TRIAL AND ERROR MATHEMATICS I: DIALECTICAL AND QUASIDIALECTICAL SYSTEMS
Jacopo Amidei, Duccio Pianigiani, Luca San Mauro, Giulia Simi, Andrea Sorbi · The Review of Symbolic Logic · 2016
Abstract We define and study quasidialectical systems, which are an extension of Magari’s dialectical systems, designed to make Magari’s formalization of trial and error mathematics more adherent to the real mathematical practice of revision: our proposed extension follows, and in several regards makes more precise, varieties of empiricist positionsà laLakatos. We prove several properties of quasidialectical systems and of the sets that they represent, called quasidialectical sets. In particular, we prove that the quasidialectical sets are ${\rm{\Delta }}_2^0$ sets in the arithmetical hierarchy. We distinguish between “loopless” quasidialectal systems, and quasidialectical systems “with loops”. The latter ones represent exactly those coinfinite c.e. sets, that are not simple. In a subsequent paper we will show that whereas the dialectical sets areω-c.e., the quasidialectical sets spread out throughout all classes of the Ershov hierarchy of the ${\rm{\Delta }}_2^0$ sets.