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I want to know whether for all infinite cardinals $\alpha$, a chain in $(P(\alpha),\subseteq)$ with cardinality greater than $\alpha$ necessarily exists in ZFC. For $\alpha=\aleph_0$, I know using Dedekind cuts, a chain of size $2^{\aleph_0}$ exists. This answer shows a chain of size $\aleph_2$ exists when $\alpha=\aleph_1$. But I don't know if the same is true for larger cardinalities.

Lucenaposition
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    Assuming GCH, then the same proof as the CH case of the linked answer shows the answer is yes. – David Gao Jun 29 '24 at 04:38
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    in James E. Baumgartner's paper Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic. 10 (1976) 401–439, he defines a property $D(\kappa,\lambda)$ which is equivalent (Theorem 2.1(b)) to the assertion that $P(\kappa)$ contains a chain of cardinality $\lambda$. As I haven't read the paper, I don't know if it answers your question. Of course the paper is almost 50 years old, so the results may have been improved. – bof Jun 29 '24 at 04:44

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