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SUMMARY:Darío García (Universidad de los Andes)
DTSTART:20230921T180000Z
DTEND:20230921T190000Z
DTSTAMP:20260423T021357Z
UID:OLS/133
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/133/">Ps
 eudofiniteness and measurability of the everywhere infinite forest</a>\nby
  Darío García (Universidad de los Andes) as part of Online logic seminar
 \n\n\nAbstract\nA structure M is said to be pseudofinite if every first-or
 der sentence that is true in M has a finite model\, or equivalently\, if M
  is elementarily equivalent to an ultraproduct of finite structures. For t
 his kind of structures\, the fundamental theorem of ultraproducts ( Los' T
 heorem) provides a powerful connection between finite and infinite sets\, 
 which can sometimes be used to prove qualitative properties of large finit
 e structures using combinatorial methods applied to non-standard cardinali
 ties of definable sets.\n\nThe concept of measurable structures was define
 d by Macpherson and Steinhorn in [2] as a method to study infinite structu
 res with strong conditions of finiteness and definability for the sizes of
  definable sets. The most notable examples are the ultraproducts of asympt
 otic classes of finite structures (e.g.\, the class of finite fields or th
 e class of finite cyclic groups). Measurable structures are supersimple of
  finite SU-rank\, but recent generalizations of this concept are more flex
 ible and allow the presence of structures whose SU-rank is possibly infini
 te.\n\nThe everywhere infinite forest is the theory of an acyclic graph G 
 such that every vertex has infinite degree. It is a well-known example of 
 an omega-stable theory of infinite rank. In this talk we will take this st
 ructure as a motivating example to introduce all the concepts mentioned ab
 ove\, showing that it is pseudofinite and giving a precise description of 
 the sizes of their definable sets. In particular\, these results provide a
  description of forking and U-rank for the infinite everywhere forest in t
 erms of certain pseudofinite dimensions\, and also show that it is a gener
 alized measurable structure that can be presented as the ultraproduct of a
  multidimensional exact class of finite graphs. These results are joint wo
 rk with Melissa Robles\, and can be found in [1].\n\nReferences:\n\n[1] Da
 río García and Melissa Robles. Pseudofiniteness and measurability of the
  everywhere infinite forest. Available at arXiv: https://arxiv.org/pdf/230
 9.00991.pdf\n\n[2] Dugald Macpherson and Charles Steinhorn. One-dimensiona
 l asymptotic classes of finite structures\, Transactions of the American M
 athematical Society\, vol. 360 (2008)\n
LOCATION:https://researchseminars.org/talk/OLS/133/
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