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If you say something is decidable unqualified then that means it's decidable by a Turing
Machine. There is no way that membership of the language $ \{a\}^\omega$ over the alphabet $ \Sigma=\{a,b\}$ is decidable by a Turing machine. The articles on similar topics seem to make similar dubious or
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It's perfectly standard usage to extend the concept of decidability to other models of computation. See paragraph 2 of
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How can membership of $ \omega$ -regular languages be decidable?
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