Two Questions on "Set Theory"

Question

Q1

Prove that the set $\mathbb{R}^+$ of the positive reals can be written as the union of two non-empty sets, say $A$ & $B$ , both these set are closed unnder addition.

Q2

$\aleph_\omega ,\aleph_{\omega_1}$ and $\aleph_{\omega_2}$ It is true that these three are smallest singular cardinals (by order, I mean the $1^{st}$, $2^{nd}$ and $3^{rd}$).

Answer 1

The first answer is the same usual use of Zorn's lemma. Simply find the right partial order (hint, pairs of disjoint sets closed under addition).

The full details appear in Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.

The second question is plain false. Recall that $\aleph_\alpha$ is singular when $\alpha$ is a limit ordinal and $\alpha<\omega_\alpha$ (it is possible to have singular cardinal with equality, though). Therefore the first three singular cardinals would be those whose indices are the first three limit ordinals.

Remember that there are $\aleph_1$ limit ordinals smaller than $\omega_1$, so it cannot possibly be the second singular cardinal.