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Added:
Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, it is consistent with ZF that every infinite set is a countable union of sets of smaller cardinality. See \cite{Gitik}.
I have expanded out (here) your line
it is consistent with ZF that
to:
it is consistent with Zermelo-Fraenkel-set theory without the axiom of choice that
If thatâ€™s not what is meant, please expand to clarify.
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