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Degrees of Enumerations of Countable Wehner-Like Families
- Source :
- Journal of Mathematical Sciences. 256:51-60
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.
- Subjects :
- Statistics and Probability
Class (set theory)
Degree (graph theory)
Applied Mathematics
General Mathematics
010102 general mathematics
01 natural sciences
Spectrum (topology)
010305 fluids & plasmas
Combinatorics
0103 physical sciences
Enumeration
Countable set
Family of sets
0101 mathematics
Turing
computer
Finite set
computer.programming_language
Mathematics
Subjects
Details
- ISSN :
- 15738795 and 10723374
- Volume :
- 256
- Database :
- OpenAIRE
- Journal :
- Journal of Mathematical Sciences
- Accession number :
- edsair.doi...........f4f259cbd297b53177c80fb43d1c3ddc