The classical continuum without points

Stewart Shapiro

14 Mar 2013, 4pm-5:30pm, Manchester 227

We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary actual infinity. Also, in contrast to intuitionistic analysis, Bishop’s constructive analysis, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our “gunky line” into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence of “indecomposability” from a non-punctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and the interval analogue of Dedekind completeness (least-upper-bound principle), and that they determine an isomorphism with the Dedekind-Cantor structure of R as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.