The classical continuum without points

Geoffrey Hellman, Stewart Shapiro

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

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 a 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, smooth infinitesimal analysis, and Eret Bishop's (1967) constructivism, 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 nonpunctiform 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.

Original languageEnglish (US)
Pages (from-to)488-512
Number of pages25
JournalReview of Symbolic Logic
Volume6
Issue number3
DOIs
StatePublished - Sep 2013

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