Essential hereditary undecidability

Abstract

In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below R. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in the converse direction, lax interpretability that interacts in a good way with essential hereditary undecidability. We introduce the class of $\Sigma ^0_1$-friendly theories and show that $\Sigma ^0_1$-friendliness is sufficient but not necessary for essential hereditary undecidability. Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to show that there is no interpretability minimal essentially hereditarily undecidable theory.

1 Introduction

Robinson’s Arithmetic Q has a wonderful property, to wit essential hereditary undecidability. This means that, if a theory is compatible with Q, it is undecidable (and even hereditarily undecidable). This property is very useful as a tool to prove that theories are undecidable. A classical example of this method is Tarski’s proof of the undecidability of group theory. See Tarski, Mostowski and Robinson’s [1]. As we will see, Q shares this property with many other theories ...¹

In this paper, we study essential hereditary undecidability. The paper is partly an exposition of some of the literature, but it also contains original results and analyses. We provide a number of examples of essentially hereditarily undecidable theories, both from the literature and new.

We connect the notion with two reduction relations: interpretability and lax interpretability (i.e., converse essential tolerance). Lax interpretability will be introduced in the present paper. Specifically, we show that essential hereditary undecidability is upward preserved under lax interpretability. We generalise Lax interpretability to a relation between sets of theories. We study the interaction of the Tarski–Mostowski–Robinson theory R with lax interpretability. We show that R is mutually laxly interpretable with the set of theories of false $\Sigma ^0_1$-sentences.

We develop the notion of $\Sigma ^0_1$-friendliness. The notion is defined in terms of the interaction of a theory with $\Sigma ^0_1$-sentences and is, thus, very much in the spirit of the present paper. The notion is in the same niche as effective inseparability for which it is a sufficient condition. It implies (but is not implied by) essential hereditary undecidability. The notion provides a rather natural and pleasantly applicable sufficient condition for many of the results of the present paper.

Finally, we demonstrate that there is no interpretability minimal essentially hereditarily undecidable theory. The proof is a minor adaptation of the proof that there is no interpretability minimal essentially undecidable theory in [3].

2 Basics

In this section we provide the basic facts and definitions needed in the rest of the paper.

2.1 Theories and interpretations

A theory is, in this paper, an RE theory of classical predicate logic in finite signature. A theory is given by an index of an RE axiom set. Here we confuse the sentences of a theory with numbers. We will usually work with a bijective Gödel numbering of the sentences. We adapt the Gödel numbering in each case to the signature at hand.

We write $U\subseteq _\textsf{e}V$ for: U and V are theories in the same language and the set of theorems of U is contained in the set of theorems of V. We use $=_\textsf{e}$ for: U and V are theories in the same language and U and V prove the same theorems.

If U is a theory, we write $U_{\mathfrak {p}}$ for the set of its theorems and $U_{\mathfrak {r}}$ for the set of its refutable sentences, i.e., $U_{\mathfrak {r}} {:}{=} \{ \varphi \mid U \vdash \lnot \, \varphi \}$.

We take $\mathfrak {id}_{U}$ to be the finitely axiomatised theory of identity for the signature of U. This theory is a built-in feature of predicate logic. However, if we work with interpretations, we need to check that it holds for the equivalence relation posing as the identity of the interpreted theory.

An interpretation K of a theory U in a theory V is based on a translation $\tau$ of the U-language into the V-language. Translations are most naturally thought of as translations between relational languages. A translation of a language with terms proceeds in two steps. First we follow a standard algorithm to translate the language with terms into a purely relational language and then we apply a translation as described below. A translation for the relational case commutes with the propositional connectives. In some broad sense, it also commutes with the quantifiers but here there are a number of extra features.

• Translations may be more-dimensional: we allow a variable to be translated to an appropriate sequence of variables.

• We may have domain relativisation: we allow the range of the translated quantifiers to be some domain definable in the V-language.

• We may even allow the new domain to be built up from pieces of possibly, different dimensions.

A further feature is that identity need not be translated to identity but can be translated to a congruence relation. Finally, we may also allow parameters in an interpretation. To handle these the translation may specify a parameter-domain $\alpha$.

We can define the obvious identity translation of a language in itself, composition of translations and a disjunctive translation $\tau {\langle \varphi \rangle }\nu$. E.g., in case $\tau$ and $\nu$ have the same dimension and are non-piecewise, the domain of $\tau {\langle \varphi \rangle }\nu$ becomes

$$\begin{aligned}(\varphi \wedge \delta _\tau (\vec x)) \vee (\lnot \,\varphi \wedge \delta _\nu (\vec x)).\end{aligned}$$

We refer the reader for details to e.g. [4].

An interpretation is a triple ${\langle U,\tau ,V \rangle }$, where $\tau$ is a translation of the U-language in the V-language such that, for all $\varphi$, if $U \vdash \varphi$, then $V \vdash \varphi ^\tau$.²

We write:

• $K:U \lhd V$ for: K is an interpretation of U in V.

• $U \lhd V$ for: there is a K such that $K:U \lhd V$. We also write $V\rhd U$ for: $U \lhd V$.

• $U \lhd _\textsf{loc} V$ for: for every finitely axiomatisable sub-theory $U_0$ of U, we have $U_0 \lhd V$.

• $U \lhd _\textsf{mod} V$ for: for every V-model $\mathcal {M}$, there is a translation $\tau$ from the U-language in the V-language, such that $\tau$ defines an internal U-model $\mathcal {N} = {\widetilde{\tau }}(\mathcal {M})$ of U in $\mathcal {M}$.

• We write $U \bowtie V$ for: $U \lhd V$ and $V\lhd U$. Similarly, for the other reduction relations.

Given two theories U and V we form in the following way. The signature of W is the disjoint union of the signatures of U and V with an additional fresh zero-ary predicate P. The theory W is axiomatised by the axioms $P \rightarrow \varphi$ if $\varphi$ is a U-axiom and $\lnot \, P \rightarrow \psi$ if $\psi$ is a V-axiom. One can show that is the infimum of U and V in the interpretability ordering $\lhd$. This result works for all choices of our notion of interpretation.

2.2 Arithmetical theories

The theory R, introduced in [1], is a primary example of an essentially hereditarily undecidable theory. For various reasons, we will work with a slightly weaker version of R. See Remark 1 below for a brief explanation of the difference and some background. The language of R, in our variant, is the arithmetical language ${\mathbb {A}}$ with 0, S, $+$, $\times$ and <.

Here are the axioms of R. The underlining stands for the usual unary numeral function.

R1.:

$\vdash {\underline{m}} + {\underline{n}} = {\underline{m+n}}$

R2.:

$\vdash {\underline{m}} \times {\underline{n}} = {\underline{m\times n}}$

R3.:

$\vdash {\underline{m}} \ne {\underline{n}}$, for $m\ne n$

R4.:

$\vdash x< {\underline{n}} \rightarrow \bigvee _{i< n} x= {\underline{i}}$

R5.:

$\vdash x< {\underline{n}} \vee x= {\underline{n}} \vee {\underline{n}} < x$

Remark 1

The original version of R in [1] did not have < in the language. Tarski, Mostowski and Robinson used $\le$ as a defined symbol with the following definition: $u \le t$ iff $\exists w\; w+u=t$. Their axioms are the obvious adaptation of the above ones with $\le$ in stead of <. One can employ an even weaker theory $\textsf{R}_0$, where one drops Axiom R5 and strengthens R4 by replacing the implication by a bi-implication. See, e.g., [5] for a discussion. Vaught, in his paper [6], employs an even weaker variant of $\textsf{R}_0$. We note that there is a mistake in Vaught’s formulation of his axioms. They need a strengthening to make everything work.

An important tool in the present paper is the theory of a number. There are various ways to develop this. E.g., we can treat the numerical operations as partial functions. Here we will employ a version using total functions. This version 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.³ The language of TN is the arithmetical language ${\mathbb {A}}$. Here are the axioms of TN.

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.:

$\vdash x=0 \vee \exists y\; x=\textsf{S}y$

TN5.:

$\vdash \textsf{S}x \not < x$

TN6.:

$\vdash x<y \rightarrow (x<\textsf{S}x \wedge y \not < \textsf{S}x)$

TN7.:

$\vdash x+0=x$

TN8.:

$\vdash x+\textsf{S}y =\textsf{S}(x+y)$

TN9.:

$\vdash x \times 0 = 0$

TN10.:

$\vdash x\times \textsf{S}y = x\times y + x$

We note that, by substituting x for y, $\textsf{TN}6$ implies $x \not < x$. So, < satisfies the axioms of a linear strict ordering with minimum 0. If follows from this fact in combination with $\textsf{TN}6$ again, that < is a discrete ordering and that S does give the order successor when applied to a non-maximal element. Moreover, by $\textsf{TN}5$, if there is a maximal element, then S maps it to itself. It follows that a model of TN is a discrete linear ordering with minimum 0. So, it either represents a finite ordinal or starts with a copy of $\omega$. Moreover, on a finite domain the successor function will behave normally, except on the maximum ${\mathfrak {m}}$, where we have $\textsf{S}{\mathfrak {m}} = {\mathfrak {m}}$.

Remark 2

The structure ${\mathbb {Z}}$ is a model of all axioms except TN1. Moreover, $\omega +1$, where we cut off all operations at $\omega$ in the obvious way, is a model of all axioms except $\textsf{TN}4$. Finally, ${\mathbb {Z}}_2$ with the ordering generated by $0<1$ is a model of all axioms except $\textsf{TN}5$.

We will be interested in the theory of a witness of a $\Sigma ^0_1$-sentence $\sigma$. There is a minor problem here. Even if the witness exists as a non-maximal element of a model, the value of a term may stick out. We can avoid this in several ways. We discussed one such way in our paper [4]. We follow the same strategy in the present paper. We define pure $\Delta _0$-formulas as follows:

• $\delta {:}{:}{=} \bot \mid \top \mid u< v \mid 0=u \mid \textsf{S}u=v \mid u+v=w \mid u\times v=w \mid \lnot \, \delta \mid$$(\delta \wedge \delta ) \mid (\delta \vee \delta ) \mid (\delta \rightarrow \delta ) \mid \forall u \mathbin {<}v\, \delta \mid \exists u\mathbin {<}v\, \delta \mid \forall u \mathbin {\le }v\, \delta \mid \exists u\mathbin {\le }v\, \delta$.

Here the bounded quantifiers are the usual abbreviations.

A pure $\Sigma ^0_1$-formula is of the form $\exists \vec u\, \delta$, where $\delta$ is pure $\Delta _0$. In [4], we showed that every ordinary $\Sigma ^0_1$-sentence can always be rewritten modulo $\textsf{EA}+{\text {B}}\Sigma _1$-provable equivalence to a pure one. We call something a 1-$\Sigma ^0_1$-formula if it starts with precisely one single existential quantifier.

A subtlety occurs in the treatment of substitution: consider a pure $\Sigma ^0_1$-formula $\sigma$ and, e.g., a substitution of a numeral in it, $\sigma [x{:}{=} {\underline{n}}\,]$. Here we will always assume that the result of substitution is rewritten to an appropriate pure $\Sigma ^0_1$ normal form.

We need the notion $z\models \varphi$, where z is considered as a number that models TN, where the arithmetical operations are cut off at z. We define $z\models \varphi$ by $\varphi ^{\mathfrak {tr}(z)}$, where $\mathfrak {tr}(z)$ is a parametric translation from the arithmetical language to the arithmetical language which is defined as follows:

• the domain of $\mathfrak {tr}(z)$ is the set of x such that $x \le z$,

• $\textsf{Z}^{\mathfrak {tr}(z)}(w) :\leftrightarrow w=0$,

• $\textsf{S}^{\mathfrak {tr}(z)}(x,w) :\leftrightarrow (\textsf{S}x \le z \wedge w=\textsf{S}x) \vee (\textsf{S}x \not \le z \wedge w=z)$,

• $\textsf{A}^{\mathfrak {tr}(z)}(x,y,w) :\leftrightarrow (x+y \le z \wedge w=x+y) \vee (x+y \not \le z \wedge w=z)$,

• $\textsf{M}^{\mathfrak {tr}(z)}(x,y,w) :\leftrightarrow (x\times y \le z \wedge w=x\times y) \vee (x\times y \not \le z \wedge w=z)$.

Let $\sigma {:}{=} \exists \vec x\, \delta$, where $\delta$ is pure $\Delta _0$ with at most the $\vec x$ free. Here $\vec x {:}{=} x_0,\dots , x_{n-1}$. We define:

• $\sigma ^{\mathfrak {q}} {:}{=} \exists z\, (\exists x_0\mathbin {<}z\dots \exists x_{n-1}\mathbin {<}z\; \delta \wedge z\models \bigwedge \textsf{TN})$.

We note that $\sigma ^{\mathfrak {q}}$ is equivalent with $\exists z\; z\models (\bigwedge \textsf{TN} \wedge \exists \textbf{x}\, (\delta \wedge \bigwedge _{i\mathbin {<}n} \textsf{S}x_i \ne x_i))$.

We will confuse $\sigma ^{\mathfrak {q}}$ with the theory axiomatised by this sentence. Since everything relevant to the evaluation of the sentence happens strictly below z, the pure $\Sigma ^0_1$-sentence $\sigma$ in the context of $(\cdot )^{\mathfrak {q}}$ has its usual arithmetical meaning. We have:

Theorem 1

• i. $\sigma ^{\mathfrak {q}} \vdash \sigma$.

• ii. Suppose $\sigma$ is true. Then, $\textsf{R}\vdash \sigma ^{\mathfrak {q}}$.

• iii. Suppose $\sigma$ is false. Then $\sigma ^{\mathfrak {q}} \vdash \textsf{R}$.

Proof

(Proof Sketch) Item (i) uses essentially the reasoning of the usual proof of $\Sigma ^0_1$-completeness. Item (ii) follows from the fact that in a model of $\sigma ^{\mathfrak {q}}$ the relevant bounding witness for $\sigma$ is non-standard so that all standard numbers are below it. The fact that this witness satisfies TN enforces that the standard numbers have the properties given by R.

In the paper [7] these facts are verified in great detail, though in a slightly modified version. $\square$

We will employ witness comparison notation. Suppose $\alpha$ is of the form $\exists x\, \alpha _0(x)$ and $\beta$ is of the form $\exists y\, \beta (y)$. We define:

• $\alpha <\beta {:}{=} \exists x\, (\alpha _0(x) \wedge \forall y \mathbin {\le }x\, \lnot \, \beta _0(y))$.

• $\alpha \mathbin {\le }\beta {:}{=} \exists x\, (\alpha _0(x) \wedge \forall y \mathbin {<}x\, \lnot \, \beta _0(y))$.

• If $\gamma$ is $\alpha < \beta$, then $\gamma ^\bot$ is $\beta \le \alpha$.

• If $\delta$ is $\alpha \le \beta$, then $\delta ^\bot$ is $\beta < \alpha$.

We note that witness comparisons between pure 1-$\Sigma ^0_1$-formulas are again pure 1-$\Sigma ^0_1$-formulas. The following insights are immediate.

Theorem 2

Suppose $\sigma$ and $\sigma '$ are pure 1-$\Sigma ^0_1$-sentences. Then,

• i. If $\sigma \le \sigma '$, then ${\sigma '}^{\mathfrak {q}} \vdash \sigma$.

• ii. $(\sigma \le \sigma ')^{\mathfrak {q}} \vdash \lnot \, (\sigma ' < \sigma )$ and $(\sigma < \sigma ')^{\mathfrak {q}} \vdash \lnot \, (\sigma ' \le \sigma )$.

• iii. Suppose $\sigma \le \sigma '$. Then, $(\sigma '< \sigma )^{\mathfrak {q}}$ is inconsistent. Suppose $\sigma < \sigma '$. Then, $(\sigma '\le \sigma )^{\mathfrak {q}}$ is inconsistent.

• iv. Suppose $\sigma$ is true. Then, if we allow piecewise interpretations, we have $\top \rhd \sigma ^{\mathfrak {q}}$. If we do not allow piecewise interpretations, we still have $(\exists x\, \exists y \, x\ne y) \rhd \sigma ^{\mathfrak {q}}$.

Remark 3

The reader of this paper will develop some feeling for the subtleties involved in our strategy to handle $\Sigma ^0_1$-sentences in the context of theories of a number. See “Appendix B” for an illustration of these subtleties.

In the paper [7] by Taishi Kurahashi and Albert Visser, a somewhat different approach to obtain sentences with the good properties of the $\sigma ^{\mathfrak {q}}$ is worked out. This paper verifies many details in a careful way.

A different idea for the treatment of the theory of the witness of a $\Sigma ^0_1$-sentence is to work with the usual definition of $\Sigma ^0_1$, but demand that the maximum element, if there is one, is larger that a suitable function of the Gödel number of $\sigma$ and the maximum of the witnesses.

One can also develop theories of a number using partial functions. This leads again to different possibilities to define the $\sigma ^{\mathfrak {q}}$-like sentences. See, e.g., [8] for an attempt to treat theories of a number in this style.

In a yet different approach, one develops finite versions of set theory. This idea is already discussed in [6]. See [9] for a beautiful way of realising the idea.

2.3 Recursive boolean isomorphism

Two theories U and V are recursively boolean isomorphic iff, there is a bijective recursive function $\varPhi$, considered as a function from the sentences of the U language to the V-language, such that:

• i. $\varPhi$ commutes with the boolean connectives. (So, $\varPhi (\bot )= \bot$ and $\varPhi (\varphi \wedge \psi )= (\varPhi (\varphi ) \wedge \varPhi (\psi ))$, etcetera.)

• ii. $U \vdash \varphi$ iff $V \vdash \varPhi (\varphi )$.

We note that it follows that, e.g.,

$$\begin{aligned} \varPhi ^{-1}(\varphi '\wedge \psi ')= & {} \varPhi ^{-1}(\varPhi \varPhi ^{-1}(\varphi ') \wedge \varPhi \varPhi ^{-1}(\psi ')) \\= & {} \varPhi ^{-1}\varPhi (\varPhi ^{-1}(\varphi ') \wedge \varPhi ^{-1}(\psi ')) \\= & {} \varPhi ^{-1}(\varphi ') \wedge \varPhi ^{-1}(\psi ') \end{aligned}$$

So $\varPhi ^{-1}$ is indeed the inverse isomorphism.

The demands on recursive boolean isomorphism are rather stringent. So it is good to know that the presence of an object satisfying far weaker demands implies the presence of a recursive boolean isomorphism. We note that the recursive boolean isomorphism was directly defined in terms of the syntactic objects. It witnesses, of course, an isomorphism in the ordinary sense between the Lindenbaum algebras of the relevant theories. We define a more lax notion of witness for the recursiveness of such a Lindenbaum isomorphism.

Let us write $\vdash$ of U-derivability and $\vdash '$ for $U'$-derivability. We also write $\sim$ for U-provable equivalence and $\sim '$ for $U'$-provable equivalence. We let $\varphi ,\psi ,\dots$ range over U-sentences and $\varphi ',\psi ',\dots$ over $U'$-sentences.

Let us say that an RE relation ${\mathcal {E}}$ between numbers, considered as a relation between U- and $U'$-sentences, witnesses a recursive Lindenbaum isomorphism iff we have:

• a. For all $\varphi$, there are $\chi$ and $\chi '$ such that $\varphi \sim \chi \mathrel {\mathcal {E}}\chi '$;

• b. For all $\varphi '$, there are $\chi$ and $\chi '$ such that $\chi \mathrel {\mathcal {E}}\chi '\sim ' \varphi '$;

• c. If $\varphi _0 \mathrel {\mathcal {E}}\varphi '_0$ and $\varphi _1 \mathrel {\mathcal {E}}\varphi '_1$, then $\varphi _0\sim \varphi _1$ iff $\varphi '_0\sim \varphi '_1$;

• d. If $\varphi \sim \varphi '$ and $\psi \sim \psi '$ and $(\varphi \wedge \psi ) \sim \chi \mathrel {\mathcal {E}} \chi '$, then $\chi ' \sim ' (\varphi '\wedge \psi ')$. Similarly, for the other boolean connectives.

The dual form of (d) follows from (a,c,d). Suppose $\varphi \mathrel {\mathcal {E}} \varphi '$ and $\psi \mathrel {\mathcal {E}}\psi '$ and $\chi \mathrel {\mathcal {E}} \chi '\sim ' (\varphi '\wedge \psi ')$. By (a) and (d), we can find $\rho$ and $\rho '$ such that

$$\begin{aligned} (\varphi \wedge \psi ) \sim \rho \mathrel {\mathcal {E}} \rho ' \sim ' (\varphi '\wedge \psi '). \end{aligned}$$

It follows that $\rho ' \sim \chi '$, and, hence, by (c), that $\rho \sim \chi$.

It is easy to see that if $\mathcal {E}$ witnesses recursive Lindenbaum isomorphism, then so does ${\sim } \circ {\mathcal {E}} \circ {\sim '}$.

Let us say that a sentence is a pseudo-atom iff it is either atomic or if it has a quantifier as main connective.

Theorem 3

Suppose $\mathcal {E}$ witnesses recursive Lindenbaum isomorphism between U and $U'$. Then, we can effectively find from an index of $\mathcal {E}$ an index of a recursive boolean isomorphism $\varPhi$ between U and $U'$.

Proof

This is by a straightforward back-and-forth argument. Suppose $\mathcal {E}$ witnesses recursive Lindenbaum isomorphism. Without loss of generality we may assume that $\mathcal {E}={\sim } \circ {\mathcal {E}} \circ {\sim '}$. Let us employ enumerations of sentences that enumerate boolean sub-sentences before sentences.

We construct $\varPhi$ in steps. Suppose we already have constructed

$$\begin{aligned} (\varphi _0,\varphi '_0),\, \dots ,\, (\varphi _{k-1},\varphi '_{k-1}). \end{aligned}$$

(Here k may be 0.) Suppose k is even. Let $\varphi _k$ be the first sentence in the enumeration of the U-sentences not among the $\varphi _i$, for $i<k$. In case $\varphi _k$ is a pseudo-atom, we take $\varphi '_k$ the first pseudo-atom in the enumeration of the $\psi '$ such that $\varphi _k \mathrel {\mathcal {E}} \psi '$. It is easy to see that there will always be such a pseudo-atom since we can always add vacuous quantifiers to a sentence. If $\varphi _k$ is, e.g., a conjunction, it will be of the form $(\varphi _i \wedge \varphi _j)$, for $i,j<k$, and we set $\varphi _k' {:}{=} (\varphi '_i \wedge \varphi '_j)$. The case that k is odd, is, of course, the dual case.

Clearly, this construction indeed delivers a recursive boolean isomorphism. $\square$

Let us write $U\approx U'$ for U is recursively boolean isomorphic to $U'$. An important insight is that $\approx$ is a bisimulation w.r.t. theory extension (in the same language). This means that:

• zig If $U \approx V$ and $U'\supseteq _\textsf{e}U$, then there is a $V' \supseteq _\textsf{e}V$, such that $U' \approx V'$;

• zag If $U \approx V$ and $V'\supseteq _\textsf{e}V$, then there is a $U' \supseteq _\textsf{e}U$, such that $U' \approx V'$.

Theorem 4

$\approx$ is a bisimulation for $\subseteq _\textsf{e}$.

Proof

We prove the zig case. Zag is similar. Suppose $U \approx V$ and $U' \supseteq _\textsf{e}U$. Let $\varPhi$ be a witnessing isomorphism. We define $V'$ as $\{ \varPhi (\varphi ) \mid \varphi \in U' \}$. We have:

$$\begin{aligned} V' \vdash \varPhi (\psi )\Leftrightarrow & {} \exists U'_0 \subseteq _\textsf{e} U'\;\; (U'_0 \text { is finite and }V \vdash \bigwedge _{\varphi \in U'_0} \varPhi (\varphi ) \rightarrow \varPhi (\psi )) \\\Leftrightarrow & {} \exists U'_0 \subseteq _\textsf{e} U'\;\; (U'_0 \text { is finite and } V \vdash \varPhi (\bigwedge _{\varphi \in U'_0} \varphi \rightarrow \psi ))\\\Leftrightarrow & {} \exists U'_0 \subseteq _\textsf{e} U'\;\; (U'_0 \text { is finite and }U \vdash \bigwedge _{\varphi \in U'_0} \varphi \rightarrow \psi )\\\Leftrightarrow & {} U' \vdash \psi \end{aligned}$$

$\square$

Suppose ${\mathcal {P}}$ is a property of theories. We say that U is essentially ${\mathcal {P}}$ if all consistent RE extensions (in the same language) of U are ${\mathcal {P}}$. We say that U is hereditarily ${\mathcal {P}}$ if all consistent RE sub-theories of U (in the same language) are ${\mathcal {P}}$. We say that U is potentially ${\mathcal {P}}$ if some consistent RE extension (in the same language) of U is ${\mathcal {P}}$.

If ${\mathcal {R}}$ is a relation between theories the use of essential and hereditary and potential always concerns the first component aka the subject. Thus, e.g., we say that U essentially tolerates V meaning that U essentially has the property of tolerating V. Tolerance itself is defined as potential intepretation. So U essentially tolerates V if U essentially potentially interprets V.

The following insight follows immediately from Theorem 4.

Theorem 5

Suppose $\mathcal {P}$ is a property of theories that is preserved by $\approx$. Then, so is the complement of $\mathcal {P}$ and the property of being essentially $\mathcal {P}$. Moreover, if $\mathcal {Q}$ is also a property of theories preserved by $\approx$, then so is the intersection of $\mathcal {P}$ and $\mathcal {Q}$.

We will see that we do not have an extension of Theorem 5 to include hereditariness.

Remark 4

Of course, the development above of recursive boolean isomorphism is very incomplete. It should be embedded in a presentation of appropriate categories. However, in the present paper, we restrict ourselves to the bare necessities.

Remark 5

Recursive boolean isomorphism is implied by sentential congruence (the interpretation equivalent of elementary equivalence). However, it is not preserved by mutual interpretability.

Here is a truly substantial result due to Mikhail Peretyat’kin: [10, Theorem 7.1.3]

Theorem 6

(Peretyat’kin) Suppose U is an RE theory with index i. Then, there is a finitely axiomatised theory $A {:}{=} \textsf{pere}(i)$ such that there is a recursive boolean isomorphism $\varPhi$ between U and A. Moreover, A and an index of $\varPhi$ can be effectively found from i.

There is a much simpler result that is also useful. We need a bit of preparation to formulate it. The result is due to Janiczak [11]. See also [3]. Let Jan be the theory in the language with one binary relationsymbol E with the following (sets of) axioms.⁴

J1.:

E is an equivalence relation.

J2.:

There is at most one equivalence class of size precisely n

J3.:

There are at least n equivalence classes with at least n elements.

We define $\textsf{A}_n$ to be the sentence: there exists an equivalence class of size precisely $n+1$. It is immediate that the $\textsf{A}_n$ are mutually independent over Jan.

Theorem 7

(Janiczak) Over Jan, every sentence is equivalent with a boolean combination of the $\textsf{A}_n$.

Jan will not be recursively boolean isomorphic to propositional logic with countably infinitely many propositional variables in our narrow sense, since, in Jan, there will be sentences equivalent to e.g. $\textsf{A}_0$ that are not identical to a boolean combination of $\textsf{A}_i$. However, Jan will be recursively Lindenbaum isomorphic to propositional logic. This does not contradict Theorem 3, since that theorem only applies to theories of predicate logic.

Let U be any theory. Remember that we work with a bijective coding for the U-sentences. We define $\textsf{jprop}(U)$ by $\textsf{Jan}$ plus all sentences of the form $\textsf{A}_{\varphi \wedge \psi } \leftrightarrow (\textsf{A}_{\varphi } \wedge \textsf{A}_{\psi })$, plus similar sentences for the other boolean connectives, plus all $\textsf{A}_\varphi$, whenever $U \vdash \varphi$. Clearly, we can effectively find an index of $\textsf{jprop}(U)$ from an index of U. We can easily se that:

Theorem 8

U is recursively boolean isomorphic with $\textsf{jprop}(U)$.

Proof

We define $\varphi \mathrel {\mathcal {E}} \varphi '$ iff $\varphi ' = \textsf{A}_\varphi$. It is easily seen that $\mathcal {E}$ witnesses recursive Lindenbaum isomorphism between U and $\textsf{jprop}(U)$. So, U and $\textsf{jprop}(U)$ are recursively boolean isomorphic, by Theorem 3. $\square$

2.4 Incompleteness and undecidability

We write $\textsf{W}_i$ for the RE set with index i. We define the following notions. We assume in all cases that U is consistent and RE.

• U is recursively inseparable iff $U_{\mathfrak {p}}$ and $U_{\mathfrak {r}}$ are recursively inseparable.

• U is effectively inseparable iff $U_{\mathfrak {p}}$ and $U_{\mathfrak {r}}$ are effectively inseparable. This means that there is a partial recursive function $\varPhi$ such that, whenever $U_{\mathfrak {p}}\subseteq \textsf{W}_i$, $U_{\mathfrak {r}}\subseteq \textsf{W}_j$, and $\textsf{W}_i \cap \textsf{W}_j = \emptyset$, we have $\varPhi (i,j)$ converges and $\varPhi (i,j) \not \in \textsf{W}_i\cup \textsf{W}_j$. We can easily show that $\varPhi$ can always taken to be total.

• U is effectively essentially undecidable, iff, there is a partial recursive $\varPsi$, such that, for every consistent RE extension V of U with index i, we have $\varPsi (i)$ converges and $\varPsi (i) \not \in V_{\mathfrak {p}} \cup V_{\mathfrak {r}}$.

The second and third of these notions turn out to coincide. This was proven by Marian Boykan Pour-El. See [12].

Theorem 9

(Pour-El) A theory is effectively inseparable iff it is effectively essentially undecidable.

Clearly, recursively inseparable implies essentially undecidable. Andrzej Ehrenfeucht, in his paper [13], provides an example of an essentially undecidable theory that is not recursively inseparable. So there is no non-effective equivalent of Theorem 9.

The next theorem is due to Marian Boykan Pour-El and Saul Kripke. See [14, Theorem 2].

Theorem 10

(Pour-El & Kripke) Consider any two effectively inseparable theories $U_0$ and $U_1$. Then, $U_0$ and $U_1$ are recursively boolean isomorphic. Moreover, an index of the isomorphism can be found effectively from the indices of the theories and the indices of the witnesses of effective inseparability.

The following result is [1, Chapter I, Lemma, p15] and [1, Chapter I, Theorem 1].

Theorem 11

(Tarski, Mostowski, Robinson) Suppose the theory U is decidable. Then, U has a complete decidable extension $U^*$. In other words, decidable theories are potentially complete. As a direct consequence, potential decidability and potential completeness coincide, or, equivalently, essential undecidability and essential incompleteness are extensionally the same.

Caveat emptor: If we, e.g., restrict ourselves to finite extensions, the equivalence between essential undecidability and essential incompleteness fails. So, it is good to recognise these as different notions even if they are extensionally the same.

The next result is fundamental is the study of hereditariness. It is [1, Chapter I, Theorem 5].

Theorem 12

(Tarski, Mostowski, Robinson) Suppose the theory U is decidable and $\varphi$ is a sentence in the U-language. Then, $U+\varphi$ is also decidable.

3 Essential hereditary undecidability: a first look

In this section, we collect the basic facts about Essential Hereditary Undecidability and provide a selection of examples.

3.1 Characterisations

We give with two pleasant characterisations of essential hereditary undecidability.

Theorem 13

A theory U is essentially hereditarily undecidable iff, for every W in the U-language, if $U+W$ is consistent, then W is undecidable.

Proof

This is immediate since W is consistent with U iff, for some consistent V, we have $U \subseteq _\textsf{e}V\supseteq _\textsf{e}W$. $\square$

We note that, more generally, U is essentially hereditarily $\mathcal {P}$ iff, for every W in the U-language, if $U+W$ is consistent, then W is $\mathcal {P}$.⁵

We say that V tolerates U if V potentially interprets U. In other words, V tolerates U iff there is a consistent $V'\supseteq _\textsf{e}V$ such that $V' \rhd U$. Equivalently, V tolerates U iff, there is a translation $\tau$ of the U-language into the V-language such $V+U_{\mathfrak {p}}^\tau$ is consistent. Finally, V tolerates U iff, there is a translation $\tau$ of the U-language into the V-language such $V+\mathfrak {id}_{U}^{\tau }+U^\tau$ is consistent.⁶

Theorem 14

Suppose U is consistent. The theory U is essentially hereditarily undecidable iff every V that tolerates U is undecidable.

Proof

We treat the argument for the parameter-free case. The case with parameters only requires a few obvious adaptations.

Suppose U is essentially hereditarily undecidable and $V+\mathfrak {id}_{U}^{\tau }+U^\tau$ is consistent. Let W be the theory in the U-language axiomatised by $\{ \varphi \mid V+\mathfrak {id}_{U}^{\tau }\vdash \varphi ^\tau \}$.

We find that $W \vdash \varphi$ iff $V +\mathfrak {id}_{U}^{\tau }\vdash \varphi ^\tau$ and that $U+W$ is consistent. Hence W is not decidable. Suppose that V is decidable. Then, $V+\mathfrak {id}_{U}^{\tau }$ is decidable and so is W. Quod non. So V is undecidable.

The other direction is immediate. $\square$

3.2 Essential hereditary incompleteness

Clearly, incompleteness is not the same as undecidability. However, essential incompleteness is the same as essential undecidability (by Theorem 11). On the other hand, incompleteness is always preserved to sub-theories. So, a fortiori, essential hereditary incompleteness coincides with essential incompleteness, which coincides with essential undecidability. For example, the decidable theory Jan has an essentially incomplete extension. So, essential hereditary incompleteness and essential hereditary undecidability do not coincide.

3.3 Closure properties

We prove closure of the essentially hereditarily undecidable theories under interpretability infima.

Theorem 15

• a. Suppose $U_0$ and $U_1$ are essentially undecidable. Then is essentially undecidable.

• b. Suppose $U_0$ and $U_1$ are essentially hereditarily undecidable. Then is essentially hereditarily undecidable.

Proof

We just treat (b). Let P be the 0-ary predicate that ‘chooses’ between $U_0$ and $U_1$ in and let $\textsf{e}_i$ be the identical translation of the $U_i$-language into the -language. Suppose W is consistent with U. Clearly, at least one of $U +W +P$ or $U+W+ \lnot \,P$ is consistent. Suppose $U +W +P$ is consistent. It follows that W tolerates $U_0$ as witnessed by the interpretation of $U_0$ in $U+W+P$ based on $\textsf{e}_0$. So W is undecidable. The other case is similar. $\square$

We show that the essentially hereditarily undecidable theories are upwards closed under interpretability.

Theorem 16

Suppose U is consistent and essentially hereditarily undecidable and $V\rhd U$. Then V is essentially hereditarily undecidable.

Proof

Suppose that U is essentially hereditarily undecidable and U is interpretable in V, say via K. Suppose further that W is a theory in the V-language that is decidable and consistent with V. Let $Z {:}{=} \{ \varphi \mid W+ \mathfrak {id}_{V}^{K} \vdash \varphi ^K \}$. It is easy to see that Z is decidable and consistent with V. Quod non.

Our proof is easily adapted to the case with parameters. $\square$

Theorem 30 of this paper will be a strengthening of this result.

3.4 Hereditary undecidability

If a theory tolerates an essentially hereditarily undecidable theory, then it is not just undecidable, but hereditarily undecidable.

Theorem 17

Suppose U is essentially hereditarily undecidable and that V tolerates U. Then V is hereditarily undecidable.

Proof

This is immediate from the fact that toleration is downward preserved in both arguments. $\square$

It would be great when the above theorem had a converse. However, the example below shows that this is not the case. The example is a minor variation of Theorem 3.1 of [2].

Example 1

(Hanf). We provide an example of a theory that is hereditarily undecidable but does not tolerate any essentially undecidable theory (and, so, a fortiori does not tolerate an essentially hereditarily undecidable theory). We consider Putnam’s example of a theory that is undecidable such that all its complete extensions are decidable. See [15, Section 6].

We start by specifying a theory in the language of identity. Let:

• ${\widetilde{n}} {:}{=} \exists x_0\dots \exists x_{n-1}\,(\bigwedge _{i<j<n} x_i \ne x_j \wedge \forall y \, \bigvee _{k<n}y=x_k)$.

Let $\mathcal {X}$ be any non-recursive set. We take: $\textsf{I}_{{\mathcal {X}}} {:}{=} \{ \lnot \, {\widetilde{n}} \mid n\in {\mathcal {X}} \}$. Clearly, $\textsf{I}_{{\mathcal {X}}}$ is non-recursive.

The theory $\textsf{I}_{{\mathcal {X}}}$ has the following complete extensions: ${\widetilde{n}}$, for $n \not \in {\mathcal {X}}$ and $\{ \lnot \,{\widetilde{n}} \mid n\in \omega \}$. So there are no non-recursive complete extensions. The theory $\textsf{I}_{\mathcal X}$ cannot be consistent with an essentially undecidable U in the same language (and, hence cannot tolerate an essentially undecidable V), since $\textsf{I}_{{\mathcal {X}}}+U$ would have a complete and recursive extension.

We now apply Theorem 6 (Peretyat’kin’s result), to obtain a finitely axiomatised theory $\textsf{J}_{{\mathcal {X}}}$ that is recursively boolean isomorphic to $\textsf{I}_{{\mathcal {X}}}$. Clearly, $\textsf{J}_{{\mathcal {X}}}$ will inherit the undecidability and the lack of non-recursive complete extensions from $\textsf{I}_{{\mathcal {X}}}$. Since, $\textsf{J}_{{\mathcal {X}}}$ is finitely axiomatised and undecidable, it will be hereditarily undecidable.

We note that the original theory $\textsf{I}_{{\mathcal {X}}}$ extends the theory of pure identity in the language of pure identity. So, $\textsf{I}_{{\mathcal {X}}}$ itself is not hereditarily undecidable.

Example 2

We show that there are theories that are essentially undecidable and hereditarily undecidable but not essentially hereditarily undecidable.⁷

Suppose U is essentially hereditarily undecidable and V is essentially undecidable but not hereditarily undecidable.

By Theorem 15(a), we find that is essentially undecidable.

Suppose W is a decidable sub-theory of . Let P be the fresh 0-ary predicate used in the definition of . Then, $W+ P$ is a sub-theory of , which is, modulo derivability, $U+P$ in the extended language. Moreover, $W+P$ is decidable. It follows that the consequences of $W+P$ in the U-language are decidable. But these consequences are a sub-theory of U. A contradiction. So, is hereditarily undecidable.

Finally, let Z be a decidable sub-theory of V. We extend the signature of Z to the signature of and add the axiom $\lnot \, P$ plus axioms of the form $\forall \vec x\, R(\vec x)$, for all predicates R of the U-signature. The resulting theory $Z'$ is a definitional extension of Z and, thus, decidable. Clearly, $Z'$ is consistent with . So is not essentially hereditarily undecidable.

3.5 Essentially hereditarily undecidable theories

In this subsection, we give an overview of some essentially hereditarily undecidable theories.

A first insight is given by Theorem 12 and [1, Chapter I, Theorem 6].

Theorem 18

(Tarki, Mostowski, Robinson) Suppose the theory A is finitely axiomatizable. If A is undecidable, then it is hereditarily undecidable. If A is essentially undecidable, then A is essentially hereditarily undecidable.

Theorems 18, 6 and 8 give us immediately the following insight:

Theorem 19

Suppose U is an (essentially) undecidable theory. Then, there are (essentially) undecidable theories $U_0$ and $U_1$ that are recursively boolean isomorphic to U of which the first is (essentially) hereditarily undecidable and the second has a decidable sub-theory. Indices for $U_0$ and $U_1$ can be effectively found from an index of U. Specifically, we can take $U_0 {:}{=} \textsf{pere}(i)$, where i is an index of U and $U_1 {:}{=} \textsf{jprop}(U)$.

The use of Theorem 6 delivers many examples of (essentially) hereditarily undecidable theories, Here is, for example, Theorem 3.3 of [2].

Theorem 20

(Hanf) Let d be any non-zero RE Turing degree. Then there is a finitely axiomatised essentially hereditarily undecidable theory A of degree d.

Proof

By the results of [16], there is an essentially undecidable RE theory U of degree d. Say it has index i. Clearly, $\textsf{pere}(i)$ fills the bill. $\square$

Using the ideas of [3], we can even arrange it so that the Turing degree of every theory that interprets the theory A of Theorem 20 is $\ge d$.

The next example is due to Cobham. This result is presented in [6]. See also [4] for an alternative presentation. We will prove the result in Sect. 4.

Theorem 21

(Cobham) The theory R is essentially hereditarily undecidable.

We have the following corollary of Theorem 20.

Corollary 22

There are essentially hereditarily undecidable theories that do not interpret R and, hence, there are essentially hereditarily undecidable theories strictly below R.

Proof

Suppose d is an RE Turing degree strictly between 0 and $0'$. By Theorem 20, we can find an essentially hereditarily undecidable theory A of RE degree d. If $A \rhd \textsf{R}$, then the degree of A would be $0'$, so, $A\mathrel {\not \! \rhd }\textsf{R}$. By Theorem 15 in combination with Theorem 21, the theory is essentially hereditarily undecidable. Moreover, since $A\mathrel {\not \! \rhd }\textsf{R}$, the theory B is strictly below R.

A different and more natural example is the theory $\textsf{PA}^{-}_\textsf{scat}$ of Sect. 6. $\square$

A well-trodden path is the construction of essentially undecidable theories using recursively inseparable sets. We give the basic lemma.

Lemma 23

Suppose $\varPhi$ is a recursive function from the natural numbers to the sentences of U. Let ${\mathcal {X}},{\mathcal {Y}}$ be a pair of recursively inseparable sets. Suppose $\varPhi$ maps ${\mathcal {X}}$ to $U_{\mathfrak {p}}$ and ${\mathcal {Y}}$ to the $U_{\mathfrak {r}}$. Then, U is essentially undecidable.

From the proof of Theorem 3.2 of [2] we can extract the following analogue of Lemma 23 for the case of essentially hereditarily undecidable theories.

Lemma 24

(Hanf) Let U be a consistent RE theory and let $U_0$ be a finitely axiomatised sub-theory of U. Suppose $\varPhi$ is a recursive function from the natural numbers to the sentences of U. Let ${\mathcal {X}},{\mathcal {Y}}$ be a pair of recursively inseparable sets. Suppose $\varPhi$ maps ${\mathcal {X}}$ to $U_{0\mathfrak {p}}$ and ${\mathcal {Y}}$ to $U_{\mathfrak {r}}$. Then, U is essentially hereditarily undecidable.

Proof

Let $U,U_0,\varPhi ,{\mathcal {X}},{\mathcal {Y}}$ be as in the statement of the theorem. Suppose W is a theory in the language of U that is consistent with U. Suppose W is decidable. By Theorem 12, we find that $W^*{:}{=} W+U_0$ is decidable. Moreover, $W^*$ is consistent with U. We have:

$$\begin{aligned} n \in {\mathcal {X}}\Rightarrow & {} U_0 \vdash \varPhi (n) \\\Rightarrow & {} W^*\vdash \varPhi (n).\\ m\in {\mathcal {Y}}\Rightarrow & {} U \vdash \lnot \, \varPhi (m) \\\Rightarrow & {} W^*\nvdash \varPhi (m). \end{aligned}$$

It follows that $\{ k \mid W^*\vdash \varPhi (k) \}$ is decidable and separates ${\mathcal {X}}$ and ${\mathcal {Y}}$. A contradiction. $\square$

As we will see, in Sect. 4, the essential hereditary undecidability of the salient theory R is directly connected with the essential hereditary undecidability of certain finitely axiomatised theories. The following example, due to Hanf in [2, Theorem 3.2], shows that there are very un-R-like essentially hereditarily undecidable theories.

Example 3

(Hanf). We produce an essentially hereditarily undecidable RE theory U that does not tolerate any finitely axiomatisable essentially undecidable theory A.

Let ${\mathcal {X}}$ and ${\mathcal {Y}}$ be recursively inseparable sets. Let $V {:}{=} \textsf{Jan} + \{ \textsf{A}_n \mid n\in {\mathcal {X}} \}$. Let B be $\textsf{pere}(i)$, where i is an index of V, and let $\varPsi$ be the boolean isomorphism from V to B. We define $\textsf{B}_i{:}{=} \varPsi (\textsf{A}_i)$ and $U {:}{=} B + \{ \lnot \, \textsf{B}_j \mid j \in {\mathcal {Y}} \}$. By Lemma 24, the theory U is essentially hereditarily undecidable.

Suppose U tolerates a finitely axiomatised essentially undecidable theory A. Then, some finite theory C in the language of U is consistent with U and interprets A. Clearly, C must itself be essentially undecidable. Now $\varPsi ^{-1}(C)$ is equivalent to a boolean combination of the $\textsf{A}_i$ over V, so C is equivalent to a boolean combination of the $\textsf{B}_i$ over B. Let the set of the i so that $\textsf{B}_i$ occurs in this boolean combination be ${\mathcal {F}}$. Let $W {:}{=} B +C + \{ \textsf{B}_i \mid i \not \in {\mathcal {F}} \}$. Clearly, W is consistent and decidable. A contradiction with the fact that C is essentially undecidable.

We note that we can get our example in any desired non-zero RE Turing degree by choosing the appropriate ${\mathcal {X}}$ and $\mathcal Y$.

Remark 6

In [17], we show the following. Suppose U is consistent, RE, and effectively Friedman-reflexive. Then, U is essentially hereditarily undecidable. Discussion of Friedman Reflexivity is outside the scope of this paper. See Remark 13 for one further comment.

4 Essential tolerance and lax interpretability

In this section we study a reduction relation that interacts very well with essential hereditary undecidability. We will prove a number of theorems that illustrate these connections.

4.1 Basic definitions and facts

Suppose U is a consistent RE theory. We remind the reader that U tolerates V, or $U \Rsh V$, iff U potentially interprets V, in other words, if for some consistent RE theory $U'\supseteq _\textsf{e}U$, we have $U'\rhd V$. We find that U essentially tolerates V iff U essentially potentially interprets V, explicitly: iff, for all consistent RE theories $U'\supseteq _\textsf{e}U$, there is a consistent RE theory $U'' \supseteq _\textsf{e}U'$, such that $U'' \rhd V$. We write for U essentially tolerates V.

We note that essential tolerance is analogous to the converse of interpretability. In other words, ‘essentially tolerates’ is analogous to ‘interprets’. We will call the converse of essential tolerance: lax interpretability.

Below we establish that essential tolerance is a bona fide reduction relation—unlike tolerance that fails to be transitive.

Remark 7

The notion of tolerance was introduced in [1] under the name of weak interpretability. We like ‘tolerates’ more since it is more directly suggestive of the intended meaning. Japaridze uses tolerance in a more general sense. See [18] and [19], or the handbook paper [20].

Example 4

We illustrate the intransitivity of tolerance. In fact, our counterexample shows a bit more.

Presburger Arithmetic essentially tolerates Predicate Logic in the language with a binary relation symbol. Predicate Logic in the language of a binary relation symbol tolerates full Peano Arithmetic. However, Presburger Arithmetic does not tolerate Peano Arithmetic.

Remark 8

The definition of suggests several variations, where we demand that some promised ingredients are effectively found from appropriate indices. We will not explore such variations in the present paper.

Remark 9

Robert Vaught, in his paper [6] introduces a notion that we would like to call parametrically local interpretability or $\lhd _\textsf{pl}$. This notion interacts in desirable ways with essential hereditary undecidability. We discuss the relationship between and $\lhd _\textsf{pl}$ in “Appendix A”. We show that , a slightly improved version of , satisfies: if $U \rhd _\textsf{pl}V$, then . Moreover, for our purposes, retains all the good properties of .

The first two insights are that lax interpretability is (strictly) between two good notions of interpretability, to wit, model interpretability and local interpretability.

Theorem 25

If $U \rhd _\textsf{mod} V$, then .

Proof

Suppose $U \rhd _\textsf{mod} V$. Let $U'$ be a consistent theory with $U' \supseteq _\textsf{e}U$. Consider any model ${\mathcal {M}}$ of $U'$. There is an $\mathcal {M}$-internal model of V, say, given by translation $\tau$. Let $U'' {:}{=} U'+ \{ \psi ^\tau \mid V \vdash \psi \}$. Clearly, $U''$ is consistent and RE and $U'' \rhd V$ as witnessed by $\tau$. $\square$

In Sect. 6, we develop the theory ${\textsf{PA}^-_\textsf{scat}}$. This theory is a sub-theory of R. We have , but ${\textsf{PA}^-_\textsf{scat}}\mathrel {\not \! \rhd }_\textsf{mod} \textsf{R}$. This tells us that the inclusion of model interpretability in lax interpretability is strict.

Openquestion 1

Are there sequential U and V such that we have , but $U \mathrel {\not \! \rhd }_\textsf{mod}V$?

We turn to the comparison of lax and local interpretability.

Theorem 26

If , then $U \rhd _\textsf{loc} V$.

Proof

Suppose . Let $V_0$ be a finitely axiomatised sub-theory of V. Let $\varphi$ be a single axiom of $V_0$ which includes $\mathfrak {id}_{V}$. Suppose $U \mathrel {\not \! \rhd }V_0$. Consider the theory $U' {:}{=} U + \{ \lnot \, \varphi ^\tau \mid \tau : \Sigma _V \rightarrow \Sigma _U \}$.⁸ The theory $U'$ is consistent since, if not, U would prove a finite disjunction of sentences of the form $\varphi ^\tau$. Say the translations involved are $\tau _0,\dots ,\tau _{n-1}$. We define:

$$\begin{aligned} \tau ^*{:}{=} \tau _0{\langle \varphi ^{\tau _0} \rangle }(\tau _1{\langle \varphi ^{\tau _1} \rangle }(\dots (\tau _{n-2}{\langle \varphi ^{\tau _{n-1}} \rangle }\tau _{n-1})\dots )). \end{aligned}$$

We find that $U \vdash \varphi ^{\tau ^*}$. Quod non. So, $U'$ is consistent. Clearly $U'$ is RE and no consistent RE extension of $U'$ can interpret the theory axiomatised by $\varphi$. But this contradicts . $\square$

Example 5

Consider a consistent finitely axiomatised sequential theory A. We do have $A \rhd _\textsf{loc} \mho (A)$. Here $\mho (A)$ is the theory $\textsf{S}^1_2+\{ \textsf{Con}_n(A) \mid n\in \omega \}$, where $\textsf{Con}_n$ means consistency w.r.t. proofs where all formulas in the proof have depth of quantifier alternations complexity $\le n$. See, e.g., [21] for more on $\mho$.

In, e.g., [22] it is verified in detail that A has a consistent RE extension ${\widetilde{A}}$ such that every interpretation of $\textsf{S}^1_2$ in ${\widetilde{A}}$ contains a restricted inconsistency statement for A. We call such an extension a Krajíček-theory based on A. Clearly, no consistent extension of ${\widetilde{A}}$ in the same language can interpret $\mho (A)$. So . This gives us our desired separating example between and $\rhd _\textsf{loc}$.

We note that in fact expresses the existence of a Krajíček extension.

Another example is as follows. Consider any complete and decidable theory U. We do have $U \rhd _\textsf{loc} \textsf{R}$. However, . Since no complete RE theory does interpret R.

It turns out that it is useful to lift to a relation between sets of theories. Specifically, this lifting will make a compact formulation possible of Theorem 31, that specifies a fundamental link between R and the theories of false $\Sigma ^0_1$-sentences. We define:

• iff for all $U\in {\mathcal {X}}$ and for all consistent RE theories $U'\supseteq _\textsf{e}U$, there is a consistent RE theory $U''\supseteq _\textsf{e}U'$ and a $V\in {\mathcal {Y}}$, such that $U'' \rhd V$.

We note that is equivalent to . We will write for , etcetera.

Theorem 27

• a. Suppose ${\mathcal {X}} \subseteq {\mathcal {X}}'$ and and ${\mathcal {Y}}' \subseteq {\mathcal {Y}}$. Then, .

• b. We have: and iff .

• c. The relation between sets of theories is transitive. As a consequence, as a relation between theories is transitive.

Proof

We just treat (c). Suppose . Consider $U \in {\mathcal {X}}$ and let $U'$ be any consistent RE extension of U. Let $U''$ be a consistent RE extension of $U'$ such that $U'' \rhd V$, for some $V \in \mathcal Y$. Say, we have $K:U'' \rhd V$. Let $V' {:}{=} \{ \psi \mid U'' \vdash \psi ^K \}$. We find that $V'$ is a consistent RE extension of V. Let $V''$ be a consistent extension of $V'$ such that $V'' \rhd W$, for some $W\in {\mathcal {Z}}$.

We consider $U^*{:}{=} U'' + \{ \psi ^K \mid V'' \vdash \psi \}$. Clearly $U^*$ is RE, $U^*\supseteq _\textsf{e}U'$ and $U^*\rhd V'' \rhd W$, so $U^*\rhd W$. We claim that $U^*$ is consistent. If not, there would be a $\psi$ such that $V'' \vdash \psi$ and $U'' \vdash (\lnot \, \psi )^K$. It follows, by the definition of $V'$, that $V' \vdash \lnot \, \psi$ and, hence, that $V'' \vdash \lnot \, \psi$, contradicting the fact that $V''$ is consistent. Thus, $U^*\supseteq _\textsf{e}U'$ and W are our desired witnesses. $\square$

We write $U \Rsh {\mathcal {Y}}$ for U tolerates some element of ${\mathcal {Y}}$. Inspection of the above proof also tells us that:

Theorem 28

Suppose $U \Rsh V$ and . Then, $U \Rsh {\mathcal {Z}}$.

Theorem 29

• i. iff and .

• ii. .

Proof

We just do (i). Claim (ii) is similar. From left-to-right is immediate, since and , and, hence, and . So, we are done by transitivity.

Let . Suppose and . Let $Z'\supseteq _\textsf{e}Z$ be RE and consistent. The theory $Z'$ is either consistent with P or with $\lnot P$. Suppose it is consistent with P. Let $U'$ be the set of U-sentences that follow from $Z'+P$. Clearly, $U'\supseteq _\textsf{e}U$ and $U'$ is RE and consistent. So, there is a $U''\supseteq _\textsf{e}U'$ that is RE and consistent such that $U'' \rhd W$. We take $Z''$ the theory axiomatised by $Z'+P+U''$ in the Z-language. Clearly, $Z''\supseteq _\textsf{e}Z'$ and $Z''$ is consistent and RE and $Z'' \rhd W$. The argument in case $\lnot \, P$ is consistent is similar. $\square$

We note that the above theorem tells us that the embedding functor of interpretability into lax interpretability preserves infima.

Remark 10

We define as follows. is the result of taking the disjoint union of the signatures of U and V and taking as axioms $\varphi \vee \psi$, whenever $U \vdash \varphi$ and $V \vdash \psi$. It is easy to see that gives representatives of the infimum for $\lhd _\textsf{mod}$, , and $\lhd _\textsf{loc}$.

Openquestion 2

It would be good to have a counterexample that shows that is not generally an infimum for $\lhd$.

Openquestion 3

The new notion of lax interpretability raises many questions. E.g.: is there a good supremum for lax interpretability? And: does the embedding functor of interpretability into lax interpretability have a right or left adjoint? Specifically, is there a $\varPhi$ such that $U \lhd \varPhi V$ iff (right adjoint); is there a $\varPsi$ such that $\varPsi U \lhd V$ iff (left adjoint)?

4.2 Essential hereditary undecidability meets lax interpretability

We start with the main insight concerning the relation between Essential Hereditary Undecidability and Lax Interpretability.

Theorem 30

Let U be RE and consistent.

• i. Suppose ${\mathcal {V}}$ is a class of essentially undecidable theories and . Then, U is essentially undecidable.

• ii. Suppose ${\mathcal {V}}$ is a class of essentially hereditarily undecidable theories and . Then, U is essentially hereditarily undecidable.

Proof

Ad (i). Suppose ${\mathcal {V}}$ is a class of essentially undecidable theories and . Suppose U has a consistent decidable extension W. Then, W has a decidable consistent complete extension $W^*$. It follows that $W^*\rhd V$, for some V in $\mathcal {V}$. Quod impossibile.

Ad (ii). Suppose ${\mathcal {V}}$ is a class of essentially hereditarily undecidable theories and . Suppose W is an RE theory in the language of U and suppose $U' {:}{=} U \cup W$ is consistent. We have to show that W is undecidable.

Let $U''$ be a consistent RE extension of $U'$ such that $K:U'' \rhd V$, for some $V\in {\mathcal {V}}$. Consider the theory $Z{:}{=} \{ \psi \mid W+\mathfrak {id}_{V}^{K} \vdash \psi ^K \}$. We have $Z \vdash \psi$ iff $W+\mathfrak {id}_{V}^{K} \vdash \psi ^K$. Clearly, Z is interpretable in V via $\tau$. So, Z is consistent with V. If W were decidable then so would Z, contradicting the fact that V is essentially hereditarily undecidable.⁹$\square$

Remark 11

Inspection of the example provided by Ehrenfeucht in [13], shows that his construction provides an example where , each element of $\mathcal {V}$ is recursively inseparable (if we wish, even effectively inseparable), but U is not recursively inseparable. The theory U of the example is essentially undecidable.

We now turn to the result that motivates looking at classes of theories as relata of . Let ${\mathfrak S}$ be the set of all theories $\sigma ^{\mathfrak {q}}$, where $\sigma$ is a false pure $\Sigma ^0_1$-sentence and $\sigma ^{\mathfrak {q}}$ is consistent.¹⁰

Theorem 31

We have .

Proof

From left-to-right. Let $U'$ be a consistent RE extension of R. Clearly, $U' \vdash \sigma ^{\mathfrak {q}}$, for all true pure $\Sigma ^0_1$-sentences $\sigma$. So, if no $\sigma ^{\mathfrak {q}}\in {\mathfrak S}$, would be consistent with $U'$, we could decide $\Sigma ^0_1$-truth. Quod non. Consider any such $\sigma ^{\mathfrak {q}}$ that is consistent with $U'$. Let $U'' {:}{=} U'+\sigma ^{\mathfrak {q}}$. Clearly, $U'' \rhd \sigma ^{\mathfrak {q}}$.

From right-to-left. Consider $\sigma ^{\mathfrak {q}}\in {\mathfrak S}$. Clearly, $\sigma ^{\mathfrak {q}} \rhd \textsf{R}$ and we are easily done. $\square$

Of course, the extension $U''$ in the proof of Theorem 31 can be found effectively from an index of $U'$. We outline one way to do it.

Proof

(Sketch of an alternative proof of Theorem 31) Let be $U'$-provability. By the Gödel Fixed Point Lemma, we can find a pure $\Sigma _1$-sentence $\jmath$ that is equivalent to .¹¹ Suppose $\jmath$ were true, then we have both and , contradicting the consistency of $U'$. So $\jmath$ is false and $\jmath ^\textsf{q}$ is consistent with $U'$. We take $U'' {:}{=} U'+\jmath ^\textsf{q}$. $\square$

Since all $\sigma ^{\mathfrak {q}}\in {\mathfrak S}$ extend R, they are essentially undecidable. Moreover, since the $\sigma ^{\mathfrak {q}}$ are finitely axiomatised, they are essentially hereditarily undecidable. It follows from Theorem 30, that R is essentially hereditarily undecidable. So, this gives us a proof of Theorem 21.

Openquestion 4

Suppose . Does it follow that U is recursively inseparable?

Let ${\mathfrak F}$ be the set of all finitely axiomatised essentially hereditarily undecidable theories. Example 3 shows that there is an essentially hereditarily undecidable theory U such that .

5 $\Sigma ^0_1$-friendliness and $\Sigma ^0_1$-representativity

In this section, we have a brief look at a rather natural property of theories that implies essential hereditary undecidability.

Consider a consistent RE theory U and a recursive function $\varPhi$ from pure 1-$\Sigma ^0_1$-sentences to U-sentences. We give three possible properties of $\varPhi$. Let $\sigma$ range over pure 1-$\Sigma ^0_1$-sentences.

$\Sigma$1.:

If $\sigma$ is true, then $U \vdash \varPhi (\sigma )$.

$\Sigma$2.:

$(U+\varPhi (\sigma ))\rhd \sigma ^{\mathfrak {q}}$.

$\Sigma$3.:

Suppose $\sigma \le \sigma '$. Then, $U \vdash \lnot \, \varPhi (\sigma '<\sigma )$. Similarly, suppose $\sigma < \sigma '$. Then, $U \vdash \lnot \, \varPhi (\sigma '\le \sigma )$.

We say that U is $\Sigma ^0_1$-friendly iff, there is a recursive $\varPhi$ satisfying $\Sigma 1$ and $\Sigma 2$. We say that U is $\Sigma ^0_1$-representative if there is a recursive $\varPhi$ satisfying $\Sigma 1$ and $\Sigma 3$.

The next theorem is, in a sense, a generalisation of the First Incompleteness Theorem. We just need $\Sigma 1$.

Theorem 32

Consider a consistent RE theory U and a recursive function $\varPhi$ from pure 1-$\Sigma ^0_1$-sentences to U-sentences. Suppose $\varPhi$ satisfies $\Sigma 1$. Let $U_i$ be a recursive sequence of consistent RE extensions of U. Then, we can effectively find a false pure 1-$\Sigma ^0_1$-sentence $\jmath$, such that $\varPhi (\jmath )$ is consistent with each of the $U_i$.

Proof

We stipulate the conditions of the theorem. We can find a pure $\Delta _0$-formula $\pi (i,p,\varphi )$ such that $U_i\vdash \varphi$ iff $\exists p\; \pi (i,p,\varphi )$. Let

$$\begin{aligned} {\vartriangle }\varphi {:}{=} \exists u\, \exists i\mathbin {<}u\, \exists p\mathbin {<}u\;\, \pi (i,p,\varphi ). \end{aligned}$$

Using the Gödel Fixed Point Construction, we find a pure 1-$\Sigma ^0_1$-sentence $\jmath$ such that $\jmath$ is true iff ${\vartriangle }\lnot \, \varPhi (\jmath )$. Suppose $\jmath$ is true. Then, $U \vdash \varPhi (\jmath )$ and, for some i, we have $U_i \vdash \lnot \, \varPhi (\jmath )$, contradicting the consistency of $U_i$. Thus, $\jmath$ is false and consistent with each of the $U_i$. $\square$

Remark 12

(Kripke). Let School be the theory in the language of arithmetic (without < as primitive) of all true closed equations. Saul Kripke shows that, for any RE extension U of School there is a co-Diophantine true sentence $\gamma$, such that $U \nvdash \gamma$. Note that this result also works in the context of decidable theories. We get Kripke’s version of the First Incompleteness Theorem from Theorem 32 by taking $\varPhi$ to be the transformation promised by Matiyasevich’s theorem that sends a pure 1-$\Sigma ^0_1$-sentence to a purely existential sentence.

E.g., it follows that there is a Diophantine equation that has solutions in all finite rings and in some non-standard model of PA, but no solutions in ${\mathbb {N}}$.

Lemma 33

Every $\Sigma ^0_1$-friendly theory U is $\Sigma ^0_1$-representative.

Proof

Let $\varPhi$ witness that U is $\Sigma ^0_1$-friendly. We prove $\Sigma 3$. Let $\sigma$ and $\sigma '$ be pure 1-$\Sigma ^0_1$-sentences. We have: $(U+\varPhi (\sigma '<\sigma )) \rhd (\sigma '< \sigma )^{\mathfrak {q}}$. Suppose $\sigma \le \sigma '$. Since, by Theorem 2, $[\sigma '< \sigma ]$ is inconsistent, we find that $U\vdash \lnot \,\varPhi (\sigma '<\sigma )$. The other case is similar. $\square$

Theorem 34

Suppose U is RE, consistent, and $\Sigma ^0_1$-friendly. Then, , and, hence, .

Proof

We note that any consistent RE extension of a $\Sigma ^0_1$-friendly RE theory is again $\Sigma ^0_1$-friendly. So it is sufficient to show that U tolerates a theory $\sigma ^{\mathfrak {q}}$, for false pure 1-$\Sigma ^0_1$-sentences $\sigma$.

Suppose U does not tolerate any false $\sigma ^{\mathfrak {q}}$. If $\sigma$ is true, we have $U \vdash \varPhi (\sigma )$. Suppose $\sigma$ is false. We have $(U+\varPhi (\sigma )) \rhd \sigma ^{\mathfrak {q}}$. So, if $U+\varPhi (\sigma )$ were consistent, then U would tolerate $\sigma ^{\mathfrak {q}}$. Quod non, ex hypothesi. So, $U \vdash \lnot \,\varPhi (\sigma )$. Since U is RE, we can now decide the halting problem. Quod impossibile. $\square$

We note that we can effectively find a sentence $\sigma$ such that $\varPhi (\sigma )$ is consistent with U from indices for U and $\varPhi$. Let be a pure 1-$\Sigma ^0_1$-formula representing the U-provability of $\lnot \,\varPhi (s)$. Then, we can take $\sigma$ to be $\jmath$, (a pure 1-$\Sigma ^0_1$ version of) the Gödel fixed point that is equivalent to . It is easy to see that $U+\varPhi (\jmath )$ is indeed consistent.

Remark 13

Inspecting [17], we find that every consistent RE effectively Friedman-reflexive theory U is $\Sigma ^0_1$-friendly. We can take . Further discussion of this, is outside the scope of the present paper.

The theory R is $\Sigma ^0_1$-friendly via the mapping $\sigma \mapsto \bigwedge \sigma ^{\mathfrak {q}}$. So this again shows that .

It turns out that $\Sigma ^0_1$-representativity coincides with a familiar notion.

Theorem 35

Consider a consistent RE theory U. Then, U is $\Sigma ^0_1$-representative iff U is effectively inseparable.

Proof

Suppose U is RE and consistent.

Suppose U is $\Sigma ^0_1$-representative as witnessed by $\varPsi$. Let $\mathcal {X}_0$ and $\mathcal {X}_1$ be any pair of effectively inseparable sets. Let $\sigma _0(x)$ be a pure 1-$\Sigma ^0_1$-formula that represents $\mathcal {X}_0$ and let $\sigma _1(x)$ be a pure 1-$\Sigma ^0_1$-formula that represents $\mathcal {X}_1$. We write $\sigma _i({\underline{n}})$ for a pure 1-representation of the result of substituting ${\underline{n}}$ in $\sigma _i$. We define

$$\begin{aligned} \varTheta (n) {:}{=} \varPsi (\sigma _0({\underline{n}}) \le \sigma _1(\underline{n})). \end{aligned}$$

Suppose $n\in \mathcal {X}_0$. Then, $\sigma _0({\underline{n}}) \le \sigma _1({\underline{n}})$ is true and, hence, $U \vdash \varTheta (n)$. Suppose $n \in \mathcal {X}_1$. Then, $\sigma _1({\underline{n}}) < \sigma _0({\underline{n}})$ is true, and, hence, $U \vdash \lnot \,\varTheta (n)$.

For the converse, suppose U is effectively inseparable. Then, by [14, Theorem 2], we find that there is a recursive boolean isomorphism $\varPsi$ from R to U. We can take $\varPsi$ restricted to pure 1-$\Sigma ^0_1$-sentences as the function witnessing the $\Sigma ^0_1$-representativity of U. $\square$

The first part of the proof of Theorem 35 can also be done via a Rosser argument. We have to be somewhat more careful with the details if we follow that road. We will give the argument in “Appendix B”.

Openquestion 5

It would be quite interesting to replace the $\sigma ^{\mathfrak {q}}$ in our definitions of friendliness and representativity by some other class of theories. However, the demands on the $\sigma ^{\mathfrak {q}}$ use both witness comparison and truth. So, it is not at all obvious here what more general analogues could be.

Example 6

At this point the time is ripe to give some separating examples. We consider properties: P1: undecidable, P2: essentially undecidable, P3: essentially hereditarily undecidable, P4: recursively inseparable, P5: effectively inseparable, P6: $\Sigma ^0_1$-friendly. We first give the list and then the description of the examples below it.

Example	P1	P2	P3	P4	P5	P6
$U_0$	−	−	−	−	−	−
$U_1$	$+$	−	−	−	−	−
$U_2$	$+$	$+$	−	−	−	−
$U_3$	$+$	$+$	$+$	−	−	−
$U_4$	$+$	$+$	−	$+$	−	−
$U_5$	$+$	$+$	−	$+$	$+$	−
$U_6$	$+$	$+$	$+$	$+$	$+$	−
$U_7$	$+$	$+$	$+$	$+$	$+$	$+$

• a. We can take $U_0$ be any decidable theory like Presburger Arithmetic.

• b. We can take $U_1$ e.g. the theory of groups, which, by results of Tarski [1, Chapter III] and Szmielev [23], is hereditarily but not essentially undecidable.

• c. We can take $U_2$ to be Ehrenfeucht’s theory which is essentially undecidable, but neither hereditarily undecidable, nor recursively inseparable. See [13].

• d. We can take $U_3$ to be a finitely axiomatised theory that is recursively boolean isomorphic to $U_2$. This theory is essentially hereditarily undecidable, but not recursively inseparable.

• e. Let d be an RE Turing degree with $0< d<0'$. Suppose $\mathcal A$, ${\mathcal {B}}$ is a recursively inseparable pair of RE sets as constructed by Shoenfield, where the Turing degree of ${\mathcal {A}}$ is d and the Turing degree of ${\mathcal {B}}$ is $\le d$. See [16] or [3]. Let

  $$\begin{aligned} U_4{:}{=} \textsf{Jan}+\{ \textsf{A}_n \mid n\in {\mathcal {A}} \}+ \{ \lnot \, \textsf{A}_n \mid n\in {\mathcal {B}} \}. \end{aligned}$$

  Then, $U_4$ is recursively inseparable, but cannot be effectively inseparable. Also, since $U_4$ contains a decidable sub-theory in the same language it cannot be essentially hereditarily undecidable.

• f. We define $U_5$ like $U_4$ only now we take ${\mathcal {A}}$ and ${\mathcal {B}}$ to be effectively inseparable.

• g. We can take $U_6$ to be the theory of Hanf’s example for the case that the recursively inseparable sets on which the construction is based are effectively inseparable. This is Example 3 in this paper.

• h. We can take $U_7$ to be, e.g., Q.

We note that, in all cases where it is not immediately obvious that $\textsf{P}i$ implies $\textsf{P}j$, our list shows that there is an example that satisfies $\textsf{P}i$ but does not satisfy $\textsf{P}j$.

Openquestion 6

Is there a finitely axiomatised and effectively inseparable theory that is not $\Sigma ^0_1$-friendly?

6 Separating model-interpretability and lax interpretability

In this section, we introduce the theory ${\textsf{PA}^-_\textsf{scat}}$ and prove some of its salient properties. In ${\textsf{PA}^-_\textsf{scat}}$ the numbers do not form the familiar neat Dedekindean linear order, but are scattered to several possibly disjoint parts of the structure. The theory ${\textsf{PA}^-_\textsf{scat}}$ will be an example of a $\Sigma ^0_1$-friendly theory U such that U is a sub-theory of R and , but $U \mathrel {\not \! \rhd }_\textsf{mod} \textsf{R}$.

Remark 14

Our theory ${\textsf{PA}^-_\textsf{scat}}$ is closely related to Vaught’s theory S. See [6].

We define the theory $\textsf{PA}^-_\textsf{scat}$ as follows. It has the relational signature of arithmetic with < minus the zero. Let $\textsf{TN}(a)$ be the realtional version of TN, where we replace the constant 0 by a parameter a. The idea is that in $\textsf{PA}^-_\textsf{scat}$ we have many zero’s. We interpret $z \models \textsf{TN}(a)$ in the obvious way. We write:

• ${\widetilde{n}}(a) := \exists z\, \exists x_0< z \dots \exists x_{n-1}< z \,\exists x_n\, (a=x_0 \wedge z= x_n \; \wedge$ ${} \bigwedge _{i<n} x_i\textsf{S} x_{i+1} \wedge \bigwedge _{i<j<n+1}x_i \ne x_j \wedge z \models \textsf{TN}(a))$.

The theory $\textsf{PA}^-_\textsf{scat}$ is axiomatised by the axioms $\exists a\;{\widetilde{n}}(a)$ for $n\in \omega$.

Let $\textsf{R}_\textsf{succ}$ be the theory in the language with 0 and S, axiomatised by ${\underline{n}} \ne {\underline{m}}$, where $n\ne m$. We prove the following theorem.

Theorem 36

We have:

• a. $\textsf{R} \supseteq _\textsf{e}{\textsf{PA}^-_\textsf{scat}}$.

• b. ${\textsf{PA}^-_\textsf{scat}}$ is $\Sigma ^0_1$-friendly and, hence, .

• c. $\textsf{PA}^-_\textsf{scat}\mathrel {\not \! \rhd }_\textsf{mod}\textsf{R}_\textsf{succ}$.

We note that it follows that and $\textsf{R} \rhd {\textsf{PA}^-_\textsf{scat}}$, and, hence, .

We prove our theorem via a sequence of lemmas. The following lemma is clear.

Lemma 37

$\textsf{R} \supseteq _\textsf{e}{\textsf{PA}^-_\textsf{scat}}$.

For any pure 1-$\Sigma ^0_1$-sentence, $\sigma$ we define $\sigma ^{\mathfrak {q}}_x$ as the theory of $\sigma$ with zero replaced by the free parameter x on the domain of the y such that $x\le y$. We show the following lemma.

Lemma 38

The theory $\textsf{PA}^-_\textsf{scat}$ is $\Sigma ^0_1$-friendly. Hence, and .

Proof

It is easy to see that the mapping $\sigma \mapsto \exists x\, \sigma ^{\mathfrak {q}}_x$ fulfils the conditions for $\varPhi$ in the definition of $\Sigma ^0_1$-friendlyness.¹²$\square$

To prove Theorem 36(c), we need a counter-model. We define ${\mathbb {N}}_\textsf{scat}$, the model of the scattered numbers, for the signature of $\textsf{PA}^-_\textsf{scat}$ as follows. It is the disjoint sum of the natural numbers considered as models of the theory of a number (in relational signature). Modulo isomorphism, we can also define ${\mathbb {N}}_\textsf{scat}$ concretely as follows.

• The domain of ${\mathbb {N}}_\textsf{scat}$ is the set of ${\langle n,m \rangle }$ with $m<n$.

• ${\langle n,m \rangle } < {\langle n',m' \rangle }$ iff $n=n'$ and $m<m'$.

• $\textsf{S}({\langle n,m \rangle },{\langle n',m' \rangle })$ iff $n=n'$ and $m' = \textsf{min}(m+1,n-1)$.

• $\textsf{A}({\langle n,m \rangle },{\langle n',m' \rangle },{\langle n'',m'' \rangle })$ iff $n=n'=n''$ and $m'' = \textsf{min}(m+m',n-1)$.

• $\textsf{M}({\langle n,m \rangle },{\langle n',m' \rangle },{\langle n'',m'' \rangle })$ iff $n=n'=n''$ and $m'' = \textsf{min}(m\times m',n-1)$.

• ${\langle n,m \rangle } < {\langle n',m' \rangle }$ iff $n=n'$ and $m<m'$.

Let $\sim$ be the equivalence relation on ${\mathbb {N}}_\textsf{scat}$ given by $x<y \vee y \le x$. We write $\llbracket x \rrbracket$ for (the purely syntactic representation of) the $\sim$-equivalence class of x. Let $\varphi ^{\llbracket x \rrbracket }$ be the relativisation of the quantifiers of $\varphi$ to $\llbracket x \rrbracket$.¹³ We write ${\hat{n}}$ for the submodel (of the theory of n) with as domain the $\sim$-equivalence class of size n.

Consider a finite set of free variables X. Let $\simeq$ be any equivalence relation on X. So, $\simeq$ is a purely syntactic relation. We say that a formula is $\theta$ is $\simeq$-good if it is of the form $\chi ^{\llbracket x \rrbracket }$, for some $x\in X$ and all free variables of $\theta$ are $\simeq$-equivalent to x. We say that a formula is $\simeq$-friendly iff it is a boolean combination of $\simeq$-good formulas of the form $\chi ^{\llbracket x \rrbracket }$, where $x\in X$. We define $\textsf{E}_{\simeq }$ to be the conjunction of all $x\sim x'$ in case $x\simeq x'$ and $x\not \sim x'$ in case $x\not \simeq x'$, for $x,x'\in X$. (We assume that X is part of the data for $\simeq$.)

We prove our third lemma. The lemma and its proof are well known for the case of binary disjoint sums of models. The present result is just an adaptation.

Lemma 39

Let X be some finite set of variables and let $\vec x$ be some enumeration of X. Consider a formula $\varphi$ with free variables among X and an equivalence relation $\simeq$ on X. Then, there is an $\simeq$-friendly $\psi$ such that

$$\begin{aligned} {\mathbb {N}}_\textsf{scat}\models \forall \vec x\, (\textsf{E}_{\simeq } \rightarrow (\varphi \leftrightarrow \psi )). \end{aligned}$$

Proof

The proof is by induction on $\varphi$.

We treat $\textsf{A}xyz$ as a prototypical atomic formula, where A stands for addition. In case $\textsf{A}xyz$ is $\simeq$-good, we can take $\textsf{A}xyz$ itself as $\psi$ noting that ${(\mathsf A}xyz)^{\llbracket x \rrbracket }$ is identical to $\textsf{A}xyz$. In case $\textsf{A}xyz$ is not $\simeq$-good, we can take $\psi$ to be $\bot$, seeing that $\textsf{A}xyz$ is equivalent to $\bot$ under the assumption $\textsf{E}_{\simeq }$ and $\bot = \bot ^{\llbracket x \rrbracket }$.

We treat $(\varphi _0 \wedge \varphi _1)$ as a prototypical case. In $\textsf{Th}({\mathbb {N}}_\textsf{scat})$, under the assumption $\textsf{E}_{\simeq }$, by the Induction Hypothesis, each of the $\varphi _i$ is equivalent to an $\simeq$-friendly $\psi _i$. So, $(\varphi _0 \wedge \varphi _1)$ is equivalent to $(\psi _0 \wedge \psi _1)$, which is $\simeq$-friendly.

Finally, consider the case of $\exists z\, \varphi '$. Since we always can rename bound variables, we can assume that $z\not \in X$. Let $X' {:}{=} X \cup \{ z \}$. We write ${\simeq } \sqsubset _z {\simeq '}$ if $\simeq '$ is an equivalence relation on $X'$ and $\simeq '$ restricted to X is $\simeq$.

We reason in $\textsf{Th}({\mathbb {N}}_\textsf{scat})$ under the assumption $\textsf{E}_{\simeq }$. We note that $\exists z\, \varphi '$ is equivalent to $\bigvee _{{\simeq } \sqsubset _z {\simeq '}} \exists z\,(\textsf{E}_{\simeq '} \wedge \varphi ')$. We zoom in on some $\alpha {:}{=}(\textsf{E}_{\simeq '} \wedge \varphi ')$. By the induction hypothesis, this can be rewritten as the conjunction of $\textsf{E}_{\simeq '}$ and a disjunction of formulas of the form $(\theta _0 \wedge \theta _1)$, where $\theta _0$ is $\simeq$-friendly and $\theta _1$ is $\simeq '$-good and of the form $\chi ^{\llbracket y \rrbracket }$, where $y \simeq z$. If the $\simeq '$-equivalence class of z contains at least two elements, we choose y different from z. Given our assumption that $\textsf{E}_{\simeq }$, the formula $\alpha$ is equivalent to a disjunction of formulas of the form $(\theta _0 \wedge (\textsf{E}_{\simeq '} \wedge \theta _1))$. It follows that $\exists z\ \alpha$ is equivalent to a disjunction of formulas of the form $(\theta _0 \wedge \exists z\, (\textsf{E}_{\simeq '} \wedge \theta _1))$. It clearly suffices to show that $\exists z\, (\textsf{E}_{\simeq '} \wedge \theta _1)$ is equivalent to an $\simeq$-friendly formula.

There are two cases. The $\simeq '$-equivalence class of z contains at least two elements or precisely one.

In the first case we can replace $\exists z\, (\textsf{E}_{\simeq '} \wedge \theta _1)$ by $\exists z\in \llbracket y \rrbracket \, \theta _1$, where $\theta _1$ is of the form $\chi ^{\llbracket y \rrbracket }$. Clearly, $\exists z\in \llbracket y \rrbracket \, \theta _1$ is $\simeq$-good.

In the second case, at most z occurs freely in $\theta _1$. Suppose $X=\{ x_0,\dots ,x_{k-1} \}$. Let $\theta _1$ be $\chi ^{\llbracket z \rrbracket }$. In the context of $\textsf{E}_{\simeq }$, we can rewrite $\exists z\,(\textsf{E}_{\simeq '} \wedge \theta _1)$ to the formula

$$\begin{aligned} \exists z\, (\bigwedge _{i<k} z\not \sim x_i \wedge \theta _1).\end{aligned}$$

In case $\chi ^{\llbracket z \rrbracket }$ can be fulfilled in more than k submodels corresponding to numbers, $\exists z\, (\bigwedge _{i<k} z\not \sim x_i \wedge \theta _1)$ will be true. So we can replace $\exists z\, (\bigwedge _{i<k} z\not \sim x_i \wedge \theta _1)$ by $\top$. If not $\chi (z)$ will be fulfilled in precisely ${\hat{n}}_0, \dots , {\hat{n}}_{s-1}$. Let

$$\begin{aligned} {{\mathfrak {c}}}_n(u) {:}{=} \exists v_0 \dots \exists v_{n-1}\, (\bigwedge _{i<j<n} v_i \ne v_j \wedge \forall w\, (w = u \leftrightarrow \bigvee _{i<n} w=v_i)). \end{aligned}$$

We find that $\exists z\, (\bigwedge _{i<k} z\not \sim x_i \wedge \theta _1)$ is equivalent to $\bigvee _{j<s} \bigwedge _{i<k} \lnot \, {{\mathfrak {c}}}^{\llbracket x_i \rrbracket }_{n_j}(x_i)$, which is clearly $\simeq$-friendly.¹⁴$\square$

Lemma 40

There is no inner model of $\textsf{R}_\textsf{succ}$ in ${\mathbb {N}}_\textsf{scat}$.

Proof

Since ${\mathbb {N}}_\textsf{scat}$ has at least two elements, we do not need to consider piece-wise interpretations. Moreover, every element in ${\mathbb {N}}_\textsf{scat}$ is definable. So, we can always eliminate parameters. Thus it is sufficient to prove our result for many-dimensional relativised interpretations without parameters.

Suppose we had an inner model of $\textsf{R}_\textsf{succ}$ given by an interpretion M. Say M is m-dimensional and suppose 0 is given by a formula ${{\mathfrak {z}}}( \vec x\,)$ and S by ${{\mathfrak {s}}}( \vec x, \vec y\,)$. We note that in the sequence $\vec x, \vec y$ all the variables are pairwise disjoint. There are two conventional aspects. The variables in ${{\mathfrak {s}}}$ need only be among the $\vec x, \vec y$, but not all need to occur. Secondly, the order of the variables $\vec x$, $\vec y$ as exhibited need not be given by anything in ${{\mathfrak {s}}}$.

Our proof strategy is to obtain a contradiction by finding a finite set of numbers ${\mathcal {N}}$ and an infinite sequence of pairwise different sequences length m with components in the ${\hat{n}}$ for $n\in {\mathcal {N}}$. We work in ${\mathbb {N}}_\textsf{scat}$.

• i We fix an m-sequence $\vec a$ such that ${{\mathfrak {z}}}( \vec a\,)$, in other words $\vec a$ represents $0^M$. We put the $n_i$ such that ${{\mathfrak {c}}}_{n_i}(a_i)$ in ${\mathcal {N}}$.

• ii For each equivalence relation on the elements of $\vec x, \vec y$ we add a set of numbers to ${\mathcal {N}}$. Consider a relation $\simeq$ on the elements of $\vec x, \vec y$. Under the assumption $\textsf{E}_\simeq$, we can rewrite ${{\mathfrak {s}}}$ as $\bigvee _{q< r}\bigwedge _{p<s_q}\theta _{qp}$, where $\theta _{qp}$ is $\simeq$-good, say, it is of the form $(\chi _{qp})^{\llbracket w_{qp} \rrbracket }$. We can clearly arrange it so that (i) $w_{qp}$ is always the first in the sequence $\vec x, \vec y$ of its $\simeq$-equivalence class and (ii) if $p<p'$, then $w_{qp}$ occurs strictly earlier in $\vec x, \vec y$ than $w_{qp'}$. So, if $p\ne p'$, we have $w_{qp} \not \simeq w_{qp'}$. Consider any $\theta _{qp}$ where $w_{qp}$ is one of the $y_i$. There are two possibilities.

  • I. Suppose the number of ${\hat{n}}$ in which $\chi _{qp}$ is satisfiable is $< 2m+1$. In this case we add all n such that $\chi _{qp}$ is satisfiable in ${\hat{n}}$ to ${\mathcal {N}}$.

  • II. Suppose the number of ${\hat{n}}$ in which $\chi _{qp}$ is satisfiable is $\ge 2m+1$. In this case we add the first $2m+1$ such n to ${\mathcal {N}}$.

• iii Nothing more will be in ${\mathcal {N}}$.

Let ${\mathcal {N}}^*$ be the elements in the ${\hat{n}}$, for $n\in {\mathcal {N}}$. Clearly ${\mathcal {N}}$ is finite and so is the number of elements in ${\mathcal {N}}^*$.

We are now ready and set to define our infinite sequence in order to obtain the desired contradiction. The sequence starts with $\vec a$. We note that $\vec a$ is in the domain of M and that the components of $\vec a$ are in ${\mathcal {N}}^*$. Each element of the sequence will be in the domain of M and its components will be in ${\mathcal {N}}^*$. Suppose we have constructed the sequence up to $\vec b$. Since $\vec b$ is in the domain of M, there is a $\vec c$ with ${{\mathfrak {s}}}( \vec b, \vec c\,)$. We define $\simeq$ on the elements of $\vec x, \vec y$ as follows. We will say that $b_i$ is the value of $x_i$ and $c_j$ is the value of $y_j$. Let $\vec d = \vec b, \vec c$ and $\vec v = \vec x, \vec y$. We take $v_i \simeq v_j$ iff $d_i \sim d_j$. We note that we have $\textsf{E}_\simeq [ \vec v: \vec d\,]$. We construct the formula $\bigvee _{q< r}\bigwedge _{p<s_q}\theta _{qp}$ as before for $\simeq$. So, for some $q<r$, we have $\bigwedge _{p<s_q}\theta _{qp}[ \vec v: \vec d\,]$.

Consider the variable $w_{qp}$. If it is an $x_i$, then all variables $y_i$ that are $\simeq$-equivalent to it, will have values that are $\sim$-equivalent to $b_i$. So they will be in $\mathcal N^*$. If it is a $y_i$ and we are in case (ii.I) of the construction of ${\mathcal {N}}$ the values of the variables equivalent to it will be in ${\mathcal {N}}^*$. The final case is that $w_{qp}$ is a $y_i$ and we are in case (ii.II) of the construction of ${\mathcal {N}}$. Since there are at least $2m+1$ numbers n such that ${\hat{n}}$ satisfies $\chi _{qp}$, we can always choose an $n^*$ among these numbers such that no $d_i$ is in ${\hat{n}}^*$ such that ${\hat{n}}^*$ satisfies $\theta _{qp}$. We assign to $y_i$ in the equivalence class of $w_{qp}$ the value $e_i$ so that under this assignment $\chi _{q,p}$ is satisfied. We now modify our sequence $\vec b, \vec c$ by replacing the $c_j$ by the $e_j$ for the cases where $y_j$ is in the equivalence class of $w_{qp}$. Say the new sequence is $\vec b, \vec {c'}$. We note that the new sequence has strictly less elements outside ${\mathcal {N}}^*$ and that we still have ${{\mathfrak {s}}}( \vec b, \vec {c'}\,)$. We repeat this procedure for all $w_{qp}$ that are among the $y_i$. The final sequence we obtain will only have values in ${\mathcal {N}}^*$.

By the axioms of $\textsf{R}_\textsf{succ}$, we we cannot have two elements in our sequence that are the same. A contradiction. $\square$

We end this section by describing how we can make the result work for parameter-free interpretations. A first step is to modify the definition of $\sigma ^{\mathfrak {q}}$, say, to $\sigma ^{\mathfrak {q}*}$. We remind the reader that we assume our $\sigma$ are in pure form. In the definition of $\sigma ^{\mathfrak {q}}$ we just asked for there to be a witness of $\sigma$. For $\sigma ^{\mathfrak {q}*}$ we ask that the witness w is the smallest one and that $w+1$ is the maximum element.

We now define $\textsf{PA}^-_\textsf{scat}!$ as the theory axiomatised by $\exists !x\,\sigma ^{\mathfrak {q}*}(x)$. We note that ${\mathbb {N}}_\textsf{scat}$ also satisfies $\textsf{PA}^-_\textsf{scat}!$. The new theory is not a sub-theory of $\textsf{R}_{<}$. However, the theory is locally finite, i.e., every finitely axiomatised sub-theory has a finite model. So, by the main result of [24], we have $\textsf{R} \rhd \textsf{PA}^-_\textsf{scat}!$.¹⁵ All our other arguments work with $\exists !x\,\sigma ^{\mathfrak {q}*}(x)$ replacing $\exists x\, \sigma ^{\mathfrak {q}}(x)$. We note that $\textsf{PA}^-_\textsf{scat}! + \exists !x\,\sigma ^{\mathfrak {q}*}(x)$ interprets $\sigma ^{\mathfrak {q}*}$ in a parameter-free way. E.g., the definition of the domain becomes:

$$\begin{aligned} \delta (y) {:}{=} \exists x\, (\sigma ^{\mathfrak {q}*}(x) \wedge x\le y). \end{aligned}$$

7 Non-minimality

There is no interpretability minimal essentially hereditarily undecidable theory. There is a quick proof of this due to Fedor Pakhomov and there is a slow proof. Since, the slow proof yields different information, I do reproduce it here. See also Remark 15.

Here is the quick proof. The proof is a minor adaptation of the proof of [3, Theorem 1.1] as given in Section 4.4 of that paper.

Theorem 41

There is no interpretability minimal essentially hereditarily undecidable RE theory.

Proof

Since, by Theorem 15, the essentially hereditarily undecidable theories are closed under interpretability suprema, it is sufficient to show that there is no minimum essentially hereditarily undecidable RE theory. Suppose, to obtain a contradiction, that $U^\star$ is such a minimum theory.

Let i be an index of an RE set. By a result of Shoenfield [16], we can effectively find an index j of the theory $\textsf{sh}(i)$ such that $\textsf{W}_i$ is not recursive iff $\textsf{sh}(i)$ is essentially undecidable. See also [3, Theorem 4.8]. Next, by a result of Peretyat’kin [10], we can effectively find an index k of a theory $\textsf{pere}(j)$ that is finitely axiomatized and recursively Boolean isomorphic with $\textsf{sh}(i)$. Let us call this theory $\textsf{shpe}(i)$. Since $\textsf{shpe}(i)$ is essentially undecidable and finitely axiomatized it is essentially hereditarily undecidable. Let Rec be the set of indices of recursive sets. We have:

$$\begin{aligned}{} & {} i \not \in \textsf{Rec}\; \text {iff} \;\textsf{shpe}(i) \text { is hereditarily essentially undecidable} \\{} & {} \quad \quad \qquad \;\text {iff} \textsf{shpe}(i) \rhd U^\star . \end{aligned}$$

By a result of Rogers and, independently, Mostowski, Rec is complete $\Sigma ^0_3$. See [25, Chapter 14, Theorem XVI] or [26, Corollary 4.3.6]. We have reduced the complement of Rec, a $\Pi ^0_3$-complete predicate, to an interpretability statement: a $\Sigma ^0_3$-predicate. Quod impossibile. $\square$

We prove the non-minimality result w.r.t. interpretability for essentially hereditarily undecidable theories again using the idea behind the construction from the proof of [2, Theorem 3.2 ], following the plan of the proof of [3, Theorem 1.1.] as given in Section 4.2 of that paper.

We will need a variation on Kleene’s well known construction of two effectively inseparable sets. We write $x\cdot y$ for Kleene application. For $i=0,1$, let

$$\begin{aligned}\textsf{Km}_i {:}{=} \{ {\langle n,x \rangle } \mid x\cdot {\langle n,x \rangle } \simeq i \}.\end{aligned}$$

Lemma 42

Suppose ${\mathcal {W}}$ is a recursive set. Let $\varTheta$ be a 0,1-valued recursive function such that $\varTheta (x)=1$ iff $x\in \mathcal {W}$. We can find an index c of $\varTheta$ effectively from an index i of $\mathcal {W}$. Then, for any n, we have ${\langle n,c \rangle }\in \textsf{Km}_0\setminus \mathcal {W}$ or ${\langle n,c \rangle }\in \textsf{Km}_1 \cap \mathcal {W}$.

Proof

We have:

$$\begin{aligned} {\langle n,c \rangle } \not \in \mathcal {W}\Leftrightarrow & {} c\cdot {\langle n,c \rangle } \simeq 0 \\\Leftrightarrow & {} {\langle n,c \rangle }\in \textsf{Km}_0\\ {\langle n,c \rangle } \in \mathcal {W}\Leftrightarrow & {} c\cdot {\langle n,c \rangle } \simeq 1 \\\Leftrightarrow & {} {\langle n,c \rangle }\in \textsf{Km}_1 \end{aligned}$$

$\square$

Lemma 43

Suppose $\mathcal {Z}$ is an RE set such that, for every m, there is an n such that ${\langle n,m \rangle }\in \mathcal {Z}$. Then $\textsf{Km}_0\cap \mathcal {Z}$ and $\textsf{Km}_1\cap \mathcal {Z}$ are effectively inseparable.

Proof

Suppose $\mathcal {W}$ with index i is decidable and that $\mathcal {W}$ separates $\textsf{Km}_0\cap \mathcal {Z}$ and $\textsf{Km}_1\cap \mathcal {Z}$. Let $\varTheta$ and c be as in Lemma 42. We find n such that ${\langle n,c \rangle }\in \mathcal {Z}$. By Lemma 42, we have ${\langle n,c \rangle }\in \textsf{Km}_0\setminus \mathcal {W}$ or ${\langle n,c \rangle }\in \textsf{Km}_1 \cap \mathcal {W}$. In the first case ${\langle n,c \rangle }\in (\textsf{Km}_0\cap \mathcal {Z})\setminus \mathcal {W}$. Quod non, by our assumption that $\mathcal {W}$ separates $\textsf{Km}_0\cap \mathcal {Z}$ and $\textsf{Km}_1\cap \mathcal {Z}$. In the second case, we have ${\langle n,c \rangle }\in \textsf{Km}_1\cap \mathcal {Z} \cap \mathcal {W}$, again contradicting the assumption. $\square$

Theorem 44

Consider any essentially undecidable RE theory U. Then, we can effectively find (an index of) an essentially hereditarily undecidable RE theory V such that $V\mathrel {\not \! \rhd }U$. Moreover, we can take V to be effectively inseparable.

Proof

Let $T {:}{=} \textsf{Jan}+\{ \textsf{A}_n \mid n\in \textsf{Km}_0 \}$. Let s be an index of T. We take $A {:}{=} \textsf{pere}(s)$. So, A is finitely axiomatised and recursively boolean isomorphic to T. Let $\varPhi$ be the witnessing recursive isomorphism from V to A and let $\textsf{B}_i {:}{=} \varPhi (\textsf{A}_i)$. Clearly, over A, every sentence is provably equivalent to a boolean combination of the $\textsf{B}_i$.

Let $\textsf{C}_{n,0},\dots , \textsf{C}_{n,2^n-1}$ be an enumeration of all conjunctions of $\pm \textsf{B}_i$, for $i<n$. Suppose U is an essentially undecidable RE theory. Let $\upsilon _0,\upsilon _1,\dots$ be an effective enumeration of the theorems of U. Let $\tau _0,\tau _1,\dots$ be an effective enumeration of all translations from the U-language into the A-language.¹⁶

Consider n, $\tau _i$ and $\textsf{C}_{n,j}$, for $j<2^n$. Let $\textsf{V}_{n,j} {:}{=} A+\textsf{C}_{n,j}+\{ \textsf{B}_k \mid k\ge n \}$. Clearly, $\textsf{V}_{n,j}$ is either inconsistent or consistent and complete. We claim that, for some k, we have $\textsf{V}_{n,j} \vdash \lnot \, \upsilon ^{\tau _i}_k$. Suppose this were not the case. Then, $\textsf{V}_{n,j}$ is consistent and $\tau _i$ carries an interpretation of U in $\textsf{V}_{n,j}$, but this is impossible since $\textsf{V}_{n,j}$ is decidable and U is essentially undecidable.

Thus, we can effectively find a number $p_{n,i,j}$ as follows. We find the first k such that $\textsf{V}_{n,j} \vdash \lnot \, \upsilon ^{\tau _i}_k$. Then, we reduce, the sentence $\upsilon ^{\tau _i}_k$ to a boolean combination of $\textsf{B}_s$ over A. Let $p_{n,i,j}$ the smallest number of the form ${\langle r,n \rangle }$ that is strictly larger than the s such that $\textsf{B}_s$ occurs in this boolean combination.

We define $\eta (n,i)$ to be the maximum of the $p_{n,i,j}$, for $j<2^n$. Let $\varPsi (0) {:}{=} 0$ and let $\varPsi (k+1){:}{=} \eta (\varPsi (k),k)$. Clearly, $\varPsi$ is recursive and strictly increasing. Let $\mathcal Z$ be the range of $\varPsi$. The set ${\mathcal {Z}}$ is obviously recursive.

We define $V: = A+\{ \lnot \,\textsf{B}_i \mid i \in \textsf{Km}_1 \cap {\mathcal {Z}} \}$. Suppose, to obtain a contradiction, that we have $K:V \rhd U$. Let the underlying translation of K be $\tau _{n^*}$.

Clearly, there is a $j^*$ such that $V+\textsf{C}_{\varPsi (n^*),j^*}$ is consistent. (This is a non-constructive step.) By construction, there is a $\varphi$ with $U \vdash \varphi$ and $\textsf{V}_{\varPsi (n^*),j^*}\vdash \lnot \, \varphi ^K$. Moreover, $A \vdash \varphi ^K \leftrightarrow \rho$, where $\rho$ is a boolean combination of $\textsf{B}_s$ with $s<\varPsi (n^*)$. We note that no $\lnot \, \textsf{B}_r$ with $\varPsi (n^*)< r < \varPsi (n^*+1)$ occurs in the axiomatisation of V. By our assumption on K, we have $V \vdash \varphi ^K$ and, so, $V \vdash \rho$. It follows that

$$\begin{aligned} A+\{ \lnot \,\textsf{B}_i \mid i \in \textsf{Km}_1 \cap {\mathcal {Z}} \text { and } i<\varPsi (n^*+1) \} \vdash \rho . \end{aligned}$$

Hence, also $\textsf{V}_{\varPsi (n^*),j^*} \vdash \rho$, i.e., $\textsf{V}_{\varPsi (n^*),j^*} \vdash \varphi ^K$, A contradiction.

We verify that V is essentially hereditarily undecidable and effectively inseparable. We note that $\varXi$ with $\varXi (n) {:}{=} \textsf{B}_n$ maps $\textsf{Km}_0$ into $A_{\mathfrak {p}}$ and $\textsf{Km}_1\cap \mathcal {Z}$ into $V_{\mathfrak {r}}$. It is immediate from Lemma 43 that $\textsf{Km}_0$ and $\textsf{Km}_1\cap \mathcal {Z}$ are effectively inseparable, hence so is V. Finally, by Lemma 24, we find that V is essentially hereditarily undecidable. $\square$

Since the essentially hereditarily undecidable RE theories are closed under interpretability infima, we again obtain Theorem 41 from Theorem 44.

Remark 15

Yong Cheng notes that the argument of [3, Section 4.2] yields more information concerning the possible classes for which the no minimality result holds. See [27]. His insights can be applied to our case. For example, we find that Theorem 44 tells us that there is no interpretability minimal element among essentially hereditarily undecidable theories that is also effectively inseparable.

Appendices

Appendix A: Parametrically local interpretability

In this appendix, we present a reduction notion, to wit c-lax interpretability, that improves upon both lax interpretability and parametrically local interpretability w.r.t. the treatment of essential hereditary undecidability.

1.1 Appendix A. 1: Parameters and finite axiomatisability

Our preferred treatment of interpretations with parameters uses a parameter domain $\pi$. Let V be the interpreting theory and let U be the interpreted theory and let $\tau$ be the translation. We demand that (a) $V \vdash \exists \vec p\, \pi (\vec p)$ and (b) if $U \vdash \varphi$, then we have $V \vdash \forall \vec q\, (\pi (\vec q) \rightarrow \varphi ^{\tau ,\vec q})$.

In case U is finitely axiomatised, we can take a short-cut and drop the parameter domain. Let $\alpha$ be the conjunction of the axioms of U plus the axioms of the theory of identity. We ask: (a$'$) $V \vdash \exists \vec p\; \alpha ^{\tau , \vec p}$.

We note that (a) and (b) immediately imply (a$'$). Conversely, if we have (a$'$), we can define a parameter domain $\pi ^*(\vec p) :\leftrightarrow \alpha ^{\tau ,\vec p}$ and use the equivalence between axioms- and theorems-interpretability to obtain (a) and (b) for $\pi ^*$.

We note that, if we start with a parameter-domain $\pi$ that satisfies (a) and (b), then $\pi ^*$ is an extension of $\pi$.

1.2 Appendix A.2: Parametrically local interpretability

We say that V parametrically locally interprets U, or $V \rhd _\textsf{pl} U$, iff, there is a parametric translation $\tau$ of the U-language to the V-language, such that, whenever $U \vdash \varphi$, then $V \vdash \exists \vec p\; \varphi ^{\tau ,\vec p}$. It is easy to see that $\rhd _\textsf{pl}$ is transitive, using the known properties of parametric interpretability.

Theorem 45

Let U be a theory in a language $\mathcal {L}$ extended with constants $\vec c$. Let $U^*$ be the theory axiomatised by the $\mathcal {L}$-consequences of U. Then, $U \bowtie _\textsf{pl} U^*$.

Proof

We interpret U in $U^*$ via $\tau$ which is the identical translation on the vocabulary of $\mathcal {L}$ and which interprets constant $c_i$ as parameter $p_i$. $\square$

1.3 Appendix A.3: c-Essential undecidability

Consider a theory U. We write $U^{(\vec c)}$ for the theory U with its signature expanded with constants $\vec c$. Tarski in [1] calls this an inessential extension.

We write $U \subseteq _\textsf{c} V$ for: for some $\vec c$, we have $U^{(\vec c)} \subseteq _\textsf{e}V$. We use c-essentially in the obvious way.

Theorem 46

• a. U is c-essentially undecidable iff U is essentially undecidable.

• b. U is c-essentially hereditarily undecidable iff U is essentially hereditarily undecidable.

Proof

We prove (a). The left-to-right direction is trivial. We treat right-to-left. Suppose U is essentially undecidable. Consider a consistent $U'\supseteq _\textsf{c} U$. Suppose $U'$ is decidable. Let $U^*$ be the theory given by the $U'$-theorems in the U-language. Then, $U^*\supseteq _\textsf{e}U$ and $U^*$ is consistent and decidable. Quod impossibile.

We prove (b). The left-to-right direction is trivial. We treat right-to-left. Suppose U is essentially hereditarily undecidable. Suppose $U'\supseteq _\textsf{c} U$ and $W \subseteq _\textsf{e}U'$. Let ${U'}^*$ be the set of consequences of $U'$ in the U-language and let $W^*$ be the set of consequences of W in the U-language. Then $U \subseteq _\textsf{e}{U'}^{*}$ and $W^*\subseteq _\textsf{e}{U'}^{*}$. If W would be decidable, then so would be $W^*$. Quod impossibile. $\square$

The next theorem combines Theorems 4 and 5 of [1, Part I].

Theorem 47

Suppose U is decidable and $\varphi$ is a sentence in the $U^{(\vec c)}$-language. Then $U^{(\vec c)}+\varphi$ is decidable.

Proof

We have $U^{(\vec c)}+\varphi \vdash \psi$ iff $U \vdash \forall \vec w\, (\varphi [\vec c:\vec w] \rightarrow \psi [\vec c:\vec w])$. $\square$

1.4 Appendix A.4: c-Lax interpretability

We define V c-laxly interprets U, or , iff, for all consistent $V'\supseteq _\textsf{c} V$, there is a consistent $V''\supseteq _\textsf{c} V'$, such that $V''\rhd U$.

Theorem 48

iff, for all $V'\supseteq _\textsf{e}V$, there is a $V'' \supseteq _\textsf{c} V$, such that $V''\rhd U$.

Proof

Left-to-right is trivial. We treat right-to-left. We assume the right-hand-side of our equivalence. Suppose $V'\supseteq _\textsf{c} V$ and $V'$ is consistent. Let $\vec c$ be the relevant new constants. We define ${V'}^*$ as the theory given by the consequences of $V'$ in the V-language. Let $Z\supseteq _\textsf{c} {V'}^*$ be a consistent theory, such that $Z \rhd U$. Clearly, we may choose the new constants of Z, say $\vec d,$ distinct from $\vec c$. We take $V'' {:}{=} {V'}^{(\vec d)}\cup Z^{(\vec c)}$. Clearly, $V'' \supseteq _\textsf{c} V'$ and $V'' \rhd U$.

We claim that $V''$ is consistent. If not, for some $\psi (\vec c,\vec d)$, we would have ${V'}^{(\vec d)} \vdash \psi (\vec c,\vec d)$ and $Z^{(\vec c)} \vdash \lnot \, \psi (\vec c,\vec d)$. It follows that ${V'}^*\vdash \exists \vec v\, \forall \vec w\,\psi (\vec v,\vec w)$, So, $Z \vdash \exists \vec v\, \forall \vec w\,\psi (\vec v,\vec w)$. On the other hand, it follows that $Z \vdash \forall \vec v\, \lnot \, \psi (\vec v,\vec d)$. This contradicts the fact that Z is consistent. $\square$

We prove that essential hereditary undecidability is preserved over c-lax interpretability.

Theorem 49

Suppose and U is essentially hereditarily undecidable. Then V is essentially hereditarily undecidable.

Proof

Suppose and U is essentially hereditarily undecidable. Suppose W is a decidable theory in the same language as V that is consistent with V. We have $V\subseteq _\textsf{e}V\cup W =: V'$. So, there is a consistent $V'' \supseteq _\textsf{c} V'$ such that $V'' \rhd U$. Say the witnessing constants are $\textbf{c}$ and the witnessing translation is $\tau$. Let $Z {:}{=} W^{(\vec c)} + \mathfrak {id}^\tau$. Then, by Theorem 47, Z decidable. Since $Z\subseteq _\textsf{e}V''$, so $T{:}{=} \{ \varphi \mid Z \vdash \varphi ^\tau \}$ is consistent with U. Since Z is decidable T is decidable. Quod non. $\square$

Theorem 50

Suppose $V \rhd _\textsf{pl} U$. Then, .

Proof

Suppose $V \rhd _\textsf{pl} U$. Let $\tau$ be the witnessing translation. Consider any consistent $V' \supseteq _\textsf{e}V$. We find that $V'$ proves $\exists \vec p\; \varphi ^{\tau ,\vec p}$, for all $\varphi$ such that $U \vdash \varphi$. We take a sequence of fresh constants $\vec c$ of the length of $\vec p$. Then, by compactness, $V'' {:}{=}V' + \{ \varphi ^{\tau , \vec c} \mid U \vdash \varphi \}$ is consistent and, clearly, $V'' \supseteq _\textsf{c} V'$ and $V'' \rhd U$. $\square$

Appendix B: An alternative proof of Theorem 35

We show that if U is $\Sigma ^0_1$-representative, then it is effectively inseparable. Let $\varPhi$ witness that U is $\Sigma ^0_1$-representative. Suppose ${\mathcal {X}}_0$ and $\mathcal X_1$ are disjoint RE sets. Via a bijective Gödel numbering, we can view these as sets of U-sentences. Suppose that $U_{\mathfrak {p}} \subseteq \mathcal {X}_0$ and $U_{\mathfrak {r}} \subseteq {\mathcal {X}}_1$.

Suppose $\exists u \, \delta _i(u,x)$ represents $\varPhi (x)\in \mathcal X_i$ where $\delta _i$ is a pure $\Delta _0$-formula.

We need a special form of the Gödel fixed point construction. We substitute a formula $\rho$ in $\sigma _0(v) < \sigma _1(v)$ in the following way. First we substitute the numeral of the Gödel number of $\rho$ in each of the $\sigma _i$. Then, we rewrite each of the resulting formulas to pure 1-$\Sigma ^0_1$-form and, subsequently, we apply witness comparison. Suppose Sub is this function. Let $\textsf{sub}(x,y,z)$ be a pure 1-$\Sigma ^0_1$-representation of the graph of Sub. Say $\textsf{sub}(x,y,z)$ is $\exists w\, \nu (w,x,y,z)$, where $\nu$ is pure $\Delta _0$ and the witness w will always majorise x, y and z. We consider the formula $\varphi (v)$:

$$\begin{aligned} (\exists a \, \exists w\mathbin {<}a \, \exists z\mathbin {<}a\, (\nu (w,v,v,z) \wedge \exists u\mathbin {<}a\, \delta _1(u,z))) \;< \\ (\exists b \, \exists w\mathbin {<}a \, \exists z\mathbin {<}b\, (\nu (w,v,v,z) \wedge \exists u\mathbin {<}b\, \delta _0(u,z))). \end{aligned}$$

Let $\rho {:}{=} \textsf{Sub}(\varphi ,\varphi )$. We note that $\rho$ is of the literal form $\rho _1 < \rho _0$, where the $\rho _i$ are pure 1-$\Sigma ^0_1$-sentences.

Suppose $\varPhi (\rho ) \in {\mathcal {X}}_i$. Then $\rho _i$. So either $\rho$ or $\rho ^\bot$. If we have $\rho$, then $\varPhi (\rho ) \in {\mathcal {X}}_1$. Moreover, by $\Sigma 1$, we find $U\vdash \varPhi (\rho )$, and, hence, $\varPhi (\rho )\in {\mathcal {X}}_0$. Quod impossibile. If we have $\rho ^\bot$, then $\varPhi (\rho ) \in \mathcal X_0$. Moreover, by $\Sigma 3$, we have $U \vdash \lnot \, \varPhi (\rho )$ and, hence, $\varPhi (\rho ) \in {\mathcal {X}}_1$. Quod impossibile iterum. Ergo, the formula $\varPhi (\rho )$ is in none of the $\mathcal {X}_i$.

Remark 16

Inspection of the proof shows that in order to derive Rosser’s theorem we do not need the formalisation of the Fixed Point Lemma inside the given theory.

Remark 17

The above proof can be somewhat simplified by employing a self-referential Gödel numbering. See, e.g., [28] or [29]. However, even with this strategy, there will be some detail on how to handle substitution of numerals.