My instinct tells me no. I would like to imagine, since most real numbers are uncomputable, there exist two real uncomputable numbers $\alpha,\beta$ such that $\alpha+\beta=1$ but I certainly don't know how to show it.
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1$\begingroup$ The computable numbers are a field, therefore an subgroup of $(\Bbb R,+)$. The complement of a subgroup is never non-empty and addition-closed. $\endgroup$– Sassatelli GiulioCommented Sep 5 at 1:11
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1$\begingroup$ @SassatelliGiulio: Is it or isn't it addition-closed? Your sentence can mean either. $\endgroup$– ptsCommented Sep 5 at 13:12
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3$\begingroup$ @pts Whichever is true is mine $\endgroup$– Sassatelli GiulioCommented Sep 5 at 14:23
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1$\begingroup$ @SassatelliGiulio: pts's question was legitimate! I would have said: "The complement of a subgroup can never be both non-empty and addition-closed." $\endgroup$– TonyKCommented Sep 5 at 16:59
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The set of non-computable numbers is not closed under addition. Take some non-computable number $a$. Then $1-a$ and $1+a$ are also non-computable, but their sum is 2, a computable number.
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$\begingroup$ Ahh, so simple. Thank you. I was led to this question because I was wondering whether the set of non computable numbers could function as a field, or a n-dimensional vector space. But I guess not $\endgroup$ Commented Sep 5 at 1:06
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