1 Prelude

Robinson’s Arithmetic Q is an old friend. The first time I met it was when I studied the First Incompleteness Theorem in Boolos and Jeffrey’s wonderful book Computability and Logic.Footnote 1 The choice of Q and its smaller brother R to prove the First Incompleteness Theorem is beautiful, since these theories seem to be about as weak as one can get to prove this result.Footnote 2 On the other hand, it is a bit awkward to go on from there to prove the Second Incompleteness Theorem, where we need internal verification of some principles. Computability and Logic also contains nice exercises illustrating how easy it is to manufacture various counter-models to show non-provability in Q.

Later I was pleased to discover that there are—in a sense—two proofs of the Second Incompleteness Theorem for Q itself. There is a truly beautiful result by Bezboruah and Shepherdson (1976). They show that, under some very reasonable assumptions, the theory \(\mathsf{PA}^-\) does not prove the consistency of any theory. Hence, a fortiori, Q does not prove its own consistency. Bezboruah and Shepherdson’s proof does depend on rather specific assumptions about the coding. Also it does not generalize to stronger theories. What is more, it tells us nothing about the question whether Q can prove its consistency on some definable cut. Later, Pavel Pudlák proved a strong version of the Second Incompleteness Theorem (see Pudlák 1985; Hájek and Pudlák 1993) that I would formulate as follows. Let U be any consistent recursively enumerable theory and let N be an interpretation of Q in U. Then, U does not prove its own consistency relativized to N. It seems to me that one should say that Bezboruah and Shepherdson on the one hand and Pudlák on the other proved quite different results which share some consequences. I always thought that the two proofs should provide a good case study for a philosophical enquiry into the problem of theorem individuation.

Still later I read Tarski et al. (1953) which proves the essential undecidability of Q. And after that there was also Nelson (1986) where the possibilities for bootstrapping in Q are used with impressive, almost magical, results.

The present paper is my tribute to Q. As is fitting among friends, I will not only praise it, but also discuss, in a respectful way, some of its weaknesses.

2 Introduction

The theory Q was introduced by Robinson (1950). He introduced it as a simplification of an earlier finitely axiomatized, essentially undecidable theory due to Mostowski and Tarski (1949). The system became widely known via the book Undecidable Theories by Tarski et al. (1953). It is given by the following axioms:

  • Q1. \(\vdash \mathsf{S}x = \mathsf{S}y \rightarrow x=y\)

  • Q2. \(\vdash \mathsf{S}x \ne 0\)

  • Q3. \(\vdash x=0 \vee \exists y\quad x=\mathsf{S}y\)

  • Q4. \(\vdash x+0=x\)

  • Q5. \(\vdash x+\mathsf{S}y = \mathsf{S}(x+y)\)

  • Q6. \(\vdash x\cdot 0 = 0\)

  • Q7. \(\vdash x\cdot \mathsf{S}y = x\cdot y + x\)

Robinson shows that Q is essentially undecidable, which tells us that any consistent theory that interprets Q is undecidable. Since Q is finitely axiomatized, it follows that any theory that weakly interprets Q,Footnote 3 i.o.w. any theory that is consistent with some translation of Q, is undecidable.

The theory Q is, in many senses, a natural theory. What would be the quintessential weak arithmetic? Well, we want to have at least the basic properties of zero and the successor function: zero is not a successor and the successor relation is total, functional and injective. Then, we want the recursion equations for addition and multiplication. However, the resulting theory is a subtheory of the theory of the positive reals including zero. Thus, it has a decidable extension. So, the theory cannot binumerate the recursive functions. We can repair that by adding the axiom that zero and successor are jointly surjective. This final axiom seems to have a certain harmony with the other successor axioms: we have axioms Q1 and Q2 to state that zero and successor are jointly injective and we have an axiom, to wit Q3, that articulates that they are jointly surjective. Thus, from the standpoint of motivation of the axioms, Q seems to be a well-balanced theory.

Clearly, Q is very weak. However, if we are just interested in the property of essential undecidability, we can go much weaker. There is the theory R, also introduced in Tarski et al. (1953). Here are the axioms of R, where underlining stands for the usual unary numeral function.

  • R1. \(\vdash {\underline{m}} + {\underline{n}} = {\underline{m+n}}\)

  • R2. \(\vdash {\underline{m}} \cdot {\underline{n}} = {\underline{m\cdot n}}\)

  • R3. \(\vdash {\underline{m}} \ne {\underline{n}} \),   for \(m\ne n\)

  • R4. \(\vdash x\le {\underline{n}} \rightarrow \bigvee _{i\le n} x= {\underline{i}}\)

  • R5. \(\vdash x\le {\underline{n}} \vee {\underline{n}} \le x\)

This theory is not only essentially undecidable but also has the property that any theory that weakly interprets it is undecidable. See Vaught (1962). Since we build up the needed machinery anyway, we will provide a proof of this last result in Sect. 3.

We can even consider weaker theories than R. See Vaught (1962) and Jones and Shepherdson (1983). See also Visser (2014) for more information.

The theory R is not finitely axiomatizable. Moreover, every finitely axiomatized subtheory of it is finitely satisfiable and, hence, has a decidable extension. So perhaps Q is the weakest finitely axiomatized theory that is essentially undecidable and, say, extends R? Alas, no such luck. We refer the reader to Vítězslav Švejdar’s paper in Švejdar (2007) where finitely axiomatized systems are studied that are weaker than Q. These systems do not extend R but with a minor modification they do: we have to set the partial functions in Švejdar’s paper when they are not defined to some default value.

The systems studied by Švejdar are mutually interpretable with Q, so perhaps Q is still minimal, among finitely axiomatized theories, with respect to interpretability? Not so, for any finitely axiomatized subtheory A of Q that extends R, we can find a finitely axiomatized subtheory B of A that extends R and such that B does not interpret A. This can be shown by a minor adaptation of the methods of Friedman (2007). We show how to do this in Sect. 3. So, the prospect of characterizing Q, or some closely related theory, as the weakest (in a suitable sense) finitely axiomatized theory with such and such a property, seems pretty dim.

A quite different and beautiful feature of Q is that it interprets fairly strong theories like \(I\varDelta _0+\varOmega _1\) on a definable cut. It follows from this that we have the second incompleteness theorem for all extensions of Q. This feature also holds for the still weaker theories studied by Švejdar that interpret Q on a definable cut. Regrettably, this does not seem to help us with the characterization problem.

From one perspective, Q seems rather natural, from another it does not. It lacks many desirable properties. As has been shown in Visser (2008) and Jeřábek (2012), Q is not a pair theory. In Sect. 5 of this paper we will show that it is not even a poly-pair theory. We will also show, in Sect. 6, that Q does not have the Pudlák property. The negation of the Pudlák property for Q tells us that there are two interpretations of \(\mathsf{S}^1_2\) in Q that do not verifiably have definably isomorphic cuts.

It is interesting to compare Q with its bigger brother \(\mathsf{PA}^{-}\). The theory \(\mathsf{PA}^{-}\) is the theory of commutative, discretely ordered semi-rings with a minimal element plus the subtraction axiom (\(\mathsf{PA}^{-}14\) below). It is employed as the basic arithmetic, e.g. in the textbook (Kaye 1991). The theory is given by the following axioms:

  • \(\mathsf{PA}^{-}1. \vdash x+0=x\)

  • \(\mathsf{PA}^{-}2. \vdash x+y = y+x\)

  • \(\mathsf{PA}^{-}3. \vdash (x+y)+z = x+(y+z)\)

  • \(\mathsf{PA}^{-}4. \vdash x\cdot 1 = x\)

  • \(\mathsf{PA}^{-}5. \vdash x\cdot y = y\cdot x\)

  • \(\mathsf{PA}^{-}6. \vdash (x\cdot y)\cdot z = x\cdot (y\cdot z)\)

  • \(\mathsf{PA}^{-}7. \vdash x\cdot (y+z) = x\cdot y + x\cdot z\)

  • \(\mathsf{PA}^{-}8. \vdash x \le y \vee y \le x\)

  • \(\mathsf{PA}^{-}9. \vdash (x\le y \wedge y \le z) \rightarrow x\le z\)

  • \(\mathsf{PA}^{-}10. \vdash x+1\not \le x\)

  • \(\mathsf{PA}^{-}11. \vdash x \le y \rightarrow (x=y \vee x+1\le y)\)

  • \(\mathsf{PA}^{-}12. \vdash x \le y \rightarrow x+z\le y+z\)

  • \(\mathsf{PA}^{-}13. \vdash x\le y \rightarrow x\cdot z \le y\cdot z\)

  • \(\mathsf{PA}^{-}14. \vdash x\le y \rightarrow \exists z \;\; x+z=y\)

Emil Jeřábek’s version in Jeřábek (2012) does not have the subtraction axiom \(\mathsf{PA}^{-}14\). Thus Jeřábek’s version is a universal theory. As noted by Jeřábek his version interprets the stronger version with subtraction axiom on a definable cut. The weak version does not extend Q but the strong version does.

The theory \(\mathsf{PA}^-\) has been shown to be sequential by Emil Jeřábek in his paper (Jeřábek 2012) (even in the weaker form without the subtraction axiom).

Bezboruah and Shepherdson (1976) show that, under some very reasonable assumptions, \(\mathsf{PA}^-\) does not prove the consistency of any theory.

Victor Pambuccian studies number theoretical theorems over \(\mathsf{PA}^{-}\). See his papers (Pambuccian 2008, 2014, 2015).

The theory Q interprets much stronger theories than \(\mathsf{PA}^{-}\), like \(I\varDelta _0 + \varOmega _1\) (see, e.g. Hájek and Pudlák 1993). Hence, a fortiori, Q is mutually interpretable with \(\mathsf{PA}^-\) (both the strong and the weak version). Using ideas of Per Lindström, one may show that Q is even mutually faithfully interpretable with \(\mathsf{PA}^-\). One can also demonstrate that the Lindenbaum algebras of Q and \(\mathsf{PA}^-\) are recursively isomorphic. This means that there is a recursive function of sentences that induces an isomorphism of Lindenbaum algebras. The result is a special case of the theorem of Marian Pour-El and Saul Kripke that the Lindenbaum algebras of all recursively enumerable theories that interpret Q are recursively isomorphic. See Pour-El and Kripke (1967).

Are these the best samenesses that we can get between these theories? In Sect. 7, we will show that the two theories are not sententially congruent. So, at least in terms of traditional notions of sameness, we cannot do better than recursive isomorphism of Lindenbaum algebras on the one hand, and mutual faithful interpretability on the other.

The main technical tool of the paper is a theorem that tells us that, in a sense, Q can be split into two disjoint parts. This result is proved in Sect. 4. The proof is an adaptation of an earlier result in Visser (2014). One might say that the progress of the present paper is to provide a better understanding of what the result of Visser (2014) really means.

We end the paper with some concluding remarks in Sect. 8.

2.1 How to read the paper

In the appendices, I present basic materials needed for understanding the paper. In the main text there are references to the appendices when needed. Section 3 can be read independently of the other sections. Section 4 is the basic preliminary for Sects. 57. The Sects.  57 are pairwise independent of each other.

3 Between R and Q

In this section, we endeavour to make the idea of characterizing Q using an appropriate minimality claim less plausible. Perhaps it is better to say: if there is a characterization of Q as the minimal theory such that ..., then it cannot take such and such a form. Specifically, we show that, for any finitely axiomatised consistent theory A such that \(\mathsf{R} \subseteq A\), there is a finitely axiomatised B such that \(\mathsf{R} \subseteq B \subseteq A\) and \(B \mathrel {\not \! \rhd }A\).Footnote 4 The result is just a rather direct application of ideas from Friedman (2007). So, we do not claim great originality here.

The following nice version of the theory of a number was developed by Johannes Marti, Nal Kalchbrenner, Paula Henk and Peter Fritz in Interpretability Project Report of 2011, the report of a project they did under my guidance in the Master of Logic in Amsterdam.Footnote 5 \({}^{,}\) Footnote 6 We call it the theory of a number since, in our intended applications, the fact that it is satisfied by the structure associated with a finite, non-zero ordinal is central.

  • TN1. \(\vdash x \not < 0\)

  • TN2. \(\vdash (x< y \wedge y<z ) \rightarrow x < z\)

  • TN3. \(\vdash x< y \vee x= y \vee y < x\)

  • TN4. \(\vdash x=0 \vee \exists y\quad x=\mathsf{S}y\)

  • TN5. \(\vdash \mathsf{S}x \not < x\)

  • TN6. \(\vdash x<y \rightarrow (x<\mathsf{S}x \wedge y \not < \mathsf{S}x)\)

  • TN7. \(\vdash x+0=0\)

  • TN8. \(\vdash x+\mathsf{S}y =\mathsf{S}(x+y)\)

  • TN9. \(\vdash x \cdot 0 = 0\)

  • TN10. \(\vdash x\cdot \mathsf{S}y = x\cdot y + x\)

Since \(\mathsf{TN}6\) implies \(x \not < x\), a model of TN is a linear ordering that either represents a finite ordinal or starts with a copy of \(\omega \).

We call a \(\varDelta _0\)-formula pure if (i) all bounding terms are variables and (ii) all occurrences of terms are in subformulas of the form \(\mathsf{S}x=y\), \(x+y=z\) and \(x \cdot y = z\). We call a \(\varSigma _1\)-sentence pure if it is of the form \(\exists \mathbf {x}\, S_0\mathbf {x}\), where \(S_0\) is a pure \(\varDelta _0\)-sentence.

We can transform an arbritrary \(\varSigma _1\)-sentence S into a pure \(\varSigma _1\)-sentence \(S^\circ \), for example, in the following way. We start with S. We treat bounded quantifiers for the moment as if they were given with the language and not defined. First we replace all implications \((A \rightarrow B)\) by \((\lnot A \vee B)\) and all bi-implications by \(((\lnot A \vee B) \wedge (\lnot B \vee A))\). Next we push all negations inside in the usual manner. We replace:

  • \(\forall x < t\) by \(\exists z\quad (t = z \wedge \forall x< z \ldots )\),

  • \(\exists x < t\) by \(\exists z\quad (t = z \wedge \exists x< z \ldots )\),

  • \(\lnot t_0 = t_1\) by \(\exists z \exists w\quad (t_0 =z \wedge t_1 = w \wedge \lnot z = w)\),

  • \(\lnot t_0 < t_1\) by \(\exists z \exists w\quad ( t_0 = z \wedge t_1 = w \wedge \lnot z < w)\).

In this way all term occurrences are on positive places. At this point we apply the usual term-unwinding algorithm to our formula using a small scope interpretation. We note that this will translate an atomic formula to a block of existential quantifiers followed by an boolean combination of atomic formulas. Finally, we bring all unbounded existential quantifiers to the front in the usual manner replacing, e.g. \(\forall x< y \exists z \ldots \) by \(\exists w \forall x<y \exists z <w \ldots \). The resulting formula is \(S^\circ \), which is clearly pure and equivalent to the original formula (say, over \(\mathsf{PA}^{-}\) plus \(\varSigma _1\)-collection).Footnote 7

We note that the transformation \(S \mapsto S^\circ \) that we described is clearly elementary. Hence it exists in EA. Inspecting the transformation, we see that \(S^\circ \) implies S in predicate logic.

Let \(S := \exists \mathbf {y} S_0\mathbf {y}\), where \(S_0\) is a pure \(\varDelta _0\)-formula. We define:

$$\begin{aligned}{}[S] := \mathsf{TN} + \exists x \exists \mathbf {y} < x S_0\mathbf {y}. \end{aligned}$$

Using the machinery of theories of a number, we can reprove Cobham’s result that any recursively enumerable theory that weakly interprets R is undecidable. Suppose U is recursively enumerable and \(U + \mathsf{R}^\tau \) is consistent. Consider a pure \(\varSigma _1\)-sentence S. Let \(S^\star \) be the sentence that says:

$$\begin{aligned} \exists x \left( \mathsf{S}x \models [S] \wedge \forall y < x \mathsf{S}y \not \models [S]\right) . \end{aligned}$$

Since [S] is finitely axiomatised, we can write out \(S^\star \) in the obvious way. Consider the set \({\mathcal {S}}\) of all S such that \(U+(\mathsf{R} + S^\star )^\tau \) is consistent. Clearly, \({\mathcal {S}}\) contains all true (pure) \(\varSigma _1\)-sentences. If \({\mathcal {S}}\) did not contain false (pure) \(\varSigma _1\)-sentences, then this would make \(\varSigma _1\)-truth decidable. Hence, there is a false (pure) \(\varSigma _1\)-sentence \(S_1\) such that \(U+(\mathsf{R} + S_1^\star )^\tau \) is consistent. We can use \(S_1\) to build a translation \(\tau _0\) such that \(U+[S_1]^{\tau _0}\) is consistent. Since \([S_1]\) is a finitely axiomatised extension of R, it follows that U is undecidable.

To prove our main result we need the following result that was first verified in detail in the Interpretability Project Report by Marti, Kalchbrenner, Henk and Fritz. The basic idea behind the result is present in Friedman’s (2007).

We remind the reader of witness comparison notation. Suppose A is of the form \(\exists x A_0(x)\) and B is of the form \(\exists y B(y)\). We define:

  • \(A<B := \exists x (A(x) \wedge \forall y \le x \lnot B(y))\).

  • \(A \le B := \exists x (A(x) \wedge \forall y < x \lnot B(y))\).

  • If C is \(A< B\), then \(C^\bot \) is \(B \le A\).

  • If D is \(A \le B\), then \(D^\bot \) is \(B < A\).

Theorem 1

Let S and \(S^{\prime }\) be pure \(\varSigma _1\)-sentences. We have:

  1. (a)

    Suppose S is true. Then, if we allow piecewise interpretations, we have \(\top \rhd [S]\).

    If we do not allow piecewise interpretations, we still have \((\exists x \exists y x\ne y) \rhd [S]\).

  2. (b)

    If \(S \le S^{\prime }\), then \([S^{\prime }] \vdash S\).

Proof

Ad (a): We note that if S is true, then [S] has a finite model. Any theory with a finite model is interpretable with a piecewise interpretation in predicate logic. If we do not allow piecewise interpretations, we can obtain the same effect using a multidimensional interpretation, assuming that we have at least two distinct elements.

Ad (b). Consider any model \({\mathcal {M}}\) of \([S^{\prime }]\). Without loss of generality we can identify the initial elements of \({\mathcal {M}}\) with \(0, 1, 2, \ldots \) It is easily shown that for any pure \(\varDelta _0\)-formula \(A\mathbf {x}\) we have \(A\mathbf {n}\) is true iff \({\mathcal {M}} \models A \mathbf {n}\), provided the \(\mathbf {n}\) are natural numbers in \({\mathcal {M}}\) and are not the top elements.Footnote 8 Suppose m is the smallest witness of S. Then we have \({\mathcal {M}} \models \lnot S_0^{\prime } \mathbf {k}\), for all \(\mathbf {k} < m\). Since \(S^{\prime }\) is witnessed by a non-top element, m is non-top and we have \({\mathcal {M}} \models S_0m\), and hence \({\mathcal {M}} \models S\). \(\square \)

We now have the materials to prove the main theorem of the present section.

Theorem 2

Suppose \(\mathsf{R} \subseteq A\), where A is finitely axiomatized and consistent. Then, there is a finitely axiomatized B such that \(\mathsf{R} \subseteq B \subseteq A\) and \(B \mathrel {\not \! \rhd }A\).

Proof

Suppose \(\mathsf{R} \subseteq A\), where A is finitely axiomatized and consistent. By the Gödel Fixed Point Lemma, we define R with:

$$\begin{aligned} \mathsf{EA} \vdash R \leftrightarrow \left( \left[ R^\circ \right] \rhd A\right) \le \left( A \rhd \left[ R^\circ \right] \right) . \end{aligned}$$

Suppose \([R^\circ ] \rhd A\). It follows that either R or \(R^\bot \).

In the first case, we find \(R^\circ \). By Theorem 1(a), we find that \(\top \rhd [R^\circ ]\) and, hence, that \(\top \rhd A\). Quod non, since A extends R and since any theory interpretable in predicate logic has finite models.

In the second case, it follows that \((R^\bot )^\circ \) and \(\lnot R^\circ \). Hence, \((R^\bot )^\circ \le R^\circ \) and, hence, by Theorem 1(b), \([R^\circ ] \vdash (R^\bot )^\circ \). Since \(R^\circ \) implies R in predicate logic and similarly for \((R^\bot )^\circ \) and \(R^\bot \), we find that \([R^\circ ]\) proves both R and \(R^\bot \). So, \([R^\circ ] \vdash \bot \). From \(R^\bot \), we also have \(A \rhd [R^\circ ]\), thus it follows that \(A\rhd \bot \). Quod non, since A is consistent.

We may conclude that \([R^\circ ] \mathrel {\not \! \rhd }A\) and hence that \((\bigwedge [R^\circ ] \vee A) \mathrel {\not \! \rhd }A\). Since R implies \([R^\circ ] \rhd A\), it also follows that R is false. Thus, \([R^\circ ]\) extends R. Let \(B := (\bigwedge [R^\circ ] \vee A) \). We find: \(\mathsf{R} \subseteq B \subseteq A\) and \(B\mathrel {\not \! \rhd }A\). \(\square \)

4 Decomposition of Q

The theory Q has many unexpected extensions. We develop one here that is especially useful for proving negative results about Q. We will use it in the subsequent sections. The construction is an adaptation of a result of Visser (2014). One could say that only in the present paper the full meaning of the construction of Visser (2014) is unfolded.

The theory \(\mathsf{Q}^\#\) in the language of Q is axiomatized by the following principles.

  • Q

  • \(\forall x \forall y ((\mathsf{S}x \ne x \wedge \mathsf{S}y \ne y ) \rightarrow \mathsf{S}(x+y) \ne x+y)\)

  • \(\forall x \forall y ((\mathsf{S}x \ne x \wedge \mathsf{S}y \ne y ) \rightarrow \mathsf{S}(x \cdot y) \ne x \cdot y)\)

  • \(\exists a \forall x ( x+a = a \wedge a+x = a)\)

    It is easy to see that this a is unique. We call it \(\infty \).

  • \(\exists x (\mathsf{S}x=x \wedge x \ne \infty )\)

  • \(\forall x \forall y ((x \ne \mathsf{S}x \wedge y = \mathsf{S}y ) \rightarrow (x +y = y \wedge y + x = y))\)

  • \(\forall x \forall y ((x = \mathsf{S}x \wedge y = \mathsf{S}y ) \rightarrow (x +y = y \vee x + y = \infty ))\)

  • \(\forall x \forall y (y = \mathsf{S}y \rightarrow x \cdot y = \infty )\)

  • \(\forall x \forall y ((x = \mathsf{S}x \wedge y \ne \mathsf{S}y) \rightarrow x \cdot \mathsf{SS}y = x+x)\)

Theorem 3 below will have the immediate consequence that \(\mathsf{Q}^\#\) is consistent.

Let \(\mathsf{Q}^\circ \) be Q plus the axiom \(\forall x \mathsf{S}x \ne x\). Let \(\mathsf{CQC}^2\) be predicate logic with identity and one binary predicate symbol R. Let \({\mathbbm {1}}\) be the theory in the language of identity that states that there is at most one object. The operation \(\boxplus \) is defined and discussed in the “Sums” section of Appendix 1. The notion of synonymy is defined in the “Provable equivalence of interpretations” section of Appendix 1. We have:

Theorem 3

The theory \(\mathsf{Q}^\#\) is synonymous to the theory \(\mathsf{Y} := \mathsf{Q}^\circ \boxplus \mathsf{CQC}^2 \boxplus {\mathbbm {1}}\).

Proof

We define \(K:\mathsf{Y} \rightarrow Q^{\#}\) as follows. We write Z for the unary relational representation of zero, S for the binary relational representation of successor, A for the ternary relational representation of addition, and M for the ternary relational representation of multiplication.

  • \(\delta _K(x) :\leftrightarrow x=x\)

  • \(x =_K y :\leftrightarrow x=y\)

  • \(\triangle _{0K}(x) :\leftrightarrow \mathsf{S}x \ne x\)

  • \(\triangle _{1K}(x) :\leftrightarrow \mathsf{S}x = x \wedge x \ne \infty \)

  • \(\triangle _{2K}(x) : \leftrightarrow x = \infty \)

  • \(\mathsf{Z}_Kx :\leftrightarrow \mathsf{Z}x\)

  • \(\mathsf{S}_Kxy :\leftrightarrow \triangle _{0K}(x) \wedge \mathsf{S}x=y\)

  • \(\mathsf{A}_Kxyz :\leftrightarrow \triangle _{0K}(x) \wedge \triangle _{0K}(y) \wedge x+y=z\)

  • \(\mathsf{M}_Kxyz :\leftrightarrow \triangle _{0K}(x) \wedge \triangle _{0K}(y) \wedge x\cdot y = z\)

  • \(R_Kxy :\leftrightarrow \triangle _{1K}(x) \wedge \triangle _{1K}(y) \wedge x+y = \infty \)

It is easy to see that the specified translation does indeed deliver the desired interpretation. We define \(M:\mathsf{Q}^\# \rightarrow \mathsf{Y}\) as follows. To make the interpretation readable we use functional notation on the side of the interpreting theory. The reader should keep in mind that, e.g. S is only defined on \(\triangle _0\). We write \(\infty \) for the unique inhabitant of \(\triangle _2\). Below the itemized definition, we repeat the definitions of addition and multiplication in more readable tabular form.

  • \(\delta _M(x) :\leftrightarrow x=x\)

  • \(x=_M y :\leftrightarrow x=y\)

  • \(\mathsf{Z}_Mx :\leftrightarrow x=0\)

  • \(\mathsf{S}_Mxy :\leftrightarrow \mathsf{S}x= y \vee (\lnot \triangle _0(x) \wedge x=y)\)

  • \(\mathsf{A}_Mxyz :\leftrightarrow x+y=z \ \vee \)

    $$\begin{aligned}&(\triangle _0(x) \wedge \triangle _1(y) \wedge z=y) \; \vee \\&(\triangle _1(x) \wedge \triangle _0(y) \wedge z=x) \; \vee \\&(\triangle _1(x) \wedge \triangle _1(y) \wedge Rxy \wedge z = \infty ) \\&(\triangle _1(x) \wedge \triangle _1(y) \wedge \lnot \, Rxy \wedge z = y) \; \vee \\&( (x=\infty \vee y = \infty ) \wedge z = \infty ) \end{aligned}$$
  • \(\mathsf{M}_Mxyz :\leftrightarrow (y=0 \wedge z=0) \vee (y=1 \wedge \mathsf{A}_M(0,x,z))\, \vee \)

    $$\begin{aligned}&\exists u (y = \mathsf{SS}u \wedge ((\triangle _0(x) \wedge x\cdot y = z) \vee \\&(\triangle _1(x) \wedge ((Rxx \wedge z= \infty ) \vee (\lnot Rxx \wedge z = x) )) \vee \\&(x = \infty \wedge z = \infty ))) \vee ((\triangle _1(y) \vee y = \infty ) \wedge z=\infty ) \end{aligned}$$

Here is the diagrammatic version of the definitions of addition and multiplication. In the diagrams, n ranges over \(\triangle _0\) and x ranges over \(\triangle _1\).

The verification that the translation we specified does indeed carry an interpretation of \(\mathsf{Q}^\#\) is immediate. We treat two sample cases of the verification of \(M \circ K =_0 \mathsf{ID}_{Y}\). We have:

$$\begin{aligned} \mathsf{Y} \vdash (\mathsf{A}xyz)^{KM}&\leftrightarrow&\left( \mathsf{S}x \ne x \wedge \mathsf{S}y \ne y \wedge x+y = z\right) ^M \\&\leftrightarrow&\triangle _0(x) \wedge \triangle _0(y) \wedge (\mathsf{A}xyz \vee (\ldots )) \\&\leftrightarrow&\mathsf{A}xyz \end{aligned}$$

The last step uses the fact that the \((\ldots )\) implies that \(\lnot (\triangle _0(x) \wedge \triangle _0(y))\).

We treat one sample case to illustrate that \(K \circ M =_0 \mathsf{ID}_{\mathsf{Q}^\#}\).

$$\begin{aligned}&\mathsf{Q}^\# \vdash (x+y=z)^{MK} \leftrightarrow (x+y=z \vee \\&\quad \quad (\triangle _0(x) \wedge \triangle _1(y) \wedge z=y) \vee \\&\quad \quad (\triangle _1(x) \wedge \triangle _0(y) \wedge z=x) \vee \\&\quad \quad (\triangle _1(x) \wedge \triangle _1(y) \wedge Rxy \wedge z = \infty ) \\&\quad \quad (\triangle _1(x) \wedge \triangle _1(y) \wedge \lnot \, Rxy \wedge z = y) \vee \\&\quad \quad ((x=\infty \vee y = \infty ) \wedge z = \infty ))^K \\&\quad \leftrightarrow (\mathsf{S}x \ne x \wedge \mathsf{S} y \ne y \wedge x+y=z) \vee \\&\quad \quad (\mathsf{S}x \ne x \wedge \mathsf{S}y = y \wedge y \ne \infty \wedge z=y) \vee \\&\quad \quad ( \mathsf{S}x= x \wedge x\ne \infty \wedge \mathsf{S}y \ne y \wedge z=x) \vee \\&\quad \quad (\mathsf{S} x = x \wedge x\ne \infty \wedge \mathsf{S}y = y \wedge y \ne \infty \wedge \\&\quad \quad x+y = \infty \wedge z = \infty ) \\&\quad \quad (\mathsf{S} x = x \wedge x\ne \infty \wedge \mathsf{S}y = y \wedge y \ne \infty \wedge \\&\quad \quad x+y \ne \infty \wedge z = y) \vee \\&\quad \quad ( (x=\infty \vee y = \infty ) \wedge z = \infty ) \\&\quad \leftrightarrow x+y=z \end{aligned}$$

Of course, the last step is by verifying that if, e.g. \(\mathsf{S} x = x \wedge x\ne \infty \wedge \mathsf{S}y = y \wedge y \ne \infty \), then \( x+y = \infty \wedge z = \infty \) iff \(x+y=z\), and similarly for the other cases. \(\square \)

Theorem 4

\(\mathsf{Q} \rhd \mathsf{Q}^{\#}\).

Proof

The theorem is a direct consequence of the fact that \(\mathsf{Q} \rhd \mathsf{Q}^\circ \) via a definable cut, \(\mathsf{Q} \rhd \mathsf{CQC}^2\) and \(\mathsf{Q} \rhd {\mathbbm {1}}\) in combination with Theorem 3, noting that \(\boxplus \) is (an implementation of) the supremum in the degrees of interpretability. \(\square \)

5 Q is not a poly-pair theory

In this section we show that Q is not a poly-pair theory. We explain the notion of poly-pair theory in Appendix 2. In this same appendix we provide various basic facts about poly-pair theories. These basics are mainly derived from our paper (Visser 2013).

Theorem 5

Q is not a poly-pair theory.

Proof

Suppose Q were a poly-pair theory. By our result of Sect. 4, the theory Y is bi-interpretable with \(\mathsf{Q}^\#\). Since, by the results of Appendix 2, the property of being a poly-pair theory is upwards preserved under theory extension and is preserved under bi-interpretations, is sufficient to show that Y is not a poly-pair theory.

Suppose Y is a poly-pair theory. Consider any model \({\mathcal {M}}\) of Y in which the relation R is empty and in which the domain of the second component is infinite. By the results in Appendix 2 the interpretation that witnesses the fact that Y is poly-pair can be taken to be parameter-free, but we do not need to use the result. Suppose the parameters of our interpretation are \(\mathbf {p}\) and let the dimension be m.

Consider two m-sequences \(\mathbf {a}\) and \(\mathbf {b}\) in the second component. We assume that the elements of \(\mathbf {a},\mathbf {b},\mathbf {p}\) are pairwise distinct. Let \(\mathbf {c}\) be a pair (according to the interpretation) containing \(\mathbf {a}\) and \(\mathbf {b}\). Some element d of \(\mathbf {a},\mathbf {b}\) does not occur in \(\mathbf {c}\). Let e in the second component be disjoint from \(\mathbf {a},\mathbf {b},\mathbf {c}, \mathbf {p}\). Let \(\sigma \) be the operation of interchanging d and e. Clearly, \(\sigma \) is an automorphism of our model that leaves \(\mathbf {p}\) fixed. So \(\mathbf {c}\) is also a pair of \(\sigma \mathbf {a}\) and \(\sigma \mathbf {b}\). Since either \(\sigma \mathbf {a} \ne \mathbf {a}\) or \(\sigma \mathbf {b} \ne \mathbf {b}\), this contradicts the defining property of pairing. \(\square \)

6 The Pudlák property

The Pudlák property of a theory U in its classical formulations says: (i) there is an interpretation \(N^\star :\mathsf{S}^1_2 \rightarrow U\) and (ii) whenever \(N:\mathsf{S}^1_2 \rightarrow U\) and \(N^{\prime }: \mathsf{S}^1_2 \rightarrow U\), then there is a U-definable, U-verifiable isomorphism F between certain U-definable, U-verifiable cuts I of N and J of \(N^{\prime }\). We take our cuts to be downwards closed w.r.t. < and closed under S, \(+\), \(\times \) and \(\omega _1\).

To keep our treatment simple we only consider the case of parameter-free interpretations. The case with parameters is briefly discussed in Remark 1.

For our purposes, it is nicer to view the Pudlák property in the light of the category \(\mathsf{INT}_1\) of bi-interpretability. See Appendix 1 for an introduction to bi-interpretability and \(\mathsf{INT}_1\).

We define a functor A from \(\mathsf{INT}_1\) to the category of preorders. Here we allow the empty preorder. Consider any theory U. We send U to the structure \(\mathsf{A}(U)\). The elements of \(\mathsf{A}(U)\) are interpretations \(N:\mathsf{S}^1_2 \rightarrow U\) modulo i-isomorphism. The structure \(\mathsf{A}(U)\) has a binary preordering \(\preceq \) defined by \(N \preceq N^{\prime }\) iff there is a U-definable, U-verifiable initial embedding F from N to \(N^{\prime }\). We note that the existence of such an embedding is independent of the choice of the syntactical representatives of N and \(N^{\prime }\). If \(K :U \rightarrow V\), then \(\mathsf{A}(K):\mathsf{A}(U) \rightarrow \mathsf{A}(V)\) is defined by \(\mathsf{A}(K)(N) := K \circ N\). We note \(\mathsf{A}(K)\) does indeed preserve \(\preceq \).

We remind the reader that a preorder is downward directed if for every x and y, there is a z with \(z \le x\) and \(z \le y\).

The Pudlák property for U is equivalent to: \(\mathsf{A}(U)\) is non-empty and downward directed.

We remind the reader of Pavel Pudlák’s well-known result from Pudlák (1985).

Theorem 6

(Pudlák) Sequential theories have the Pudlák property.

We show that the converse of Pudlák’s result does not hold in Appendix 3. Specifically, we show that, if U has the Pudlák Property, then so does \(U \boxplus \mathsf{EQ}\). Here EQ is the pure theory of equality in the minimal signature. It follows that, e.g. \(\mathsf{S}^1_2 \boxplus \mathsf{EQ}\) has the Pudlák Property. However, the methods of Sect. 7 show that \(\mathsf{S}^1_2 \boxplus \mathsf{EQ}\) is not sequential.

Remark 1

What is the Pudlák Property in the case with parameters? It seems that there are lots of options. Instead of systematically looking at various versions, I will just give the one that I think is most attractive. For the basic definitions and notations concerning parameters the reader is referred to the “Adding parameters” section of Appendix 1.

First we generalize \(\preceq \) to the case with parameters. We define:

  • \(N \preceq N^{\prime }\) iff, for some F, we have:

    \(U\vdash \forall \mathbf {q} (\alpha _{N^{\prime }}(\mathbf {q}) \rightarrow \exists \mathbf {p} \; (\alpha _N(\mathbf {p}) \wedge F(\mathbf {p}, \mathbf {q}): N^{\mathbf {p}} \preceq {N^{\prime }}^{\mathbf {q}}))\).

Here \((F(\mathbf {p}, \mathbf {q}): N^{\mathbf {p}} \preceq {N^{\prime }}^{\mathbf {q}})\) means that F is a formula representing an initial embedding of \(N^{\mathbf {p}}\) in \({N^{\prime }}^{\mathbf {q}}\), where domain and range of F are cuts. We note that we could allow F to have some extra parameters of its own, so that the formula \((F(\mathbf {p}, \mathbf {q}): N^{\mathbf {p}} \preceq {N^{\prime }}^{\mathbf {q}})\) would become \(\exists \mathbf {r} (F(\mathbf {p}, \mathbf {q}, \mathbf {r}): N^{\mathbf {p}} \preceq {N^{\prime }}^{\mathbf {q}})\). However, we will refrain from doing so.

The Pudlák Property now takes a simple form: (i) There is an \(N^\star :\mathsf{S}^1_2 \rightarrow U\) and (ii) for every \(N:\mathsf{S}^1_2 \rightarrow U\), there is a parameter-free \(N_0:\mathsf{S}^1_2 \rightarrow U\), such that \(N_0 \preceq N\). This holds even in the case that the direct interpretation that witnesses the sequentiality of U itself contains parameters. We can rephrase this version of the Pudlák property as follows: \(\mathsf{A}(U)\) is not empty, and the parameter-free interpretations are coinitial in \(\mathsf{A}(U)\).

We note that the two interpretations N and \(N^{\prime }\) in the formulation of the parameter-free case can be subsumed under a single interpretation with parameters \(N{\langle x=0 \rangle } N^{\prime }\). So, the Pudlák Property with parameters includes the one without parameters.

Let us call my version of the Pudlák Property with parameters: the strong Pudlák Property. Sequential theories have the strong Pudlák Property. This result holds even in the case that U is sequential via a direct interpretation with parameters. The result also holds in the poly-sequential case. We refer the reader to Visser (2013), for details, specifically to the second proof of Theorem 5.2 of that paper.

I have not worked out the full development of the case with parameters. However note that we show that Q fails to have the weaker property, so a fortiori it fails to have the stronger property.

We collect some basic insights. Since the homomorphic image of a downward directed preorder is downward directed, we have:

Theorem 7

Suppose \(K:U \rightarrow V\) and U has the Pudlák Property and \(\mathsf{A}(K)\) is surjective. Then, V has the Pudlák property.

We show that A applied to an instance of the extension relation is surjective.

Theorem 8

Suppose V is an extension of U in the same language. Let \(\mathsf{emb}_{UV}\) be the identical embedding. Suppose further that \(\mathsf{A}(U)\) is non-empty. Then, \(\mathsf{A}(\mathsf{emb}_{UV})\) is surjective.

Proof

Suppose \(N_0 \in \mathsf{A}(U)\) and \(N \in \mathsf{A}(V)\). Let \(N_1 := N {\langle (\bigwedge \mathsf{S}^1_2)^N \rangle } N_0\).Footnote 9 Then, clearly \(N_1 \in \mathsf{A}(U)\). Moreover, \(\mathsf{emb}_{UV}(N_1) = N\). \(\square \)

We remind the reader of the following. Consider a category \({\mathcal {C}}\). Suppose \(f: x\rightarrow y\) and \(g: y \rightarrow x\) and \(g\circ f = \mathsf{id}_x\). In this case, we call f a section or split monomorphism. We call g a retraction or split epimorphism. The object x is in this situation a retract of y.

We show that A applied to a retraction is surjective.

Theorem 9

Suppose \(K:U \rightarrow V\) is a retraction, then \(\mathsf{A}(K)\) is a retraction and hence surjective.

Proof

Any functor preserves retractions. So \(\mathsf{A}(K)\) is a retraction in the category of preorders. It follows that \(\mathsf{A}(K)\) is surjective. \(\square \)

Question 1

We note that both retractions and theory extensions are epimorphisms in \(\mathsf{INT}_1\). Does A preserve epimorphisms? (Clearly an epimorphism in the category of preorders is surjective.)

In Visser (2006), we proved that, in \(\mathsf{INT}_0\), epimorphisms can always be split in first a theory extension and then an isomorphism. So, a fortiori, in \(\mathsf{INT}_0\), epimorphisms are preserved by the \(\mathsf{INT}_0\)-analogue of A.

Theorem 10

Q does not have the Pudlák Property.

Proof

By Theorem 3, we have extensions \(\mathsf{Q}^\#\) and \(\mathsf{Q}^\circ \) of Q such that \(\mathsf{Q}^\#\) is synonymous to \(\mathsf{Y} := \mathsf{Q}^\circ \boxplus \mathsf{CQC}^2 \boxplus {\mathbbm {1}}\). We extend \(\mathsf{CQC}^2\) to AS with R in the role of \(\in \).Footnote 10 Let \(A: = \mathsf{Q}^\circ \boxplus \mathsf{AS} \boxplus {\mathbbm {1}}\).

Suppose Q has the Pudlák Property. By Theorems 8 and 9, the property is preserved from Q to \(\mathsf{Q}^\#\), from \(\mathsf{Q}^\#\) to Y, and from Y to A.

Consider interpretations \(N:\mathsf{S}^1_2 \rightarrow Q^\circ \) and \(M:\mathsf{S}^1_2 \rightarrow \mathsf{AS}\). Let \(N^*:= \mathsf{in}_0 \circ N : \mathsf{S}^1_2 \rightarrow A\) and let \(M^*:= \mathsf{in}_1 \circ M: \mathsf{S}^1_2 \rightarrow A\). Suppose there is an embedding F in A of a cut of \(N^*\) into a cut of \(M^*\). By Theorem 15, we have:

$$\begin{aligned} A,\bigwedge _i \triangle _0(x_i), \bigwedge _\ell \triangle _1(y_\ell ) \vdash F(\mathbf {x}, \mathbf {y}) \leftrightarrow \bigvee _j (D_j(\mathbf {x}) \wedge E_j(\mathbf {y})). \end{aligned}$$

Here the \(D_j\) are formulas in the range of \(\mathsf{in}_0\) and the \(E_j\) are formulas in the range of \(\mathsf{in}_1\). Suppose \(\mathbf {x}\) is in \(D_j\). In that case all \(\mathbf {y}\) in \(E_j\) are in the F-image of \(\mathbf {x}\). Hence, \(E_j\) is closed under \(=_{M^*}\). We may conclude that the range of F is standardly finite modulo \(=_{M^*}\). Quod non. \(\square \)

7 \(\mathsf{PA}^{-}\) and Q

In this section we show that \(\mathsf{PA}^{-}\) is not sententially congruent with Q.Footnote 11 In a sense, we could have written this section without even mentioning \(\mathsf{PA}^-\), since the result that \(\mathsf{PA}^-\) and Q are not sententially congruent follows from a much stronger result proven here. However, since the theories \(\mathsf{PA}^-\) and Q seem to be so close together, I feel that the specific result concerning \(\mathsf{PA}^-\) and Q speaks more to the imagination than the stronger result from which it follows.

To prove our main result, we need a few lemmas. We will be interested in retractions in the category \(\mathsf{INT}_3\) (see the “Five categories” section of Appendix 1). This takes the following form: we have interpretations \(K:U\rightarrow V\) and \(M:V \rightarrow U\) such that, for all U-sentences A, \(U \vdash A \leftrightarrow A^{KM}\). In this case K is a section or split monomorphism.

A basic insight is that the section relation has the forward or zig property w.r.t. theory-extension in \(\mathsf{INT}_3\). This is illustrated by the following diagram.

figure a

Theorem 11

The section relation in \(\mathsf{INT}_3\) has the forward or zig property with respect to theory extension.

Proof

Suppose \(K: U \rightarrow V\) is a section. We suppose that M is an inverse of K, so \(M: V \rightarrow U\) and \(M\circ K = \mathsf{ID}_U\). We suppose also \(U \subseteq U^{\prime }\). We define \(V^{\prime }:= \{ A \in \mathsf{sent}_V \mid U^{\prime } \vdash A^M \}\). Clearly, we have an interpretation \(M^{\prime }:V^{\prime } \rightarrow U^{\prime }\) based on the same translation as M. We have:

$$\begin{aligned} U^{\prime } \vdash B\Rightarrow & {} U^{\prime } \vdash B^{KM} \\\Rightarrow & {} V^{\prime } \vdash B^K \end{aligned}$$

Hence there is an interpretation \(K^{\prime }\) based on the same translation as K such that \(K^{\prime }:U^{\prime } \rightarrow V^{\prime }\). We find that \(K^{\prime }\) is a section, since \(B^{K^{\prime }M^{\prime }}\) is only dependent on the underlying translations, and hence strictly identical to \(B^{KM}\). \(\square \)

We need a sufficient store of incomparable extensions of given finitely axiomatised theories. We remind the reader that a theory U tolerates or weakly interprets a theory V if, for some translation \(\tau \), the theory \(U+V^\tau \) is consistent. Note that we take the identity axioms for \(\varSigma _V\) including \(\exists x x=x\) to be part of V. The following theorem can probably be much improved, but it is what we need for the current application.

Theorem 12

Suppose A and B are finitely axiomatized theories that tolerate \(\mathsf{S}^1_2\). Then, there are finitely axiomatized theories \(A^\star \supseteq A\) and \(B^\star \supseteq B\), that are incomparable w.r.t. \(\lhd \), i.e. \(A^\star \mathrel {\not \! \rhd }B^\star \) and \(B^\star \mathrel {\not \! \rhd }A^\star \).

Proof

Suppose \(\tau \) witnesses that A tolerates \(\mathsf{S}^1_2\) and \(\nu \) witnesses that B tolerates \(\mathsf{S}^1_2\). We take \(A^{\prime } := A +(\mathsf{S}^1_2)^\tau \) and \(B^{\prime } := B +(\mathsf{S}^1_2)^\nu \). So there is an N based on \(\tau \) such that \(N:\mathsf{S}^1_2 \rightarrow A^{\prime }\) and there is an M based on \(\nu \) such that \(M:\mathsf{S}^1_2 \rightarrow B^{\prime }\). By the Gödel Fixed Point Lemma, we find R such that:

We take \(A^\star := A^{\prime } +R^N\) and \(B^\star := B^{\prime } + \lnot R^M\). Suppose \(A^\star \rhd B^\star \). It follows that R or \(R^\bot \). In case we have R, we find, by \(\varSigma _1\)-completeness, that \(A^{\prime } \rhd \bot \). Quod non. If we have \(R^\bot \), it follows that \((B^{\prime }+ \lnot R^M)\rhd (A^{\prime }+R^N)\), by the fixed point equation. By \(\varSigma _1\)-completeness, we have \(B^{\prime } \rhd \bot \). Quod non. We may conclude that \(A^\star \mathrel {\not \! \rhd }B^\star \).

The proof that \(B^\star \mathrel {\not \! \rhd }A^\star \) is similar. \(\square \)

The following theorem gives the basic simple insight concerning the unsplittability of connected theories. For the definition of connected see the “Sums” section of Appendix 1.

Theorem 13

Suppose U and V are incomparable w.r.t. local interpretability. Then \(U\boxplus V\) is not connected. It follows that no connected theory W can be mutually locally interpretable with \(U \boxplus V\).

Proof

Suppose that U and V are incomparable w.r.t. local interpretability and that \(U \boxplus V\) is connected. Since \((U \boxplus V)\rhd _\mathsf{loc} (U \boxplus V)\), it follows, by connectedness, that either U locally interprets \(U \boxplus V\) or V locally interprets \(U\boxplus V\). Suppose U locally interprets \(U\boxplus V\). Then, \(U \rhd _\mathsf{loc} (U \boxplus V) \rhd _\mathsf{loc} V\). So, \(U \rhd _\mathsf{loc} V\). Quod non. The assumption that V locally interprets \(U \boxplus V\) leads similarly to a contradiction.

\(\square \)

We are now ready to prove the main result of this section.

Theorem 14

Q cannot be an \(\mathsf{INT}_3\)-retract of a sequential theory.

Proof

We work in \(\mathsf{INT}_3\). Suppose U is sequential and \(\mathsf{Q}\) is a retract of U. We derive a contradiction.

We have \(\mathsf{Q} \subseteq \mathsf{Q}^\#\) and, hence by Theorem 11, we can find a theory \(V \supseteq U\) such that \(\mathsf{Q}^\#\) is a retract of V. Since \(\mathsf{Q}^\#\) is synonymous with Y, the theory \(\mathsf{Q}^\#\) is a fortiori, sententially congruent to Y. It follows that Y is an \(\mathsf{INT}_3\)-retract of V.

We can interpret \(\mathsf{S}^1_2\) in \(\mathsf{Q}^\circ \), so \(\mathsf{Q}^\circ \) tolerates \(\mathsf{S}^1_2\). We can extend \(\mathsf{CQC}^2\) to the weak set theory AS which interprets \(\mathsf{S}^1_2\). So, \(\mathsf{CQC}^2\) tolerates \(\mathsf{S}^1_2\). It follows that \( \mathsf{CQC}^2 \boxplus {\mathbbm {1}}\) tolerates \(\mathsf{S}^1_2\).Footnote 12 Let \(A \supseteq \mathsf{Q}^\circ \) and \(B \supseteq \mathsf{CQC}^2 \boxplus {\mathbbm {1}}\) be the mutually incomparable theories promised by Theorem 12. By Theorem 11, we can find a theory \(W \supseteq V\) such that \(A \boxplus B\) is a retract of W.

It follows that \(A \boxplus B\) is mutually locally interpretable with W. Moreover, A and B are incomparable w.r.t. local interpretability, since they are finitely axiomatized. This contradicts the result of Theorem 13.

figure b

We may conclude that Q is not a retract of a sequential theory. \(\square \)

From our theorem, we immediately have that Q and \(\mathsf{PA}^-\) are not sententially congruent.

We note that the only property we used in our proof of sequential theories is the fact that sequential theories are closed under theory-extension-in-the-same-language. Thus for any class \({\mathcal {X}}\) of connected theories, such that \({\mathcal {X}}\) is closed under theory-extension, we have that Q cannot be an \(\mathsf{INT}_3\)-retract of \({\mathcal {X}}\).

8 Concluding remarks

The paper shows that the Pudlák Property and being a poly-pair theory are not preserved under mutual (faithful) interpretability. This provides two examples of good properties of theories that are not preserved under mutual (faithful) interpretability.Footnote 13 It illustrates the usefulness of having more refined notions of sameness of theories.

Connectedness is preserved under mutual interpretability and even under mutual local interpretability. It is a notion of non-splittability. As we have seen the theory Q is splittable in a sense, but this splittability is under the radar of the notion of connectedness, since Q is connected. This discrepancy suggests that it might be interesting to explore more refined versions of connectedness that would exclude Q but include \(\mathsf{PA}^-\).

We note that our results in Sect. 7 imply that Q is connected but has a non-connected extension, to wit (a theory synonymous to) a theory of the form \(A \boxplus B\), where A and B are finitely axiomatized theories that are mutually incomparable w.r.t. relative interpretability. The natural class of sequential theories, however, is upwards closed under theory-extension. So, one may wonder if a more refined version of connectedness would have the property of being preserved under theory-extension.