Trial and error mathematics I: dialectical and quasi-dialectical systems

We define and study quasi-dialectical 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 positions a` la Lakatos. We prove severalproperties of quasi-dialectical systems and of the sets that they represent, called quasi-dialecticalsets. In particular, we prove that the quasi-dialectical sets are Delta-0-2 sets in the arithmetical hierarchy. We distinguish between “loopless” quasi-dialectal systems, and quasi-dialectical 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 are omega-c.e., the quasi-dialectical sets spread out throughout all classes of the Ershov hierarchy of the Delta-0-2 sets.