Computability and Complexity pp 623-632 | Cite as

# A Note on the Differences of Computably Enumerable Reals

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## Abstract

We show that given any non-computable left-c.e. real \(\alpha \) there exists a left-c.e. real \(\beta \) such that \(\alpha \ne \beta +\gamma \) for all left-c.e. reals and all right-c.e. reals \(\gamma \). The proof is non-uniform, the dichotomy being whether the given real \(\alpha \) is Martin-Löf random or not. It follows that given any universal machine *U*, there is another universal machine *V* such that the halting probability \(\Omega _U\) of *U* is not a translation of the halting probability \(\Omega _V\) of *V* by a left-c.e. real. We do not know if there is a uniform proof of this fact.

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