Is this countable? If it is, why?For purposes of proving $\phi(\omega_1)$ true, select $\omega_1$ as the order type for $S$:

$$S = \{1\}_1, \{2, 3\}_2, \{4,5,6\}_3, \dots, \{\omega, \omega +1, \omega +2, \dots \}_{\omega}, \{\omega \cdot 2, \omega \cdot 2 +1, \omega \cdot 2 + 2, \dots, \omega \cdot 3\}_{\omega + 1}, \dots, \text{ (inclusive of all ordinals in } \omega_1 \text{)}$$

If it is not, then why is the following true:

Then we have an uncountable set, $\omega_1$, being equal to a countable union of countable sets.