Inductively defined sets of reals

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Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Monotone inductive definitions over the continuum

Cenzer

1976

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Models for recursion theory

Moldestad

Normann

1976

Several results in the theory of recursion in higher types indicate that the effect of a higher type functional on the lower types does not reflect the high type, i.e. the same effect could be obtained by functionals of relatively low type. The two main results here are:Plus -1 - Theorem (G. Sacks [6] for k = 1, [7] for k > 1). Let H be a normal functional of type ≥ k + 1. Then there exists a normal functional F of type k + 1 such that k-sc(F) = k-sc(H), i.e. the same subsets of tp(k − 1) are recursive in F and H.Plus - 2 - Theorem (L. Harrington [1]). Let H be a normal functional of type ≥ k + 2. Then there exists a normal functional F of type k +2 such that k-en(H) = k-en(F), i.e. the same subsets of tp(k − 1) are semirecursive in F and H.The results in this paper also indicate that higher types cannot have too much influence on lower types. The key is the Skolem-Löwenheim theorem. Among the results we mention:(1) Let n < m. A ⊆ tp(n) × tp(m) be Kleene-semicomputable. Let x ∈ B ⇔ ∀y∈tp(m), ⟨x, y⟩ ∈ A. Then B is . This result may be relativized to a functional of type n + 1.(2) Let k0 be the type-k-functional that is constant zero. Let F be a functional of type < k. Then, for i ≤ k −2, i-sc(F ,k0) = i-sc(F), i-en(F, k0) = ∀tp(i)(i-en(F)).

Cores of Π₁¹ sets of reals

Blass

Cenzer

1974

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A classical result of descriptive set theory expresses every co-analytic subset of the real line as the union of an increasing sequence of Borel sets, the length of the chain being at most the first uncountable ordinal ℵ1 (see [5], [8]). An effective analog of this theorem, obtained by replacing co-analytic (Π11) and Borel (Δ11) with their lightface analogs, would represent every Π11 subset of the real line as the union of a chain of Δ11 sets. No such analog is true, however, because some Δ11 sets are not the union of their Δ11 subsets. For example, the set W, consisting of those reals which code well-orderings (in some standard coding) is Π11, but, by the boundedness principle ([3], [9]), any Δ11 subset of W contains codes only for well-orderings shorter than ω1, the first nonrecursive ordinal. Accordingly, we define the core of a Π11 set to be the union of its Δ11 subsets; clearly this is the largest subset of the given Π11 set for which an effective version of the classical representation could exist.In §1, we develop the elementary properties of cores of Π11 sets. For example, such a core is itself Π11 and can be represented as the union of a chain of Δ11 sets in a natural way; the chain will have length at most ω1. We show that the core of a Π11 set is “almost all” of the set, while on the other hand there are uncountable Π11 sets with empty cores.