The Arithmetical Hierarchy in the Setting of Ω 1 Computability

The arithmetical hierarhy in the setting of !1 Jacob Carson, Jesse Johnson, Julia F. Knight, Karen Lange, Charles McCoy CSC, John Wallbaum University of Notre Dame 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. We do this in two different ways. We then give analogues of the well-known results from [2] and [4] on relatively intrinsically arithmetical relations. To do this, we define computable Σα and computable Πα formulas, for countable ordinals α. We do this also in two different ways, corresponding to our two definitions of the arithmetical hierarchy. We prove, for both approaches, that a relation R on a computable structure A is rela- 0 tively intrinsically Σα if and only if it is defined in A by a computable Σα formula. 1 Introduction We consider computability in the setting where countable ordinals play the role of natural numbers, and countable sets play the role of finite sets. It will be beneficial for us to assume that every subset of !1 is amenable for L!1 . So we assume V = L, or at least P (!) ⊆ L. The reason for this assumption is that, since L!1 is the domain for our model of computation, it will be important for our purposes that for all A ⊆ L!1 , L!1 will be closed under the function x 7! A\x. In the remainder of the introduction, we review some basic definitions and results from [1]. In Section 2, we define the arithmetical hierarchy in two different ways, one resembling the standard definition of the hyperarithmetical hierarchy, and the other resembling the definition of the effective Borel hierarchy. In Section 3, we define computable infinitary formulas, allowing countable tuples of variables. We do this in two different ways, corresponding to the different definitions of the arithmetical hierarchy. In Section 4, we prove the main result, saying that a relation R on a computable structure A is relatively intrinsically Σα if and only if it is defined in A by a computable Σα formula. There are two versions of the result, one for each set of definitions. This main result is an analogue to a result in the standard computability setting in [2] and [4]. Finally, 1 in Section 5, we explain why the second set of definitions seems preferable.1 1.1 Basic definitions Below, we first say what it means for a set or relation 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 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. n • A relation R ⊆ (L!1 ) is computable if R and the complementary relation :R 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 in- volve countable ordinal steps. Computable functions are defined by recursion on ordinals|the Σ1 definition for a function f says that there is 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. 1.2 Ordinals and sets Results of Gödelgive us a computable 1 − 1 function g from the countable ordinals onto L!1 such that the relation g(α) 2 g(β) is computable. The function g gives us 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 a more explicit argument, see [1]. 1.3 Arity of relations and functions We have been thinking of relations and functions of finite arity. However, we may allow relations and functions of arity α, for any α < !1. We extend the definition as follows. Definition 2. Suppose R is a relation of arity α < !1. • R is c.e. if fβ : g(β) 2 Rg is Σ1-definable; i.e., the set of ordinal codes for sequences in R is c.e. • R is computable if it is both c.e. and co-c.e. α • A function f :(L!1 ) ! L!1 is computable if its graph fa; f(a): a 2 domfg is a c.e. relation. 1We are grateful to Joe Mileti for helpful discussions of this. 2 1.4 Indices for c.e. sets • We have a c.e. set C of codes for pairs ('; c), each 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. 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 ; 2). Notation: We write Wα for the c.e. set with index α. 1.5 Stage at which an element enters a c.e. set 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 2 Wα,β, if Lβ contains x, the parameters c, and witnesses making the formula '(c; x) true. Remark. The relation x 2 Wα,β is computable. The relation is Σ1 definable, and so is the complementary relation|we use Lβ as a bound on the quantifiers. 1.6 Universal enumeration of c.e. sets, and K Proposition 1.1. There is a c.e. set U ⊆ !1 × !1 consisting of the pairs (α; β) such that β 2 Wα. We define the halting set, just as in the standard setting. Definition 5 (Halting set). The halting set is K = fα : α 2 Wαg. Just as in the standard setting, K is c.e. and not computable. 1.7 Relative computability We define relative computability in a natural way. Definition 6. Let X ⊆ !1 and let R be a relation. • R is c.e. relative to X if it is Σ1-definable in (L!1 ; 2;X). • R is computable relative to X if R and the complementary relation :R 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 2 and a predicate symbol for X), and '(c; x) defines R in (L!1 ; 2;X). 3 X Notation: We write Wα for the c.e. set with index α relative to X. Proposition 1.2. There is a c.e. set U of the codes for triples (σ; α; β) such that σ 2 2ρ (for some countable ordinal ρ), and for all X with characteristic X function extending σ, β 2 Wα . A proof appears in [1]. 1.8 Jumps We define the jump of a set X ⊆ !1 so that it is clearly complete. 0 X X Definition 8. The jump of X is X = ⊕α<ω1 Wα = f(α; x): x 2 Wα g. As in the standard setting, X0 is c.e. relative to X, and any set c.e. relative to X is m-reducible to X0. We iterate the jump function through countable levels as follows. • X(0) = X, • X(α+1) = (X(α))0, • for limit α, X(α) is the set of codes for pairs (β; x) such that β < α and x 2 X(β). 0 (n−1) Notation. For convenience, we write ∆n for ; for 1 ≤ n < !, and we 0 (α) write ∆α for ; for countable ordinals α ≥ !. 2 Arithmetical hierarchy|two definitions In this section, we give two different definitions of the arithmetical hierarchy. 2.1 First approach In our first definition, we imitate the approach used in defining the effective Borel hierarchy [7]. Definition 9. Let R be a relation on !1. 0 0 • R is Σ1 if it is c.e.; R is Π1 if the complementary relation, :R, is c.e. 0 • For countable α > 1, R is Σα if it is a c.e. union of relations, each of 0 [ which is Πβ for some β < α. That is, R = Ri, where for each i, there c:e: is some β < α such that R is Π0 ; R is Π0 if it is a c.e. intersection of i i βi α 0 relations, each of which is Σβ for some β < α. 0 0 0 • R is ∆α if it is both Σα and Πα. 4 0 Remarks. In general, when we form a Σα relation, we are taking the union of 0 an uncountable c.e. family of lower complexity relations. A relation is Πα just 0 in case the complementary relation, :R, is Σα. 0 0 We may assign indices to the Σα and Πα sets in a natural way, for α ≥ 1.

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