Cohesive Powers of Linear Orders

2019/01/15 by Rumen Dimitrov, Valentina Harizanov, Andrey Morozov +3 · 1 citation
Mathematics · #math.LO #msc:03C57 #msc:03D45 #msc:03C20

paper · pdf · doi:10.1007/978-3-030-22996-2_15

arxiv created 2019/01/15 · arxiv updated 2019/08/28

Abstract

Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers Π_C% L for familiar computable linear orders L. If % L is isomorphic to the ordered set of natural numbers ℕ and has a computable successor function, then Π_CL is isomorphic to ℕ+ℚ× ℤ. Here, + stands for the sum and × for the lexicographical product of two orders. We construct computable linear orders L₁ and L₂ isomorphic to ℕ, both with noncomputable successor functions, such that Π_CL₁\mathbb is isomorphic to ℕ+% ℚ× ℤ, while Π_CL₂ is not. While cohesive powers preserve all Π₂⁰ and Σ₂⁰ sentences, we provide new examples of Π₃⁰ sentences Φ and computable structures % M such that M\vDash Φ while Π_CM% \vDash \urcorner Φ.

Cited by

• Tennenbaum-like theorems for cohesive powers