On \(1\)-degrees inside \(m\)-degrees (Q2062067)

This paper gives a partial answer to a question of \textit{P. Odifreddi} [Bull. Am. Math. Soc., New Ser. 4, 37--86 (1981; Zbl 0484.03024)]: If a c.e. \(m\)-degree contains more than one \(1\)-degree, must it contain an infinite antichain of \(1\)-degrees (i.e. a chain of pairwise incomparable degrees)? \textit{A. N. Degtev} [Algebra Logic 12, 78--89 (1974; Zbl 0338.02023); translation from Algebra Logika 12, 143--161 (1973)] proved this true when the \(m\)-degree is c.e. In this paper, the author proves that the answer is ``yes'' for \(\Delta^0_2\) \(m\)-degrees. It is still unknown whether there is a c.e. \(m\)-degree with one least and infinitely many minimal \(1\)-degrees.