Hyperarithmetical hierarchy

In recursion 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.

The central focus of hyperarithmetic theory is the sets of natural numbers known as hyperarithmetic sets. There are three equivalent ways of defining this class of sets; the study of the relationships between these different definitions is one motivation for the study of hyperarithmetical theory.

The first definition of the hyperarithmetic sets uses the analytical hierarchy. A set of natural numbers is classified at level ${\displaystyle \Sigma _{1}^{1}}$ of this hierarchy if it is definable by a formula of second-order arithmetic with only existential set quantifiers and no other set quantifiers. A set is classified at level ${\displaystyle \Pi _{1}^{1}}$ of the analytical hierarchy if it is definable by a formula of second-order arithmetic with only universal set quantifiers and no other set quantifiers. A set is ${\displaystyle \Delta _{1}^{1}}$ if it is both ${\displaystyle \Sigma _{1}^{1}}$ and ${\displaystyle \Pi _{1}^{1}}$. The hyperarithmetical sets are exactly the ${\displaystyle \Delta _{1}^{1}}$ sets.

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