Describing free groups. II: \(\Pi ^{0}_{4}\) hardness and no \(\Sigma _{2}^{0}\) basis (Q2844727)
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scientific article; zbMATH DE number 6199337
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Describing free groups. II: \(\Pi ^{0}_{4}\) hardness and no \(\Sigma _{2}^{0}\) basis |
scientific article; zbMATH DE number 6199337 |
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19 August 2013
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free groups
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index set
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computability-theoretic complexity
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0.84401935
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0.8371508
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0.8363968
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0.83543575
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0.83472407
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Describing free groups. II: \(\Pi ^{0}_{4}\) hardness and no \(\Sigma _{2}^{0}\) basis (English)
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For definitions and notations see Part I [\textit{J. Carson} et al., Trans. Am. Math. Soc. 364, No. 11, 5715--5728 (2012; Zbl 1302.03045)]. The study of free groups from a computability-theoretic point of view is continued. It is shown that the descriptions for \(F_\infty\) given in the first part are the best possible. Namely it is proved that \(I(F_\infty)\) and \(I(\mathrm{FrGr})\) are \(m\)-complete \(\Pi^0_4\) and there is a computable copy \(F\) of \(F_\infty\) with no \(\Sigma^0_2\) basis.
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