Essential Hereditary Undecidability

In this paper we study \emph{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 \ehu\ theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below {\sf R}. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation \emph{essential tolerance}, or, in the converse direction, \emph{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 hereditarilyundecidable theory.


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 [TMR53].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 study the interaction of the Tarski-Mostowski-Robinson theory R with lax interpretability.We show that, in a sense that generalises mutual lax interpretability, R is equal to the false Σ 0 1 -sentences.We develop the notion of Σ 0 1 -friendliness and show that it is sufficient but not necessary for essential hereditary undecidability.
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 [PMV22].

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 ⊆ 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 = 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 p for the set of its theorems and U r for the set of its refutable sentences, i.e., U r := {ϕ | U ⊢ ¬ ϕ}.
We take 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 τ 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 α.
We can define the obvious identity translation of a language in itself, composition of translations and a disjunctive translation τ ϕ ν.E.g., in case τ and ν have the same dimension and are non-piecewise, the domain of τ ϕ ν becomes (ϕ ∧ δ τ ( #x )) ∨ (¬ ϕ ∧ δ ν ( #x )).
We refer the reader for details to e.g.[Vis17].An interpretation is a triple U, τ, V , where τ is a translation of the U -language in the V -language such that, for all ϕ, if U ⊢ ϕ, then V ⊢ ϕ τ . 2  We write: • K : U V for: K is an interpretation of U in V .
• U V for: there is a K such that K : U V .We also write V U for: U V .
• U loc V for: for every finitely axiomatisable sub-theory U 0 of U , we have U 0 V .• U mod V for: for every V -model M, there is a translation τ from the U -language in the V -language, such that τ defines an internal U -model N = τ (M) of U in M. • We write U ⊲⊳ V for: U V and V U .Similarly, for the other reduction relations.Given two theories U and V we form W := U V 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 → ϕ if ϕ is a U -axiom and ¬ P → ψ if ψ is a V -axiom.One can show that U V is the infimum of U and V in the interpretability ordering .This result works for all choices of our notion of interpretation.
2.2.Arithmetical Theories.The theory R, introduced in [TMR53], 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 2.1 below for a brief explanation of the difference and some background.The language of R, in our variant, is the arithmetical language A with 0, S, +, × and <.
Here are the axioms of R. The underlining stands for the usual unary numeral function.
Remark 2.1.The original version of R in [TMR53] did not have < in the language.Tarski, Mostowski and Robinson used ≤ as a defined symbol with the following definition: u ≤ t iff ∃w w + u = t.Their axioms are the obvious adaptation of the above ones with ≤ in stead of <.One can employ an even weaker theory R 0 , where one drops Axiom R5 and strengthens R4 by replacing the implication by a bi-implication.See, e.g., [JS83] for a discussion.Vaught, in his paper [Vau62], employs an even weaker variant of 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 2 In case we have parameters with parameter-domain α this becomes: ). See also Appendix A for a discussion of the use parameters for interpretations of finitely axiomatised theories.
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. 3The language of TN is the arithmetical language A.Here are the axioms of TN.TN1.
We note that, by substituting x for y, TN6 implies x < x.So, < satisfies the axioms of a linear strict ordering with minimum 0. If follows from this fact in combination with TN6 again, that < is a discrete ordering and that S does give the order successor when applied to a non-maximal element.Moreover, by TN5, 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 ω.Moreover, on a finite domain the successor function will behave normally, except on the maximum m, where we have Sm = m.
Remark 2.2.The structure Z is a model of all axioms except TN1.Moreover, ω + 1, where we cut off all operations at ω in the obvious way, is a model of all axioms except TN4.Finally, Z 2 with the ordering generated by 0 < 1 is a model of all axioms except TN5.❍ We will be interested in the theory of a witness of a Σ 0 1 -sentence σ.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 [Vis17].We follow the same strategy in the present paper.We define pure ∆ 0 -formulas as follows: • Here the bounded quantifiers are the usual abbreviations.
A pure Σ 0 1 -formula is of the form ∃ #u δ, where δ is pure ∆ 0 .In [Vis17], we showed that every ordinary Σ 0 1 -sentence can always be rewritten modulo EA + BΣ 1provable equivalence to a pure one.We call something a 1-Σ 0 1 -formula if it starts with precisely one single existential quantifier.
A subtlety occurs in the treatment of substitution: consider a pure Σ 0 1 -formula σ and, e.g., a substitution of a numeral in it, σ[x := n ].Here we will always assume that the result of substitution is rewritten to an appropriate pure Σ 0 1 normal form.We need the notion z |= ϕ, where z is considered as a number that models TN, where the arithmetical operations are cut off at z.We define z |= ϕ by ϕ tr(z) , where tr(z) is a parametric translation from the arithmetical language to the arithmetical language which is defined as follows: • the domain of tr(z) is the set of x such that x ≤ z, Let σ := ∃ #x δ, where δ is pure ∆ 0 with at most the #x free.Here #x := x 0 , . . ., x n−1 .We define: We note that σ q is equivalent with ∃z z |= ( TN ∧ ∃ x (δ ∧ i<n Sx i = x i )).
We will confuse σ q with the theory axiomatised by this sentence.Since everything relevant to the evaluation of the sentence happens strictly below z, the pure Σ 0 1 -sentence σ in the context of (•) q has its usual arithmetical meaning.
The following result is easily verified.
We will employ witness comparison notation.Suppose α is of the form ∃x α 0 (x) and β is of the form ∃y β(y).We define: We note that witness comparisons between pure 1-Σ 0 1 -formulas are again pure 1-Σ 0 1 -formulas.The following insights are immediate.
If we do not allow piecewise interpretations, we still have (∃x ∃y x = y) σ q .
Remark 2.5.The reader of this paper will develop some feeling for the subtleties involved in our strategy to handle Σ 0 1 -sentences in the context of theories of a number.See Appendix B for an illustration of these subtleties.
In Taishi Kurahashi's paper [Kur22] a somewhat different approach to obtain sentences with the good properties of the σ q is worked out.Kurahashi's paper verifies many details in a careful way.
A different idea for the treatment of the theory of the witness of a Σ 0 1 -sentence is to work with the usual definition of Σ 0 1 , but demand that the maximum element, if there is one, is larger that a suitable function of the Gödel number of σ 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 σ q -like sentences.See, e.g., [Vis12] 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 [Vau62].See [Pak19] 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 Φ, considered as a function from the sentences of the U language to the V -language, such that: i. Φ commutes with the boolean connectives, so, e.g., We note that it follows that, e.g., 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.
Let us say that an RE relation E between numbers, considered as a relation between U -and U ′ -sentences, witnesses a recursive Lindenbaum isomorphism iff we have: a.For all ϕ, there are χ and χ ′ such that ϕ ∼ χ E χ ′ ; b.For all ϕ ′ , there are χ and the other boolean connectives.
It is easy to see that if E witnesses recursive Lindenbaum isomorphism, then so does 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 2.6.Suppose E witnesses recursive Lindenbaum isomorphism between U and U ′ .Then, we can effectively find from an index of E an index of a recursive boolean isomorphism Φ between U and U ′ .
Proof.This is by a straightforward back-and-forth argument.Suppose E witnesses recursive Lindenbaum isomorphism.Without loss of generality we may assume that E = ∼ • E • ∼ ′ .Let us employ enumerations of sentences that enumerate boolean sub-sentences before sentences.
We construct Φ in steps.Suppose we already have constructed (Here k may be 0.) Suppose k is even.Let ϕ k be the first sentence in the enumeration of the U -sentences not among the ϕ i , for i < k.In case ϕ k is a pseudo-atom, we take ϕ ′ k the first pseudo-atom in the enumeration of the ψ ′ such that ϕ k E ψ ′ .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 ϕ k is, e.g., a conjunction, it will be of the form (ϕ i ∧ ϕ j ), for i, j < k, and we set ϕ ′ k := (ϕ ′ i ∧ ϕ ′ j ).The case that k is odd, is, of course, the dual case.
Clearly, this construction indeed delivers a recursive boolean isomorphism.❑ Let us write U ≈ U ′ for U is recursively isomorphic to U ′ .An important insight is that ≈ is a bisimulation w.r.t.theory extension (in the same language).This means that: Proof.We prove the zig case.Zag is similar.Suppose U ≈ V and U ′ ⊇ e U .Let Φ be a witnessing isomorphism.We define V ′ as {Φ(ϕ) | ϕ ∈ U ′ }.We have: Suppose P is a property of theories.We say that U is essentially P if all consistent RE extensions (in the same language) of U are P.We say that U is hereditarily P if all consistent RE sub-theories of U (in the same language) are P.We say that U is potentially P if some consistent RE extension (in the same language) of U is P.
If 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 2.7.
Theorem 2.8.Suppose P is a property of theories that is preserved by ≈.Then, so is the complement of P and the property of being essentially P.Moreover, if Q is also a property of theories preserved by ≈, then so is the intersection of P and Q.
We will see that we do not have an extension of Theorem 2.8 to include hereditariness.
Remark 2.9.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 2.10.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: [Per97, Theorem 7.1.3]Theorem 2.11 (Peretyat'kin).Suppose U is an RE theory with index i.Then, there is a finitely axiomatised theory A := pere(i) such that there is a recursive boolean morphism Φ between U and A. Moreover, A and an index of Φ 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 [Jan53].See also [PMV22].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 A n to be the sentence: there exists an equivalence class of size precisely n + 1.It is immediate that the A n are mutually independent over Jan.
Theorem 2.12 (Janiczak).Over Jan, every sentence is equivalent with a boolean combination of the A n .
Jan will not be recursively boolean isomorphic to propositional logic with countably propositional variables in our narrow sense, since, in Jan, there will be sentences equivalent to e.g.A 0 that are not identical to a boolean combination of A i .However, Jan will be recursively Lindenbaum isomorphic to propositional logic.
Let U be any theory.Remember that we work with a bijective coding for the U -sentences.We define jprop(U ) by Jan plus all sentences of the form A ϕ∧ψ ↔ (A ϕ ∧ A ψ ), plus similar sentences for the other boolean connectives, plus all A ϕ , whenever U ⊢ ϕ.Clearly, we can effectively find an index of jprop(U ) from an index of U .We find: Theorem 2.13.U is recursively boolean isomorphic with jprop(U ).
Proof.We define ϕ E ϕ ′ iff ϕ ′ = A ϕ .It is easily seen that E witnesses recursive Lindenbaum isomorphism between U and jprop(U ).So, U and jprop(U ) are recursively isomorphic, by Theorem 2.6.❑ 4 Our theory differs slightly from the theory considered by Janiczak in that we added J3.We did this to make the characterisation in Theorem 2.12 as simple as possible.
2.4.Incompleteness and Undecidability.We write 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 p and U r are recursively inseparable.
• U is effectively inseparable iff U p and U r are effectively inseparable.This means that there is a partial recursive function Φ such that, whenever U p ⊆ W i , U r ⊆ W j , and W i ∩ W j = ∅, we have Φ(i, j) converges and Φ(i, j) ∈ W i ∪ W j .We can easily show that Φ can always taken to be total.• U is effectively essentially undecidable, iff, there is a partial recursive Ψ , such that, for every consistent RE extension V of U with index i, we have The second and third of these notions turn out to coincide.This was proven by Marian Boykan Pour-El.See [BPE68].
Theorem 2.14 (Pour-El).A theory is effectively inseparable iff it is effectively essentially undecidable.
Clearly, recursively inseparable implies essentially undecidable.Andrzej Ehrenfeucht, in his paper [Ehr61], provides an example of an essentially undecidable theory that is not recursively inseparable.So there is no non-effective equivalent of Theorem 2.14.
The next theorem is due to Marian Boykan Pour-El and Saul Kripke.See [BPEK67, Theorem 2].Theorem 2.16 (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 [TMR53, Chapter I, Theorem 5].
Theorem 2.17 (Tarski, Mostowski, Robinson).Suppose the theory U is decidable and ϕ is a sentence in the U -language.Then, U + ϕ is also decidable.

Essential Hereditary Undecidability: A First Look
In this section, we collect the basic facts about Essential Hereditary Undecidability and provide a selection of examples.
Theorem 3.1.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 ⊆ e V ⊇ e W . ❑ We note that, more generally, U is essentially hereditarily P iff, for every W in the U -language, if U + W is consistent, then W is P. 5We say that V tolerates U if V potentially interprets U .In other words, V tolerates U iff there is a consistent V ′ ⊇ e V such that V ′ U .Equivalently, V tolerates U iff, there is a translation τ of the U -language into the V -language such V + U τ p is consistent.Finally, V tolerates U iff, there is a translation τ of the U -language into the V -language such V + id τ U + U τ is consistent.6Theorem 3.2.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 The other direction is immediate.❑ 3.2.Essential Hereditary Incompleteness.Clearly, incompleteness is not the same as undecidability.However, essential incompleteness is the same as essential undecidability (by Theorem 2.16).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.
Theorem 3.3.a. Suppose U 0 and U 1 are essentially undecidable.Then U 0 U 1 is essentially undecidable.b.Suppose U 0 and U 1 are essentially hereditarily undecidable.Then U 0 U 1 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 U := U 0 U 1 and let e i be the identical translation of the U i -language into the U 0 U 1 -language.Suppose W is consistent with U .Clearly, at least one of U + W + P or U + W + ¬ 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 e 0 .So W is undecidable.The other case is similar.❑ We show that the essentially hereditarily undecidable theories are upwards closed under interpretability.
Theorem 3.4.Suppose U is consistent and essentially hereditarily undecidable and V 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 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.❑ Theorem 4.15 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 3.5.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 closed in both arguments.❑ 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 [Han65].
Example 3.6.(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 [Put57, Section 6].
We start by specifying a theory in the language of identity.Let: Let X be any non-recursive set.We take: The theory I X has the following complete extensions: n, for n ∈ X and {¬ n | n ∈ ω}.So there are no non-recursive complete extensions.The theory I X cannot be consistent with an essentially undecidable U in the same language (and, hence cannot tolerate an essentially undecidable V ), since I X + U would have a complete and recursive extension.
We now apply Theorem 2.11 (Peretyat'kin's result), to obtain a finitely axiomatised theory J X that is recursively boolean isomorphic to I X .Clearly, J X will inherit the undecidability and the lack of non-recursive complete extensions from I X .Since, J X is finitely axiomatised and undecidable, it will be hereditarily undecidable.
We note that the original theory I X extends the theory of pure identity in the language of pure identity.So, I X itself is not hereditarily undecidable.❍ Example 3.7.We show that there are theories that are essentially undecidable and hereditarily undecidable but not essentially hereditarily undecidable. 7Suppose U is essentially hereditarily undecidable and V is essentially undecidable but not hereditarily undecidable.
By Theorem 3.3(a), we find that U V is essentially undecidable.Suppose W is a decidable sub-theory of U V .Then, W + P is a sub-theory of (U V ) + P , i.e., 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, U V is hereditarily undecidable.
Finally, let Z be a decidable sub-theory of V .We extend the signature of Z to the signature of U V and add the axiom ¬ P plus axioms of the form ∀ #x R( #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 U V .So U V 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 2.17 and [TMR53, Chapter I, Theorem 6].
Theorem 3.8 (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 3.8, 2.11 and 2.13 give us immediately the following insight: Theorem 3.9.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 := pere(i), where i is an index of U and U 1 := jprop(U ).
The use of Theorem 2.11 delivers many examples of (essentially) hereditarily undecidable theories, Here is, for example, Theorem 3.3 of [Han65].
Theorem 3.10 (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 [Sho58], there is an essentially undecidable RE theory U of degree d.Say it has index i.Clearly, pere(i) fills the bill.❑ Using the ideas of [PMV22], we can even arrange it so that the Turing degree of every theory that interprets the theory A of Theorem 3.10 is ≥ d.
The next example is due to Cobham.This result is presented in [Vau62].See also [Vis17] for an alternative presentation.We will prove the result in Section 4. Theorem 3.11 (Cobham).The theory R is essentially hereditarily undecidable.
We have the following corollary of Theorem 3.10.
Corollary 3.1.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 3.10, we can find an essentially hereditarily undecidable theory A of RE degree d.If A R, then the degree of A would be 0 ′ , so, A R. By Theorem 3.3 in combination with Theorem 3.11, the theory B := A R is essentially hereditarily undecidable.Moreover, since A R, the theory B is strictly below R.
A different and more natural example is the theory PA − scat of Section 6. ❑ A well-trodden path is the construction of essentially undecidable theories using recursively inseparable sets.We give the basic lemma.
Lemma 3.1.Suppose Φ is a recursive function from the natural numbers to the sentences of U .Let X , Y be a pair of recursively inseparable sets.Suppose Φ maps X to U p and Y to the U r .Then, U is essentially undecidable.
From the proof of Theorem 3.2 of [Han65] we can extract the following analogue of Lemma 3.1 for the case of essentially hereditarily undecidable theories.Lemma 3.2 (Hanf).Let U be a consistent RE theory and let U 0 be a finitely axiomatised sub-theory of the U .Suppose Φ is a recursive function from the natural numbers to the sentences of U .Let X , Y be a pair of recursively inseparable sets.Suppose Φ maps X to U 0p and Y to U r .Then, U is essentially hereditarily undecidable.
Proof.Let U, U 0 , Φ, X , 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 2.17, we find that W * := W + U 0 is decidable.Moreover, W * is consistent with U .We have: It follows that {k | W * ⊢ Φ(k)} is decidable and separates X and Y.A contradiction. ❑ As we will see, in Section 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 [Han65, Theorem 3.2], shows that there are very un-R-like essentially hereditarily undecidable theories.
Example 3.12.(Hanf).We produce an essentially hereditarily undecidable RE theory U that does not tolerate any finitely axiomatisable essentially undecidable theory A.
Let X and Y be recursively inseparable sets.Let V := Jan + {A n | n ∈ X }.Let B be pere(i), where i is an index of V , and let Ψ be the boolean isomorphism from V to B. We define B i := Ψ (A i ) and U := B + {¬ B j | j ∈ Y}.By Lemma 3.2, 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 Ψ −1 (C) is equivalent to a boolean combination of the A i over V , so C is equivalent to a boolean combination of the B i over B. Let the set of the i so that B i occurs in this boolean combination be F .Let 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 X and Y. ❍ In [Vis22], we show that effectively Friedman-reflexive theories are essentially hereditarily undecidable.We state it here as a theorem.The theorem will be a direct consequence of Theorem 5.4 of this paper.Theorem 3.13.Suppose U is consistent, RE, and effectively Friedman-reflexive.Then, U is essentially hereditarily undecidable.

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 V , iff U potentially interprets V , in other words, if for some consistent RE theory U ′ ⊇ e U , we have U ′ V .We find that U essentially tolerates V iff U essentially potentially interprets V , explicitly: iff, for all consistent RE theories U ′ ⊇ e U , there is a consistent RE theory U ′′ ⊇ e U ′ , such that U ′′ V .We write U ◮ V 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 relationunlike tolerance that fails to be transitive.
Remark 4.1.The notion of tolerance was introduced in [TMR53] 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 [DJ92] and [DJ93], or the handbook paper [JdJ98].❍ Example 4.2.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 4.3.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 4.4.Robert Vaught, in his paper [Vau62] introduces a notion that we would like to call parametrically local interpretability or pl .This notion interacts in desirable ways with essential hereditary undecidability.We discuss the relationship between ◭ and pl in Appendix A. We show that ◭ c , a slightly improved version of ◭, satisfies: if U pl V , then U ◮ c V .Moreover, for our purposes, ◭ c 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.
Proof.Suppose U mod V .Let U ′ be a consistent theory with U ′ ⊇ e U .Consider any model M of U ′ .There is an M-internal model of V , say, given by translation τ .Let U ′′ := U ′ + {ψ τ | V ⊢ ψ}.Clearly, U ′′ is consistent and RE and U ′′ V as witnessed by τ .❑ In Section 6, we develop the theory PA − scat .This theory is a sub-theory of R. We have PA − scat ◮ R, but PA − scat mod R.This tells us that the inclusion of model interpretability in lax interpretability is strict.
Open Question 4.6.Are there sequential U and V such that we have U ◮ V , but U mod V ?❍ We turn to the comparison of lax and local interpretability.
Proof.Suppose U ◮ V .Let V 0 be a finitely axiomatised sub-theory of V .Let ϕ be a single axiom of V 0 which includes id V .Suppose U V 0 .Consider The theory U ′ is consistent since, if not, U would prove a finite disjunction of sentences of the form ϕ τ .Say the translations involved are τ 0 , . . ., τ n−1 .We define: We find that U ⊢ ϕ τ * .Quod non.So, U ′ is consistent.Clearly U ′ is RE and no consistent RE extension of U ′ can interpret the theory axiomatised by ϕ.But this contradicts U ◮ V .❑ Example 4.8.Consider a consistent finitely axiomatised sequential theory A. We do have A loc (A).Here (A) is the theory S 1 2 + {Con n (A) | n ∈ ω}, where Con n means consistency w.r.t.proofs where all formulas in the proof have depth of quantifier alternations complexity ≤ n.See, e.g., [Vis11] for more on .
In, e.g., [Vis14a] it is verified in detail that A has a consistent RE extension A such that every interpretation of S 1 2 in A contains a restricted inconsistency statement for A. We call such an extension a Krajíček-theory based on A. Clearly, no consistent extension of A in the same language can interpret (A).So A ◮ (A).This gives us our desired separating example between ◮ and loc .
We note that A ◮ (A) 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 loc R.However, U ◮ R. Since no complete RE theory does interpret R. ❍ It turns out that it is useful to lift ◮ to a relation between sets of theories.We define: • X ◮ Y iff for all U ∈ X and for all consistent RE theories U ′ ⊇ e U , there is a consistent RE theory U ′′ ⊇ e U and a V ∈ Y, such that 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 X ◮ Y ◮ Z.Consider U ∈ X and let U ′ be any consistent RE extension of U .Let U ′′ be a consistent RE extension of U ′ such that U ′′ V , for some V ∈ Y. Say, we have K : and U * V ′′ W , so U * W .We claim that U * is consistent.If not, there would be a ψ such that V ′′ ⊢ ψ and U ′′ ⊢ (¬ ψ) K .It follows, by the definition of V ′ , that V ′ ⊢ ¬ ψ and, hence, that V ′′ ⊢ ¬ ψ, contradicting the fact that V ′′ is consistent.Thus, U * ⊇ e U ′ and W are our desired witnesses.❑ We write U Y for U tolerates some element of Y. Inspection of the above proof also tells us that: Proof.We just do (i).Claim (ii) is similar.From left-to-right is immediate, since U (U V ) and V (U V ), and, hence, U ◮ (U V ) and V ◮ (U V ).So, we are done by transitivity.
Let Z := U V .Suppose U ◮ W and V ◮ W .Let Z ′ ⊇ e Z be RE and consistent.The theory Z ′ is either consistent with P or with ¬P .Suppose it is consistent with P .Let U ′ be the set of U -sentences that follow from Z ′ +P .Clearly, U ′ ⊇ e U and U ′ is RE and consistent.So, there is a U ′′ ⊇ e U ′ that is RE and consistent such that U ′′ W .We take Z ′′ the theory axiomatised by Z ′ + P + U ′′ in the Z-language.Clearly, Z ′′ ⊇ e Z ′ and Z ′′ is consistent and RE and Z ′′ W .The argument in case ¬ P is consistent is similar.❑ We note that the above theorem tells us that the embedding functor of interpretability into lax interpretability preserves infima.
Remark 4.12.We define ∨ as follows.U ∨ V is the result of taking the disjoint union of the sigmatures of U and V and taking as axioms ϕ ∨ ψ, whenever U ⊢ ϕ and V ⊢ ψ.It is easy to see that ∨ gives representatives of the infimum for mod , ◭, and loc .❍ Open Question 4.13.It would be good to have a counterexample that shows that ∨ is not generally an infimum for .❍ Open Question 4.14.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?❍ 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 4.15.Let U be RE and consistent.i. Suppose V is a class of essentially undecidable theories and U ◮ V.Then, U is essentially undecidable.ii.Suppose V is a class of essentially hereditarily undecidable theories and U ◮ V.
Then, U is essentially hereditarily undecidable.
Proof.Ad (i).Suppose V is a class of essentially undecidable theories and U ◮ V. Suppose U has a consistent decidable extension W . then, W has a decidable consistent complete extension W * .It follows that W * V , for some V in V. Quod impossibile.
Ad (ii).Suppose V is a class of essentially hereditarily undecidable theories and U ◮ V. Suppose W is an RE theory in the language of U and suppose Clearly Z is a sub-theory of V .If W were decidable then so would Z, contradicting the fact that V is essentially hereditarily undecidable.9❑ Remark 4.16.Inspection of the example provided by Ehrenfeucht in [Ehr61], shows that his construction provides an example where U ◮ V, each element of 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 a relata of .Let S be the set of all theories σ q , where σ is a false pure Σ 0 1 -sentence and σ q is consistent.10Theorem 4.17.We have R S.
Proof.From left-to-right.Let U ′ be a consistent RE extension of R. Clearly, U ′ ⊢ σ q , for all true pure Σ 0 1 -sentences σ.So, if no σ q ∈ S, would be consistent with U ′ , we could decide Σ 0 1 -truth.Quod non.Consider any such σ q that is consistent with U ′ .Let U ′′ := U ′ + σ q .Clearly, U ′′ σ q .
From right-to-left.Consider σ q ∈ S. Clearly, σ q R and we are easily done.❑ Of course, the extension U ′′ in the proof of Theorem 4.17 can be found effectively from an index of U ′ .We outline one way to do it.
Sketch of an alternative proof of Theorem 4.17.Let be U ′ -provability.By the Gödel Fixed Point Lemma, we can find a pure Σ 1 -sentence  that is equivalent to ¬  q . 11Suppose  were true, then we have both  q and ¬  q , contradicting the consistency of U ′ .So  is false and  q is consistent with U ′ .We take U ′′ := U ′ +  q .❑ Since all σ q ∈ S extend R, they are essentially undecidable.Moreover, since the σ q are finitely axiomatised, they are essentially hereditarily undecidable.It follows from Theorem 4.15, that R is essentially hereditarily undecidable.So, this gives us a proof of Theorem 3.11.
Open Question 4.18.Suppose U ◮ S. Does it follow that U is recursively inseparable?❍ Let F be the set of all finitely axiomatised essentially hereditarily undecidable theories.Example 3.12 shows that there is an essentially hereditarily undecidable theory U such that U ◮ F.

Σ 0
1 -friendliness and Σ 0 1 -representativity In this section, we have a brief look at a rather natural property of theories that implies essential hereditary undecidability.
Theorem 5.1.Consider a consistent RE theory U and a recursive function Φ from pure 1-Σ 0 1 -sentences to U -sentences.Suppose Φ satisfies Σ1.Let U i be a recursive sequence of consistent RE extensions of U .Then, we can effectively find a false pure 1-Σ 0 1 -sentence , such that Φ() is consistent with each of the U i .Proof.We stipulate the conditions of the theorem.We can find a pure ∆ 0 -formula π(i, p, ϕ) such that U i ⊢ ϕ iff ∃p π(i, p, ϕ).Let △ϕ := ∃u ∃i < u ∃p < u π(i, p, ϕ).
Using the Gödel Fixed Point Construction, we find a pure 1-Σ 0 1 -sentence  such that  is true iff △¬ Φ().Suppose  is true.Then, U ⊢ Φ() and, for some i, we have U i ⊢ ¬ Φ(), contradicting the consistency of U i .Thus,  is false and consistent with each of the U i .❑ Remark 5.2.(Kripke).We immediately get Kripke's version of the First Incompleteness Theorem from Theorem 5.1.Let School be the theory in the language of arithmetic (without < as primitive) of all true closed equations.We get Kripke's result by setting U := School and Φ the transformation promised by Matiyasevich's theorem that sends a pure 1-Σ 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 N. ❍ Lemma 5.1.Every Σ 0 1 -friendly theory U is Σ 0 1 -representative.Proof.Let Φ witness that U is Σ 0 1 -friendly.We prove Σ3.Let σ and σ ′ be pure 1-Σ 0 1 -sentences.We have: (U + Φ(σ ′ < σ)) [σ ′ < σ].Suppose σ ≤ σ ′ .Since, by Theorem 2.4, [σ ′ < σ] is inconsistent, we find that U ⊢ ¬ Φ(σ ′ < σ).The other case is similar.❑ Theorem 5.3.Suppose U is RE, consistent, and Σ 0 1 -friendly.Then, U ◮ S, and, hence, U ◮ R.
Proof.We note that any consistent RE extension of a Σ 0 1 -friendly RE theory is again Σ 0 1 -friendly.So it is sufficient to show that U tolerates a theory σ q , for false pure 1-Σ 0 1 -sentences σ.Suppose U does not tolerate any false σ q .If σ is true, we have U ⊢ Φ(σ).Suppose σ is false.We have (U + Φ(σ)) σ q .So, if U + Φ(σ) were consistent, then U would tolerate σ q .Quod non, ex hypothesi.So, U ⊢ ¬ Φ(σ).Since U is RE, we can now decide the halting problem.Quod impossibile. ❑ We note that we can effectively find a sentence σ such that Φ(σ) is consistent with U from indices for U and Φ.Let ¬ Φ(s) be a pure 1-Σ 0 1 -formula representing the U -provability of ¬ Φ(s).Then, we can take σ to be , (a pure 1-Σ 0 1 version of) the Gödel fixed point that is equivalent to ¬ Φ().It is easy to see that U + Φ() is indeed consistent.
The following result employs the notions and notations of [Vis22].
Proof.We can take Φ(σ) := σ ❑ We note that Theorems 5.3 and 5.4 immediately give Theorem 3.13.The theory R is Σ 0 1 -friendly via the mapping σ → σ q .So this again shows that R ◮ S.
It turns out that Σ 0 1 -representativity coincides with a familiar notion.Theorem 5.5.Consider a consistent RE theory U .Then, U is Σ 0 1 -representative iff U is effectively inseparable.
Proof.Suppose U is RE and consistent.
Suppose U is Σ 0 1 -representative as witnessed by Ψ .Let X 0 and X 1 be any pair of effectively inseparable sets.Let σ 0 (x) be a pure 1-Σ 0 1 -formula that represents X 0 and let σ 1 (x) be a pure 1-Σ 0 1 -formula that represents X 1 .We write σ i (n) for a pure 1-representation of the result of substituting n in σ i .We define For the converse, suppose U is effectively inseparable.Then, by [BPEK67, Theorem 2], we find that there is a recursive boolean isomorphism Ψ from R to U .We can take Ψ restricted to pure 1-Σ 0 1 -sentences as the function witnessing the Σ 0 1representativity of U .❑ The first part of the proof of Theorem 5.5 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.
Open Question 5.6.It would be quite interesting to replace the σ q in our definitions of friendliness and representativity by some other class of theories.However, the demands on the σ q use both witness comparison and truth.So, it is not at all obvious here what more general analogues could be.❍ Example 5.7.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: Σ 0 1 -friendly.We first give the list and then the description of the examples below it.
We can take U 0 be any decidable theory like Presburger Arithmetic.
Finally, consider the case of ∃z ϕ ′ .Since we always can rename bound variables, we can assume that z ∈ X.Let X ′ := X ∪ {z}.We write ≃ z ≃ ′ if ≃ ′ is an equivalence relation on X ′ and ≃ ′ restricted to X is ≃.
We reason in Th(N scat ) under the assumption E ≃ .We note that ∃z ϕ ′ is equivalent to ≃ z ≃ ′ ∃z (E ≃ ′ ∧ϕ ′ ).We zoom in on some α := (E ≃ ′ ∧ϕ ′ ).By the induction hypothesis, this can be rewritten as the conjunction of E ≃ ′ and a disjunction of formulas of the form (θ 0 ∧θ 1 ), where θ 0 is ≃-friendly and θ 1 is ≃ ′ -good and of the form χ ⌈y⌉ , where y ≃ z.If the ≃ ′ -equivalence class of z contains at least two elements, we choose y different from z.Given our assumption that E ≃ , the formula α is equivalent to a disjunction of formulas of the form (θ 0 ∧ (E ≃ ′ ∧ θ 1 )).It follows that ∃z α is equivalent to a disjunction of formulas of the form (θ 0 ∧ ∃z (E ≃ ′ ∧ θ 1 )).It clearly suffices to show that ∃z (E ≃ ′ ∧ θ 1 ) is equivalent to an ≃-friendly formula.
There are two cases.The ≃ ′ -equivalence class of z contains at least two elements or precisely one.
Proof.Since N scat has at least two elements, we do not need to consider piece-wise interpretations.Moreover, every element in N 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 R succ given by an interpretion M .Say M is m-dimensional and suppose 0 is given by a formula z( #x ) and S by s( #x , #y ).We note that in the sequence #x , #y all the variables are pairwise disjoint.There are two conventional aspects.The variables in s need only be among the #x , #y , but not all need to occur.Secondly, the order of the variables #x , #y as exhibited need not be given by anything in s.
Our proof strategy is to obtain a contradiction by finding a finite set of numbers N and an infinite sequence of pairwise different sequences length m with components in the n for n ∈ N .We work in N scat .
i.We fix an m-sequence #a such that z( #a ), in other words #a represents 0 M .We put the n i such that c ni (a i ) in N .
ii.For each equivalence relation on the elements of #x , #y we add a set of numbers to N .Consider a relation ≃ on the elements of #x , #y .Under the assumption E ≃ , we can rewrite s as q<r p<sq θ qp , where θ qp is ≃-good, say, it is of the form (χ qp ) ⌈wqp ⌉ .We can clearly arrange it so that (i) w qp is always the first in the sequence #x , #y of its ≃-equivalence class and (ii) if p < p ′ , then w qp occurs strictly earlier in #x , #y than w qp ′ .So, if p = p ′ , we have w qp ≃ w qp ′ .Consider any θ qp where w qp is one of the y i .There are two possibilities.I. Suppose the number of n in which χ qp is satisfiable is < 2m + 1.In this case we add all n such that χ qp is satisfiable in n to N .II.Suppose the number of n in which χ qp is satisfiable is ≥ 2m + 1.In this case we add the first 2m + 1 such n to N .iii.Nothing more will be in N .
Let N * be the elements in the n, for n ∈ N .Clearly N is finite and so is the number of elements in N * .
We are now ready and set to define our infinite sequence in order to obtain the desired contradiction.The sequence starts with #a .We note that #a is in the domain of M and that the components of #a are in N * .Each element of the sequence will be in the domain of M and its components will be in N * .Suppose we have constructed the sequence up to #b .Since #b is in the domain of M , there is a #c with s( #b , #c ).We define ≃ on the elements of #x , #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 #d = #b , #c and #v = #x , #y .We take v i ≃ v j iff d i ∼ d j .We note that we have We construct the formula q<r p<sq θ qp as before for ≃.So, for some q < r, we have p<sq θ qp [ #v : #d ].Consider the variable w qp .If it is an x i , then all variables y i that are ≃-equivalent to it, will have values that are ∼-equivalent to b i .So they will be in N * .If it is a y i and we are in case (ii.I) of the construction of N the values of the variables equivalent to it will be in N * .The final case is that w qp is a y i and we are in case (ii.II) of the construction of N .Since there are at least 2m + 1 numbers n such that n satisfies χ qp , we can always choose an n * among these numbers such that no d i is in n * such that n * satisfies θ qp .We assign to y i in the equivalence class of w qp the value e i so that under this assignment χ q,p is satisfied.We now modify our sequence #b , #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 #b , #c ′ .We note that the new sequence has strictly less elements outside N * and that we still have s( #b , #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 N * .
By the axioms of R succ , we we cannot have two elements in our sequence that are the same.A contradiction. ❑ We end this section by describing how we can make the result work for parameterfree interpretations.A first step is to modify the definition of σ q , say, to σ q * .We remind the reader that we assume our σ are in pure form.In the definition of σ q we just asked for there to be a witness of σ.For σ q * we ask that the witness w is the smallest one and that w + 1 is the maximum element.
We now define PA − scat ! as the theory axiomatised by ∃!x σ q * (x).We note that N scat also satisfies PA − scat !.The new theory is not a sub-theory of R < .However, the theory is locally finite, i.e., every finitely axiomatised sub-theory has a finite model.
Theorem 2.15 (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 [TMR53, Chapter I, Lemma, p15] and [TMR53, Chapter I, Theorem 1].

7
I thank Taishi Kurahashi for his suggestion to replace my previous concrete example by the current more general class of examples.