Hierarchical incompleteness results for arithmetically definable extensions of fragments of arithmetic

There has been a recent interest in hierarchical generalisations of classic incompleteness results. This paper provides evidence that such generalisations are readibly obtainable from suitably hierarchical versions of the principles used in the original proof. By collecting such principles, we prove hierarchical versions of Mostowski's theorem on independent formulae, Kripke's theorem on flexible formulae, and a number of further generalisations thereof. As a corollary, we obtain the expected result that the formula expressing"$T$ is $\Sigma_n$-ill"is a canonical example of a $\Sigma_{n+1}$ formula that is $\Pi_{n+1}$-conservative over $T$.

In this paper I argue that such hierarchical generalisations can often be obtained from the original proofs by replacing certain principles used in the proofs by appropriately formulated hierarchical versions, while the essence of the arguments remains the same. The hierarchical principles, once appropriately formulated, are in turn often provable by appropriate generalisations of the core concepts employed in the proofs of the ordinary ones, but even so, there is no single source to which to turn for them. Both Smoryński (1985) and Beklemishev (2005) give good partial accounts of the syntactical side, and Poizat (2000) gives a hierarchical perspective on the basic model theory of arithmetic, including model-theoretic proofs of hierarchical versions of Gödel's first and second incompleteness theorems. Still, I find certain aspects lacking. With this in mind, one aim of this paper is to collect a number of principles that may be useful to the reader who herself wishes to prove hierarchical incompleteness results without having to reinvent the wheel.
These principles are then put to use to prove a number of general incompleteness results for arithmetically definable extensions of fragments of PA. The goal is not to prove the sharpest or most general results (in fact some of the results follow from each other), but rather to exemplify how the hierarchical principles enter into more or less well-known proof methods. Even so, the results presented here improve on some results of Chao & Seraji (2018), Kikuchi & Kurahashi (2017), and Salehi & Seraji (2017), and sharpen some of Blanck (2017), Hamkins (2018), Lindström (1984), and Woodin (2011). These sharpenings are in terms of gauging the amount of induction needed for the proofs, bringing the (in this particular sub-field largely ignored) fragments-of-arithmetic perspective to attention.
In order to state the results in a more general form, I have chosen to consider only extensions of I∆ 0 +exp, although under the somewhat unusual name IΣ 0 +exp. This allows for formulating results for extensions of, e.g., IΣ n + exp, ensuring that partial satisfaction predicates are well behaved even for n = 0. While it is sometimes possible to push the background theory below IΣ 0 +exp, I have refrained from doing so, to instead focus on the more general hierarchical picture.

Notation and conventions
The expressions ∃x ≤ tφ(x) and ∀x ≤ tφ(x) are used as shorthand for ∃x(x ≤ t ∧ φ(x)) and ∀x(x ≤ t → φ(x)), where t is some term in the language of arithmetic. The initial quantifiers of these formulae are bounded and a formula containing only bounded quantifiers is a bounded formula. Let ∆ 0 = Σ 0 = Π 0 be the class of bounded formulae.
The arithmetical hierarchy is defined as follows. A formula is Σ n+1 iff it is of the form ∃x 1 . . . x m π(x 1 , . . . , x m ) where π is a Π n formula (that may contain other variables than x 1 , . . . , x m ). Similarly, a formula is Π n+1 iff it is of the form ∀x 1 . . . x m σ(x 1 , . . . , x m ) where σ is a Σ n formula. ∆ n (M) (∆ n (T)) is the set of Σ n formulae that are equivalent to Π n formulae in a given model M (theory T). Throughout the paper Γ denotes either Σ n+1 or Π n+1 , and we always assume only that n ≥ 0.
Theories are understood as sets of sentences, thought of as the set of nonlogical axioms of the theory. IΣ n is the theory obtained by adding induction for Σ n formulae to Robinson's arithmetic Q, while IΣ 0 +exp is Q plus Σ 0 -induction plus an axiom stating that the exponentiation function is total. We assume that all theories denoted T, etc., are consistent, arithmetically definable, extensions of IΣ 0 + exp.
T is Γ-sound iff for all Γ sentences γ, if T ⊢ γ, then N |= γ. The converse implication is sometimes known as Γ-completeness: hence T is Γ-complete iff for all Γ sentences γ, if N |= γ, then T ⊢ γ. S is Γ-conservative over T iff for all Γ sentences γ, if S ⊢ γ, then T ⊢ γ.
We rely on a coding of finite sets and sequences in IΣ 0 + exp, as developed by Hájek & Pudlák (1993, Chapter I.1). The set Σ 0 (X) is obtained by adding atomic formulae of the form t ∈ X (where t is any term) and closing under propositional connectives and bounded quantifiers. The set Σ 1 (X) is obtained from Σ 0 (X) in the usual manner.

Preliminary principles
This section presents a number of suitably formulated principles that are useful in proving hierarchical incompleteness results of the kind given in Theorem 1.1. The basic versions of these principles can be found scattered across the literature, and none of them should be too surprising to the reader familiar with, e.g., Hájek & Pudlák (1993), Kaye (1991), Lindström (2003), and Smoryński (1985).
The essence of the generalisations presented in this paper is that r.e. theories are replaced by Σ n+1 -definable ones, and that the base theory is pushed down as far as it will go below PA. On some occasions the theories have to satisfy additional constraints such as Σ n -soundness or Π n -completeness for the generalisation to go through, and whenever this is the case that will be pointed out explicitly.
3.1. Basics, representability, recursion theory. Many textbooks in metamathematics rely on some version of the fact that Robinson's arithmetic Q is Σ 1complete: it proves all true Σ 1 sentences. This has the consequence that every r.e. set can be numerated in Q by a Σ 1 formula, which in turn allows for representing the theorem set of Q in Q and proving the first incompleteness theorem. Many proofs of these facts rely on ∆ 0 being closed under bounded quantification, and Q deciding all ∆ 0 sentences.
Since the aim of this paper is to generalise incompleteness results to Σ n+1definable theories rather than r.e. ones, there is a need for establishing a similar correspondence between Σ n+1 -definable sets and theories sufficient to represent them. On these higher levels, the roles of ∆ 0 and Q are played by the classes Σ 0 (Σ n ) and the theories IΣ n + Th Πn (N), respectively. Establishing this relationship is the goal of this subsection.
While the usual formulation of Gödel's theorem pertains to r.e., consistent theories, the hierarchically formulated Theorem 1.1 is stated for Σ n+1 -definable, Σ nsound theories. In light of the preceding fact, consistency and Σ 0 -soundness are equivalent, so the ordinary statement fits nicely into the hierarchical statement of the theorem. However, inspection of the published proofs of Theorem 1.1 reveals that Π n -complete theories enter the argument in an indispensable way. In the proof by Kikuchi & Kurahashi (2017), Π n -completeness of T can be explicitly assumed, since in the case where T is not Π n -complete, it fails to prove all true Π n sentences and can hardly prove all true Π n+1 sentences. By contrast, the proof by Salehi & Seraji (2017) bypasses assuming Π n -completeness of T by instead constructing a Π n+1 sentence that is independent of the Π n -complete T + Th Πn (N), a theory whose consistency is guaranteed by the Σ n -soundness of T. That sentence is, a fortiori, also independent of T.
The most conservative generalisation of "r.e., consistent" is therefore "Σ n+1definable, Π n -complete, and consistent" rather than "Σ n+1 -definable, Σ n -sound". As is clear from the Theorem 1.1, the assumption of Π n -completeness of T is sometimes excessively strong, and mere Σ n -soundness is indeed enough for some further applications as well. In those cases, the proofs rely on the consistency of T + Th Πn (N), as in the proof by Salehi & Seraji (2017). In other cases, however, the Π n -completeness is indispensable for the generalisation to go through. 2 The analogy with regards to soundness and completeness therefore goes as follows. Since Q is Π 0 -complete, it is also Σ 1 -complete, and since it is also consistent it follows that Q is Σ 0 -sound and therefore also Π 1 -sound. Now consider Th Πn (N): This theory is Π n -complete and therefore also Σ n+1 -complete; it is consistent and therefore Π n+1 -sound.
The role of IΣ n in the analogy between Q and IΣ n + Th Πn (N) becomes clear through the next fact. Its proof will take up most of the remainder of this subsection. (1) Every Σ n -(or Π n -) definable relation on ω is binumerated by a Σ n (or Π n ) formula in Q + Th Πn (N).
(2) Every Σ n+1 -definable relation on ω is numerated by a Σ n+1 formula in IΣ n + Th Πn (N). (3) Every function from ω k to ω that is recursive in ∅ (n) is strongly representable by a Σ n+1 formula (with particularly nice properties) in IΣ n + exp + Th Πn (N). 3 The first item is immediately seen to be true, since Q + Th Πn (N) is consistent, Σ n+1 -complete, and Π n+1 -sound. The second and third items are elaborated on below, but first we need a few more stepping stones to help out in the constructions. (1) For every Γ formula γ(x, y), we can effectively find a Γ formula ξ(x) such that Q ⊢ ∀x(ξ(x) ↔ γ(x, ξ )).
(2) For every Γ formula γ(x, y), we can effectively find a Γ formula ξ(x) such that, for each k, Bibliographical remark. Item 1. of the above is essentially due to Montague (1962, Lemma 1). The proof by Ehrenfeucht & Feferman (1960, Lemma 1) of 2. actually suffices to prove 1.; see also Smoryński (1981) for a discussion of the development of the diagonal lemma.
2 One of the referees pointed out that even Πn-completeness is sometimes not enough for a straightforward hierarchical generalisation to hold: see, e.g., Theorem 11 of Kurahashi (2018) for an example. 3 As is well known, IΣ 1 proves the totality of the exponential function, so the additional axiom exp is only required in the case n = 0 to make sure that the partial satisfaction predicates used to establish the particularly nice properties are well-behaved. A similar remark applies also to many of the following facts, and to the statements of some of the theorems in Section 4.
Convention. In light of Craig's trick, we may assume that any Σ n+1 -definable theory is in fact Π n -defined. In what follows, we write Prf T (x, y) to denote (ambiguously) any formula Prf τ (x, y) where τ is a Π n binumeration of T in IΣ n +Th Πn (N), and moreover, this formula can be assumed to be Π n . Consequently, Pr T (x) is Σ n+1 and Con T is Π n+1 in IΣ n .
(1) In IΣ n , both Σ n and Π n are closed under bounded quantifiers. 5 (2) IΣ n ⊢ IΣ 0 (Σ n ). ( All the pieces are now in place to prove a, sometimes useful, lemma from which Fact 3.2.2 follows immediately. The proof highlights the steps where induction and the additional truth from the standard model is required. For the statement of the lemma, recall that a relation X is correctly numerated by φ in T if φ numerates X in T, and for all k 1 , . . . , k n , T ⊢ φ(k 1 , . . . , k n ) iff φ(k 1 , . . . , k n ) is true.
Recall that if T is Σ n+1 -definable, then there is a deductively equivalent Π n definition of T, and Prf T (x, y) is therefore equivalent to a Π n formula in IΣ n . Then is a Σ 0 (Σ n ) formula, and is therefore, by Fact 3.5, equivalent to a ∆ n+1 formula in IΣ n . It follows that φ is equivalent to a Σ n+1 formula in IΣ n .
for all p. It follows that Suppose, for a contradiction, that ¬R(k 1 , . . . , k m ). Then for all i, It follows that IΣ n + Th Πn (N) ⊢ ¬∃z < pπ(k 1 , . . . , k m , z). We get For the other direction, suppose T ⊢ φ(k 1 , . . . , k m ), and let p be the least such proof. Then Suppose further that there is no i < p such that π(k 1 , . . . , k m , i). Then, as before, T ⊢ ¬φ(k 1 , . . . , k m ), a contradiction. Thus there is an i < p such that is true, and therefore φ(k 1 , . . . , k m ) is true, as desired.
Bibliographical remark. The correct representability of Σ 1 relations in Q is due to Ehrenfeucht & Feferman (1960). The proof above mimics Lindström's (2003) version of Shepherdson's (1961) proof of the same result.
Fact 3.2.2 follows directly from Lemma 3.6, and it only remains to give an argument for Fact 3.2.3. For this we need two more facts, the first being a version of Post's theorem.
Fact 3.8 (The selection theorem). For each Σ n+1 formula φ with exactly the variables x 1 , . . . , x k free, there is a Σ n+1 formula Sel{φ} with exactly the same free variables, such that: ( Proof sketch. We may assume that any given Σ n+1 formula φ(x 1 , . . . , x k ) is on the form ∃wπ(x 1 , . . . , x k , w) with π ∈ Π n . Let (w) i denote the ith element of the ordered pair coded by w, and define Sel{φ}(x 1 , . . . , x k ) to be the formula By Fact 3.5, this formula is equivalent in IΣ n to a Σ n+1 formula. Using Σ ninduction, it is easy to check that it has the three desired properties.
Bibliographical remark. Smoryński (1985, Theorem 0.6.9) provides a proof of the selection theorem for n = 0, and the generalisation is straightforward. The numeration below of partial recursive functions is also based on the treatment in Section 0 of his book.
Convention. The partial n-recursive function with index e, ϕ n e , can now be defined to be the function whose graph is defined by Sel{Sat Σn+1 }(e, y 1 , . . . , y k , z) in N. The resulting enumeration is acceptable in the sense of Rogers (1967). Whenever convenient, ϕ n e (m 1 , . . . , m i ) = k is used as a shorthand for Sel{Sat Σn+1 }(e, m 1 , . . . , m i , k). To wrap up this subsection, we sketch a proof of a formalised version of the second recursion theorem.
Fact 3.9 (Formalised second recursion theorem). Let f : ω 2 → ω be n-recursive. There is an e ∈ ω such that formula strongly representing f in IΣ n + exp + Th Πn (N). Let, by Fact 3.3, γ(y, z) be a formula such that Then e = γ is as desired.
The recursion theorem is usually deployed in the following manner. Define an n-recursive function f (z, x) in stages, using z as a parameter; the resulting function may differ depending on the choice of z. By the recursion theorem, there is then an n-index e such that ϕ n e (x) coincides with f (e, x). This legitimates self-referential constructions where an index of f is being used in the construction of f itself.
3.2. Strong provability predicates. A central piece in the proof of Gödel's incompleteness theorem for r.e. theories T is the use of formal provability predicates Pr T (x), expressing "x is provable in T". In the current setting, with Σ n+1 -definable theories in focus, the corresponding strong provability predicates Pr T,Σn+1 (x) express "x is provable in T from true Σ n+1 sentences". 6 To define these predicates, we first need to introduce partial satisfaction predicates.
6 Similar provability predicates, Pr T,Πn (x) appear in the literature under the name strong provability, n-provability, and oracle provability (Ignatiev, 1993;Beklemishev, 2005;Visser, 2015;Kolmakov & Beklemishev, 2019). The difference is usually only a matter of taste, since the theories T + Th Πn (N) and T + Th Σ n+1 (N) are deductively equivalent, and under reasonable assumptions this is reflected also in the formalised notions Pr T,Πn (x) and Pr T,Σ n+1 (x). However, the relationship between the related notions Pr k T,Πn (x) and Pr k T,Σ n+1 (x) (introduced below) is not as immediate, since there k bounds the length of the proof of x and the Gödel number of the additional true Πn or Σ n+1 sentence used in the proof. Even though every Σ n+1 -provable Fact 3.10 (Partial satisfaction predicates). For each k and Γ, there is a k + 1-ary Γ formula Sat Γ (x, x 1 , . . . , x k ) such that for every Γ formula φ(x 1 , . . . , x k ), Hence there is also a Γ formula Tr Γ (x) such that for every Γ sentence φ, Bibliographical remark. Modern proofs of this Fact are due to Kaye (1991) and Hájek & Pudlák (1993). The use of partial satisfaction predicates, however, goes back to Hilbert & Bernays (1939).
Fact 3.13 (Löb conditions). If T is a Σ n+1 -definable extension of IΣ 0 + exp, then, for each m ≥ 0, and for all sentences φ, ψ, Similar statements also hold for formulae: The stronger background theory used in items L1. and L1'. is enough to ensure the numerability of the Σ n+1 -definable theory T. For r.e. theories T, IΣ 0 + exp suffices.
We now turn our attention to the bounded provability predicates that feature prominently in the sequel. Consider again the Σ n+1 formula Pr T,Σn+1 (x) for a Σ n+1 -definable T. We may assume that there is a Π n formula π(x, y) such that IΣ n ⊢ Pr T,Σn+1 (x) ↔ ∃yπ(x, y).
sentence has a Πn-proof, this proof might be much longer than the original one. We opt for the Σ n+1 versions, since it helps in the proof Fact 3.22 below, and since it makes (some of) the indices align nicely.

Let Pr k
T,Σn+1 (x) be the formula ∃z ≤ kπ(x, z). 7 Then, by Fact 3.5, this formula is Π n in IΣ n , and therefore decidable in IΣ n + exp + Th Πn (N). 8 As a consequence, we have strong reflection properties for the bounded proof predicates.
Fact 3.15 (Formalised small reflection). Let T be a Σ n+1 -definable, consistent extension of IΣ n+1 . Then we have: Bibliographical remark. Verbrugge & Visser (1994) show how the small reflection principle can be formalised in (theories weaker than) IΣ 0 + exp. I am grateful to one of the refeees for pointing out a crucial error in an earlier statement of this Fact.

3.3.
Model theory of arithmetic. The remainder of this section concerns the model theory of arithmetic, building up to a characterisation of Π n -conservativity in the spirit of Orey, Hájek, Guaspari, and Lindström. A first step towards that goal is the following miniaturisation of the arithmetised completeness theorem in the style of McAloon (1978, Theorems 1.7 and 2.2).
Fact 3.16 (The arithmetised completeness theorem). Fix m ≤ n. If M |= IΣ n+1 , and T is a theory that is Σ n+1 -definable in M such that M |= Con T,Πm , then there is a Σ m -elementary end-extension of M satisfying T.
Here, a tree is a set of finite binary sequences that is closed under taking initial segments. A branch through a tree is a subtree that is linearly ordered under the relation "being an initial segment of". See Hájek & Pudlák (1993, Chapter I.3(b)) for more details. A definition of the class LL n+1 used in the statement of the low basis theorem can be found in Hájek & Pudlák (1993, Chapter I.2(d)). The only properties of LL n+1 sets that are used in the proof of the arithmetised completeness theorem is that IΣ n+1 proves induction for Σ 1 (LL n+1 ) sets, and that every set recursive in an LL n+1 set is itself LL n+1 (Hájek & Pudlák, 1993, I.2.78-79).
For the proof of the arithmetised completeness theorem, we also rely on IΣ n+1 being able to define formalised versions of syntactic and semantic notions such 7 In the notation of, e.g., Lindström & Shavrukov (2008), Pr k T,Σ n+1 (x) would be written k : Pr T,Σ n+1 (x). 8 The additional axiom exp is again only required for n = 0, to handle the partial truth definition occuring in Pr T,Σ n+1 . Remarks of this type will hereafter be omitted.
as "formula", "term", "theory", "satisfaction", and "model", so that the relevant constructions can be carried out within IΣ n+1 itself. The reader is again referred to Hájek & Pudlák (1993), especially Chapter I.4, for a detailed development of these concepts. The generalisation to Σ n+1 -definable theories is straightforward, and fits safely within IΣ n+1 .
Proof of the arithmetised completeness theorem. Fix m ≤ n. Let M |= IΣ n+1 and let T be a theory Σ n+1 -definable in M such that M |= Con T,Πm . Reason in M: Since T is Σ n+1 and we have IΣ n+1 , we may assume T to be Π n -defined and Henkinised. Let ψ 0 , ψ 1 , . . . be an enumeration of all sentences. Define a dyadic tree T by s ∈ T iff there is no p ≤ s such that p is a proof of contradiction in T from the true Π m sentences plus {ψ (s)i i : i < l(s)}. The proof relation for T + Th Πm (M) is Π n in M, so the tree T is at most ∆ n+1 in M. Then IΣ n+1 suffices to show that T is unbounded, so by Fact 3.17 there is an ThenT is recursive in the LL n+1 branch B, and is therefore LL n+1 itself. A term model K satisfying T + Th Πm (M) can then be read offT in the usual way. Finally, note that K is recursive inT and therefore LL n+1 . Using induction for Σ 1 (LL n+1 ) (provided by IΣ n+1 ), we can now define an LL n+1 embedding f of M onto an initial segment of K by letting f (0) = 0 K and f (x+1) = the K-successor of f (x).
The next few facts are used in the proof of the generalised Orey-Hájek characterisation.
Bibliographical remark. This refinement of Friedman's (1973) embedding theorem for n = 0 is due to Ressayre (1987 We are now ready to prove the final fact: an excerpt of a generalisation of the Orey-Hájek-Guaspari-Lindström characterisation of interpretability. For extensions of PA, the equivalence of 1. and 3. is due to Guaspari (1979, Theorem 6.5(1)). The equivalence of 1. and 2. for finitely axiomatisable theories seems to have been known to experts for some time, while the equivalence of 2. and 3. for r.e. extensions of fragments of PA is stated without proof by Blanck & Enayat (2017, Theorem 2.11). With the previous facts of this section in place, the generalisation to Σ n+1 -definable theories presents no further difficulties.
Fact 3.22 (OHGL characterisation). Let S and T be Σ n+1 -definable, consistent extensions of IΣ n+1 , and suppose that S is also Π n -complete. The following are equivalent: (1) S is Π n+1 -conservative over T; (2) for all k ∈ ω, T ⊢ Con k S,Σn+1 ; (3) every countable model M of T with S ∈ SSy(M) has a Σ n -elementary extension to a model of S.
Proof. 1. ⇒ 2. Suppose that S is Π n+1 -conservative over T. By Fact 3.14 and Π n -completeness of S, S ⊢ Con k S,Σn+1 for all k ∈ ω. But Con k S,Σn+1 is at most Π n+1 , so T ⊢ Con k S,Σn+1 for all k ∈ ω. 2. ⇒ 3. Suppose T ⊢ Con k S,Σn+1 for all k ∈ ω. Let M be a countable model of T and suppose that S ∈ SSy(M). Since T extends IΣ n+1 , it follows that Con k

S,Σn+1
is at most Π n+1 , and we can use overspill to get M |= Con c S,Σn+1 for some nonstandard c ∈ M . By Fact 3.20, there is a submodel M 0 |= PA of M, all of whose elements are below c, and there is some non-standard a below c that codes Th Σn+1 (M). This ensures that M 0 |= Con S+{m:mεa} , so Fact 3.16 guarantees the existence of an end-extension K of M 0 , satisfying S + {m : mεa} and therefore also S + Th Σn+1 (M).
At this point, the situation is that SSy(M) = SSy(M 0 ) = SSy(K), M and K are countable, and Th Σn+1 (M) ⊆ Th Σn+1 (K). Then Fact 3.21 ensures that M can be embedded as a Σ n -elementary initial segment of K.
3. ⇒ 1. Prove the contrapositive statement by assuming that S is not Π n+1conservative over T. Then there is a Π n+1 sentence π such that S ⊢ π but T + ¬π is consistent. Let M |= T + ¬π . If K |= S were a Σ n -elementary extension of M, then K would satisfy both π and ¬π, a contradiction.

Applications
The goal of this section is to prove a handful of hierarchical incompleteness results, using the tools we reviewed in the previous one. The first such result stems from Mostowski (1961, Theorem 2), who proved that whenever {T i : i ∈ ω} is an r.e. family of consistent, r.e. theories extending Q, then there is a Π 1 formula that is simultaneously independent over these theories. Here, we understand the concept of an independent formula in the following way: Definition 4.1. A formula ξ(x) is independent over T if, for every g : ω → {0, 1}, the theory T + {ξ(k) g(k) } is consistent. Recall that ξ(k) 0 = ¬ξ(k) and ξ(k) 1 = ξ(k).
While one of Mostowski's accomplishments was the simultaneous independence over a whole r.e. family of theories, this aspect of his result is deliberately ignored here. Instead, we focus on how to construct formulae independent over Σ n+1definable theories.
Theorem 4.2. Let T be a Σ n+1 -definable, Σ n -sound extension of IΣ n + exp. Then there is a Σ n+1 formula ξ(x) that is independent over T.
Proof. Define a function f (x) by the stipulation that f (m) = k iff there is a proof p of ϕ n m (m) = k in T, and for each q < p and k 0 ≤ k, q is not a proof of ϕ n m (m) = k 0 in T.
Since T is Σ n+1 -definable, there is a deductively equivalent Π n -definition of T. Hence, the relation f (x) = y is r.e. in ∅ (n) , and therefore partial n-recursive. Let e be an n-index for f , and let ξ(x) be the formula ∃z(ϕ n e (e) = z ∧ Sat Σn+1 (z, x)). Since T extends IΣ n , we may assume that both ϕ n x (y) = z and ξ(x) are equivalent in T to Σ n+1 formulae. The proof that ξ(x) is as desired has two parts. The first part shows that T + ϕ n e (e) = k is consistent for each k ∈ ω. Suppose, for a contradiction, that T + ϕ n e (e) = k is inconsistent for some k ∈ ω. We may assume that k is the least such number. Then T ⊢ ϕ n e (e) = k with some minimal proof p, so f (e) = ϕ n e (e) = k by definition. With T extending IΣ n , we have T + Th Πn (N) ⊢ ϕ n e (e) = k by Fact 3.2.3. But then T + Th Πn (N) is inconsistent, which by Fact 3.1 contradicts the assumption that T is Σ n -sound. Hence the theory T + ϕ n e (e) = k is consistent for any choice of k ∈ ω. In this final part of the proof, we show that ξ(x) is independent over T. Let g be any function from ω to {0, 1} and let X = {ξ(k) g(k) : k ∈ ω}. Let Y be any finite subset of X, and let Z be the set {k : ξ(k) ∈ Y }. Let ζ(x) := xεa, where a is a code for the finite set Z; then ζ binumerates Z in Q.
The Gödel-Rosser incompleteness theorem 1.1 for arithmetically definable theories follows immediately from the result above. While the generalisation to arithmetically definable theories is new, the basic idea of this proof is due to Kripke (1962, Corollary 1.1), who used it to rederive Mostowski's result from his own theorem on the existence of flexible formulae. Here, we understand flexibility in the following sense: The definitions used by Kripke obscure the original content of his theorem, but, in hindsight, his proof yields that for every consistent, r.e. extension T of IΣ 0 + exp, there is a Σ n+1 formula that is flexible for Σ n+1 over T. Striving for some unification, we derive a hierarchical version of Kripke's theorem by generalising a result of Lindström's (1984, Proposition 2); which in turn is a generalisation of both Mostowski's and Kripke's results, as well as of Scott's famous lemma used to realise countable Scott sets as standard systems of models of PA (1962).
Since T is Σ n+1 -definable, there is a deductively equivalent Π n -definition of T. Hence, the relation f (x, y) = z is r.e. in ∅ (n) , and therefore partial n-recursive. Let e be an n-index for f . Let Seq φ (x) be the formula where l(x) denotes the length of x (this is the formula "x is a l(x)-piece of φ"). Whenever φ(x) is Σ m , Seq φ (x) is Σ 0 (Σ m ), and since T ⊢ IΣ m , it is ∆ m+1 in T. Let, by Fact 3.3, γ(x) be such that T ⊢ ∀x(γ(x) ↔ ∃s∃z(Seq φ (s) ∧ ϕ n e (s, γ ) = z ∧ Sat Σm+1 (z, x))). Since T ⊢ IΣ m and m ≥ n, the formula strongly representing f in T + Th Πn (N) is equivalent to a Σ n+1 formula in T. It follows that γ(x) is equivalent to a Σ m+1 formula in T.
Suppose, for a contradiction, that there is a g : ω → {0, 1} and a σ(x) ∈ Σ m+1 such that T g is Σ n -sound, but T g + ∀x(γ(x) ↔ σ(x)) is inconsistent. If there are more than one such σ(x) for a given g, consider the one with the least Gödel number. There is then an initial subsequence s of g, of length k + 1 for some k, such that p is a proof of ¬∀x(γ(x) ↔ σ(x)) in T + φ(0) (s)0 + · · · + φ(k) (s) k . Let s be the initial subsequence of g corresponding to the least such p.
By choosing φ(x) as ⊤ in the construction above, we obtain the expected hierarchical version of Kripke's theorem. A similar, but not entirely correct, claim is made by Blanck (2017, Theorem 4.8).
Corollary 4.5. Let T be a Σ n+1 -definable, Σ n -sound extension of IΣ n + exp. For all m ≥ n, there is a Σ m+1 formula γ(x) that is flexible for Σ m+1 over T.
Mostowski's theorem for r.e. extensions of IΣ 0 + exp then follows immediately by using the method described in the proof of Theorem 4.2. A similar argument also yields Scott's lemma.
The next objective is to show how the hierarchical version of Kripke's theorem can be formalised in Π n -complete extensions of IΣ n+1 . A similar, but not entirely correct, claim is made by Blanck (2017, Theorem 5.8). The present proof is a minor modification of an argument of Blanck (2017, Theorem 5.1).
The next theorem has a different flavour than the earlier ones, and is a generalisation of Woodin's theorem on the universal algorithm (2011); see also Blanck & Enayat (2017, Theorem 3.1). A version for r.e. extensions of PA is independently due to Hamkins (2018, Theorem 18), and the proof presented here uses a method that I learned from Shavrukov. The particular Solovay-style construction used in the proof is similar to the ones used by Berarducci (1990) and Japaridze (1994).
Proof. The set W e is defined (in IΣ n+1 + Th Πn (N) and in the real world) as follows, using the formalisation of the recursion theorem (Fact 3.9). At the same time, an auxiliary function r(x) is defined.
Stage 0: Set W e,0 = ∅, and r(0) = ∞. 9 Stage x + 1: Suppose r(x) = m. There are two cases: Case A: s is a finite set such that s ⊇ W e,x , k < m, and x witnesses a Σ n+1 formula σ(s) such that k is a proof in T + Th Σn+1 (N) of ∀t(σ(t) → W e = t). Should there be more than one eligible candidate for either k or s, then choose the least such k, and then the least s corresponding to that k. Then set W e,x+1 = s and r(x + 1) = k. Case B: Otherwise, set W e,x+1 = W e,x and r(x + 1) = m.
Note also that T ⊢ R > k for all k ∈ ω. To show this, fix k ∈ ω and argue in T: Suppose R ≤ k. Let y be minimal such that r(y + 1) = R. Then W e = W e,y+1 = s for some s such that R is a proof in T+Th Σn+1 (N) of ∀t(σ(t) → W e = t), where σ(s) is a true Σ n+1 formula. But, by Fact 3.14, since ∀t(σ(t) → W e = t) is proved from a true Σ n+1 sentence with a proof not exceeding k, it must be true. Since σ(s) is true, W e = s is also true, and the contradiction proves R > k. To prove 2., argue for the contrapositive statement in IΣ n+1 + Th Πn (N): If W e = s = ∅, then Pr m T,Σn+1 ( ∀t(σ(t) → W e = t) ) for some m and some true Σ n+1 formula σ(s). The relation s ⊆ W e,x is ∆ n+1 by Fact 3.5.3, so Pr T,Σn+1 ( ṡ ⊆ W e ) follows by Fact 3.12. Now reason inside Pr T,Σn+1 : There is some u = W e with u ⊇ s, so by construction, σ ′ (u) is true, and Pr k T,Σn+1 ( ∀t(σ ′ (t) → W e = t) ) for some k ≤ m and some Σ n+1 formula σ ′ (u). Apply Fact 3.15, and continue reasoning inside Pr T,Σn+1 : Then ∀t(σ ′ (t) → W e = t) and σ ′ (u), so W e = u. Then Pr T,Σn+1 ( ∃u(W e = u∧W e = u) ), so Fact 3.13 gives ¬Con T,Σn+1 as desired. To prove 3., first fix m ∈ ω. By Fact 3.14, there is a proof k in T of ∀t(Pr m T,Σn+1 ( W e =ṫ ) → W e = t).

Now reason in T:
Consider any finite s ⊇ W e , and suppose x is a proof ≤ m of W e = s in T + Th Σn+1 (N). Then s ⊇ W e,x+1 , and therefore r(x + 1) ≤ k by construction of r(x + 1): here Pr m T,Σn+1 ( W e =ṡ ) is IΣ n+1equivalent to a true Σ n+1 sentence playing the role of σ(s). But k < R ≤ r(x + 1), and the contradiction proves Con m T+We=ṡ,Σn+1 . Therefore for all m ∈ ω, T ⊢ ∀s ⊇ W e Con m T+We=ṡ,Σn+1 . For the final part of the proof, let M be any countable model of T, and let s be any M-finite set such that M |= W e ⊆ s. Since T is a Σ n+1 -definable, Π ncomplete extension of IΣ n+1 , Facts 3.10 and 3.19 imply that T+Th Σn+1 (M)+W e = s ∈ SSy(M). Since T ⊢ Con m T+We=ṡ,Σn+1 for all m ∈ ω, Fact 3.22 guarantees the existence of a Σ n -elementary extension of M satisfying T+W e = s, which concludes the proof of the theorem.
Proof. Every countable model of T has a Σ n -elementary extension satisfying W e = ∅, and therefore also T+ ¬Con T,Σn+1 by the Theorem. By Fact 3.22, the conclusion follows. 10 The set W e defined in Theorem 4.8 can be used to prove results of a more Kripkean variety, by using the information contained in W e as codes for other sets. The next result is of this kind, and improves on Theorem 7.21 of Blanck (2017) by generalising to arithmetically definable theories. A version for r.e. extensions of PA is independently due to Hamkins (2018, Theorem 22(1)), who also noted that there is a very short proof of it from Theorem 4.8. 10 As pointed out by one of the referees, adapting Kreisel's original proof of the Π 1conservativity of ¬Con T over T is a simpler way to establish Corollary 4.9 than going via Theorem 4.8.
Proof sketch. It is straightforward to adapt the construction in the proof of Theorem 4.8 to produce an M-finite binary sequence S e , rather than a set (Woodin, 2011;Blanck & Enayat, 2017;Hamkins, 2018). Assume an enumeration of Σ m+2 formulae in which every Σ m+2 formula occurs infinitely often, and that every finite binary sequence codes such a formula. Let γ(x) be the formula ∃z(S e = z ∧ Sat Σm+2 (z, x)).
Since S e = z is at most Σ n+2 and m ≥ n, it follows that γ(x) is Σ m+2 . Pick any σ(x) ∈ Σ m+2 , let M be any countable model of T and let s be S e as calculated within M. By assumption on the enumeration of Σ m+2 formulae, there is an M-finite sequence t ⊇ s such that t codes σ(x) . By the sequence version of Theorem 4.8, there is a Σ n -elementary extension K of M in which S e = t. Then γ(x) coincides with σ(x) in K, and therefore is as desired.
The question remains to which extent m + 2 can be replaced by m + 1 in the statement of Theorem 4.10. Some partial answers are already available: Theorem 4.8 gives a positive answer restricted to Σ m+1 formulae σ(x) whose extension is M-finite and for which M |= ∀x(γ(x) → σ(x)), while Theorem 4.6 can be seen as giving a partial positive answer that is restricted to models of T + Con T,Σn+1 . Blanck (2017, Chapter 7.4) lists several other partial answers to this question in a setting where T is an r.e. extension of PA and n = 0. By using the principles of Section 3 of the present paper, those constructions can be easily modified to give equally unsatisfactory answers in the present setting. The salient remaining question is as follows: Question. Let T be a Σ n+1 -definable, Π n -complete, and consistent extension of IΣ n+1 . Is there a Σ n+1 formula γ(x) such that: (1) IΣ n+1 + Th Πn (N) ⊢ Con T,Σn+1 → ∀x¬γ(x); (2) for every σ(x) ∈ Σ n+1 , every countable model of T + ∀x(γ(x) → σ(x)) has a Σ n -elementary extension satisfying T + ∀x(γ(x) ↔ σ(x))?

Discussion
By inspecting the results proved in Section 4, we see two classes of Σ n+1 -definable theories emerging: (1) Σ n -sound extensions of IΣ n + exp; and (2) Π n -complete, consistent extensions of IΣ n+1 . As suggested by Theorem 1.1, theories in the first class are strong enough for some applications. These include the results of Salehi & Seraji (2017) and Kikuchi & Kurahashi (2017), together with Theorems 4.2 and 4.4 (and their corollaries) of the present paper. This success relies on the fact that Σ n -soundness of T guarantees the consistency of T + Th Πn (N), in which the n-recursive functions can be strongly represented by a formula that is Σ n+1 in the presence of Σ n -induction.
The second class of theories is required to prove results on Σ n -elementary extensions of models of Σ n+1 -definable theories, for example results on partial conservativity via the OHGL characterisation (Theorem 4.6 and onwards). In these cases, Π n -completeness of T ensures that every model M of T is a Σ n -elementary extension of the standard model, which in the presence of Σ n+1 -induction suffices for T to be Σ n+1 -definable in M by using Craig's trick. Σ n+1 -induction is also used to prove the arithmetised completeness theorem for Σ n+1 -definable theories, which is indispensable for constructing the Σ n -elementary extensions.
The Facts listed in Section 3 should be enough to derive hierarchical generalisations for arithmetically definable extensions of fragments of PA of many of the theorems in, e.g., Lindström's classic Aspects of Incompleteness (2003). As suggested by the results in the present paper, some of these generalisations would apply only to Π n -complete theories, while in other cases mere Σ n -soundness would do. Others might not be prone to such generalisations at all, as shown by Kurahashi (2018, Theorem 11) and pointed out to me by one of the referees. In fact, it would be interesting to see which of the results in, say, the first 5 chapters of Aspects (where the results do not depend on T being essentially reflexive) that are prone to such generalisations, using these principles.