The Classical continuum without Points

We develop a point-free construction of the classical onedimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quanti…cation. In some respects this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of classical "actual in…nity". Also, in contrast to intuitionistic, Bishop, and smooth in…nitesimal 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 that they determine an isomorphism with the Dedekind-Cantor structure of R as a complete, separable, ordered …eld. 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 e¤orts related to our own.