The Arithmetical Hierarchy in the Setting of $\omega_1$

Abstract

We continue work from [1] on computable structure theory in the setting of ω1, 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 [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.