On Cohesive Powers of Linear Orders

Journal of Symbolic Logic 88 (3):947-1004 (2023)
  Copy   BIBTEX

Abstract

Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$\omega $,$\zeta $, and$\eta $denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$\omega $. If$\mathcal {L}$is a computable copy of$\omega $that is computably isomorphic to the usual presentation of$\omega $, then every cohesive power of$\mathcal {L}$has order-type$\omega + \zeta \eta $. However, there are computable copies of$\omega $, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$\omega + \zeta \eta $. For example, we show that there is a computable copy of$\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 $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$, where$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$denotes the shuffle of the order-types inXand the order-type$\omega + \zeta \eta + \omega ^*$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X)$.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,923

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Generalized cohesiveness.Tamara Hummel & Carl G. Jockusch - 1999 - Journal of Symbolic Logic 64 (2):489-516.
Generalized Cohesiveness.Tamara Hummel & Carl Jockusch - 1999 - Journal of Symbolic Logic 64 (2):489-516.
Coloring linear orders with Rado's partial order.Riccardo Camerlo & Alberto Marcone - 2007 - Mathematical Logic Quarterly 53 (3):301-305.
Countably categorical coloured linear orders.Feresiano Mwesigye & John K. Truss - 2010 - Mathematical Logic Quarterly 56 (2):159-163.
An Undecidable Linear Order That Is $n$-Decidable for All $n$.John Chisholm & Michael Moses - 1998 - Notre Dame Journal of Formal Logic 39 (4):519-526.
The Block Relation in Computable Linear Orders.Michael Moses - 2011 - Notre Dame Journal of Formal Logic 52 (3):289-305.
Extending partial orders to dense linear orders.Theodore A. Slaman & W. Hugh Woodin - 1998 - Annals of Pure and Applied Logic 94 (1-3):253-261.
A cohesive set which is not high.Carl Jockusch & Frank Stephan - 1993 - Mathematical Logic Quarterly 39 (1):515-530.

Analytics

Added to PP
2023-03-14

Downloads
13 (#1,061,253)

6 months
7 (#486,539)

Historical graph of downloads
How can I increase my downloads?