Trial and error mathematics II: dialectical sets and quasi-dialectical sets, their degrees, and their distribution within the class of limit sets.

This paper is a continuation of a previous paper, where we have introduced the quasi-dialectical systems, which are abstract deductive systems designed to provide, in line with Lakatos’ views, a formalization of trial and error mathematics more adherent to the real mathematical practice of revision than Magari’s original dialectical systems. In this paper we prove that the two models of deductive systems (dialectical systems and quasi-dialectical systems) have in some sense the same information content, in that they represent two classes of sets (the dialectical sets, and the quasi-dialectical sets, respectively), which have the same Turing degrees (namely, the computably enumerable Turing degrees), and the same enumeration degrees (namely, the Pi-0-1 enumeration de- grees). Nonetheless, dialectical sets and quasi-dialectical sets do not coincide. Even restricting our attention to the so-called loopless quasi-dialectical sets, we show that the quasi-dialectical sets properly extend the dialectical sets. As both classes consist of Delta-0-2 sets, the extent to which the two classes differ is conveniently measured using the Ershov hierarchy: indeed, the dialectical sets are ω-computably enumerable (close inspection also shows that there are dialectical sets which do not lie in any finite level; and in every finite level n ≥ 2 of the Ershov hierarchy there is a dialectical set which does not lie in the previous level); on the other hand, the quasi-dialectical sets spread out throughout all classes of the hierarchy.