Seminar
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| Location: | SLMath: Online/Virtual |
Hilbert’s Tenth Problem was the only decision problem among his twenty-three problems. Precise mathematical theory of (in)computability and its interaction with number theory led to the negative solution of the problem. The seminar will focus on modern topics on computability-theoretic phenomena in number-theoretic and other algebraic and model-theoretic structures.
To participate in this seminar, please register here: https://www.msri.org/seminars/25206
Cohesive Powers Of Linear Orders
To participate in this seminar, please register here: https://www.msri.org/seminars/25206
Abstract:
A cohesive power of a computable structure is an effective analog of an ultrapower, where a cohesive set acts as an ultrafilter. We investigate the following question. Fix a cohesive set C and a computably presentable structure A, and consider the various computable copies B of A. How do the cohesive powers of B by C vary as B varies?
Let omega, zeta, and eta denote the respective order-types of (N, <), (Z, <), and (Q, <). We take omega as our computably presentable structure, and we consider the cohesive powers of its computable copies. If L is a computable copy of omega that is computably isomorphic to the standard presentation (N, <), then all of L’s cohesive powers have order-type omega + (zeta x eta), which is familiar as the order-type of countable non-standard models of PA.
We show that it is possible to realize a variety of order-types other than omega + (zeta x eta) as cohesive powers of computable copies of omega (that are necessarily not computably isomorphic to the standard presentation). For example, we show that if C is a co-c.e. cohesive set, then there is a computable copy L of omega where the cohesive power of L by C has order-type omega + eta. More generally, we show that it is possible to achieve order-types of the form omega + certain shuffle sums as cohesive powers of computable linear orders of type omega.
This work is joint with Rumen Dimitrov, Valentina Harizanov, Andrey Morozov, Alexandra Soskova, and Stefan Vatev.
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Slides
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Cohesive Powers Of Linear Orders
| H.264 Video | 25186_28684_8603_Cohesive_Powers_of_Linear_Orders.mp4 |