The Relation Between Logic, Set Theory and Topos Theory as it Is Used by Alain Badiou

Abstract

AbstractTopos theory plays, in Alain Badiou’s philosophical model, the role of inner logic of mathematics, given its power to explore possible mathematical universes; whereas set theory, because of its axiomatics, plays the role of ontology. However, in category theory, which is a vaster theory, topos theory embodies a particular axiomatic choice, the fundamental consequence of which consists in imposing an internal intuitionist logic, that is a non-contradictory logic which gets rid of the principle of excluded middle. Category theory shows that the dual axiomatic choice exists, namely the one imposing, this time, a logic of the excluded middle which accepts true contradictions without deducing from them everything, and this is called a paraconsistent logic. Therefore, after recalling the basics of category and topos theory necessary to demonstrate the categorical duality of paracompleteness (i.e. intuitionism) and paraconsistency, we will be able to introduce into Badiou’s thought category theory seen as a logic of the possible ontologies, a logic which demonstrates the strong symmetry of the axioms of excluded middle and of non-contradiction.KeywordsAlain BadiouCategory theorySet theoryToposIntuitionist logicParaconsistent logicHeyting algebraBrouwer algebraInternal logicOpen subsetsClosed subsetsCategorical dualityClosed Cartesian categoryClosed co-Cartesian categoryMathematics Subject Classification (2000)Primary 00A30 · Secondary 03G3018A15