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SUMMARY:Alexandra Soskova (Sofia University St. Kliment Ohridski)
DTSTART:20230223T190000Z
DTEND:20230223T200000Z
DTSTAMP:20260423T035750Z
UID:OLS/121
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/121/">Co
 hesive Powers of Linear Orders</a>\nby Alexandra Soskova (Sofia University
  St. Kliment Ohridski) as part of Online logic seminar\n\n\nAbstract\nCohe
 sive powers of computable structures are effective analogs of ultrapowers\
 , where cohesive sets play the role of ultrafilters. The aim is also to co
 mpare and contrast properties of cohesive powers with those of classical\n
 ultrapowers. Classically\, an ultrapower of a structure is elementarily eq
 uivalent to the base structure by\n&#321\;o&#347\;'s theorem. Effectively\
 ,   &#321\;o&#347\;'s theorem holds for cohesive powers of decidable struc
 tures. For cohesive powers of $n$-decidable structures\,  &#321\;o&#347\;'
 s  theorem need only\nhold up to $\\Delta_{n+3}$-expressible sentences. In
  fact\, every $\\Sigma_{n+3}$ sentence true of an $n$-decidable\nstructure
  is also true of all of its cohesive powers\, but this is optimal in gener
 al. Classically\, ultrapowers of isomorphic structures over a fixed ultraf
 ilter are isomorphic. Effectively\,\ncohesive powers of computably isomorp
 hic computable structures over a fixed cohesive\nset are isomorphic. Howev
 er\, it is possible for isomorphic (but not computably\nisomorphic) comput
 able structures to have non-elementarily equivalent (hence non-isomorphic)
 \ncohesive powers. Classically\, the Keisler–Shelah theorem states that 
 two structures are elementarily equivalent if\nand only if there is an ult
 rafilter over which the corresponding\nultrapowers are isomorphic. Effecti
 vely\, an analogous result holds for decidable structures.\nIf the structu
 res are computable that are not necessarily decidable\, then the\neffectiv
 e version of the Keisler–Shelah theorem can fail in either direction. Cl
 assically\, for a countable language\, ultrapowers over countably incomple
 te ultrafilters are $\\aleph_1$-saturated. Effectively\, cohesive powers o
 f decidable structures are recursively saturated. Furthermore\, cohesive p
 owers of n-decidable structures are $\\Sigma_n$-recursively saturated. Mos
 t interestingly\, if the cohesive set is assumed to be co-c.e.\, then we o
 btain an additional level of saturation: cohesive powers of n-decidable st
 ructures over co-c.e.\ncohesive sets are $\\Sigma_{n+1}$-recursively satur
 ated.\n\n\nWe investigate the cohesive powers of computable linear orders\
 , with special emphasis on computable copies of $\\omega$.  If $\\mathcal{
 L}$ is a computable copy of $\\omega$ that is computably isomorphic to the
  standard presentation of $\\omega$\, then every cohesive power of $\\math
 cal{L}$ has order-type $\\omega + \\zeta\\eta$.  However\, there are compu
 table copies of $\\omega$\, necessarily not computably isomorphic to the s
 tandard presentation\, having cohesive powers not elementarily equivalent 
 to $\\omega + \\zeta\\eta$.  For example\, we show that there is a computa
 ble copy of $\\omega$ with a cohesive power of order-type $\\omega + \\eta
 $.  Our most general result is that if $X \\subseteq \\mathbb N \\setminus
  \\{0\\}$  is a Boolean combination of $\\Sigma_2$ sets\, thought of as a 
 set of finite order-types\, then there is a computable copy of $\\omega$ w
 ith a cohesive power of order-type $\\omega + \\bm{\\sigma}(X \\cup \\{\\o
 mega + \\zeta\\eta + \\omega^*\\})$\, where $\\bm{\\sigma}(X \\cup \\{\\om
 ega + \\zeta\\eta + \\omega^*\\})$ denotes the shuffle of the order-types 
 in $X$ and the order-type $\\omega + \\zeta\\eta + \\omega^*$.  Furthermor
 e\, if $X$ is finite and non-empty\, then there is a computable copy of $\
 \omega$ with a cohesive power of order-type $\\omega + \\bm{\\sigma}(X)$.\
 n\nThis is a joint work with Rumen Dimitrov\, Valentina Harizanov\, Andrey
  Morozov\, Paul Shafer  and Stefan Vatev.\n\nIt was partially  supported b
 y Bulgarian National Science Fund KP-06-Austria-04/06.08.2019\,\nFNI-SU 80
 -10-134/20.05.2022.\n
LOCATION:https://researchseminars.org/talk/OLS/121/
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