$\mathbb{R^+}$ is the disjoint union of two nonempty sets, each closed under addition. I saw Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition. and have a question related to the answer (I'm not sure if this is the right place to post it);
Why don't we just take
$\mathcal{A}=\{A\subseteq \mathbb{R^+}\ :\text{A is closed under addition and all elements of A are irrational}\}$,
$\mathcal{B}=\{B\subseteq \mathbb{R^+}\ :\text{B is closed under addition and all elements of B are rational}\}$
and order both of them with the partial order $\subseteq$ so that they will satisfy the chain condition? Then, they have maximal elements $\bigcup{\mathcal{A}}$ and $\bigcup{\mathcal{B}}$ respectively. So it remains to verify that $\bigcup{\mathcal{A}}$ is the set of all irrationals and $\bigcup{\mathcal{B}}$ is the set of all rationals. Is this approach correct?

There is a reason why you shouldn't try to explicitly construct such a decomposition. If $\mathbb{R}^{+}$ is a disjoint union of $A$ and $B$, where both $A$ and $B$ are closed under addition then neither one of $A$ and $B$ is either Lebesgue measurable or has Baire property since if $X$ is either a non meager set of reals with Baire property or $X$ has positive Lebesgue measure then $X + X$ has non empty interior. It is well known that such sets cannot be provably defined within set theory (ZFC).
