Maximum subset of $\mathbb{R}$ with a special property An additional question for the one I asked here I thought it will be better to ask it separately.
While looking at all the subsets of $\mathbb{R}$ with the following property: $a,b\in A \Rightarrow a+b\not \in A$, I thought about trying to find the maximum set of all those subsets. Is it possible to prove that there must be a maximum set that follows the property?

Yes, there must be a $\subseteq$-maximal such set -- by Zorn's lemma:
Let $\mathcal A$ be the set of all of these sets. For any chain $X \subseteq \mathcal A$, we have $\bigcup X \in \mathcal A$ as a $\subseteq$-upper bound for $X$. Hence Zorn's lemma applies.
(Note, however, that this maximal element of $(\mathcal A; \subseteq)$ is not unique -- which is why I wouldn't call it a maximum.)
