The rearrangement number

  • A. Blass, J. Brendle, W. Brian, J. D. Hamkins, M. Hardy, and P. B. Larson, “The rearrangement number,” , 2016. (manuscript under review)  
    @ARTICLE{BlassBrendleBrianHamkinsHardyLarson:TheRearrangementNumber,
    author = {Andreas Blass and J\"org Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson},
    title = {The rearrangement number},
    journal = {},
    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 = {},
    }

Abstract.  How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We show that the minimum number of permutations needed for this purpose, which we call the rearrangement number, is uncountable, but whether it equals the cardinal of the continuum is independent of the usual axioms of
set theory. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied
cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally we deal briefly with some variants concerning rearrangements by a special sort of permutations and with rearranging some divergent series to become (conditionally) convergent.

This project started with Michael Hardy’s question on MathOverflow, How many rearrangements must fail to alter the value of a sum before you conclude that none do? I had proposed in my answer that we should think of the cardinal in question as a cardinal characteristic of the continuum, the rearrangement number, since we could prove that it was uncountable and that it was the continuum under MA, and had begun to separate it from other familiar cardinal characteristics. Eventually, the research effort grew into the collaboration of this paper. What a lot of fun!

Colloquium talk at Vassar | Lecture notes talk at CUNY | the original MathOverflow question

Here are the lecture notes for an introductory talk on the topic I had given at the Vassar College Mathematics Colloquium

The rearrangement number: how many rearrangements of a series suffice to verify absolute convergence? Mathematics Colloquium at Penn, September 2016

This will be a talk for the Mathematics Colloquium at the University of Pennsylvania, Wednesday, September 14, 2016, 3:30 pm, tea at 3 pm, in the mathematics department.

UPenn Campus
Abstract. The well-known 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. But 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 exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and, time permitting, an account of Freiling’s axiom of symmetry.

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

Related MathOverflow post: How many rearrangements must fail to alter the value of a sum before you conclude that none do?

The rearrangement number: how many rearrangements of a series suffice to verify absolute convergence? Vassar Math Colloquium, November 2015

This will be a talk for the Mathematics Colloquium at Vassar College, November 10, 2015, tea at 4:00 pm, talk at 4:15 pm, Rockefeller Hall 310

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 introduced just recently, 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 exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number 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 current joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.

My notes are available here:

The rearrangement number, CUNY set theory seminar, November 2015

This will be a talk for the CUNY Set Theory Seminar on November 6, 2015.

The Riemann rearrangement theorem states that a convergent real series $\sum_n a_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $\sum_n a_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if the series is merely conditionally convergent, then one may find rearrangements for which the new sum $\sum_n a_{p(n)}$ has any desired (extended) real value or which becomes non-convergent.  In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence.

Specifically, we define the rearrangement number $\newcommand\rr{\mathfrak{rr}}\rr$ (“double-r”), a new cardinal characteristic of the continuum, to be the smallest cardinality of a set $P$ of permutations of the natural numbers, such that if a convergent real series $\sum_n a_n$ remains convergent and with the same sum after all rearrangements $\sum_n a_{p(n)}$ by a permutation $p\in P$, then it is absolutely convergent. The corresponding rearrangement number for sums, denoted $\newcommand\rrsum{\rr_{\scriptscriptstyle\Sigma}}
\rrsum$, is the smallest cardinality of a family $P$ of permutations, such that if a series $\sum_n a_n$ is conditionally convergent, then there is a rearrangement $\sum_n a_{p(n)}$, by some permutation $p \in P$, which converges to a different sum. We investigate the basic properties of these numbers, and explore their relations with other cardinal characteristics of the continuum. Our main results are that $\mathfrak{b}\leq\rr\leq\mathop{\bf non}(\mathcal{M})$, that $\mathfrak{d}\leq \rrsum$, and that $\mathfrak{b}<\rr$ is relatively consistent.

MathOverflow question | CUNY Set Theory Seminar

The continuum hypothesis and other set-theoretic ideas for non-set-theorists, CUNY Einstein Chair Seminar, April, 2015

At Dennis Sullivan’s request, I shall speak on set-theoretic topics, particularly the continuum hypothesis, for the Einstein Chair Mathematics Seminar at the CUNY Graduate Center, April 27, 2015, in two parts:

  • An introductory background talk at 11 am, Room GC 6417
  • The main talk at 2 – 4 pm, Room GC 6417

I look forward to what I hope will be an interesting and fruitful interaction. There will be coffee/tea and lunch between the two parts.

Abstract. I shall present several set-theoretic ideas for a non-set-theoretic mathematical audience, focusing particularly on the continuum hypothesis and related issues.

At the introductory background talk, in the morning (11 am), I shall discuss and prove the Cantor-Bendixson theorem, which asserts that every closed set of reals is the union of a countable set and a perfect set (a closed set with no isolated points), and explain how it led to Cantor’s development of the ordinal numbers and how it establishes that the continuum hypothesis holds for closed sets of reals. We’ll see that there are closed sets of arbitrarily large countable Cantor-Bendixson rank. We’ll talk about the ordinals, about $\omega_1$, the long line, and, time permitting, we’ll discuss Suslin’s hypothesis.

At the main talk, in the afternoon (2 pm), I’ll begin with a discussion of the continuum hypothesis, including an explanation of the history and logical status of this axiom with respect to the other axioms of set theory, and establish the connection between the continuum hypothesis and Freiling’s axiom of symmetry. I’ll explain the axiom of determinacy and some of its applications and its rich logical situation, connected with large cardinals. I’ll briefly mention the themes and goals of the subjects of cardinal characteristics of the continuum and of Borel equivalence relation theory.  If time permits, I’d like to explain some fun geometric decompositions of space that proceed in a transfinite recursion using the axiom of choice, mentioning the open questions concerning whether there can be such decompositions that are Borel.

Dennis has requested that at some point the discussion turn to the role of set theory in the foundation for mathematics, compared for example to that of category theory, and I would look forward to that. I would be prepared also to discuss the Feferman theory in comparison to Grothendieck’s axiom of universes, and other issues relating set theory to category theory.