The rearrangement and subseries numbers: how much convergence suffices for absolute convergence? Mathematics Colloquium, University of Münster, January 2019

This will be a talk for the Mathematics Colloquium at the University of Münster, January 10, 2019.

Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The subseries number is defined similarly, as the smallest number of subseries whose convergence suffices to test a series for absolute convergence. The exact values of the rearrangement and subseries numbers turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement and subseries numbers into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.

This talk is based in part on joint work with Andreas Blass, Joerg Brendle, Will Brian, myself, Michael Hardy and Paul Larson.

  • The rearrangement number.
    • A. Blass, J. Brendle, W. Brian, J. D. Hamkins, M. Hardy, and P. B. Larson, “The rearrangement number,” ArXiv e-prints, 2016. (manuscript under review)  
      @ARTICLE{BlassBrendleBrianHamkinsHardyLarson:TheRearrangementNumber,
      author = {Andreas Blass and Jörg Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson},
      title = {The rearrangement number},
      journal = {ArXiv e-prints},
      year = {2016},
      volume = {},
      number = {},
      pages = {},
      month = {},
      note = {manuscript under review},
      url = {http://jdh.hamkins.org/the-rearrangement-number},
      eprint = {1612.07830},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      abstract = {},
      keywords = {under-review},
      source = {},
      }

  • The subseries number.
    • J. Brendle, W. Brian, and J. D. Hamkins, “The subseries number,” ArXiv e-prints, 2018. (manuscript under review)  
      @ARTICLE{BrendleBrianHamkins:The-subseries-number,
      author = {Jörg Brendle and Will Brian and Joel David Hamkins},
      title = {The subseries number},
      journal = {ArXiv e-prints},
      year = {2018},
      volume = {},
      number = {},
      pages = {},
      month = {},
      note = {manuscript under review},
      url = {http://jdh.hamkins.org/the-subseries-number},
      eprint = {1801.06206},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      abstract = {},
      keywords = {under-review},
      source = {},
      }

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