Ranked structures and arithmetic transfinite recursion
Noam Greenberg, Antonio Montalbán · Transactions of the American Mathematical Society · 2007
A T R 0 \mathsf {ATR}_0 is the natural subsystem of second-order arithmetic in which one can develop a decent theory of ordinals. We investigate classes of structures which are in a sense the “well-founded part" of a larger, simpler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian p p -groups, and countable, compact topological spaces. Using computable reductions between these classes, we show that Arithmetic Transfinite Recursion is the natural system for working with them: natural statements (such as comparability of structures in the class) are equivalent to A T R 0 \mathsf {ATR}_0 . The reductions themselves are also objects of interest.