An ordinal is recursive just if there exists a decidable order on the natural numbers that is order-isomorphic to it.
Rogers, Hartley. 1987. Theory of Recursive Functions and Effective Computability. MIT Press.
@book{rogers_1987,
title = {Theory of recursive functions and effective computability},
isbn = {0-262-68052-1},
publisher = {MIT Press},
author = {Rogers, Hartley},
year = {1987},
}
Odifreddi, Piergiorgio. 1992. Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers. Elsevier.
@book{odifreddi_1992,
title = {Classical recursion theory: the theory of functions and sets of natural numbers},
isbn = {978-0-08-088659-6},
publisher = {Elsevier},
author = {Odifreddi, Piergiorgio},
year = {1992},
}