Compactness arguments with effectively closed sets for the study of relative randomness (Q2907048)
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scientific article; zbMATH DE number 6078041
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Compactness arguments with effectively closed sets for the study of relative randomness |
scientific article; zbMATH DE number 6078041 |
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Compactness arguments with effectively closed sets for the study of relative randomness (English)
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5 September 2012
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\(\Pi ^{0}_{1}\) classes
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relative Martin-Löf randomness
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K-trivials
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Turing degrees
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LR-degrees
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In the paper under review, the author presents a variety of compactness arguments with \(\Pi^0_1\)-classes which yield results about relative randomness, and in particular properties of the LR-degrees. The methods in the paper can be viewed as a twist of forcing arguments. The role of compactness is to make sure the existence of ``generic'' reals. NEWLINENEWLINENEWLINE It should be pointed out that Question 2.5 has been negatively answered by the reviewer. Question 3.6 was also negatively answered by a bunch of most recent results due to Day, Miller and others. Theorem 4.1 can also be obtained by applying the result, which is due to the author and Lewis, that there is a perfect \(\Pi^0_1\)-class in which every real is LR-below \(\emptyset'\).
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