Locally hyperarithmetical induction

We compute the closure ordinals of hyperarithmetical inductive definitions of sets of integers and of locally hyperarithmetical inductive definitions of sets of integers.


INTRODUCTION
The notion of an inductively defined set of natural numbers is motivated by the following question: how complicated a set of integers can we define by iterating a simple process? Over time, this notion has become part of the backbone of Mathematical Logic. Kreisel [12] introduced the theories ID of iterated inductive definitions which were proof-theoretically analyzed by Pohlers [17] and ultimately led to massive advances in proof theory; we mention Feferman-Sieg [6] and Rathjen [18,19]. In reverse mathematics, theories of inductive definability provide a very fine array of alternatives with which one can gauge the strength of mathematical theorems, particularly for theories of determinacy; we mention Tanaka [24,25] and Welch [27]. Moschovakis [14] began the study of classes of relations definable inductively over a structure and of how definability relates to computability in this context. This leads to connections with admissibility and other branches of generalized recursion theory (see Barwise [3], Kechris [10], Sacks [21]) and descriptive set theory (see Moschovakis [15]).
The notion is very simple: given an operator we can iterate transfinitely by setting 0 = ∅, and for every ordinal , Because there are only countably many natural numbers, there must be a least countable stage such that = +1 ; this is called the closure ordinal of and is denoted by | |. For this , we write ∞ = ⋃ ⩽ and call this the set defined inductively by . For a class of operators Γ, |Γ| is defined as the supremum of | |, for ∈ Γ. In most cases of interest, the question of what sets can be defined inductively by an operator of a given complexity reduces to the question of identifying the corresponding closure ordinal. A central question is then: for a given class Γ, what is |Γ|?
An operator is hyperarithmetical (or Δ 1 1 ) if it is Π 1 1 and, additionally, there is a Σ 1 1 operator such that for all ∈ (ℕ) we have ( ) = ( ). (1.1) The classical Kleene-Suslin theorem asserts that these are precisely the operators whose graph is Borel and, moreover, the procedure that generates from basic open sets by applying unions and complements can be described effectively. Aczel and Richter [1] proved that |Δ 1 1 | is an inadmissible ordinal, though their argument does not show what |Δ 1 1 | is. There is something unsatisfactory about the definition of a hyperarithmetical inductive operator, because it requires that (1.1) be true for all inputs, even though the value of the operator on most inputs is inconsequential to the outcome of the inductive definition itself. A more natural condition is to only require that (1.1) be true for the inputs which figure as one of the stages in the inductive definition. An operator is thus said to be locally hyperarithmetical (or Δ 1, 1 ) if it is Π 1 1 and there is a Σ 1 1 operator such that (1.1) holds for all of the form < . This definition makes sense for other classes, though, as we observe below, in all other cases we have Indeed, Equation (1.2) can easily be seen to hold for all , ∈ ℕ except the case = = 1. It is rather surprising that this does not apply in the context of hyperarithmetical sets, where we obtain: Hyperarithmetical operators are special in this sense. The closure ordinal of locally hyperarithmetical definitions is a very interesting number which we can describe both in proof-theoretic and in recursion-theoretic terms. Recall that an ordinal is said to be -stable if that is, if every Σ 1 sentence with parameters in true in is true in . Here, denotes the th stage of Gödel's constructible hierarchy. Below, given a function , we denote by the least ordinal which is ( )-stable. Similarly, we denote by the least ordinal which is ( + )-stable.
By Theorem 1, the sets of integers which are inductively definable by a locally hyperarithmetical operator are precisely the sets of integers in . The proof of Theorem 1 also immediately yields a computation of the closure ordinal of fully hyperarithmetical inductive definitions.
By Theorem 2, the sets of integers which are inductively definable by a hyperarithmetical operator are precisely the sets of integers in , where is as in the statement of the theorem. The notion of a set-recursive function (also known as -recursive) can be defined in terms of a finite collection of schemes (much like ordinary recursive functions), known as the Normann schemes. It was discovered by Normann [16] and, independently, by Moschovakis (see Kechris-Moschovakis [11]), who developed the theory of -recursion centered around that of inductive definability. Based on what has been said so far, the reader might suppose that the inductive-definability viewpoint will be more useful than the schematic viewpoint for our purposes. However, we will use neither.
Van de Wiele's theorem [26] establishes a link between weak set theories and set recursion. Recall that a set is called admissible if it is transitive and ( , ∈) is a model of Kripke-Platek set theory. It states that is a total set-recursive function if and only if for every admissible set , [ ] ⊂ and ↾ is uniformly Σ 1 -definable over (that is, the Σ 1 definition does not depend on ). Van de Wiele's proof of his theorem relied on the theory of Π 1 2 -logic, though a purely recursiontheoretic proof was given by Slaman [23]. Van de Wiele's proof, however, shows that if is a set-recursive function, then one can find a recursive dilator which dominates on every input. Conversely, recursive dilators can be completely evaluated in every admissible set, which yields the second equality in the statement of Theorem 1.
We refer the reader to Barwise [3] for background on admissibility theory and to Sacks [21] for background on generalized recursion. We also refer the reader to Rathjen [18] for more on recursively large ordinals, to Girard [7] for more on Π 1 2 -logic, to Girard-Normann [8] and Slaman [22] for some generalizations of Van de Wiele's theorem, as well as to Lubarsky [13] for an alternate proof of Slaman's [22] theorem.

LOCALLY HYPERARITHMETICAL INDUCTIVE DEFINITIONS
Below, if is an ordinal, we denote by + the least admissible ordinal greater than . We prove Theorem 1. We begin with an observation.

Lemma 4. If is a locally hyperarithmetical operator, there is a total set-recursive function on ordinals such that if is
Proof. Let be a locally hyperarithmetical operator, and let 0 and 1 be the Π 1 1 formulae exhibiting this fact. Recall that Recall that by Barwise-Gandy-Moschovakis [4], Π 1 1 questions about a countable set can be uniformly translated into Σ 1 questions about the next admissible set + . Thus, we can find Σ 1 formulae 0 , 1 such that for all and all , We define Proof. This is proved by induction. Suppose that is a an ordinal and that < ∈ , for < additively indecomposable. Then, by the pair of equivalences above and Lemma 3, we have ∈ ↔ + ⊧ 0 ( , < ) ↔ + ̸ ⊧ 1 ( , < ).
Proof. This follows from the previous claim: if belongs to some admissible set , then < ∈ , and then ( ) is the least ordinal such that for all , This ordinal is smaller than < 1 [ < ] and thus belongs to any admissible set containing < . □ It follows from the claim and Van de Wiele's theorem that is set-recursive.
Proof. Let be an ordinal which is ( )-stable and suppose that ∈ , that is, that By definition of and choice of 0 , we have ( ) ⊧ 0 ( , < ). Hence, ( ) ⊧ "there is an admissible such that 0 ( , < )." Since < can be defined over by a Σ 1 -formula, by stability we have ⊧ "there is an admissiblēsuch that 0 ( , <̄) ." By choice of 0 and 1 , we havē+ and since these are both Σ 1 formulas (thus upwards absolute) and they cannot both hold, we must Proof. Fix any ordinal-valued set-recursive function . By replacing by the function if necessary, we may assume that is nondecreasing. By Van de Wiele's theorem, if is an admissible set, then is uniformly Σ 1 -definable over . Moreover, is total on every admissible set, so it is uniformly Δ 1 ( ( ) ≠ if and only if there is some different such that ( ) = ). We define an ordinal-valued function (⋅) as follows.
Given a set , define ( ) to be the least ordinal such that one of the following holds: ∈ ( ) and, letting be least such that ∈ +1 , ( ) = ( ).
Thus, to find ( ), we look in ( ) until we either reach an admissible stage, or we reach a structure over which is definable, and then apply to that ordinal. We consider the following class of operators Γ: an operator on ℕ is Γ-definable if there is a Σ 1 formula in the language of set theory such that for every ⊂ ℕ, According to the definition, Γ operators are those that ask a Σ 1 question about ( ) and (viewed as a predicate). Note that we do not require that appear only positively in .
Proof. Let be as above and let * ( , ) be the formula asserting that ( , ) holds and, letting be least such that ( , ) ⊧ ( , ), no ordinal smaller than satisfies the definition of ( ). Since is Σ 1 , * is Σ 1 . We claim that If ∈ ( ), then ( ( ) , ) ⊧ ( , ) and, letting be least such that The following lemma shows that if is a Γ operator that inductively constructs codes for ordinal numbers (possibly among other things), then is locally Δ 1 1 .

Sublemma 10.
Let be a Γ operator such that for every < | |, there is a wellordering of ℕ of length recursive in < . Then, is locally Δ 1 1 .
Proof. Let be the Γ operator and let be the associated Σ 1 formula. We will show that the following equivalence holds: If so, then the operator̆given by agrees with on all inputs of the form < . By Barwise-Gandy-Moschovakis [4], (2.2) is Σ 1 1 , and so it follows that is locally Δ 1 1 , which finishes the proof. Let us prove (2.1). The proof of (2.1) is by induction, so we suppose that (2.1) holds for all < , so that and̆agree on all inputs of the form < , for < . Thus, for < , we have ) .
The upper formula on the right-hand side of the displayed equation is Σ 1 and the one on the bottom is Π 1 with the parameter < . It follows that belongs to every admissible set which contains < . By recursion on , it follows that admissible sets are closed under the function ↦ , for < . By hypothesis, < computes a wellordering of length , so < < 1 and thus by the observation in the previous paragraph, we have Since is total in every admissible set, the definition of (⋅) yields that ( < ) < < 1 as well, and that ( < ) = ( * ), where * is least such that < ∈ * +1 . For such a * , we have ∈ ( < ) ↔ ( ( < ) , < ) ⊧ ( , < ) ↔ ( ( * ) , < ) ⊧ ( , < ). (

2.4)
The formula is Σ 1 , so (2.4) is equivalent to the right-hand side of (2.1). □ Before moving on, let us record the following fact which we will need later. This was verified as part of the proof of Sublemma 10. Sublemma 11. The function ↦ is uniformly Δ 1 over all admissible sets.
Since the upper formula on the right-hand side is Σ 1 and the one on the bottom is Π 1 with parameter < , the claim follows. □ We consider the following "universal" Γ operator, : where is the th Σ 1 formula with one first-order free variable and one second-order free variable.
For the remainder of the argument, we shall employ a version of the Aczel-Richter [1] technique of constructing an operator Θ that generates notations for ordinals while being padded by a suitably chosen "slow" inductive definition that ensures that this process goes on for as long as possible. This slow definition will be , and Θ generating ordinals will ensure that it ends up being a locally Δ 1 1 operator. We will then show that Θ satisfies the requirements in the statement of the lemma; namely, that its closure ordinal |Θ| is (|Θ|)-stable.
Given a set , let denote the th section of , consisting of all such that ⟨ , ⟩ ∈ . We first give an intuitive definition of the operator Θ ∶ (ℕ) → (ℕ), and then follow with a more precise definition.
Θ will be made up of four components Θ 0 , Θ 1 , Θ 2 , Θ 3 . These are essentially four inductive definitions being carried out simultaneously, and some of them refer to the others. Θ is defined so that as one constructs Θ inductively, the component Θ 0 is simply (Θ < ). This serves two purposes: the first one is to use the universality of to make Θ capable of computing other Γ operators, at least when applied to sets of the form Θ < ; the second (though this is done implicitly) is to use the universality of to make Θ take a long time to converge. The component Θ 1 is the stage comparison relation on elements in Θ < 0 ; Θ 2 is the set of all triples ( , , ⃗ ) where ⃗ ∈ Θ < 2 and ∈ Θ 0 ⧵ field(Θ < 1 ), viewed each as a code for the set of all ∈ | | that satisfy a formula ( , ⃗ ) (where | | denotes the rank according to the stage comparison relation; so these are the codes for the elements definable over | | ); and the component Θ 3 consists of the membership-or-equality relation between all elements of Θ 2 (from this relation we can recover both the membership and equality relations). Hence, the essence of Θ is that it constructs codes for sets of the form , and this process goes on for as long as allowed by .

Sublemma 12. Θ is locally
Proof. Clearly Θ is Γ-definable. By construction, Θ 1 is the stage comparison relation of Θ 0 : at every stage, the points added to the field of the binary relation Θ 1 are those of the form ( , ), where is an element of Θ 0 that had not appeared in the field of Θ 1 before, and is one that had. It follows that from Θ 1 , one can uniformly compute a prewellordering of ℕ of rank , and indeed one can compute a wellordering of ℕ of length from Θ < . The result then follows from Sublemma 10. □ Sublemma 13. Let be a Σ 1 formula in the language of set theory. Then, there is a Γ operator such that for all admissible ⩽ |Θ|, where | | denotes the rank of ∈ ℕ in the relation Θ < 1 .
For the "moreover" part of the lemma, suppose that is of the form ↦ + for some recursive ordinal . Instead of defining the class Γ as before, take Γ to be the class of all operators such that for some Σ 1 formula in the language of set theory,

∈ ( ) ↔ ( ) ⊧ ( , ).
Then, each operator in Γ is easily seen to be Δ 1 1 , as is the operator Θ defined as above. The rest of the argument proceeds as above.
□ This completes the proof of Theorem 1. We make the following observation concerning the proof of Lemma 8: the proof that the operator Θ was locally Δ 1 1 used the fact that was total and set-recursive, though really we only needed the fact that was total below |Δ 1, 1 |, so we obtain the following very slight strengthening of Lemma 8: Let be a function on ordinals which is total and uniformly Σ 1 -definable over every admissible with < |Δ 1, 1 |. Then, there is a locally hyperarithmetical operator Θ such that |Θ| is (|Θ|)-stable.
It is not hard to explicitly define a Δ 1 2 truth predicate for |Δ 1, 1 | . It thus follows from Proposition 15 below that one cannot in general determine over |Δ 1, 1 | whether a formula defines a total set-recursive function on ordinals. In particular, there are many set-recursive functions on ordinals which are not total, but are total below |Δ 1, Proof. This is essentially immediate from Van de Wiele's theorem. Call a functor on the category of linear orders a pre-dilator if it preserves pullbacks and direct limits; thus, a dilator is a pre-dilator which preserves wellfoundedness. A pre-dilator is recursive if it maps finite ordinals to finite ordinals and the function which maps to ( ), for finite functions on natural numbers, is recursive. The set of recursive pre-dilators is definable over , but the set of dilators is Π 1 2 -complete, by a theorem of Girard-Ressayre [9]. Given a recursive pre-dilator , there is a Σ 1 formula ( , ) which asserts that ( ) is isomorphic to and is an ordinal. Then, is a dilator if and only if is total in every admissible set, if and only if it is the graph of a total setrecursive function on ordinals. It follows that the set of indices of dilators reduces to , so is as desired. □ Corollary 16. |Δ 1, 1 | is an inadmissible limit of admissibles.
Proof. Let = lim , where ranges over all set-recursive functions. First, note that a simple reflection argument shows that is not of the form , for if is set-recursive, so is the function + 1 ∶= ↦ ( ) + 1.
Hence, is a limit of ordinals which are ( + 1)-stable, thus a limit of admissible ordinals.
Given a Σ 1 formula which defines a partial function , denote by the least ordinal such that either does not define a total function in every admissible set contained in , or else is ( )-stable. By Corollary 14, < |Δ 1, 1 | for every such ; thus, the function mapping to yields a failure of admissibility. □
Let be the function ↦ + . Arguing as in Lemma 4, we see that if is ( )-stable, then | | ⩽ . The converse is immediate from Lemma 8. This proves the theorem.
Proof. This is immediate from Theorem 2, since the sequence { ∶ < 1 } yields a failure of admissibility. □

DISCUSSION
Let us show that the closure ordinals of hyperarithmetical definitions and locally hyperarithmetical definitions are different.
Proof. Let be the following function: if is an admissible ordinal and 1 < , then ( ) = + 1 + 1; otherwise, ( ) = 1. This is a total function uniformly Σ 1 -definable over all admissible sets, so it is set-recursive. Clearly, |Δ 1 1 | < , so it follows that The fact that follows from the fact that |Δ 1, 1 | ⩽ |Π 1 1 | (by definition), and the facts that |Π 1 1 | is admissible and |Δ 1, 1 | is not. □ By the proofs given above, the strength of locally hyperarithmetical induction comes from the fact that operators are able to "adapt" to the definition and make use of ordinals constructed along the way. For instance, they allow the possibility that we inductively construct a wellordering of length 1 , use that to compute some wellordering of length ∈ ( 1 , 2 ), and then essentially behave like an operator in the class Σ 0 ( ). If one tried to emulate this procedure with a fully hyperarithmetical operator, one would find trouble in ensuring that the operator behaves properly on inputs which code illfounded relations, whereas in the local case the burden of wellfoundedness is carried by the inductive definition. The difference between locally hyperarithmetical induction and fully hyperarithmetical induction illustrates a key feature of computing sets of natural numbers from ordinal numbers versus computing them from codes of ordinal numbers.
When working with self-dual classes of operators extending Π 1 1 , we are able to discriminate the real numbers which code ordinals from those which do not, and this is the reason why Theorem 18 does not hold in those contexts. Although this fact follows easily from known results, it is worth noting.
Since we must have it follows that these ordinals are all equal. Suppose now that ⩾ 1 and that and are not both 1. By [1, Theorem 11.1], |Δ | is the supremum of lengths of Δ -definable wellorderings of ℕ. The proof of [1, Theorem 11.1] only uses the fact that the operator is locally Δ , so the result follows. □ We would also like to point out an aspect of the definition of locally hyperarithmetical operators. These were defined as the Π 1 1 operators which agree with a fixed Σ 1 1 operator on all stages of the inductive definition. We could have considered the dual notion of all Σ 1 1 operators which agree with a fixed Π 1 1 operator on all stages of the inductive definition. This results in a class of operators which, while different, produces the same inductive definitions as the locally hyperarithmetical operators.

A C K N O W L E D G E M E N T S
This work was partially supported by FWO grant 3E017319 and FWF grant ESP-3.

J O U R N A L I N F O R M AT I O N
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