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The Classical Continuum without Points

Hellman, Geoffrey and Shapiro, Stewart (2012) The Classical Continuum without Points. In: [2012] Philosophy of Science Assoc. 23rd Biennial Mtg (San Diego, CA) > PSA 2012 Contributed Papers.

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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 logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, 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 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.


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    Item Type: Conference or Workshop Item (UNSPECIFIED)
    Keywords: continuum, punctiform, non-punctiform, decomposable, indecomposable, real numbers, classical analysis
    Subjects: Specific Sciences > Mathematics
    Conferences and Volumes: [2012] Philosophy of Science Assoc. 23rd Biennial Mtg (San Diego, CA) > PSA 2012 Contributed Papers
    Depositing User: Prof Geoffrey Hellman
    Date Deposited: 06 Nov 2012 19:14
    Last Modified: 10 Nov 2012 09:58
    Item ID: 9409
    URI: http://philsci-archive.pitt.edu/id/eprint/9409

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