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3 results

  • A BOREL MAXIMAL COFINITARY GROUP

  • Part of
  • HAIM HOROWITZ, SAHARON SHELAH
  • Journal:
    The Journal of Symbolic Logic , First View
    Published online by Cambridge University Press:
    11 December 2023, pp. 1-14
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  • We construct a Borel maximal cofinitary group.

  • CONSTRUCTING MAXIMAL COFINITARY GROUPS

  • Part of
  • DAVID SCHRITTESSER
  • Journal:
    Nagoya Mathematical Journal / Volume 251 / September 2023
    Published online by Cambridge University Press:
    30 January 2023, pp. 622-651
    Print publication:
    September 2023
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  • Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in $\mathsf {ZF}$, i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.

  • NONDECREASING FUNCTIONS, EXCEPTIONAL SETS AND GENERALIZED BOREL LEMMAS

  • Part of
  • R. G. HALBURD, R. J. KORHONEN
  • Journal:
    Journal of the Australian Mathematical Society / Volume 88 / Issue 3 / June 2010
    Published online by Cambridge University Press:
    11 June 2010, pp. 353-361
    Print publication:
    June 2010
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  • According to the classical Borel lemma, any positive nondecreasing continuous function T satisfiesT(r+1/T(r))≤2T(r) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. In this paper, we consider generalizations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.