Varieties of truth definitions

Abstract

We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence $\alpha$ which extends a weak arithmetical theory (which we take to be ${{\,\mathrm{I\Delta _{0}+\exp }\,}}$) such that for some formula $\Theta$ and any arithmetical sentence $\varphi$, $\Theta (\ulcorner \varphi \urcorner )\equiv \varphi$ is provable in $\alpha$. We say that a sentence $\beta$ is definable in a sentence $\alpha$, if there exists an unrelativized translation from the language of $\beta$ to the language of $\alpha$ which is identity on the arithmetical symbols and such that the translation of $\beta$ is provable in $\alpha$. Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not $\Sigma _2$-definable in the standard model of arithmetic. We conclude by remarking that no $\Sigma _2$-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.

Appendices

Appendices

A Characterizations of definability in terms of provability

Now we return to the problem of unifying our methods of showing comparability (Lemma 44) and incomparability (Lemma 45) of theories of the form $\mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }$, with respect to the relation of definability. We note that there is indeed a significant logical gap between the two lemmata: the sufficient condition for the definability is given in the comparability lemma in terms of provability in ${{\,\mathrm{I\Delta _{0}+\exp }\,}}$, while the necessary condition in the incomparability lemma uses a much stronger theory, ${{\,\textrm{CT}\,}}_0$. The arithmetical consequences of the latter theory of truth are exhausted only by the union of all finite iterations of uniform reflection principles over ${{\,\textrm{PA}\,}}$, the theory usually denoted by $\text {RFN}^{<\omega }({{\,\textrm{PA}\,}})$. A natural conjecture would be that the condition mentioned in the remark immediately following the comparability lemma is not only sufficient, but also necessary, i.e. we have the following equivalence for any $\alpha , \beta$ such that $\mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }, \mathrm{{CT}}^{-}{\upharpoonright }_{\beta }\in \Delta$:

As we already remarked, the left-to-right implication indeed holds. For the converse one, one could try the following strategy: assume that $\mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }\blacktriangleleft \mathrm{{CT}}^{-}{\upharpoonright }_{\beta }$ but for every $n\in \omega$ ${{\,\mathrm{I\Delta _{0}+\exp }\,}}+\exists x (\alpha (x+n)\wedge \lnot \beta (x))$ is consistent. Then by a straightforward compactness argument, one can show that there is a model ${\mathcal {M}}\models {{\,\mathrm{I\Delta _{0}+\exp }\,}}$, in which for some number $c>\omega$, the length of the initial segment determined by $\alpha$ is at least by c greater than the length of the initial segment determined by $\beta$. The same relation between $\alpha$ and $\beta$ holds in any elementary extension of ${\mathcal {M}}$. However, there is ${\mathcal {N}}$, an elementary extension of ${\mathcal {M}}$, in which there is a subset $T\subseteq N$ such that $({\mathcal {N}},T)\models {\text {CT}}^{-}$ (see e.g. [26]). Consider $T_{\beta }:= \{\sigma \in N: \sigma \in T\wedge {\mathcal {N}}\models \beta ({{\,\textrm{dpt}\,}}(\sigma ))\}$. Then $({\mathcal {N}}, T_{\beta })\models \mathrm{{CT}}^{-}{\upharpoonright }_{\beta }$. So there is a formula $\Theta (x)\in {{\,\mathrm{\mathcal {L}_{T}}\,}}$ such that $({\mathcal {N}}, T_{\beta })\models {\text {CT}}^{-}_{\alpha }[\Theta (x)/T(x)]$. In the previous sentence, and below, for $\Phi (x)\in {{\,\mathrm{\mathcal {L}_{T}}\,}}$ and any formu $\Psi (x)$,

$$\begin{aligned} \Phi [\Psi (t)/T(t)] \end{aligned}$$

denotes the result if a substitution of a formula $\Psi (t)$ for any occurrence of a formula T(t), possibly preceded by renaming the bounded variables. The last conclusion may seem contradictory, since e.g. in the standard model one can define partial truth predicates, but the truth predicate for all formulae is not definable. However, this intuition breaks down in the case of nonstadard models and truth predicates that do not have to satisfy induction axioms⁸.

Proposition 60

(Wcisło) Assume that ${\mathcal {M}}$ is a countable and recursively saturated model of ${{\,\textrm{PA}\,}}$ and $c\in M$ is a nonstadard element which is definable without parameters in ${\mathcal {M}}$. Then, there is $T\subseteq M$ such that $({\mathcal {M}}, T)\models {\text {CT}}^{-}(c)\wedge \forall x \bigl (T(x)\rightarrow {{\,\textrm{dpt}\,}}(x)\leqslant c\bigr )$ and for some formula $\Theta (x)$, $({\mathcal {M}}, T)\models {\text {CT}}^{-}[\Theta (x)/T(x)]$.

In other words, for an arbitrary countable recursively satrurated model and for an arbitrary nonstandard definable c, one can find a truth predicate for sentences of depth at most c, which can define a full truth predicate for all sentences in the sense of ${\mathcal {M}}$.

Proof

(Sketch of proof) Fix ${\mathcal {M}}$ and c as in the assumptions. Let $\varepsilon _k(x)$ be a formula defined inductively

$$\begin{aligned} \varepsilon _0(x)&:= x=x\\ \varepsilon _{k+1}(x)&:= \varepsilon _k(x)\vee \varepsilon _k(x) \end{aligned}$$

Thus the syntactic tree of $\varepsilon _k(x)$ is a full binary tree of depth k with internal vertices and root labelled by disjunctions and with the formula $x=x$ in the leaves. Let $T'\subseteq M$ be such that $({\mathcal {M}},T')\models {\text {CT}}^{-}$ and $({\mathcal {M}}, T')$ is recursively saturated (such a $T'$ exists by the conservativity of ${\text {CT}}^{-}$ over ${{\,\textrm{PA}\,}}$ and the chronic resplendency of countable recursively saturated models). Then, by Theorem 3.3 of [22], there is a $T''\subseteq M$ such that $({\mathcal {M}}, T'')\models {\text {CT}}^{-}$ and⁹

$$\begin{aligned} T':= \{a\in M: ({\mathcal {M}}, T'')\models T''[\varepsilon _c(\dot{a})]\}. \end{aligned}$$

Observe now that $\varepsilon _c(x)$ is a formula of depth c and it is definable without parameters (since it is definable from c). Hence, for $T:= \{\sigma \in M: \sigma \in T''\wedge {\mathcal {M}}\models {{\,\textrm{dpt}\,}}(\sigma )\leqslant c\}$, set $T'$ is definable without parameters in $({\mathcal {M}}, T)\models {\text {CT}}^{-}(c)\wedge \forall x \bigl (T(x)\rightarrow {{\,\textrm{dpt}\,}}(x)\leqslant c\bigr )$. $\square$

Obviously, the above example is not sufficient to disprove conjecture (*). However, we see no way to fix the proof strategy described above.

Having said that, we note that if we assume that our definitions of truth extend ${{\,\textrm{PA}\,}}$ instead of ${{\,\mathrm{I\Delta _{0}+\exp }\,}}$ (i.e. are “finitely axiomatisable modulo ${{\,\textrm{PA}\,}}$”), then we have the following version of the incomparability lemma. To motivate the formulation of the thesis, let us observe that there is a constant k such that for all $n>0$, ${{\,\textrm{Tr}\,}}_n(x)$ (see “The arithmetization of syntax and metalogic” in the preliminaries) has depth at most $k\cdot n$. Let $k_0$ be the least such k.

Proposition 61

For all $\alpha$, $\beta$ such that $\mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }, \mathrm{{CT}}^{-}{\upharpoonright }_{\beta }\in \Delta$,

$$\begin{aligned} (\mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }+{{\,\textrm{PA}\,}})\blacktriangleleft (\mathrm{{CT}}^{-}{\upharpoonright }_{\beta }+{{\,\textrm{PA}\,}})\Longrightarrow \bigl (\exists n\in \omega \ {{\,\textrm{PA}\,}}\vdash \forall x (\alpha (k_0\cdot x+n)\rightarrow \beta (x))\bigr ). \end{aligned}$$

Proof

(Sketch of proof) In the proof we deal with two satisfaction relations for models that are strongly interpretable in a fixed model ${\mathcal {M}}\models {{\,\textrm{PA}\,}}$. For a model ${\mathcal {K}}$ strongly interpretable in ${\mathcal {M}}$, ${\mathcal {K}}\models \phi$ denotes the satisfiability of $\phi$ by ${\mathcal {K}}$ according to the standard satisfaction relation (as defined outside of ${\mathcal {M}}$), and ${\mathcal {K}}\models ^{{\mathcal {M}}} \phi$ denotes the satisfiability according to the ${\mathcal {M}}$-definable satisfaction relation. We observe that whenever $\phi$ is a standard sentence and ${\mathcal {K}}$ is any model which is strongly interpretable in ${\mathcal {M}}$, then ${\mathcal {K}}\models \phi$ iff ${\mathcal {K}}\models ^{{\mathcal {M}}}\phi$ (“satisfaction is absolute”). We now pass to the argument.

As previously, suppose that $(\mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }+{{\,\text {PA}\,}})\blacktriangleleft (\mathrm{{CT}}^{-}{\upharpoonright }_{\beta }+{{\,\text {PA}\,}})$ and that there exists a nonstandard model ${\mathcal {M}}\models {{\,\textrm{PA}\,}}$ with nonstandard numbers $a,c\in M$ such that ${\mathcal {M}}\models \alpha (k_0\cdot a+c)\wedge \lnot \beta (a)$. Let $\Phi$ be such that $\mathrm{{CT}}^{-}{\upharpoonright }_{\beta }+{{\,\text {PA}\,}}\vdash \mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }[\Phi (x)/T(x)]$. Since both $\alpha$ and $\beta$ define initial segments, this means that in ${\mathcal {M}}$ the length of the initial segment defined by $\alpha$ is much greater than the length of the initial segment defined by $\beta$. Let n be such that both $\alpha$ and $\beta$ are of depth at most n and let k be the depth of $\Phi$. By the reflexivity of ${{\,\textrm{PA}\,}}$ and an easy overspill argument there is a nonstandard number d such that in ${\mathcal {M}}$ there is no proof of $0=1$ from the axioms of ${{\,\mathrm{I\Sigma _{d}}\,}}$ and true arithmetical sentences of depth at most n, i.e.

$$\begin{aligned} {\mathcal {M}}\models {{\,\textrm{Con}\,}}({{\,\mathrm{I\Sigma _{d}}\,}}+{{\,\textrm{Tr}\,}}_{n}). \end{aligned}$$

By the arithmetized completeness theorem, there is a strongly interpretable in ${\mathcal {M}}$ model ${\mathcal {N}}$ such that ${\mathcal {N}}\models ^{{\mathcal {M}}}{{\,\mathrm{I\Sigma _{d}}\,}}+{{\,\textrm{Tr}\,}}_{n}$.

In particular, ${\mathcal {M}}$ is (${\mathcal {M}}$-definably isomorphic to) an n-elementary initial segment of ${\mathcal {N}}$, hence we have ${\mathcal {N}}\models \alpha (k_0\cdot a+c)\wedge \lnot \beta (a)$. Moreover, from an external perspective, ${\mathcal {N}}\models {{\,\textrm{PA}\,}}$ (but ${\mathcal {M}}$ may not know this).

We now work in ${\mathcal {M}}$. Let e be the greatest element such that ${\mathcal {N}}\models ^{{\mathcal {M}}} \beta (e)$. Then $e\leqslant a$. For any $y\in M$, a formula ${{\,\textrm{Tr}\,}}_{y}(x)$ is a partial truth predicate for sentences of complexity y in the sense of ${\mathcal {N}}$. Even though for nonstandard y, ${{\,\textrm{Tr}\,}}_y$ is a nonstandard formula, we can make sense of it using $\models ^{{\mathcal {M}}}$. Observe that ${\mathcal {N}}\models ^{{\mathcal {M}}} {\text {CT}}^{-}_{\beta }[{{\,\textrm{Tr}\,}}_{e}(x)/T(x)]$. Put $T_e:= \{x\in N: {\mathcal {N}}\models ^{{\mathcal {M}}} {{\,\textrm{Tr}\,}}_e(x)\}$. Since $T_e$ is a definable subset of ${\mathcal {N}}$, there is an ${\mathcal {M}}$-definable satisfaction relation for the model $({\mathcal {N}}, T_e)$. Seen from the external (to ${\mathcal {M}}$) perspective, $({\mathcal {N}}, T_e)\models \mathrm{{CT}}^{-}{\upharpoonright }_{\beta }+{{\,\text {PA}\,}}$, hence also it is true that $({\mathcal {N}}, T_e)\models \mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }[\Phi (x)/T(x)]$. Since $\mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }[\Phi (x)/T(x)]$ is a standard sentence, $({\mathcal {N}}, T_e)\models ^{{\mathcal {M}}} \mathrm{{CT}}^{-}{\upharpoonright }_{\alpha }[\Phi (x)/T(x)]$. Let $\Psi := \Phi [{{\,\textrm{Tr}\,}}_e(x)/T(x)]$. Then $\Psi$ is an arithmetical formula (in the sense of ${\mathcal {M}}$) of depth at most $k_0\cdot e+k$.

It follows that ${\mathcal {N}}\models ^{{\mathcal {M}}} {{\,\textrm{CT}\,}}^-(k_0\cdot a+c)[\Psi (x)/T(x)]$, so in ${\mathcal {N}}$, $\Psi$ is a truth predicate for sentences of complexity $k_0\cdot a+c$. Let $\lambda$ be the $\Psi$-liar sentence, i.e. a sentence constructed as in the standard proof of Tarski’s theorem on the undefinability, such that ${\mathcal {N}}\models ^{{\mathcal {M}}}\lambda \leftrightarrow \lnot \Psi (\ulcorner \lambda \urcorner )$. By the inspection of the usual proof of the fixed point lemma, $\lambda$ is of depth $k_0\cdot e +l$ for some standard number l. In particular ${\mathcal {N}}\models ^{{\mathcal {M}}}\lambda \leftrightarrow \Psi (\ulcorner \lambda \urcorner )$. A contradiction. $\square$

What makes the above proof possible is the reflexivity of ${{\,\textrm{PA}\,}}$. We use it to show that every model of ${{\,\textrm{PA}\,}}$ strongly interprets a suitable model of ${{\,\textrm{PA}\,}}$.

B The lattice structure of the definability order

Theorem 42

The definability order forms a distributive lattice. The infimum and supremum operations are derived from the propositional operations as follows.

• The infimum of $\alpha$ and $\beta$ is $\alpha \vee \beta$.

• The supremum of $\alpha$ and $\beta$ is $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )$, where $\mathrm{{rs}}_{\alpha }{(x)}$ is a fixed translation that replaces non-arithmetic relational symbols in x with symbols of the same arity that are not present in $\alpha$.

Proof

The proof has three parts. First, we show that $\alpha \vee \beta$ and $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )$ are syntactically conservative over ${{\,\mathrm{I\Delta _{0}+\exp }\,}}$ definitions of truth (whenever $\alpha$ and $\beta$ are such). Then we show that $\alpha \vee \beta$ is an infimum and $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )$ is a supremum of $\alpha$ and $\beta$ (the proof also shows that the operations are well defined as operations on the equivalence classes of truth definitions modulo the relation of mutual definability, i.e. if $\alpha \blacktriangleright \!\!\blacktriangleleft \alpha '$ and $\beta \blacktriangleright \!\!\blacktriangleleft \beta '$, then $\alpha \vee \beta \blacktriangleright \!\!\blacktriangleleft \alpha '\vee \beta '$ and $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )\blacktriangleright \!\!\blacktriangleleft \alpha '\vee {{\,\textrm{rs}\,}}_{\alpha '}(\beta ')$). We complete the proof by showing that the structure satisfies distributivity axioms.

Both $\alpha \vee \beta$ and $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )$ extend ${{\,\mathrm{I\Delta _{0}+\exp }\,}}$. Given a sentence $\sigma \in {{\,\mathrm{\mathcal {L}_{{{\,\textrm{PA}\,}}}}\,}}$, suppose that $\alpha \vee \beta \vdash \sigma$, then in particular $\alpha \vdash \sigma$, so ${{\,\mathrm{I\Delta _{0}+\exp }\,}}\vdash \sigma$ by the conservativity of $\alpha$. Now suppose that $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )\vdash \sigma$, then

$$\begin{aligned} \emptyset \vdash \alpha \rightarrow ({{\,\textrm{rs}\,}}_{\alpha }(\beta )\rightarrow \sigma ). \end{aligned}$$

Since the sets of the symbols present in $\alpha$ and ${{\,\textrm{rs}\,}}_{\alpha }(\beta )$ respectively, intersect exactly at the arithmetic symbols, by Craig’s interpolation theorem (see 2.5.5 in [4]), there exists an arithmetical sentence $\tau$ such that

$$\begin{aligned} \emptyset \vdash \alpha \rightarrow \tau \ \ \text {and} \ \ \emptyset \vdash \tau \rightarrow ({{\,\textrm{rs}\,}}_{\alpha }(\beta )\rightarrow \sigma )\text {,} \end{aligned}$$

and hence $\emptyset \vdash {{\,\textrm{rs}\,}}_{\alpha }(\beta )\rightarrow (\tau \rightarrow \sigma )$. The arithmetical consequences of $\alpha$ and ${{\,\textrm{rs}\,}}_{\alpha }(\beta )$ are exactly the same, so $\emptyset \vdash {{\,\textrm{rs}\,}}_{\alpha }(\beta )\rightarrow \tau$, and therefore ${{\,\textrm{rs}\,}}_{\alpha }(\beta )\vdash \sigma$. Hence ${{\,\mathrm{I\Delta _{0}+\exp }\,}}\vdash \sigma$, since ${{\,\textrm{rs}\,}}_{\alpha }(\beta )$ is syntactically conservative over ${{\,\mathrm{I\Delta _{0}+\exp }\,}}$.

Now suppose that $\Theta _{\alpha }(x)$ and $\Theta _{\beta }(x)$ act as truth-predicates in $\alpha$ and $\beta$ respectively. Then $\Theta _{\alpha }(x)$ also acts as a truth-predicate in $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )$. As a truth-predicate in $\alpha \vee \beta$ we can use

$$\begin{aligned} (\alpha \rightarrow \Theta _{\alpha }(x))\wedge (\lnot \alpha \rightarrow \Theta _{\beta }(x)). \end{aligned}$$

We now come to the second part of the proof. We see that both $\alpha$ and $\beta$ define $\alpha \vee \beta$, as each of them proves it. Similarly, $\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )\blacktriangleright \alpha \text {, }\beta$. Now assume that $\alpha \blacktriangleright \gamma$ and $\beta \blacktriangleright \gamma$. Suppose that the former definability is given by the translation of the form ${\mathcal {L}}_{\gamma }\ni \mathfrak {R}(\overline{x})\longmapsto \varphi _{\mathfrak {R}}(\overline{x})\in {\mathcal {L}}_{\alpha }$ and that the second is given by the translation of the form ${\mathcal {L}}_{\gamma }\ni \mathfrak {R}(\overline{x})\longmapsto \psi _{\mathfrak {R}}(\overline{x})\in {\mathcal {L}}_{\beta }$. Then “$\alpha \vee \beta \blacktriangleright \gamma$” is given by the translation of the form

$$\begin{aligned} \mathfrak {R}(\overline{x})\longmapsto (\alpha \rightarrow \varphi _{\mathfrak {R}}(\overline{x})) \wedge (\lnot \alpha \rightarrow \psi _{\mathfrak {R}}(\overline{x})). \end{aligned}$$

Now assume that $\delta \blacktriangleright \alpha$ and $\delta \blacktriangleright \beta$. Suppose that the first is given by the translation of the form ${\mathcal {L}}_{\alpha }\ni \mathfrak {R}(\overline{x})\longmapsto \varphi _{\mathfrak {R}}(\overline{x})\in {\mathcal {L}}_{\delta }$ and that the second is given by the translation of the form ${\mathcal {L}}_{\beta }\ni \mathfrak {R}(\overline{x})\longmapsto \psi _{\mathfrak {R}}(\overline{x})\in {\mathcal {L}}_{\delta }$. Then “$\delta \blacktriangleright \alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )$” is given by the translation of the form:

• $\mathfrak {R}(\overline{x})\longmapsto \varphi _{\mathfrak {R}}(\overline{x})$, for $\mathfrak {R}\in {\mathcal {L}}_{\alpha }$,

• $\mathfrak {R}(\overline{x})\longmapsto \psi _{{{\,\textrm{rs}\,}}_{\alpha }^{-1}(\mathfrak {R})}(\overline{x})$, for $\mathfrak {R}\in {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\alpha }(\beta )}$.

Now we come to the third part of the proof. First, we see that “$x\cap (y\cup z)=(x\cap y)\cup (x\cap z)$” holds: given definitions of truth $\alpha$, $\beta$ and $\gamma$,

$$\begin{aligned} \alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta \vee \gamma )=(\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\beta )) \vee (\alpha \wedge {{\,\textrm{rs}\,}}_{\alpha }(\gamma )). \end{aligned}$$

Now we show that “$x\cup (y\cap z)=(x\cup y)\cap (x\cup z)$” holds, i.e. given definitions of truth $\alpha$, $\beta$ and $\gamma$,

$$\begin{aligned} \alpha \vee (\beta \wedge {{\,\textrm{rs}\,}}_{\beta }(\gamma ))\blacktriangleright \!\!\blacktriangleleft (\alpha \vee \beta )\wedge {{\,\textrm{rs}\,}}_{\alpha \vee \beta } (\alpha \vee \gamma ). \end{aligned}$$

The definability $\alpha \vee (\beta \wedge {{\,\textrm{rs}\,}}_{\beta }(\gamma ))\blacktriangleright (\alpha \vee \beta ) \wedge {{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha \vee \gamma )$ is given by the translation of the form:

• $\mathfrak {R}(\overline{x})\longmapsto \mathfrak {R}(\overline{x})$, for $\mathfrak {R}\in {\mathcal {L}}_{\alpha }\cup {\mathcal {L}}_{\beta }$,

• $\mathfrak {R}(\overline{x})\longmapsto {{\,\textrm{rs}\,}}_{\alpha \vee \beta }^{-1}(\mathfrak {R})(\overline{x})$, for $\mathfrak {R}\in {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha )}\setminus {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\gamma )}$,

• $\mathfrak {R}(\overline{x})\longmapsto {{\,\textrm{rs}\,}}_{\beta }({{\,\textrm{rs}\,}}_{\alpha \vee \beta }^{-1}(\mathfrak {R}))(\overline{x})$, for $\mathfrak {R}\in {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\gamma )}\setminus {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha )}$,

• $\mathfrak {R}(\overline{x})\longmapsto (\alpha \rightarrow {{\,\textrm{rs}\,}}_{\alpha \vee \beta }^{-1}(\mathfrak {R})(\overline{x}))\wedge (\lnot \alpha \rightarrow {{\,\textrm{rs}\,}}_{\beta }({{\,\textrm{rs}\,}}_{\alpha \vee \beta }^{-1}(\mathfrak {R}))(\overline{x}))$, for $\mathfrak {R}\in {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha )}\cap {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\gamma )}$.

We complete the proof claiming that the definability $(\alpha \vee \beta )\wedge {{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha \vee \gamma ))\blacktriangleright \alpha \vee (\beta \wedge {{\,\textrm{rs}\,}}_{\beta }(\gamma ))$ is given by the translation which maps $\mathfrak {R}(\overline{x})$ to:

• $\mathfrak {R}(\overline{x})$, for $\mathfrak {R}\in {\mathcal {L}}_{\beta }\setminus {\mathcal {L}}_{\alpha }$,

• $\bigl [{{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha )\rightarrow {{\,\textrm{rs}\,}}_{\alpha \vee \beta } (\mathfrak {R})(\overline{x})\bigr ]\wedge \bigl [\lnot {{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha )\rightarrow \bigl ((\alpha \rightarrow \mathfrak {R}(\overline{x}))\wedge (\lnot \alpha \rightarrow {{\,\textrm{rs}\,}}_{\alpha \vee \beta } ({{\,\textrm{rs}\,}}_{\beta }^{-1}(\mathfrak {R}))(\overline{x}))\bigr )\bigr ]$, for $\mathfrak {R}\in {\mathcal {L}}_{\alpha }\cap {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )}$,

• $({{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha )\rightarrow {{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\mathfrak {R})(\overline{x})) \wedge (\lnot {{\,\textrm{rs}\,}}_{\alpha \vee \beta }(\alpha )\rightarrow \mathfrak {R}(\overline{x}))$, for $\mathfrak {R}\in {\mathcal {L}}_{\alpha }\setminus {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )}$,

• ${{\,\textrm{rs}\,}}_{\alpha \vee \beta }({{\,\textrm{rs}\,}}_{\beta }^{-1}(\mathfrak {R}))(\overline{x})$, for $\mathfrak {R}\in {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )}\setminus {\mathcal {L}}_{\alpha }$.

Notice that ${\mathcal {L}}_{\alpha }\cup {\mathcal {L}}_{\beta }\cup {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )}= ({\mathcal {L}}_{\beta }{\setminus }{\mathcal {L}}_{\alpha })\oplus ({\mathcal {L}}_{\alpha }\cap {\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )}) \oplus ({\mathcal {L}}_{\alpha }{\setminus }{\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )})\oplus ({\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )} {\setminus }{\mathcal {L}}_{\alpha })$ as ${\mathcal {L}}_{\alpha }{\setminus }{\mathcal {L}}_{{{\,\textrm{rs}\,}}_{\beta }(\gamma )} \supseteq {\mathcal {L}}_{\alpha }\cap {\mathcal {L}}_{\beta }$. $\square$