The Arithmetical Hierarchy in the Setting of Ω1

The arithmetical hierarchy in the setting of !1 Jacob Carson, Jesse Johnson, Julia F. Knight, Karen Lange, Charles McCoy CSC, John Wallbaum April 29, 2013 Abstract We continue work from [1] on computable structure theory in the setting of !1, where the countable ordinals play the role of natural numbers, and countable sets play the role of finite sets. In the present paper, we define the arithmetical hierarchy through all countable levels (not just the finite levels). We consider two different ways of doing this|one based on the standard definition of the hyperarithmetical hierarchy, and the other based on the standard definition of the effective Borel hierarchy. For each definition, we define computable infinitary formulas through all countable levels, and we obtain analogues of the well-known results from [2] and 0 [4] saying that a relation is relatively intrinsically Σα just in case it is definable by a computable Σα formula. Although we obtain the same results for the two definitions of the arithmetical hierarchy, we conclude that the definition resembling the standard definition of the hyperarithmetical hierarchy seems preferable. 1 Introduction We consider computability in the setting of !1. The countable ordinals play the role of natural numbers, and countable sets play the role of finite sets. We assume V = L. This implies that all reals are present in L!1 . In fact, every subset of !1 is \amenable" for L!1 ; i.e., for all A ⊆ L!1 and all x ∈ L!1 , A ∩ x ∈ L!1 . In the remainder of the introduction, we review some basic definitions and results 0 0 from [1]. In Section 2, we define the arithmetical hierarchy|the Σα and Πα subsets of !1, for countable ordinals α. We do this in two different ways. The first is based on the standard definition of the hyperarithmetical hierarchy|a 0 0 set is Σα if it is c.e. relative to a complete ∆α oracle. The second definition is 0 based on the definition of the effective Borel hierarchy|a set is Σα if it is a c.e. 0 union of sets each of which is Πβ for some β < α. In Section 3, we define computable infinitary Σα formulas in two different ways, corresponding to the different definitions of the arithmetical hierarchy. We allow countable tuples of quantifiers, and our formulas will be from L!2;!1 . In Section 4, we give the main results, saying that for each set of definitions, a relation R on a computable structure A is relatively intrinsically Σα if and 1 only if it is defined in A by a computable Σα formula. This is the analogue of a result for the standard computability setting in [2] and [4]. Finally, in Section 5, we explain why the first set of definitions seems preferable. 1.1 Basic definitions Below, we first say what it means for a set or relation on L!1 to be computably enumerable. We then define the computable sets and relations, and the computable functions. Definition 1. n • A relation R ⊆ (L!1 ) is bounded, or ∆0, if it is defined by a finitary formula with only bounded quantifiers, (∃x ∈ y) and (∀x ∈ y), in the language with ∈ and constants from L!1 . n • A relation R ⊆ (L!1 ) is computably enumerable, or c.e., if it is defined in L!1 by a Σ1-formula '(c; x), with finitely many parameters|the formula is finitary, with only existential and bounded quantifiers, and all negations appear inside quantifiers. n • A relation R ⊆ (L!1 ) is computable if it and its complement are both computably enumerable. n • A (partial) function f ∶ (L!1 ) → L!1 is computable if its graph is c.e. When we work with these definitions, we soon see that computations involve countable ordinal steps, and that computable functions are generally defined by recursion on ordinals|the Σ1 definition for a function f says that there exists a sequence of steps leading to the value of f at a given ordinal α. Thus, it might be appropriate to use the term \recursive" instead of \computable" in this setting. Results of Gödelgive a computable 1−1 function g from the countable ordinals onto L!1 such that the relation g(α) ∈ g(β) is computable. The function g prvides ordinal codes for sets|α is the code for g(α). There is also a computable function ` taking α to the code for Lα. From this, we see that computing in !1 is essentially the same as computing in L!1 . For more details on this point, see [1]. Above, we were thinking of relations and functions of finite arity. We may allow relations and functions of arity α, where α < !1. We extend the definition as follows. Definition 2. Suppose R is a relation of arity α < !1. • R is c.e. if {β ∶ g(β) ∈ R} is Σ1-definable—the set of ordinal codes for sequences in R is c.e. • A relation of arity α is computable if it is both c.e. and co-c.e. 2 α • A function f ∶ (L!1 ) → L!1 is computable if its graph {a; f(a) ∶ a ∈ domf} is a c.e. relation. • We have a c.e. set C of codes for pairs ('; c), representing Σ1 definitions, where '(u; x) is a Σ1-formula and c is a tuple of parameters appropriate for u. Note that u and x can be countable tuples. • We have a computable function h mapping !1 onto C. Definition 3. For α < !1, α is a c.e. index for X if h(α) is the code for a pair ('; c), where '(c; x) is a Σ1 definition of X in (L!1 ; ∈). Notation: We write Wα for the c.e. set with index α. Suppose Wα is determined by the pair ('; c); i.e., '(c; x) is a Σ1 definition. Definition 4. We say that x is in Wα at stage β, and we write x ∈ Wα,β, if Lβ contains x, the parameters c, and witnesses making the formula '(c; x) true. Remark. The relation x ∈ Wα,β is computable. After all, in the current setting, countable sets, such as Lβ, appear “finite”. In the standard setting, the class of c.e. sets is closed under finite intersection. We have the analogue here. Proposition 1.1. The class of c.e. sets is closed under countable intersection. Proof. Let Γ be a countable set of countable ordinals (indices for c.e. sets), and let S = ∩γ Γ. We have x ∈ S iff (∃β)(∀γ ∈ Γ) x ∈ Wγ,β, so S is Σ1 definable in L! . 1 ∈ 1.2 Relative computability and jumps Proposition 1.2. There is a c.e. set U ⊆ !1 × !1 consisting of the pairs (α; β) such that β ∈ Wα. In the standard setting, we define the halting set K, and we prove that it is c.e. and non-computable. We then use the s − m − n theorem to show that all c.e. sets are 1-reducible to K. In the setting of !1, we could proceed in exactly the same way [1], letting K = {α ∶ α ∈ Wα}. Instead, we define a set that is obviously 1-complete. Definition 5. Let K = {(α; y) ∶ y ∈ Wα}. This set K is c.e. We have a Σ1 definition for K, saying that there exists β such that Lβ contains the pair h(α) = ('(u; y); a), and Lβ ⊧ '(a; y). The complement of K cannot be c.e., for then the set {α ∶ α ∉ Wα} would also be c.e. Relative computability is important in what follows. 3 Definition 6. Let X ⊆ !1. • A relation is c.e. relative to X if it is Σ1-definable in (L!1 ; ∈;X). • A relation is computable relative to X if it and its complement are both c.e. relative to X. • A (partial or total) function is computable relative to X if the graph is c.e. relative to X. Definition 7. A c.e. index for a relation R relative to X is an ordinal α such that h(α) = ('; c), where ' is a Σ1 formula (in the language with ∈ and a predicate symbol for X), and '(c; x) defines R in (L!1 ; ∈;X). X Notation: We write Wα for the c.e. set with index α relative to X. Proposition 1.3. There is a c.e. set U of the codes for triples (σ; α; β) such ρ that σ ∈ 2 (for some countable ordinal ρ), and for all X with characteristic X function extending σ, β ∈ Wα . A proof of the proposition above appears in [1]. We define the jump of a set X ⊆ !1 to make the universality obvious. X Definition 8. The jump of X is X = {(α; y) ∶ y ∈ Wα }. ′ As in the standard setting, for each X, X is c.e. relative to X and not computable relative to X. ′ We iterate the jump function through countable levels as follows. 0 • X = X, ( ) α 1 α • X = (X ) , ( + ) ( ) ′ α • for limit α, X is the set of codes for pairs (β; x) such that β < α and β x ∈ X . ( ) ( ) It is convenient to have, for each countable ordinal α ≥ 1, a name for a specific oracle set. 0 n 1 Notation. For finite n ≥ 1, we write ∆n for ∅ , and for countable ordinals 0 α ( − ) α ≥ !, we write ∆α for ∅ . We can relativize to a set X. For finite n ≥ 1, we 0 n 1 0 α let ∆n(X) = X , and( for) α ≥ !, we let ∆α(X) = X .

Load more