Every Nonrecursive Many-One Degree Contains Either One or Infinitely Many Finite-One Degrees

This paper proves that every nonrecursive many-one degree contains either exactly one or infinitely many finite-one degrees. This gives a negative answer to Open Question 2 of Richter, Stephan, and Zhang~\cite[p.~17]{RSZ}. Earlier work established that, for almost every AsubseteqN with respect to the standard product measure on 2^N, the degree [A]_{m} contains an infinite antichain of finite-one degrees~Cintioli. Thus the question had already received an almost-sure negative answer; the present theorem settles it for every nonrecursive many-one degree.