Foundations for real analysis traditionally comes in two kinds: classical (as per Cauchy, Weierstraß, Dedekind, etc.) and constructive (Brouwer, Weyl etc.). These are underpinned by, respectively, classical and intuitionistic logics. We seek to establish a solid non-classical foundation for real analysis relying on paraconsistent logics.
Paraconsistent logics are characterized by rejection of the universal validity of the principle ex contradictione quodlibet. It is this principle which, in classical and intuitionistic logics, causes global absurdity in the presence of local contradiction. We build on recent developments in set theory, geometry and arithmetic to investigate models of the continuum capable of supporting both an interesting structure and interesting substructure; inconsistent phenomena that arise need not lead to disaster.