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SUMMARY:Julia Knight (Notre Dame)
DTSTART:20241121T190000Z
DTEND:20241121T200000Z
DTSTAMP:20260423T035817Z
UID:OLS/158
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/158/">Co
 mplexity of well-ordered sets in an ordered Abelian group</a>\nby Julia Kn
 ight (Notre Dame) as part of Online logic seminar\n\n\nAbstract\nWe consid
 er the following three basic problems\, plus some variants.\n\n1. How hard
  is it to say of a countable well-ordering that it has type at least $\\al
 pha$?\n\n2. How hard is it to say of well-ordered sets $A\,B$ in an ordere
 d Abelian group $G$ that the set $A+B = \\{a+b:a\\in A\\ \\&\\ b\\in B\\}$
  has type at least $\\alpha$?\n\n3. How hard is it to say of a well-ordere
 d set $A$ of non-negative elements in an ordered Abelian group $G$ that th
 e set $[A]$ consisting of finite sums of elements of $A$ has type at least
  $\\alpha$? \n\n\nEach problem asks the complexity of membership a smaller
  class $K$\, assuming membership in a larger class $K^*$.  We want to meas
 ure complexity in the Borel and effective Borel hierarchies.  However\, th
 e classes $K^*$ and $K$ are not Borel.  Calvert's notions of complexity an
 d completeness <i>within</i> allow us to measure complexity in the way we 
 want\, setting upper bounds\, and showing that the bounds are sharp.  \n\n
 Authors: Chris Hall\, Julia Knight\, and Karen Lange\n
LOCATION:https://researchseminars.org/talk/OLS/158/
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