Working in ZF(wihtout choice), according to this answer: $\omega_2$ is not the countable union of countable sets.
Can $\omega_2$ be the order type of a countable union of countable sets of ordinals?
2026-04-06 05:18:54.1775452734
Can $\omega_2$ be the order type of a countable union of countable sets of ordinals?
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The property of being a countable union of countable sets is preserved by bijections (just take the images of the sets under the bijection). So, since $\omega_2$ is not a countable union of countable sets, neither is any set of cardinality $\aleph_2$.