Syntactic characterization of hyperarithmetic analysis

Develop a syntactic characterization of the class of hyperarithmetic analysis theories—namely, a proof-theoretic or axiomatic description that characterizes the class without appeal to ω-models—so that HYP(X) is captured by such syntactic conditions for every set X.

Background

Hyperarithmetic analysis is traditionally defined via ω-models HYP(X), and the paper seeks a syntactic (proof-theoretic) characterization of this class. While the author shows that RFN^{-1}(ATR) approximates HA and shares several closure properties and inclusions, a complete syntactic description is not achieved.

This longstanding gap motivates the investigation into reflection-based characterizations and related closure properties, but the paper explicitly notes the absence of a full syntactic account.