The Small-Is-Very-Small Principle

The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable cut, then it shows that the property has a very small witness: i.e. a witness below a given standard number. We draw various consequences from the central result. For example (in rough formulations): (i) Every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative w.r.t. formulas of complexity $\leq n$. (ii) Every sequential model has, for any $n$, an extension that is elementary for formulas of complexity $\leq n$, in which the intersection of all definable cuts is the natural numbers. (iii) We have reflection for $\Sigma^0_2$-sentences with sufficiently small witness in any consistent restricted theory $U$. (iv) Suppose $U$ is recursively enumerable and sequential. Suppose further that every recursively enumerable and sequential $V$ that locally inteprets $U$, globally interprets $U$. Then, $U$ is mutually globally interpretable with a finitely axiomatized sequential theory. The paper contains some careful groundwork developing partial satisfaction predicates in sequential theories for the complexity measure depth of quantifier alternations.


Introduction
Some proofs are like hollyhocks.If you are nice to them they give different flowers every year.This paper is about one such proof.I discovered it when searching for alternative, more syntactic, proofs of certain theorems by Harvey Friedman (discussed in [Smo85]) and by Jan Krajíček (see [Kra87]).The relevant theorem due to Harvey Friedman tells us that, if a finitely axiomatized, sequential, consistent theory A interprets a recursively enumerable theory U , then A interprets U faithfully.Krajíček's theorem tells us that a finitely axiomatized, sequential, consistent theory cannot prove its own inconsistency on arbitrarily small cuts.There is a close connection between these two theorems.
The quest for a syntactic proof succeeded and the results were reported in [Vis93].One advantage of having such a syntactic proof is clearly that it can be 'internalized' in the theories we study.I returned to the argument in a later paper [Vis05], which contains improvements and, above all, a better theoretical framework.In my papers [Vis14a] and [Vis15], the argument is employed to prove results about provability logic and about degrees of interpretability, respectively.
The syntactic argument in question is a Rosser-style argument or, more specifically, a Friedman-Goldfarb-Harrington-style argument.It has all the mystery of a Rosser argument: even if every step is completely clear, it still retains a feeling of magic trickery.
1.1.Contents of the Paper.In the present paper, we will obtain more information from the Friedman-Goldfarb-Harrington-style argument discussed above.In previous work, the basic conclusion of the argument is that, given a consistent, finitely axiomatized, sequential theory A, there is an interpretation M of the basic arithmetic S 1 2 in A that is Σ 0 1 -sound.In the present paper, we extend our scope from finitely axiomatized sequential theories to restricted sequential theories -this means that we consider theories with axioms of complexity below a fixed finite bound.Secondly, we replace the Σ 0 1 -soundness by the more general small-is-verysmall principle (SIVS).
The improved results have a number of consequences.In Section 4, we show that, for any n, every consistent, restricted, recursively enumerable, sequential theory has a finitely axiomatized extension that is conservative w.r.t.formulas of complexity ≤ n.In Section 5, we show that, for any n, every sequential model has an elementary extension w.r.t.formulas of of complexity ≤ n, such that the intersection of all definable cuts consists of the standard numbers.In Section 6, we indicate how results concerning Σ 0 2 -soundness can be derived from our main theorem.Finally, in Section 7, we prove a result in the structure of the combined degrees of local and global interpretability of recursively enumerable, sequential theories.We show that if a local degree contains a minimal global degree, then this global degree contains a finitely axiomatized theory.Thus, finite axiomatizability has a natural characterization, modulo global interpretability, in terms of the double degree structure. 1 Section 2 provides the necessary elementary facts.Unlike similar sections in other papers of mine, this section also contains something new.In [Vis93], I provided groundwork for the development of partial satisfaction predicates for the complexity measure depth of quantifier alternations.Our present Subsection 2.3 gives a much better treatment of the complexity measure than the one in [Vis93].Subsection 2.5 develops the facts about sequential theories and partial satisfaction predicates in greater detail than previously available in the literature.Moreover, we provide careful estimates of the complexities yielded by the various constructions.On the one hand, these subsections contain 'what we already knew', on the other hand, as I found, even if you already know how things go, it can still be quite a puzzle to get all nuts and bolts at the precise places where they have to go.Of course, the present treatment is still not fully explicit, but we are further on the road.
Section 3 contains the central result of the paper.As the reader will see, after all is said and done, the central argument is amazingly simple.The work is in creating the setting in which the result can be comfortably stated.
1 I presented this result in a lecture for the Moscow Symposium on Logic, Algebra and Computation in 2006.However, I was not able to write down the proof, since I lacked the necessary groundwork on partial satisfaction.This groundwork is provided in Section 2 of the present paper.

Basic Notions and Facts
In the present section, we provide the basics needed for the rest of the paper.As pointed out in the introduction the development of partial satisfaction predicates is done in more detail here than elsewhere.For this reason this section may also turn out to be useful for subsequent work.Of course, the reader who wants to get on quickly to more exciting stuff could briefly look over the relevant subsections and, if needed, return to them later.
2.1.Theories.In this paper we will study theories with finite signature.In most of our papers, theories are intensional objects equipped with a formula representing the axiom set.In the present paper, to the contrary, a theory is just a set of sentences of the given signature closed under deduction.This is because most of the results in the paper are extensional.
Also we do not have any constraints on the complexity of the axiom set of the theory.If a theory is finitely axiomatizable, par abus de langage, we use the variables like A and B for it, making the letters do double work: they both stand for the theory and for a single axiom.
When we diverge from our general format this will always be explicitly mentioned.
In the paper, we will meet many concrete theories, to wit AS, PA − , EA, PRA, PA.We refer the reader to the textbooks [HP93] and [Kay91] for an introduction to these theories.

Translations and Interpretations.
We present the notion of m-dimensional interpretation without parameters.There are two extensions of this notion: we can consider piecewise interpretations and we can add parameters.We will give some details on parameters in Appendix A. We will not describe piecewise interpretations here.
Consider two signatures Σ and Θ.An m-dimensional translation τ : Σ → Θ is a quadruple Σ, δ, F , Θ , where δ(v 0 , . . ., v m−1 ) is a Θ-formula and where, for any n-ary predicate P of Σ, F (P ) is a formula A( v 0 , . . ., v n−1 ) in the language of signature Θ, where v i = v i0 , . . ., v i(m−1) .Both in the case of δ and A all free variables are among the variables shown.Moreover, if i = j or k = ℓ, then v ik is syntactically different from v jℓ .
We demand that we have ⊢ F Here ⊢ is provability in predicate logic.This demand is inessential, but it is convenient to have.
There are two worries about this definition.First, what variables x i on the side of the translation A τ correspond with x i in the original formula A? The second worry is that substitution of variables in δ and F (P ) may cause variable-clashes.These worries are never important in practice: we choose 'suitable' sequences x to correspond to variables x, and we avoid clashes by α-conversion.However, if we want to give precise definitions of translations and, for example, of composition of translations, these problems come into play.The problems are clearly solvable in a systematic way, but this endeavor is beyond the scope of this paper.We allow the identity predicate to be translated to a formula that is not identity.
A translation τ is direct, if it is one-dimensional and if δ τ (x) := (x = x) and if it translates identity to identity.
There are several important operations on translations.
• We can compose translations.Suppose τ : Σ → Θ and ν : Θ → Λ.Then ν • τ or τ ν is a translation from Σ to Λ.We define: - • Let τ, ν : Σ → Θ and let A be a sentence of signature Θ.We define the disjunctive translation σ := τ A ν : Σ → Θ as follows.We take m σ := max(m τ , m ν ).We write v ↾ n, for the restriction of v to the first n variables, where n ≤ length( v). - Note that in the definition of τ A ν we used a padding mechanism.In case, for example, m τ < m ν , the variables v mτ , . . ., v mν −1 are used 'vacuously' when we have A. If we had piecewise interpretations, where domains are built up from pieces with possibly different dimensions, we could avoid padding by building the domain directly of disjoint pieces with different dimensions.
A translation relates signatures; an interpretation relates theories.An interpretation K : U → V is a triple U, τ, V , where U and V are theories and τ : Σ U → Σ V .We demand: for all theorems A of U , we have V ⊢ A τ .Here are some further definitions. • A translation τ maps a model M to an internal model τ (M) provided that M |= ∃ x δ τ ( x ).Thus, an interpretation K : U → V gives us a mapping K from MOD(V ), the class of models of V , to MOD(U ), the class of models of U .If we build a category of theories and interpretations, usually MOD with MOD(K) := K will be a contravariant functor.We use The alternative notations ✁ and ✄ are used in a context where we are interested in interpretability as a preorder or as a provability analogue.
We write: U ✁ V and U ✄ V , for: there is an interpretation K : U ✁ V .We use U ≡ V , for: U ✁ V and U ✄ V .
The arrow notations are mostly used in a context where we are interested in a category of interpretations, but also simply when they improve readability.
We write U ✁ loc V or V ✄ loc U for: for all finite subtheories U 0 of U , U 0 ✁ V .We pronounce this as: U is locally interpretable in V or V locally interprets U .We use ≡ loc for the induced equivalence relation of ✁ loc .
2.3.Complexity and Restricted Provability.Restricted provability plays an important role in the study of interpretability between sequential theories.An n-proof is a proof from axioms with Gödel number smaller or equal than n only involving formulas of complexity smaller or equal than n.To work conveniently with this notion, a good complexity measure is needed.Such a measure should satisfy three conditions.
i. Eliminating terms in favor of a relational formulation should raise the complexity only by a fixed standard number.ii.Translation of a formula via the translation τ should raise the complexity of the formula by a fixed standard number depending only on τ .iii.The tower of exponents involved in cut-elimination should be of height linear in the complexity of the formulas involved in the proof.Such a good measure of complexity together with a verification of desideratum (iii) -a form of nesting degree of quantifier alternations-is supplied in the work of Philipp Gerhardy.See [Ger03] and [Ger05].A slightly different measure is provided by Samuel Buss in [Bus15].Buss also proves that (iii) is fulfilled for his measure.In fact, Buss proves a sharper result.He shows that the bound is d + O(1) for d alternations.In the present paper, we will follow Buss' treatment.
We work over a signature Θ.The formula-classes we define are officially called Σ * n (Θ) and Π * n (Θ).However, we will suppress the Θ when it is clear from the context.Let AT be the class of atomic formulas for Θ, extended with ⊤ and ⊥.We define: Buss uses Σ n+1 and Π n+1 where we use Σ * n+1 and Π * n+1 .We employ the asterix to avoid confusion with the usual complexity classes in the arithmetical hierarchy where bounded quantifiers also play a role.Secondly, we modified Buss' inductive definition a bit in order to get unique generation histories.For example, Buss adds Π * n to Σ * n+1 in stead of ∀v Π * n .In addition our Σ * 0 and Π * 0 are empty, where Buss' corresponding classes consist of the quantifier-free formulas.
Here is the parse-tree of ∀x (∀y ∃z P xyz → ∃u ∃v Qxuv) as an element of Σ * 4 .
Σ * 4 : ∀x The extensional equivalence, for n > 0 of our definition to Buss's is immediate from the following: Fact 2.1.The quantifier-free formulas are in The proof is by five simple inductions.We define: We have: n .This measure is very close to the measure that was employed in [Vis93].We recursively define this measure by taking ρ := ρ ∃ , where ρ ∃ is defined as follows: • ρ(A) := ρ ∃ (A), ρ 0 (A) := max(ρ ∃ (A), ρ ∀ (A)).We verify the basic facts about ρ.
Proof.The proof is by induction on A. We treat the case that A = ∃v B. We have: ρ ∀ (∃v B) = ρ ∃ (B) + 1 = ρ ∃ (∃v B) + 1.Note that this does not use the induction hypothesis.
Theorem 2.5.Let τ : Σ → Θ.We have: Proof.The proof is by induction on A. The case of the atoms is trivial.
We treat the case of implication and ρ ∃ .Suppose A is B → C. We have: The other cases concerning the propositional connectives are similar.
We treat the case for universal quantification and ρ ∃ .Suppose A is ∀v B. We have: The remaining cases for the quantifiers are similar or easier.✷ 2.4.Sequential Theories.The notion of sequentiality is due to Pavel Pudlák.
To define sequentiality we use the auxiliary theory AS + (Adjunctive Set Theory with extras).The signature A of AS + consists of unary predicate symbols N and Z, binary predicate symbols ∈, E, ≤, <, S, ternary predicate symbols A and M.
AS + 1 We have a set of axioms that provide a relative interpretation N of S 1 2 in AS + , where N represents the natural numbers, E represents numerical identity, Z stands for zero modulo E, A stands for addition modulo E, and M stands for multiplication modulo E. AS + 2 ⊢ ∃x ∀y y ∈ x, An important point is that we do not demand extensionality for our sets.A manysorted version of AS + would be somewhat more natural.We refrain from developing it in this way here to avoid the additional burden of working with interpretations between many-sorted theories.
A theory is sequential iff it interprets the theory AS + via a direct interpretation S. We call such an S a sequence scheme.
It is possible to work with an even simpler base theory.The theory AS is given by the following axioms.
Remark 2.6.We could work in a somewhat richer class of theories, the polysequential theories.See [Vis13].
Let's say that an interpretation is m-direct, if it is m-dimensional, if its domain consists of all m-tuples of the original domain, and if identity is interpreted as component-wise identity.A theory U is m-sequential, if there is an m-direct interpretation of AS in U .A theory is polysequential, if it is m-sequential for some m ≥ 1.Note that if we want the AS + format, the interpretation of the natural numbers should also be chosen to be m-dimensional for the given m.The development given in the present paper also works with minor adaptations in the polysequential case.
It is known that there are polysequential theories that are not sequential.However, I only have an artificial example.Every polysequential theory is polysequential without parameters, where a sequential theory may essentially need an interpretation with parameters to witness its sequentiality.(One raises the dimension to 'eat up' the parameters.)Polysequential theories are closed under bi-interpretability.Moreover, every polysequential theory is bi-interpretable with a sequential one.

Satisfaction & Reflection.
In this subsection, we develop partial satisfaction predicates for sequential theories with some care.We prove the corresponding partial reflection principles.This subsection is rather long because it provides many details.The impatient reader could choose to proceed to Theorem 2.17, since that is the main result of the subsection that we will use in the rest of the paper.
Consider any signature Θ.We extend the signature A of AS + in a disjoint way with Θ to, say, A + Θ.Call the resulting theory (without any new axioms) AS + (Θ).
We work towards the definition of partial satisfaction predicates, We provide a series of definitions illustrative of what we need to get off the ground.
).We can easily show that for all u and v there is a w such that pair(u, v, w) and, whenever pair(u, v, w) and pair(u ′ , v ′ , w), then u = u ′ and v = v ′ .Note that there may be several w such that pair(u, v, w).
Note that we adapt the notion of function to our non-extensional pairing.We demand that there is at most one witnessing pair for a given argument.This choice makes resetting a function on an argument where it is defined a simple operation: we subtract one pair and we add one.
So outside of ndom we set the value of α to a default value.In this way we made it a total function on the natural numbers.
We will use α, β to range over functions with numerical domains (elements of nfun).
We note that AS + (Θ) proves that the 'operation' α → α[a : y] is total and that any output sequences are extensionally the same.We employ a usual efficient coding of syntax in the interpretation N of S 1 2 .We have shown above how to formulate things in order to cope with the fact that each number, set, pair, function and sequence can have several representatives.The definition of satisfaction would be completely unreadable if we tried to adhere to this high standard.Hence, we will work more informally pretending, for example, that each number has just one representative.An assignment will simply be a numerical function where we restrict our attention to the codes of variables in the domain.
A sat-sequence is a triple of the form i, α, A , where: i is + or − (coded as, say, 1 and 0), α is an assignment and A is a formula.An a-sat-sequence is a sat-sequence i, α, A , where A set X is good if, for all numbers b the virtual class X ∩ K b exists as a set.We have: Lemma 2.7.(AS + (Θ)).The good sets are closed under the empty sets, singletons, union, intersection and subtraction and are downwards closed w.r.t. the subset ordering.
Proof.We leave the easy proof to the reader.✷ We count two sat-sequences i, α, A and j, β, B as extensionally equal if (i) i and j are E-equal, (ii) α and β have the same functional behaviour on the natural numbers N, and (iii) A and B are E-equal.We say that a sequence σ is of the form i, α, A if it is extensionally equal to a sequence τ with τ 0 = i, τ 1 = α and τ 2 = A. We define n + 1-adequacy and sat n by external recursion on n.We define sat 0 (+, α, A) :↔ ⊥ and sat 0 (−, α, A) :↔ ⊤.We define sat n+1 (i, α, A) iff, for some n + 1-adequate set X, we have i, α, A ∈ X.A set X is n + 1-adequate, if it is good, if its elements are sat-sequences, and if it satisfies the following clauses: Similarly, for other atomic formulas including ⊤ and ⊥.
b.If a sequence of the form −, Similarly, for other atomic formulas including ⊤ and ⊥. c.If a sequence of the form +, Note that if σ and τ are extensionally equal and if X is n-adequate and σ is in X, then the result of replacing σ in X by τ is again n-adequate.
We will often write α |= i n A for: sat n (i, α, A).The relation sat n , when restricted to K n , will have a number of desirable properties.We write sat * n for sat n ∩ K n .We note that we have implicitly given a formula Φ 0 (X , i, α, A), where X is a second order variable with: Thus, for some fixed standard c 0 , we have ρ(sat n+1 (u, v, w)) = ρ(sat n (u, v, w)) + c 0 .It follows that ρ(sat n (u, v, w)) = c 0 n + c 1 , for some fixed standard number c 1 .
Remark 2.8.We note that X occurs twice in the formula Φ 0 (X , k, α, A) described above.This has no effect on the growth of the complexity of the formula sat n but it makes the number of symbols of the formula sat n grow exponentially in n.For the purposes of this paper, this is good enough.However, a slightly more careful rewrite of our definition reduces the number of occurrences of X to one.As a consequence, we can get the number of symbols of sat n linear in n.So, the code of sat n will be bounded by a polynomial in n, assuming we use an efficient Gödel numbering.
Our next step is to verify in AS + (Σ) some good properties of n + 1-adequacy and sat n .We first show that n-adequacy is preserved under certain operations.
Theorem 2.9.The n-adequate sets are closed under unions and under intersection with the virtual class of a-sat-sequences, for any a.
Proof.Closure under unions is immediate given that good sets are closed under unions.Closure under restriction to a-sat-sequences is immediate by the definition of good and the fact that our formula classes are closed under subformulas.✷ We prove a theorem connecting sat * k and sat * n , for k < n.We remind the reader that K k is the virtual class of all k-sat-sequences and sat The proof is by external induction on n.The case that n = 0 is trivial.
We note that Y is a set by Theorem 2.9.We claim that Y is k-adequate.It is clear that Y satisfies all clauses for a k-adequate set automatically except (l) and (m).Let's zoom in on (l).Suppose a sequence of the form −, β, ∃v B is in Y .From this it follows that k = 0. Since X is n + 1-adequate, it follows that: ¬ sat n (+, β, ∃v B).By the induction hypothesis, we find that ¬ sat k−1 (+, β, ∃v B).Hence, the clause for k-adequacy is fulfilled.Clause (m) is similar.
The argument that W is also n + 1-adequate, is analogous to the argument above.✷ We note that in the proof of Theorem 2.10, we could as well do the induction on k.This observation is important in case we study models with a full satisfaction predicate.In this context, we can replace n by a non-standard number and still get our result for standard k.
In the following theorem we prove the commutation conditions for sat n+1 .
Proof.We will treat the illustrative clauses (e), (k) and (l).In the first two cases the right-to-left direction is trivial.We note that the commutation conditions are inherited by sat * n+1 , provided that the formulas in the conditions belong to the right classes.

Ad (e). Suppose
The commutation conditions proven in Theorem 2.11 are not yet full commutation conditions.The defect is in the clauses (l) and (m).Let's zoom in on (m): To get the desired commutation condition, we would like to move from γ |= − n+1 B to γ |= + n+1 B. We have seen in Theorem 2.10 that to make our predicates behave in expected ways, it is better to consider the formulas in their 'intended range To prove this we need induction, which we do not have available in AS + (Θ).The solution is to move to a cut.To realize this, we define a second measure of complexity ν (depth of connectives) as follows: We define: • J † n+1 is the virtual class of all numbers x such that, for all α and for all
Proof.Downwards closure is immediate.

By definition, we have α |=
The case for atomic C follows by previous reasoning.Suppose, for example, that The other unary and binary propositional connectives are similar.Now suppose C is of the form ∃v D and It follows that n = 0. We have, by Theorem 2.10: Curiously, this step does not not use the fact that x is in J † n+1 .The case of ∀ is similar. ✷ Hence, we also have the full Tarskian commutation clauses for these C.
We note that J † n+1 is Φ 1 (n + 1, sat n+1 , x), for a standard formula Φ 1 (y, X , x).Thus the ρ-complexity of J † n+1 is linear in n where the relevant linear term is of the form c 0 n + c 2 .
We still miss an important ingredient.Let i, α, A be a sat-sequence.Suppose α and β assign the same values to the free variables in A. Do we have α To prove such a thing we need induction.We would like to have even more than this, since we want to check the validity of the inference rules for the quantifiers.We define the property Q n+1 by: • The formula A has the property Q n+1 if the following holds.Consider any sat-sequence i, α, A .Suppose w is free for v in A. Let B be (of the form) A[v := w].Suppose further that the functions α[u] = β[u] for all free variables u of A, except possibly v, and that We allow that v and w are equal and that v does not occur in A. Both degenerate cases tell us that Q n+1 (A) implies that α |= i n+1 A iff β |= i n+1 A, whenever α and β agree on the free variables of A.
We can now proceed in two ways to construct a cut that gives us the desired property for the formulas of ν-complexity in the cut.One way does not involve the Σ * n and the Π * n and one way does involve them.The second way yields a more efficient construction of the cut.For completeness, we explore both ways.
We first address the first way.We define: Theorem 2.13 (AS + (Θ)).The virtual class J • n is closed under 0, successor, and is downwardly closed w.r.t.≤.
The other propositional cases are similar.
We treat the case of the existential quantifier, the case of the universal quantifier being similar.Let A be of the form ∃z C. Suppose w is free for v in A. Let B be (of the form) A[v := w].Suppose further that α and β assign the same values to the free variables of A except v and We first address the |= + -case.The argument splits into two subcases.First we have the case that z is (of the form) v, we find that B is of the form A. Hence, replacing z by v, we have: Finally we address the |= − -case.We have: We turn to the second approach.We define:
Proof.The cases of closure under 0 and downwards closure are trivial.Suppose x is in J ⋆ n+1 .The cases of the propositional connectives use the same argument as we saw in the proof of theorem 2.13.We turn to the case of the existential quantifier, the case of the universal quantifier being dual.Suppose C is in Γ x ∩ ∆ * n+1 .The case of |= + is again the same as we saw in the proof of Theorem 2.13.We consider the case of |= − .Suppose ∃v C is in ∆ * n+1 .In this case ∃v C must be in Σ * n .We have: n+1 ∃v D The first and the fifth step use the commutation conditions for ∃.The second and the fourth step use Theorem 2.10.The third step uses the previous case for |= + .✷ We note that the definition of J ⋆ n+1 is of the form Φ 3 (n, Sat).So, its ρ 0 -complexity is estimated by a linear term of the form c 0 n + c 6 .Here the use of J ⋆ n+1 has an advantage over J • n , since construction of the J • m gives us a linear complexity but conceivably with a higher constant as coefficient of n.
Let us take stock of what we accomplished.We have defined virtual classes J ⋆ n+1 that are closed under 0 and S and that are downwards closed such that for all formulas A in Γ J ⋆ n+1 ∩ ∆ * n+1 , we have, for all α, that α x .Also, we have developed virtual classes J • n+1 and J ⋆ n+1 such that all A in Γ J • n+1 , and, similarly, all A in Γ J ⋆ n+1 ∩ ∆ n+1 have the property Q n+1 defined above.So, if we take J ‡ n+1 either J ⋆ n+1 ∩ J • n+1 or J ⋆ n+1 ∩ J ⋆ n+1 then J ‡ n+1 is progressive and all elements of Ξ n+1 := Γ J ‡ n+1 ∩ ∆ n+1 have both good properties.Let us choose for J ⋆ in the definition of Ξ n+1 , so that its ρ-complexity is estimated by c 0 n + c 7 .
We summarize the result in a theorem.
We write A n A for the formalization of A ⊢ n A, where A codes a finite set of formulas and ⊢ n is provability in predicate logic where we restrict ourselves in the proof to Ξ n+1 -formulas.We choose to code the set of formulas in the natural numbers.This is a bit unnatural since AS(Θ) contains sets as first-class citizens.However, if we code sets of formulas in the sets provided by AS(Θ) directly we do not know, for example, that ass(p) the set of assumptions of a proof p is a set.Of course, this problem can be evaded by shortening N in such a way that any set coded in the natural numbers maps to first-class set.If the reader prefers this other road, we think it is sufficiently clear how to adapt the results below to this alternative approach.We write p : A n A for: p is the code of a n-proof witnessing A n A. We write Λ n,y for the class of n-proofs p where the number of steps of p is ≤ y.On the semantical side, we define, for A ∪ {A} ⊆ Ξ n+1 : We work in AS + (Θ).We write ass(p) for the assumption set of (proof code) p.Let Y n be the class of y such that, for all p ∈ Λ n,y , if p : ass(p) n A, then ass(p) |= n A.
Theorem 2.16.The virtual class Y n is downwards closed under ≤, contains 0, and is closed under successor.
Proof.Downwards closure under ≤ is trivial.We show that Y n is progressive.We follow the system for Natural Deduction in sequent style as given in [TS00, Subsection 2.1.4].By Theorem 2.15, the propositional cases are immediate.We will treat the introduction and the elimination rule of the universal quantifier.This follows mainly the usual text book proof.For the convenience of the reader, we repeat the property Q n+1 : • The formula C has the property Q n+1 if the following holds.Consider any sat-sequence i, α, C .Suppose u is free for z in C. Suppose further that α[a] = β[a] for all free variables a of A except z and that We treat the case of universal generalization.Let w be substitutable for v in A and suppose w does not occur freely in the elements of A ∪ {A}.Suppose we have ( †) Consider any α and suppose α |= + n+1 A ′ , for all A ′ in A .We want to show that α |= Inspecting the construction of Y n we see that it is of the form Φ 4 (n, sat n , J ‡ n+1 ), where Φ 4 (x, X , Y) is a fixed formula.Thus ρ(Y n ) is estimated by c 0 n + c 8 .
We now have a refined result involving separate restrictions on ρ on ν and on the length of the proofs.For other applications this refinement may be useful, however, in the present paper, we will simply demand that our proofs are in a cut ℑ n (Θ) that is obtained by taking the intersection of J ‡ n+1 and Y n and shortening to obtain downwards closure and closure under 0, S, +, × and ω 1 .Since the shortening procedure only adds a standardly finite depth to the input formula J ‡ n+1 ∩ Y n , ρ(ℑ n (Θ)) will have complexity c 0 n + c 9 .Moreover, when p is in ℑ n (Θ), then ipso facto its length is in Y n and its ν-complexity is in J ‡ n+1 .We write [A ⊢ n A] for provability in predicate logic involving only ∆ * n -formulas where the proof is constrained to be in the cut J.We write [A ⊢ J n A] when the witness for [A ⊢ n A] is constrained to the cut J.We write ✷ Θ,n A for [∅ ⊢ n A], and For sentences A, we will write true Θ,n (A) for ∀α sat n (+, α, A).
Theorem 2.17.We can find an ω 1 -cut ℑ n (Θ) such that ρ(ℑ n (Θ)) is of order c 0 n + c 9 and such that: As a special case, we have:

Small-Is-Very-Small Principles
In this section we present the central argument of this paper.It is a simple Rosser argument.The bulk of the work has already been done in creating the setting for the result.I choose to give the pure argument in Theorem 3.1 rather than proceed immediately to the somewhat more complicated Theorem 3.2.The more complicated version is needed for application in model theory.
First some preliminaries and notations, in order to avoid too heavy notational machinery.
We will work in sequential theories U of signature Θ with sequence scheme S. So, S : AS + dir −→ U .We can lift S to a direct interpretation S Θ : AS + (Θ) dir −→ U by translating Θ identically. 4e remind the reader that N : S 1 2 → AS + .We will write N := S • N : S 1 2 → U .So, e.g., δ N = N S .We write ℑ n for (ℑ n (Θ)) SΘ .When we write numerals n these are always numerals w.r.t.N .We note that the numerals really are eliminated using the term elimination algorithm.However, this elimination just gives an overhead on 1 in ρ 0 -complexity.
Let η be a Σ b 1 -formula defining a set of axioms.We write ✷ η for provability from the axioms in η.We write ✷ J η for the result of restricting the witnesses for ✷ η to J. We write ✷ η,n for the result of restricting the formulas in a witnessing proof to ∆ * n .5 Formulas like ✷ J η,n have the obvious meanings.We suppress the information about the signature Θ, which should be clear from the context.In case η = (x = A ), we write ✷ A for ✷ η . 6 We will employ witness comparison notation: Theorem 3.1.Let A be a finitely axiomatized sequential theory in a language with signature Θ with sequence scheme S. Consider any sentence B in the language of A of the form B := ∃x ∈ δ N B 0 (x).Let n := max(ρ 0 (A), ρ 0 (B) + c 10 , ρ 0 (S) + c 10 ).
Here c 10 is a fixed finite constant that does not depend on A, B and S. We will determine c 10 below.Suppose A ⊢ ∃x ∈ ℑ n B 0 (x).Then, for some k, we have A ⊢ ∃x ≤ N k B 0 (x), or, equivalently, A ⊢ q≤k B 0 (q).constrained to be ≤ n.In this paper this extra demand is not made.
6 Clearly, this introduces an ambiguity.E.g., does ✷ ⊤ mean provability form all sentences or from the axiom ⊤?However, what we intend will be always clear from the context, Proof.We work under the conditions of the theorem.Using the Gödel Fixed Point Lemma, we find a sentence R such that A ⊢ R ↔ B ≤ ✷ N A,n R. We need that ρ 0 (R) ≤ n.
Here the +3 is due to the additional quantifiers.We note that, if we unravel the numerals n wide scope, we even just need +2.So, we can take c 10 := max(ρ0(sub), ρ0(prov)) + 3.
Reason in A. We have ∃x ∈ ℑ n B 0 (x).In case ¬ ✷ ℑn A,n R, we have R. Suppose ✷ ℑn A,n R. By reflection, as guaranteed by Theorem 2.17, we find R. So, in both cases, we may conclude that R. We leave A again.Thus, we have shown (i) A ⊢ R. By cut-elimination, we find: A ⊢ n R. Hence, (ii) for some k, we find A ⊢ proof N A,n (k, R).Combining (i) and (ii), we we may conclude that A ⊢ ∃x ≤ N k B 0 (x), or, equivalently, A ⊢ q≤k B 0 (q).✷ We note that, due to the use of cut-elimination, we need the totality of superexponentiation in the metatheory.Such theorems usually leave watered-down traces in weaker metatheories.We do not explore such possibilities in the present paper.
The above argument has some analogies with Harvey Friedman's beautiful proof that, in a constructive setting, the disjunction property implies the existence property.See [Fri75].I analyzed this argument in [Vis14b], having the benefit of many perceptive remarks by Emil Jeřábek.One surprising aspect of the above proof is that the minimization principle is not used.Joost Joosten pointed out to me in conversation that the closely related Friedman-Goldfarb-Harrington Theorem also can be proven without using minimization.
For our model theoretic applications we need a variant of Theorem 3.1 that adds domain constants.We allow for the domain constants the exceptional position that they are real constants rather than unary predicates posing as constants.
Theorem 3.2.Consider a finite set of domain constants C. Let A 0 be any finitely axiomatized sequential theory with signature Θ and sequence scheme S. Let A 1 := A 1 ( c ) be any sentence in the language with signature Θ + C. Let A := A 0 ∧ A 1 .
Consider any sentence B( c ) in the language of signature Θ + C of the form B( c Then, for some k, we have that Proof.We work under the conditions of the theorem.We find a sentence R( c Since A and R are standard and since we have A( c ), we find R( c ).Thus, we have shown (i) A( c ) ⊢ R( c ).By cut-elimination, we find: A( c ) ⊢ n R( c ).Hence, (ii) for some k, we have A( c ) ⊢ proof N A( c ),n (k, R( c )). Combining (i) and (ii), we get A( c ) ⊢ ∃x ≤ N k B 0 (x, c ), or, equivalently, A( c ) ⊢ q≤k B 0 (q, c ). ✷ We call a theory U restricted if, for some m all its axioms are in ∆ * m .Theorem 3.3.Suppose A 0 is a finitely axiomatized sequential theory in signature Θ with sequence scheme S. Let m be any number such that m ≥ ρ 0 (A 0 ).Let C be a set of domain constants: C is allowed to have any cardinality.Let U be a restricted theory bounded by m in the language of signature Θ + C extending A 0 .The theory U may have any complexity.
Suppose U ⊢ ∃x ∈ ℑ n B 0 (x).Then, for some k, we have Proof.The theorem is immediate from Theorem 3.2, using compactness.✷ It is of course trivial to take the contraposition of Theorem 3.3.However this contraposion has some heuristic value.So we state it here as a separate theorem.
Theorem 3.4.Suppose A 0 is a finitely axiomatized sequential theory in signature Θ with sequence scheme S. Let m be any number such that m ≥ ρ 0 (A 0 ).Let C be a set of domain constants: C is allowed to have any cardinality.Let U be a restricted theory bounded by m in the language of signature Θ + C extending A 0 .The theory U may have any complexity.Consider any formula C(x).Let n := max(m, ρ 0 (C) + c 10 , ρ 0 (S) + c 10 ).
If the theory U + {C(q) | q ∈ ω} is consistent, then the theory U + ∀x ∈ ℑ n C(x) is consistent.
Proof.We apply Theorem 3.3 to ∃x ∈ δ N ¬ C(x) and take the contraposition.✷

A Conservativity Result
We can use the machinery we built up to prove a Lindström-style result on conservative extensions.
Suppose U is a restricted, sequential, recursively enumerable theory with sequence scheme S. Let p be a bound for the complexity of the axioms of U .By Craig's trick, we can give a Σ b 1 -axiomatization of U .Say the Σ b 1 -formula representing the axioms is η.Suppose A 0 is a finite subtheory of U such that S makes A 0 sequential.Clearly U can be axiomatized by This representation of the axiom set leads immediately to the following theorem.
Theorem 4.1.Suppose U is a restricted, sequential, recursively enumerable theory.Consider any number m.Then there is a finitely axiomatized sequential theory A in the same language that extends U and is ∆ * m -conservative over U .Proof.By our above observations there is a finitely axiomatized sequential theory A 0 and a formula B(x) such that U can be axiomatized as A 0 + {B(q) | q ∈ ω}.
We take A := A 0 + ∀x ∈ ℑ n B(x).We note that A is a finitely axiomatized extension of U .Consider any C ∈ ∆ * m .Suppose U C. Then, the theory In other words, we find A C. ✷ We have the following corollary.
Corollary 4.2.Suppose U is a restricted, sequential, recursively enumerable theory.Suppose further that D is a finite extension of U such that U D. Then, there is a finite extension Proof.Let m be a ρ 0 -bound on U and on D. Let A be the sentence promised in Theorem 4.1 for ∆ * m .Let Since, as is well-known, the finitely axiomatized sequential theories in the signature of U are dense w.r.t.⊣, it follows that we can add to the statement of the Corollary that U D ′ : in case the D ′ provided the theorem would happen to axiomatize U , we simply replace it by a D ′′ strictly between the original D ′ and D.
Here is one more corollary.
Corollary 4.3.Consider any finitely axiomatized, sequential theory A in signature Θ. Suppose that for some of class of Θ-sentences Ω we have a definable predicate TRUE such that, for any Ω-sentence B, we have A ⊢ B ↔ TRUE( B ).Let X be any recursively enumerable set of Ω-sentences.Then there is a finite extension A + of A + X such that A + is Ω-conservative over A + X.
Example 4.4.Let U be any recursively enumerable extension of PA.Then, there is a finite extension A of ACA 0 such that the arithmetical consequences of A are precisely the consequences of U .Similarly for the pair ZF and GB.This result was previously proven by Robert van Wesep in his paper [Wes13].
Example 4.5.By Parsons' result IΣ 1 is Π 0 2 -conservative over PRA.8 Since, over EA, we have Σ m -truth predicates.It follows that, for every m, we have an finite extension A m of PRA that is Σ m -conservative.We can easily arrange that these extensions become strictly weaker when m grows.
We refer the reader to [PV18] for a number of results in the same niche using a different methodogy.

Standardness Regained
Finiteness is Predicate Logic's nemesis.However hard Predicate Logic tries, there is no way it can pin down the set of standard numbers.What happens when we invert the question?Are there theories that interpret some basic arithmetic that do not have models in which the standard numbers are interpretable?The answer is a resounding yes.For example, PA+incon(PA) has no models in which the standard numbers are interpretable.More generally, consider any recursively enumerable consistent theory U with signature Θ. Suppose the signature of arithmetic is Ξ.Then, the theory U + {( (S 1 2 ) τ → incon τ (U )) | τ : Ξ → Θ} is consistent and does not have any models that have an internal model isomorphic to the standard numbers.9 The situation changes when we put some restriction on the complexity of the axioms of the theory.The classical work concerning this idea the beautiful paper by Kenneth McAloon [McA78].McAloon shows that arithmetical theories with axioms of restricted complexity that are consistent with PA always have a model in which the standard integers are definable.McAloon's work was further extended by Zofia Adamowicz, Andrés Cordón-Franco and Felix Lara-Martín.See [ACL16].
Our aim in this paper is to find an analogue of McAloon's Theorem that works for all sequential theories.We prove a result that is more general in scope but, at the same time, substantially weaker in its statement.We show that any consistent restricted sequential theory U has a model in which the intersection of all definable cuts is isomorphic to the standard natural numbers.This intersection is not generally itself definable in the model.We will show that the intersection of all definable cuts is a good notion that, for sequential theories, is not dependent on the original choice of the interpretation of number theory.
5.1.The Intersection of all Definable Cuts.In this subsection, we establish that the intersection of all definable cuts is a good notion.
Consider a sequential model M. Let N be a M-internal model satisfying S 1 2 .Let J M,N be the intersection of all M-definable N -cuts in M.
Now consider two M-internal models N and N ′ satisfying S 1 2 .By a result of Pavel Pudlák ([Pud85]), there is an M-definable isomorphism F between an Mdefinable cut I of N and an M-definable cut I ′ of N ′ .It is easily seen that F restricted to J M,N is an isomorphism between J M,N and J M,N ′ .
Suppose G and H are two M-definable partial functions between N and N ′ such that the restrictions of G and H to J M,N commute with zero and successor.Then it is easy to see that G and H are extensionally equal isomorphisms between J M,N and J M,N ′ .Thus, in a sense, there is a unique definable isomorphism between J M,N and J M,N ′ .
The above observations justify the notation J M for J M,N modulo isomorphism.We note that, in the definition of J M it does not matter whether we allow parameters in the definition of the cuts.Every cut with parameters has a parameterfree shortening.Suppose I is an N -cut that is given by I(x, b ).Then, defines a cut I * that is a shortening of I.
Remark 5.1.What happens if the sequence scheme S itself involves parameters?In [Vis13], it is shown that these parameters can eliminated by raising the dimension of the interpretation.Since the standard development of an interpretation of S 1 2 in a sequential theory does not involve parameters, it follows that even in a sequential theory with a sequence scheme involving parameters, there is an interpretation of 5.2.ω-models.Before proceeding, we briefly reflect on the notion of ω-model.The common practice is to say, e.g., that M is an ω-model of ZF if the von Neumann numbers of M are (order-)isomorphic to ω.Of course, there are other interpretations M of arithmetic in ZF.However, we have the feature that if the M -numbers are isomorphic to ω, then so are the von Neumann numbers -but not vice versa.If we consider GB in stead of ZF we do not know whether this feature is preserved.It is conceivable that, in some model, a definable cut of the von Neumann numbers is isomorphic to ω and the von Neumann numbers are not.
It seems to me that the proper codification of the common practice would be to say that an ω-model is not strictly a model but a pair M, M of a model and an interpretation M of a suitable arithmetic in M such that M (M) is isomorphic to ω.
Of course there is the option of existentially quantifying out the choice of the interpretation of arithmetic.Let's say that M is an e-ω-model, if for some M , M, M is an ω-model.
Finally, in the sequential case, there is a third option.We define: a sequential model M is an i-ω-model if J M is isomorphic to the standard numbers.(i stands for: intersection.)In other words, M is an i-ω-model if, for some interpretation M of S 1 2 , for every non-standard element a, there is an M-definable M -cut I such that I < a.
We have the following property of i-ω-models.
Theorem 5.8.Suppose M is a sequential i-ω-model and M : S 1 2 ✁ M. Let X be a parametrically definable class of M -numbers.Suppose ω ⊆ X. (We confuse the standard part of M with ω.) Then there is a M-definable M -cut J such that J ⊆ X.
Proof.Suppose ω ⊆ X.Consider the class Y := {a ∈ N | ∀b ≤ a b ∈ X}.In case Y is closed under successor we can shorten it to a definable cut, and we are done.In case Y is not closed under successor, there is an a 0 such that ∀b ≤ a 0 b ∈ X but Sa 0 ∈ X.By our assumption ω < a 0 .Hence there must be a definable cut I with ω ≤ I < a 0 .So, I ⊆ X. ✷ 5.3.The Main Result.If we are content with the countable case, our main result is a simple application of the Omitting Types Theorem.We first give this easier proof.
Theorem 5.9.Let U be a consistent restricted sequential theory.Here U may be of any complexity.We allow countably many constants in U .Then, U has a model M in which J M is isomorphic to the standard natural numbers.
Proof.We fix a sequence scheme S for U .We work with the interpretation N of S 1 2 provided by this scheme.Suppose that in all countable U -models the type is realized.Then, by the Omitting Types Theorem, there is a formula A(x), such that, for a fresh constant c, we have (i) U + A(c) is consistent and (ii) U + A(c) ⊢ c = n, for each n ∈ ω, and (iii) U + A(c) ⊢ c ∈ ℑ n , for each n ∈ ω.We apply Theorem 3.4 to (i) and (ii) obtaining that, for some n * , the theory U + A(c) + ∀x ∈ ℑ n * c = x is consistent.However, this directly contradicts (i) and (iii).
We may conclude that there is a countable model M in which T (x) is omitted.Clearly, this tells us that J M is isomorphic to the standard numbers.✷ We proceed to prove the stronger version of our theorem where the restriction to countability is lifted.We first prove a Lemma.
Lemma 5.10.Let M be any sequential model of signature Θ with domain M .Let S be a sequence scheme for M. As usual, N is the interpretation of S 1 2 given by the sequence scheme.
Let k be any number.We note that the ρ 0 -complexity of the axioms of AS + is a fixed number, say s.So the sequentiality of M is witnessed by the satisfaction of a sentence D of complexity below s+ρ 0 (S).Let n := max(k, s+ρ 0 (S), 1+c 10 , ρ 0 (S)+ c 10 ).
Then, M has a sequential ∆ * k -elementary extension K with sequence scheme S such that ℑ K n ∩ M = ω.Proof.Without loss of generality we may assume that k ≥ s + ρ 0 (S), so that is consistent (for n as given in the statement of the Lemma).If not, then for some nonstandard m 0 , . . ., m ℓ−1 in M , we have Let m be the minimum of the m i .We find Γ ⊢ m ∈ ℑ n .On the other hand, the theory Γ + 0 < m, 1 < m, . . . is consistent.Hence, by Theorem 3.4, we have that Γ + ℑ n < m is consistent.A contradiction.
Let K be a model of Γ * .Clearly, in K, we have that ℑ n is below all non-standard elements inherited from M (but, of course, not necessarily below new non-standard elements).Also K is, by construction, a ∆ * n -elementary extension.Finally, since we have chosen n ≥ s + ρ 0 (S), the model K is again sequential with the same sequence scheme. ✷ With the Lemma in hand, we can now prove the promised theorem using a limit construction.
Theorem 5.11.Let M be any sequential model.Then, for any k, M has a ∆ * kelementary extension K in which J K is (isomorphic to) ω.
Proof.Let S be a sequence scheme for M. We work with the numbers N provided by this scheme.Let s as in Lemma 5.10.We take: • n 0 := max(k, s + ρ 0 (S)).
• n j+1 := max(ρ 0 (ℑ j ) + 1, s + ρ 0 (S), 1 + c 10 , ρ 0 (S) + c 10 ).(We note that ρ 0 (ℑ j ) ≈ c 0 j + c 9 and that for j > 0, we have n j+1 := ρ 0 (ℑ j ) + 1.) We construct a chain of models M i .Let M 0 = M. Suppose we have constructed M j .We now take as M j+1 a model that is a ∆ * nj+1 -elementary extension of M j such that ℑ Mj+1 nj ∩ M j = ω.Let K be the limit of (M i ) i∈ω .Consider any non-standard element a in N (K).We have to show that there is a K-definable cut below it.Suppose a occurs in M j .We have, by the construction of our sequence, that ( †) ℑ Mj+1 nj < a.By the fact that all M s , with s > j + 1 are ∆ * nj+1 -elementary extensions of M j+1 , it follows that ( †) is preserved to the limit: ℑ K nj +1 < a. ✷ From Theorem 5.11, we have immediately the desired strengthening of Theorem 5.9.
Theorem 5.12.Let U be a consistent restricted sequential theory.Here U may be of any complexity.We allow a number of constants in U of any cardinality.Then U has a model M in which J M is isomorphic to the standard natural numbers.
Remark 5.13.From Theorem 5.12, we retrace our steps and derive a less explicit form of Theorem 3.4.Let U be a restricted sequential theory and consider any C(x).Suppose Let M be a model of V in which J M is isomorphic to the standard natural numbers.By Theorem 5.8, there is a definable M-cut J so that, in M, we have ∀x ∈ J C(x).Now J is a definable cut in M, but it need not automatically be a cut in U .There is a standard trick to remedy that.We define J • := J cut(J) N .Clearly, J • is a definable cut in U .Moreover, in the context of M, the cuts J and J • coincide.By the above considerations, it follows that U + ∀x ∈ J • Cx is consistent, where J • is a U -definable cut.
Remark 5.14.Consider any model M. We define DEF(M) as the class of (parametrically) definable classes of M. We define DEF − (M) as the class of classes over M that are definable without parameters.Also, DEF n (M) is the class (parametrically) definable n-ary relations and similarly for the parameter-free case.
It would seem that Theorem 5.11 gives us information about possible sequential models of the form M, DEF(M) , since J M is definable in M, DEF(M) .However, this is not so, since we have a much stronger result for the models M, DEF(M) , where M is sequential.
We assume that M has finite signature, where we may allow an infinity of constants.In each model M, DEF(M) , where M satisfies these demands, the natural numbers are definable.The argument is simple.Let comm x (X) mean that X satisfies the commutation conditions for satisfaction of ∆ * x -formulas in J M .Consider, in M, DEF(M) the class Y := {x ∈ N | ∃X comm x (X)}.Clearly, each standard x is in Y .If a non-standard number b would be in Y , the defining formula for the witnessing X would violate Tarski's Theorem of the undefinability of truth for M. 10If the sequence scheme for M is parameter-free, then the same argument works for M, DEF − (M) .If the sequence scheme contains parameters, we can make the argument work for M, DEF − n (M) , for sufficiently large n.Finally, note that if M is a non-standard model of Peano Arithmetic, then J M is simply isomorphic to M itself.Thus, adding J M (viewed as intersection of all cuts on the identical interpretation of S 1 2 ) to M does not increase the expressiveness of the language.This consideration shows that adding the definable sets can be more expressive than adding J M (as intersection of the cuts for a given interpretation of S 1 2 ).

Reflection
If we apply Theorem 3.3 to a formula of a special form, we get a reflection principle.Theorem 6.1.Consider any consistent, restricted, sequential theory U with sequence scheme S. Let N be the interpretation of the numbers provided by S. Let m 0 be the bound for U and let m 1 be any number.Let n := max(m 0 , m 1 + ρ 0 (S) + c 10 ).
Then, for every Σ 2 -sentence C of the form C = ∃x C 0 (x), where C 0 is Π 1 and ρ 0 (C) ≤ m 1 , we have: if U ⊢ ∃x ∈ ℑ n C N 0 (x), then C is true.Proof.Under the assumptions of the theorem, we suppose U ⊢ ∃x ∈ ℑ n C N 0 (x).By Theorem 3.3, there is a k such that U ⊢ q≤k C N 0 (q).Suppose C is false.Then, for each q ≤ k, we have ¬ C 0 (q).Hence, by Σ 1 -completeness, for each q ≤ k, we have U ⊢ ¬ C N 0 (q).It follows that U ⊢ ⊥.Quod non.✷ We note that the above proof uses Σ 0 1 -collection in the metalanguage.If we take the formulas still simpler we can improve the above result.We fix a logarithmic cut S 1 2 -cut J.There is a Σ 1 -truth predicate, say True, for Σ 1 -sentences, such that, for any Σ 1 -sentence S, we have S 1 2 ⊢ True(S) → S and S 1 2 ⊢ S J → True(S).(See e.g.[HP93], Part V, Chapter 5b for details.) Theorem 6.2.Consider any consistent, restricted, sequential theory U with sequence scheme S and bound m.Let n := max(m, ρ 0 (True(x)) + ρ 0 (S) + c 10 + 1).For all Σ 1 -sentences S, we have: if U ⊢ S Jℑn , then S is true.
Proof.We have: The second step is by theorem 6.1.✷

Degrees of Interpretability
In this section, we apply our results to study the joint degree structure of local and global interpretability for recursively enumerable sequential theories.The main result of this section is a characterization of finite axiomatizability in terms of the double degree structure.We will study the degree structures as partial preorderings.
7.1.The Basic Idea.The degree structures we are interested in are the degrees of global interpretability of recursively enumerable sequential theories Glob seq and the degrees of local interpretability of recursively enumerable sequential theories Loc seq .It is well known that both structures are distributive lattices.We have the obvious projection functor π from Glob seq onto Loc seq .
Let Fin be the property of global degrees of containing a finitely axiomatized theory.What we want to show is that, if we start with the pair Glob seq and Loc seq and with the projection π, then we can define Fin (using a first-order formula).
The basic idea is simple.Zoom in on a local degree of U .This degree contains a distributive lattice, say, L of global degrees.The lattice L has a maximum, to wit ✵ U .Does it have a minimum?Well, if there is a finitely axiomatized theory U 0 in L, then its global degree will automatically be the minimum.We will see that (i) not in all cases a minimum degree of L exists and (ii) if such a minimum exists it contains a finitely axiomatizable theory.In other words, the mapping φ with: is partial.However, if it has a value, this value contains a finitely axiomatized theory.So, we can define: fin(U ) iff, for all V , we have U ✁ glob V iff π(U ) ✁ loc π(V ).
Open Question 7.1.Can we define fin in Glob seq alone?Remark 7.2.In the context of local degrees of arbitrary theories with arbitrarily large signatures, Mycielski, Pudlák and Stern charactarize loc-finite as the same as compact in terms of the ✁ loc -ordering.This will not work in our context of global interpretability and recursively enumerable sequential theories.We briefly give the argument that, in our context, every non-minimal globally finite degree is non-compact.
Let A be any ✁-non-minimal, finitely axiomatized, sequential theory.Let B i be a enumeration of all finitely axiomatized sequential theories.Let C 0 := S 1 2 , We note that, by our assumption, C 0 ✁ = A. Let C n+1 := B n if C n ✁ = B n ✁ = A and C n+1 := C n otherwise.Clearly, A✄C i , for all i.Consider any sequential recursively enumerable theory U such that U ✄ C i , for all i.Without loss of generality, we may assume that the signatures of U and A are disjoint.We easily see that the theory U * axiomatized by {(D ∨ A) | D is an axiom of U } is the infimum in the degrees of global interpretability of U and A. So C i ✁ U * ✁ A. Suppose U * ✁ = A. In this case, by [Vis17, Theorem 5.3], we can find a finitely axiomatized sequential B such that U * ✁ = B ✁ = A. let B = B j .Since C j ✁ U * ✁ = B ✁ = A, we will have C j+1 := B j .A contradiction.It follows that U * ≡ A. We may conclude that A ✁ U .So, A is the supremum of the C i .Clearly, A cannot be the supremum of a finite number of the C i .Thus, A is not compact.7.2.Preliminaries.We consider the recursively enumerable theory U .By Craig's Theorem, we can give U a Σ b 1 -definable axiomatization X.Let this axiomatization is given by a Σ b 1 -formula η.We define U ↾n as the theory axiomatized by the axioms of U , as given by η that are ≤ n.
One can show that ✵ is the right adjoint of π: See e.g.[Vis11a] or [Vis17].
A theory is glob-finite iff it is mutually globally interpretable with a finitely axiomatized theory.A theory is loc-finite iff it is mutually locally interpretable with a finitely axiomatized theory.If we enrich Glob seq with a predicate Fin for the globally finite degrees, we have a first-order definition of Loc seq over this structure as follows: Here is a first basic insight.
Theorem 7.3.The theory U is a-finite, for a ∈ {glob, loc}, iff U ≡ a U ↾n, for some n.
Proof.Suppose U ≡ a V , where V is finitely axiomatized.Clearly, U ↾n ✄ V , for some n.We have: We show that any global, recursively enumerable, sequential degree contains an element that is not finitely axiomatizable.
Theorem 7.5.Consider any consistent, sequential, recursively enumerable theory U .Then there is a U • ≡ U such that U • is sequential and recursively enumerable and not finitely axiomatizable.
Proof.Consider a consistent, sequential and recursively enumerable theory U .In case U is not finitely axiomatizable, we are done, taking U • := U .Suppose U is finitely axiomatizable, say by a single sentence A. Par abus de langage, we write A also for the theory axiomatized by x = A .By Theorem 6.2, we can find M : S 1 2 ✁ A for which A is Σ 0 1 -sound.Consider: U • := A + {(con(A) → con n+1 (A)) M | n ∈ ω}.
We have, by Feferman's version of the Second Incompleteness Theorem: So U • ≡ A. Suppose U • were finitely axiomatizable.Then, we would have, for some n > 0, A + (con(A) → con n (A)) M ⊢ (con(A) → con n+1 (A)) M .

Hence A ⊢ (✷ n+1
A ⊥ → ✷ n A ⊥) M .So, by Löb's Theorem, A ⊢ (✷ n A ⊥) M , contradicting the Σ 0 1 -soundness of A w.r.t.M .✷ Here is a sufficient condition for failure to be loc-finite.We define: Here U ↾ n is defined with respect to a chosen Σ b 1 -formula η that represents the axiom set of U .We call U strongly loc-reflexive if U ✄ ✵ + U .We note that e.g.PRA is an example of a strongly loc-reflexive theory.
Theorem 7.6.Suppose that U is strongly loc-reflexive.Then, U is not loc-finite.Open Question 7.10.Every element of Loc seq contains an extension of S 1 2 (to wit an element of the form ✵ U ).Does every element of Glob seq contain an extension of S 1 2 ?

Proof.
Closure under 0 and downwards closure are trivial.We prove closure under successor by induction on n.The case of J • 0 is trivial.Suppose that J • n is closed under successor.Consider x in J • n+1 .Let C and D be in Γ x .Let A be of the form (C ∧ D).Suppose w is free for v in A. Let B be (of the form) A[v := w].Suppose further that α and β assign the same values to the free variables of A except v and β[w] = α[v].Clearly B is of the form E ∧ F , where E is of the form C[v := w] and F is of the form D[v := w].We have: in the left-to-right direction of the second step we can take δ of the form β[v : γ(v)].We can use the fact that C has ν-complexity x and x ∈ J • n+1 .The property Q n+1 is applied with v in the role of both v and w.Next we have the case that z and w are different variables.Let D be of the form C[v : w].So B is of the form ∃z D. We have: α |= + n+1 ∃z C ↔ ∃γ (α [[z]] γ and γ |= + n+1 C) ↔ ∃δ (β [[z]] δ and δ |= + n+1 D) ↔ β |= + n+1 ∃z D E.g., in the left-to-right direction of the second step we can again take δ of the form β[z : γ(z)].
and we are done.✷loc-finite glob-finite fin.axiom.Figure 1. Separating Sequential Local and Global Finiteness 7.3.Some Examples.Before formulating and proving our main result, we briefly pause to provide a few examples.Example 7.4.The theories I∆ 0 and S 2 = I∆ 0 + Ω 1 are examples of theories of which the finite axiomatizability is an open problem, but which are, by an argument of Alex Wilkie, glob-finite.They are, for example, mutually interpretable with the finitely axiomatized sequential theory AS and with the finitely axiomatized sequential theory PA − .