From the text: In this note we would like to refer to a theorem and its proof that were given in our paper “Regressive isols and comparability” [Z. Math. Logik Grundlagen Math. 22, 403–412 (1976; Zbl 0358.02062)], Theorem 3]. Both of these are incorrect. We would like to reflect on the particular result.
Let \(a\) and \(b\) be isols, and let \(a\) be a regressive isol. Let us assume that \(a\leq^*b\leq^*a+1\).
Our Theorem 3 of the paper concluded that then the isols \(b\) and \(a+b\) are also regressive isols. However, neither be so. One may find in T. McLaughlin’s book [Regressive sets and the theory of isols. Marcel Dekker Inc., New York (1982; Zbl 0484.03025), Theorem 8.3] an appropriate counter-example. In addition, one may refer to Theorem 8.6 of McLaughlin’s book to see one variety of valid conclusion that the above assumptions will yield.
Early when the paper appeared both Dr. Charles Applebaum and Dr. Thomas McLaughlin each pointed out to us the non-valid nature of the proof to our Theorem 3. I express my appreciation to each of them.