Effectively closed sets of reals are also called \Pi^0_1 classes.
In mathematical logic, the arithmetical hierarchy classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical.
In computability theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an important tool in effective descriptive set theory.
Hyperarithmetical theory -wiki
The analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers
In the mathematical field of descriptive set theory, a subset A of a Polish space is projective if it is \Sigma^1_n for some n.
In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory.