Trial and error mathematics: dialectical systems and completions of theories.

*(English)*Zbl 1444.03142Summary: This paper is part of a project that is based on the notion of a dialectical system, introduced by R. Magari [Ann. Mat. Pura Appl. (4) 98, 119–152 (1974; Zbl 0286.02051)] as a way of capturing trial and error mathematics. In [the first author et al., Rev. Symb. Log. 9, No. 2, 299–324 (2016; Zbl 1384.03079); ibid. 9, No. 4, 810–835 (2016; Zbl 1392.03017)], we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasi-dialectical systems, that enrich Magari’s systems with a natural mechanism of revision. In the present paper we consider a third class of systems, that of \(p\)-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about \(p\)-dialectical systems and the sets that they represent. Then we focus on the completions of first-order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and \(q\)-dialectical systems coincide with respect to the completions that they can represent. Yet, \(p\)-dialectical systems are more powerful; we exhibit a \(p\)-dialectical system representing a completion of Peano Arithmetic that is neither dialectical nor \(q\)-dialectical.