ClassInfo

This course is structured around an about-to-appear book from Oxford University Press, by Stewart Shapiro and myself, entitled "Varieties of Continua: from Regions to Points and Back". The book develops several new axiom systems, with theorems and proofs of results on recovering the now standard point-based classical continua going back to Dedekind and Cantor on the non-punctiform basis of regions of a space and key relations on them. Surprisingly, despite the fact that the broad outline of such a program was presented by Alfred North Whitehead early in the 20^th Century, only very recently has it been carried out in detail (where the devil resides!). As we will see early on in the course, even Alfred Tarski's ingenious efforts fell quite short of fulfilling the program. The original approach taken in our book - using mereology (theory of parts and wholes, a key tool in Tarski's efforts) along with 2d-order logic (equivalently, a logic of plural quantifiers or a weak set theory) - achieves full-scale reductions of point-based to regions-based classical continua (Rⁿ⁾, and we show how to recover metrical structure of n-dimensional Euclidean and some non-Euclidean geometries, as well. Surprisingly, our axioms - very elementary in character - suffice to derive regions-based versions of the Archimedean properties of these spaces and even the full axiom of Dedekind continuity (least upper bound principle), resulting in streamlined, unified systems. As we also demonstrate, however, our regions based concepts are definable in the classical punctiform systems, and our axioms derivable as theorems on translation, demonstrating the full mathematical equivalence of the two approaches.