Does there exist a set $X$ linearly ordered by $\subset$ where $|\bigcup X|>\sup\{|x|:x\in X\}$? I'm having trouble thinking around constructing one.
I'm fairly certain that finite sets won't work, so I tried considering sets which can be built up inductively like $\mathbb{N}$ or $\mathbb{R}$, but then I think it makes the statement an equality since the supremum covers the limit case.
Is there a set or way one can be constructed which cannot be from this inductively building up? I would think that construction would then imply that it is bigger than the ordinals, a class, and make it not possible to be a set, but albeit my cardinal skills aren't as good as they could be.