Hyperarithmetical theory

Summary

Hyperarithmetical theory studies the hyperarithmetical sets of natural numbers, those computable from an iterate of the Turing jump along a computable ordinal. The computable ordinals are those below the Church–Kleene ordinal $\omega_1^{CK}$, the least ordinal that is not the order type of a computable well-ordering of $\N$. By a theorem of Kleene, the hyperarithmetical sets are exactly the $\Delta^1_1$ sets of the analytical hierarchy. The theory connects computability theory, descriptive set theory and the study of subsystems of second-order arithmetic.