = \forall x_3\forall x_0<x_3 \big (x_0 = x_1 \wedge \exists x_2 \big (x_2 = ((0+1)+1)\cdot x_0\big )\big )$ . $\triangle $

Proof. Fix a formula $\phi $ , any assignment $\zeta \in \mathrm {Asn}(\phi )$ , and a substitution of terms $\eta $ which agrees with $\zeta $ . We reason by induction on y in the $\Delta _0(\mathcal {L}_S^+)$ -formula

$$\begin{align*}\theta(y):=\forall \psi\leq_o \phi \big(\psi\in\mathrm{dp}(y)\rightarrow S(\mathrm{cl}_{\phi}(\psi\equiv \psi[\eta{{\upharpoonright_{\psi}}}]),\zeta{{\upharpoonright_{\cdot}}})\big).\end{align*}$$

Let us observe that $S(\mathrm {cl}_{\phi }(\psi \equiv \psi [\eta {{\upharpoonright _{\psi }}}]),\zeta {{\upharpoonright _{\cdot }}})$ makes sense: each occurrence of a free variable of $\psi \equiv \psi [\eta {{\upharpoonright _{\psi }}}]$ is either an occurrence of a free variable of $\phi $ and hence the variable gets assigned a value by $\zeta {{\upharpoonright _{\psi }}}$ , or is bounded in $\phi $ and so, belonging to $\mathrm {Var}(\phi /\psi )$ , gets bounded by a quantifier occurring in a prefix of $\mathrm {cl}_{\phi }(\psi \equiv \psi [\eta {{\upharpoonright _{\psi }}}])$ .

Let z be the least variable not occurring in $\phi $ . We show that $\theta (0)$ holds. The unique sentences of depth $0$ are atomic sentences, so let us fix two terms $s,t$ and argue that

$$\begin{align*}S\big(\mathrm{cl}_{\phi}(s=t\equiv (s=t[\eta{{\upharpoonright_{s=t}}}])), \zeta{{\upharpoonright_{\cdot}}}\big)\end{align*}$$

holds. Let $c = \mathrm {Var}(\phi /\psi )$ . Consequently

$$ \begin{align*}\mathrm{cl}_{\phi}(s=t\equiv (s=t[\eta{{\upharpoonright_{s=t}}}])) = \forall z ^\frown \mathrm{bucl}(s=t\equiv (s=t[\eta{{\upharpoonright_{s=t}}}]), z, c).\end{align*} $$

Call the above sentence on the right-hand side $\xi $ . By Corollary 3.3 we know that $S(\xi , \zeta {{\upharpoonright _{\cdot }}})$ is equivalent to

$$\begin{align*}\forall \alpha\in \mathrm{Asn}\big(\mathrm{dom}(\alpha) = c\rightarrow S(s=t\equiv (s=t[\eta{{\upharpoonright_{s=t}}}]), \zeta{{\upharpoonright_{\xi}}}\cup \alpha)\big).\end{align*}$$

The last sentence holds, since, by the compositional conditions for atomic sentences and connectives, it is equivalent to the assertion that for every $\alpha \in \mathrm {Asn}$ such that $\mathrm {dom}(\alpha )=c$

($) $$ \begin{align} (s^{\zeta{\upharpoonright_{\xi}}\cup\alpha} = t^{\zeta{\upharpoonright_{\xi}}\cup\alpha}) \equiv (s[\eta{{\upharpoonright_{s=t}}}]^{\zeta{{\upharpoonright_{s=t[\eta{{\upharpoonright_{s=t}}}]}}}\cup\alpha} = t[\eta{{\upharpoonright_{s=t}}}]^{\zeta{{\upharpoonright_{s=t[\eta{{\upharpoonright_{s=t}}}]}}}\cup\alpha}). \end{align} $$

To avoid double restrictions let us abbreviate $\zeta {\upharpoonright _{\xi }}$ with $\zeta _{\xi }$ . Observe that $\zeta _{\xi }{\upharpoonright _{s=t}} = \zeta _{\xi }$ . To prove ($) it is sufficient to show that each occurrence of a variable in either s or t gets assigned the same value on both sides. This holds since if $o\in Occ(s)$ , then either $o\in \mathrm {dom}(\eta {\upharpoonright _{s=t}})$ or not. In the former case by our assumption $\zeta _{\xi }(v_o)=\zeta (v_o) = ({\eta (o)})^{\circ }$ . If $o\notin \mathrm {dom}(\eta {\upharpoonright _{s=t}})$ , then $o\notin \mathrm {dom}(\eta )$ and ${v_o\in \mathrm {dom}(\zeta _{\xi }{{\upharpoonright _{s=t[\eta {{\upharpoonright _{s=t}}}]}}}\cup \alpha )\subseteq \mathrm {dom}(\zeta _{\xi }\cup \alpha )}$ , and hence o get assigned the same value on both sides of the equivalence. Consequently

$$\begin{align*}s[\eta{{\upharpoonright_{s=t}}}]^{\zeta_{\xi}{{\upharpoonright_{s=t[\eta{{\upharpoonright_{s=t}}}]}}}\cup\alpha} = s^{\zeta_{\xi}\cup\alpha}.\end{align*}$$

The same holds for t in place of s. Now assume that the thesis holds for y and consider (an occurrence of) $\psi $ of depth $y+1$ . We shall do the case of $\psi = \psi _1\vee \psi _2$ and $\psi = \exists v \psi _3$ . We treat them simultaneously. As previously put $\xi := \mathrm {cl}_{\phi }\big (\psi \equiv (\psi [\eta {{\upharpoonright _{\psi }}}])\big )$ , $\xi _i := \mathrm {cl}_{\phi }\big (\psi _i\equiv (\psi _i[\eta {{\upharpoonright _{\psi _i}}}])\big )$ , $c = \mathrm {Var}(\phi /\psi )$ , and $c_i = \mathrm {Var}(\phi /\psi _i)$ . Applying the inductive assumption and Corollary 3.3 we have for $i\in \{1,2,3\}$

$$\begin{align*}\forall \alpha\in \mathrm{Asn}\big(\mathrm{dom}(\alpha) = c_i \rightarrow S(\psi_i\equiv (\psi_i[\eta{{\upharpoonright_{\psi_i}}}]), \zeta{{\upharpoonright_{\xi_i}}}\cup \alpha)\big).\end{align*}$$

Now observe that $\zeta {\upharpoonright _{\xi _i}}{\upharpoonright _{\psi _i}} = \zeta {\upharpoonright _{\xi _i}}$ and if $\mathrm {dom}(\alpha ) = c_i$ , then $\alpha {\upharpoonright _{\psi _i}} = \alpha $ . By this and compositional conditions, for arbitrary $\alpha $ such that $\mathrm {dom}(\alpha ) = c_i$ , the succedent of the above implication is equivalent to

(3) $$ \begin{align} S(\psi_i, \zeta{{\upharpoonright_{\xi_i}}}\cup \alpha) \equiv S(\psi_i[\eta{{\upharpoonright_{\psi_i}}}],\zeta{{\upharpoonright_{\xi_i}}}{{\upharpoonright_{\psi_i[\eta{{\upharpoonright_{\psi_i}}}]}}}\cup \alpha{{\upharpoonright_{\psi_i[\eta{{\upharpoonright_{\psi_i}}}]}}}). \end{align} $$

In the case $\psi = \psi _1\vee \psi _2$ we have $c = c_1\cup c_2$ . Now, by compositional conditions, the following are equivalent for an arbitrary assignment $\alpha $ such that $\mathrm {dom}(\alpha ) = c$ :

(4) $$ \begin{align} S(\psi_1\vee \psi_2&\equiv (\psi_1\vee\psi_2[\eta{{\upharpoonright_{\psi_1\vee\psi_2}}}]),\zeta{{\upharpoonright_{\xi}}}\cup \alpha). \end{align} $$

(5) $$ \begin{align} \left(\bigvee_{i\in\{0,1\}} S(\psi_i, \zeta{{\upharpoonright_{\xi}}}{{\upharpoonright_{\psi_i}}}\cup\alpha{{\upharpoonright_{\psi_i}}})\right) &\equiv\left( \bigvee_{i\in\{0,1\}} S(\psi_i[\eta{{\upharpoonright_{\psi_i}}}], \zeta{{\upharpoonright_{\xi}}}{{\upharpoonright_{\psi_i[\eta{{\upharpoonright_{\psi_i}}}]}}}\cup\alpha{{\upharpoonright_{\psi_i[\eta{{\upharpoonright_{\psi_i}}}]}}})\right). \end{align} $$

Now observe that $\zeta {{\upharpoonright _{\xi }}}{{\upharpoonright _{\psi _i}}} = \zeta {{\upharpoonright _{\xi _i}}}{{\upharpoonright _{\psi _i}}} = \zeta {\upharpoonright _{\xi _i}}$ . Indeed, v is a free variable in $\mathrm {cl}_{\phi }(\psi \equiv \psi [\eta {{\upharpoonright _{\psi }}}])$ ( $=\xi $ ) and a free variable in $\psi _i$ , if and only if v is a free variable in $\mathrm {cl}_{\phi }(\psi _i\equiv \psi _i[\eta {\upharpoonright _{\psi _i}}])$ ( $=\xi _i$ ) and in $\psi _i$ . Hence $\mathrm {dom}(\zeta {{\upharpoonright _{\xi }}}{{\upharpoonright _{\psi _i}}})= \mathrm {dom}(\zeta {{\upharpoonright _{\xi _i}}}{{\upharpoonright _{\psi _i}}})$ and this completes our claim. The same reasoning shows also that $\zeta {{\upharpoonright _{\xi }}}{{\upharpoonright _{\psi _i[\eta {\upharpoonright _{\psi _i}}]}}} = \zeta {{\upharpoonright _{\xi _i}}}{{\upharpoonright _{\psi _i[\eta {\upharpoonright _{\psi _i}}]}}}$ . Finally, if $\mathrm {dom}(\alpha ) = c$ , then $\mathrm {dom}(\alpha {\upharpoonright _{\psi _i}}) = c_i$ . It follows that (3) implies the above condition $(5)$ and the case of $\vee $ is done.

In the case of $\exists $ , we observe that

$$ \begin{align*} c_3 = \left\{\begin{array}{l} c\cup \{v\},\ \mathrm{ if }\ v\in\ \mathrm{FV}(\psi_3),\\ c,\ \mathrm{ otherwise.}\end{array} \right. \end{align*} $$

The following are equivalent for every $\alpha \in \mathrm {Asn}$ such that $\mathrm {dom}(\alpha ) = c$ :

(6) $$ \begin{align} S(\exists v \psi_3&\equiv (\exists v\psi_3[\eta{{\upharpoonright_{\exists v\psi_3}}}]), \zeta{{\upharpoonright_{\xi}}}\cup\alpha). \end{align} $$

(7) $$ \begin{align} &\bigg(\exists \beta\geq_v(\alpha\cup \zeta{{\upharpoonright_{\xi}}}){\upharpoonright_{\exists v \psi_3}} S(\psi_3, \beta{{\upharpoonright_{\psi_3}}})\bigg) \nonumber\\ & \quad \equiv \bigg(\exists \beta\geq_v(\alpha\cup \zeta{{\upharpoonright_{\xi}}}){{\upharpoonright_{\exists v\psi_3[\eta{{\upharpoonright_{\exists v\psi_3}}}]}}} S(\psi_3[\eta{{\upharpoonright_{\psi_3}}}], \beta{{\upharpoonright_{\psi_3[\eta{{\upharpoonright_{\psi_3}}}]}}})\bigg). \end{align} $$

(8) $$ \begin{align} &\bigg(\exists \beta\geq_v(\alpha\cup \zeta{{\upharpoonright_{\xi}}}) S(\psi_3, \beta{{\upharpoonright_{\psi_3}}})\bigg)\nonumber \\ & \quad \equiv \bigg(\exists \beta\geq_v(\alpha\cup \zeta{{\upharpoonright_{\xi}}}){{\upharpoonright_{\exists v\psi_3[\eta{{\upharpoonright_{\exists v\psi_3}}}]}}} S(\psi_3[\eta{{\upharpoonright_{\psi_3}}}], \beta{{\upharpoonright_{\psi_3[\eta{{\upharpoonright_{\psi_3}}}]}}})\bigg). \end{align} $$

Observe that $v\notin \mathrm {dom}(\zeta {{\upharpoonright _{\xi }}})\cup \mathrm {dom}(\zeta {{\upharpoonright _{\xi _3}}})$ , because every occurrence of v in $\exists v\psi _3$ is bounded in $\phi $ . Hence $\zeta {{\upharpoonright _{\xi }}} = \zeta {{\upharpoonright _{\xi _3}}}$ , and consequently $\zeta {{\upharpoonright _{\xi }}}{{\upharpoonright _{\psi _3}}} = \zeta {{\upharpoonright _{\xi _3}}}{{\upharpoonright _{\psi _3}}} = \zeta {{\upharpoonright _{\xi _3}}}$ . With this observation the proof in the case $v\notin \mathrm {FV}(\psi _3)$ is straightforward, for $(8)$ immediately reduces to (for all $\alpha $ such that $\mathrm {dom}(\alpha ) = c_3$ )

$$\begin{align*}\bigg(S(\psi_3, \alpha\cup \zeta{{\upharpoonright_{\xi_3}}})\bigg)\equiv \bigg(S(\psi_3[\eta{{\upharpoonright_{\psi_3}}}], (\alpha\cup \zeta{{\upharpoonright_{\xi_3}}}){{\upharpoonright_{\psi_3[\eta{{\upharpoonright_{\psi_3}}}]}}})\bigg).\end{align*}$$

The above is the same as our induction assumption. So we may assume that $v\in \mathrm {FV}(\psi _3)$ . Now fix $\alpha $ such that $\mathrm {dom}(\alpha ) =c$ . Suppose first that $\beta \geq _v (\alpha \cup \zeta {{\upharpoonright _{\xi _3}}})$ is such that $S(\psi _3, \beta {{\upharpoonright _{\psi _3}}})$ holds. Let us observe that in this case, $\beta {\upharpoonright _{\psi _3}} = \beta $ . Moreover $\mathrm {dom}(\alpha )\cap \mathrm {dom}(\zeta {{\upharpoonright _{\xi _3}}}) = \emptyset $ ; hence for some $\alpha '$ such that $\mathrm {dom}(\alpha ') = c_3$ , $\beta = \alpha '\cup \zeta {\upharpoonright _{\xi _3}}$ . Hence $S(\psi _3[\eta {{\upharpoonright _{\psi _3}}}], \beta {{\upharpoonright _{\psi _3[\eta {{\upharpoonright _{\psi _3}}}]}}})$ follows by induction assumption (3). It is left to show that $\beta {{\upharpoonright _{\psi _3[\eta {{\upharpoonright _{\exists v\psi _3}}}]}}}\geq _v (\alpha \cup \zeta {{\upharpoonright _{\xi }}}){{\upharpoonright _{\exists v\psi _3[\eta {{\upharpoonright _{\exists v\psi _3}}}]}}}$ . This holds since no occurrence of v is in $\mathrm {dom}(\eta {{\upharpoonright _{\psi _3}}})=\mathrm {dom}(\eta {{\upharpoonright _{\exists v\psi _3}}})$ and $\beta \geq _v (\alpha \cup \zeta {{\upharpoonright _{\xi _3}}})$ . So now assume that for some $\beta \geq _v(\alpha \cup \zeta {{\upharpoonright _{\xi _3}}}){{\upharpoonright _{\exists v\psi _3[\eta {{\upharpoonright _{\exists v\psi _3}}}]}}} S(\psi _3[\eta {{\upharpoonright _{\psi _3}}}], \beta {{\upharpoonright _{\psi _3[\eta {{\upharpoonright _{\psi _3}}}]}}})$ holds. As no occurrence of v is in $\mathrm {dom}(\eta {\upharpoonright _{\psi _3}})$ , we can infer that $S(\psi _3[\eta {{\upharpoonright _{\psi _3}}}], \beta )$ holds. Extend $\alpha $ to $\alpha '$ such that $\mathrm {dom}(\alpha ') = c_3$ and $\alpha '(v) = \beta (v)$ . Then $(\alpha '\cup \zeta {\upharpoonright _{\xi _3}}){\upharpoonright _{\psi _3[\eta {{\upharpoonright _{\psi _3}}}]}} = \beta $ and by (3) we obtain $S(\psi _3, \alpha '\cup \zeta {\upharpoonright _{\xi _3}})$ . Hence there exists $\beta '$ such that $\beta '\geq _v \alpha \cup \zeta {\upharpoonright _{\xi _3}}$ and $S(\psi _3,\beta ')$ . This ends the whole proof.□

Proof. By the previous considerations at the beginning of this section, for a fixed $\phi (v)$ , $S(\mathrm {Ind}(\phi (v)), \varepsilon )$ is equivalent to

$$\begin{align*}S(\phi[0/v], \varepsilon)\wedge \forall x\big(S(\phi(v),[x])\rightarrow S(\phi[v+1/v],[x])\big)\longrightarrow \forall x S(\phi(v), [x]).\end{align*}$$

By Lemma 3.6 and Corollary 3.7 we have

$$ \begin{align*} &S(\phi[0/v], \varepsilon) \equiv S(\phi(v), [0]),\\ &\forall x \big(S(\phi[v+1/v], [x])\equiv S(\phi(v), [x+1])\big). \end{align*} $$

Hence, finally $S(\mathrm {Ind}(\phi (v)), \varepsilon )$ is equivalent to the following axiom of $\Delta _0(\mathcal {L}_S^+)$ -induction:

$$\begin{align*}S(\phi(v), [0])\wedge \forall x\big(S(\phi(v),[x])\rightarrow S(\phi(v),[x+1])\big)\longrightarrow \forall x S(\phi(v), [x]).\\[-34pt] \end{align*}$$

□

Recall that $\phi (t/v)$ denotes the substitution of t for all (free) occurrences of v in $\phi (v)$ .

Proof. We reason in $\mathrm {CS}_0^+$ . We fix a sequent calculus for the first-order logic with equality, as in [Reference Takeuti24] (this choice is just a matter of convenienceFootnote ¹¹ ). We fix a proof p of a sentence $\phi $ and by induction on its length argue that whenever a sequent $\Gamma \Rightarrow \Delta $ occurs in p, then

(9) $$ \begin{align} S(\mathrm{cl}\left(\bigwedge\Gamma\rightarrow \bigvee \Delta\right), \varepsilon) \end{align} $$

holds, where $\bigwedge \Gamma $ and $\bigvee \Gamma $ denote (the canonically parenthesized) conjunction and disjunction over sentences from sets $\Gamma $ and $\Delta $ , respectively. To simplify the notation $\bigwedge \Gamma \rightarrow \bigvee \Delta $ will be abbreviated using the sequent notation as $\Gamma \Rightarrow \Delta $ . In the course of the induction we rely on the fact that the following sentence is provable in $\mathrm {CS}_0$ :

(DC) $$ \begin{align} \forall \alpha \in \mathrm{Asn}(\Gamma)\big(S\left(\bigvee\Gamma,\alpha\right)\equiv \exists \phi \in \Gamma S(\phi,\alpha{{\upharpoonright_{\phi}}})\big). \end{align} $$

The proof of (DC) in $\mathrm {CT}_0$ consists in a straightforward induction on the size of $\Gamma $ and a similar argument can be given in the case of $\bigwedge $ yielding a dual equivalence (see also [Reference Cieśliński3, Reference Wcisło and Łełyk25] for precise arguments). We go back to the main induction on the length of the fixed proof p. In the base step we have to establish that all initial sequents satisfy (9). These include initial sequents for equality and all sequents of the form $\phi \Rightarrow \phi $ . In both cases the proof follows the same pattern: first using Corollary 3.3 we get rid of the quantifier prefix and then verify that the formula following it is satisfied by every assignment. In the case of the initial sequents for equality we use the conditions for atomic sentences from $\mathrm {CS}^-$ , in case of $\phi \Rightarrow \phi $ we use the axioms for $\neg $ and $\vee $ .

In the induction step the cases of quantifier rules are the unique non-obvious ones. Let us consider the dictum de omni rule:

$$\begin{align*}\frac{\Gamma, \phi(t)\Rightarrow \Delta}{\Gamma, \forall v \phi(v)\Rightarrow\Delta},\end{align*}$$

where $\phi (v)$ is a formula and t is free for v in $\phi (v)$ . We may safely assume that v is a free variable in $\phi $ . For simplicity abbreviate $\phi (t/v)$ with simply $\phi (t)$ . So suppose for every $\alpha \in \mathrm {Asn}\left (\big (\Gamma +\phi (t)\big )\Rightarrow \Delta \right )$ it holds that

(10) $$ \begin{align} S\left(\big(\Gamma+\phi(t)\big)\Rightarrow\Delta, \alpha\right). \end{align} $$

Fix an arbitrary $\alpha \in \mathrm {Asn}\left (\big (\Gamma +\forall v\phi (v)\big )\Rightarrow \Delta \right )$ and assume that for every $\theta \in \Gamma \cup \{\forall v \phi (v)\}$ , $S(\theta ,\alpha {{\upharpoonright _{\cdot }}})$ holds. Consider any $\beta \in \mathrm {Asn}\left (\big (\Gamma +\phi (t)\big )\Rightarrow \Delta \right )$ such that $\beta {{\upharpoonright _{\big (\Gamma +\forall v\phi (v)\big )\Rightarrow \Delta }}}=\alpha $ . Since $S(\forall v \phi (v), \beta {{\upharpoonright _{\cdot }}})$ , then by Lemma 3.9, $S(\phi (t), \beta {{\upharpoonright _{\cdot }}})$ as well. Hence for every $\theta \in \Gamma \cup \{\phi (t)\}\ \mathrm{it\ holds\ that}\ S(\theta , \beta {{\upharpoonright _{\cdot }}})$ . By the induction assumption, for some $\psi \in \Delta $ , $S(\psi ,\beta {{\upharpoonright _{\cdot }}})$ and since $\beta {{\upharpoonright _{\psi }}} = \alpha {{\upharpoonright _{\psi }}}$ , this ends the proof.

Now consider the rule of universal generalisation

$$\begin{align*}\frac{\Gamma\Rightarrow \Delta,\phi(v)}{\Gamma\Rightarrow\Delta, \forall v\phi(v)},\end{align*}$$

where v does not occur free in the lower sequent. As previously assume that for all $\alpha \in \mathrm {Asn}\left (\Gamma \Rightarrow (\Delta +\phi (v))\right )$ ,

$$\begin{align*}S\big(\Gamma\Rightarrow \big(\Delta+\phi(v)\big),\alpha\big).\end{align*}$$

Fix an arbitrary $\beta \in \mathrm {Asn}\left (\Gamma \Rightarrow \big (\Delta +\forall v\phi (v)\big )\right )$ and arbitrary $\gamma \geq _v\beta $ and assume that for all $\psi \in \Gamma $ , $S(\psi , \gamma {{\upharpoonright _{\cdot }}})$ . Since $\gamma {{\upharpoonright _{\Gamma }}} = \beta {{\upharpoonright _{\Gamma }}}$ (v is not a free variable in $\Gamma $ ) it follows that there is $\psi \in \Delta \cup \{\phi (v)\}$ such that $S(\psi ,\gamma {{\upharpoonright _{\cdot }}})$ . If $\psi \in \Delta $ , then we are done, since $\gamma {{\upharpoonright _{\Delta }}} = \beta {{\upharpoonright _{\Delta }}}$ . Otherwise $S(\phi (v),\gamma {\upharpoonright _{\cdot }})$ and it follows that $S(\forall v\phi (v),\beta {{\upharpoonright _{\cdot }}})$ , since $\gamma $ was arbitrary.

Now, the thesis of the theorem follows, since, for arbitrary $\phi $ , if $\mathrm {Pr}_{\emptyset }^S(\phi )$ holds, then there is a sequent calculus proof of $\Gamma \Rightarrow \phi $ , where for every $\psi \in \Gamma $ we have $S(\psi ,\varepsilon )$ . Hence, by the proof above we obtain

$$\begin{align*}S\left(\mathrm{cl}\left(\bigwedge \Gamma\rightarrow \phi\right), \varepsilon\right),\end{align*}$$

and since $\bigwedge \Gamma \rightarrow \phi $ does not admit any free variables, then we have $S(\bigwedge \Gamma \rightarrow \phi ,\varepsilon )$ . The thesis follows by the conjunctive correctness and compositional conditions: since for every $\psi \in \Gamma $ we have $S(\psi ,\varepsilon )$ , then $S(\bigwedge \Gamma ,\varepsilon )$ holds and an application of Modus Ponens yields $S(\phi ,\varepsilon )$ .□

Corollary 3.11. For every Gödelized theory $\mathrm {Th}$ , we have

$$\begin{align*}\mathrm{CS}_0 + \forall x \big(\mathrm{Th}(x)\rightarrow S(x,\varepsilon)\big)\vdash \forall \phi \big(\mathrm{Pr}_{\mathrm{Th}}(\phi)\rightarrow S(\phi,\varepsilon)\big).\end{align*}$$

Hence,

$$\begin{align*}\mathrm{CS}_0 + \forall x \big(\mathrm{Th}(x)\rightarrow S(x,\varepsilon)\big)\vdash\mathrm{REF}^{\omega}(\mathrm{Th}).\end{align*}$$

Proof. The first part follows directly from Theorem 3.10. Having it, we prove the second one: by induction on n we prove that

$$\begin{align*}\mathrm{CS}_0+\forall x \big(\mathrm{Th}(x)\rightarrow S(x,\varepsilon)\big)\vdash \forall \phi\big(\mathrm{Pr}_{\mathrm{REF}^n(\mathrm{Th})}(\phi)\rightarrow S(\phi,\varepsilon)\big).\end{align*}$$

For $n=0$ this follows from the first part. Fix n and assume the thesis holds for it. By the compositional clauses for $\mathrm {CS}^-$ we have

$$\begin{align*}\mathrm{CS}_0+\forall x \big(\mathrm{Th}(x)\rightarrow S(x,\varepsilon)\big)\vdash \forall \phi\big(S(\ulcorner \mathrm{Pr}_{\mathrm{REF}^n(\mathrm{Th})}(\phi)\rightarrow \phi \urcorner, \varepsilon)\big).\end{align*}$$

Consequently, reapplying the first part of this corollary for $\mathrm {REF}^n(\mathrm {Th})$ substituted for $\mathrm {Th}$ we get the induction thesis for $n+1$ .□

The corollary below easily follows from the corollary above and Corollary 3.8.

3.4 Corollaries