The notion of 'hyperarithmetical set' is well-known (see e.g. [1,p. 18]). In the language of second-order arithmetic, this notion is used to define the notion of 'system of hyperarithmetical analysis' (see e.g. [0]). Of course, this involves a rather restrictive language.
Is there a notion of 'system of hyperarithmetical analysis' formulated in the language of third-order arithmetic, or even for all finite types?
My motivation is the connection between hyperarithmetical sets and Kleene's third-order $\exists^2$, which is essentially a discontinuous function on $2^{\mathbb{N}}$ or $[0,1]$.
References
[0] Antonio Montalban, Indecomposable linear orderings and hyperarithmetic analysis, J. Math. Log. 6 (2006), no. 1, 89–120.
[1] Steve Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, CUP, 2009.
