Let the underlying set theory be ZFC. Let $x_1 \subseteq x_2 \subseteq \dots$ and $y_1 \subseteq y_2 \subseteq \dots$ be ascending sequences of sets such that, for every $n \in \{1,2,\dots\}$, $|x_n| = |y_n|$. Is it the case that $\big|\cup_{n =1}^{\infty}x_n\big| = \big|\cup_{n =1}^{\infty}y_n\big|$? If this is not generally true, is it possible to characterize all those—or at least some interesting—cases for which this does hold? Is there a standard terminology for these cases? Can this be generalized to transfinite sequences? Does the answer change if we require that the sequences be strictly increasing, i.e. for every $n \in \{1,2,\dots\}$, $x_n \subsetneq x_{n+1}$ and $y_n \subsetneq y_{n+1}$?
from Hot Weekly Questions - Mathematics Stack Exchange
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