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The metamathematics of scattered linear orderings

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Abstract

Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II 11 -CA0 is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR0 is equivalent to a version of the Continuum Hypothesis for linear orderings, which states that every countable linear ordering either has countably many or continuum many Dedekind cuts.

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Authors and Affiliations

  1. Department of Computer Science, Boston College, 02167, Chestnut Hill, MA, USA

    P. Clote

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  1. P. Clote
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Research partially supported by NSF grant # DCR-8606165. AMS Subject Classification 03F35, 03F15, 03D55

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Clote, P. The metamathematics of scattered linear orderings. Arch Math Logic 29, 9–20 (1989). https://doi.org/10.1007/BF01630807

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  • DOI: https://doi.org/10.1007/BF01630807

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