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THE CLASSICAL CONTINUUM WITHOUT POINTS

Published online by Cambridge University Press:  25 March 2013

GEOFFREY HELLMAN  and
STEWART SHAPIRO
GEOFFREY HELLMAN*
Affiliation:
Department of Philosophy, University of Minnesota
STEWART SHAPIRO*
Affiliation:
Department of Philosophy, The Ohio State University
*
*DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF MINNESOTA, 831 HELLER HALL, 271-19TH AVENUE SOUTH, MINNEAPOLIS, MN 55455
†DEPARTMENT OF PHILOSOPHY, THE OHIO STATE UNIVERSITY, 350 UNIVERSITY HALL, 230 NORTH OVAL MALL, COLUMBUS, OH 43210
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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.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 3 , September 2013 , pp. 488 - 512
Copyright
Copyright © Association for Symbolic Logic 2013 

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