The author of my book stated

The Continuum Hypothesis:There exists no set$\displaystyle S$such that

$\displaystyle \aleph_0<|S|<c.$ where $\displaystyle c = |\mathbb{R}|$

He then went on raising a question: Is there as set $\displaystyle S$ such that $\displaystyle |S|>c$?

Following the question, he proved that $\displaystyle |A|<|P(a)|$, and at the end of the proof, he said the proof shows that there is no largest set. In particular, there is a set S with $\displaystyle |S|>c$.

Remark: I have not a clue what he meant in blue texts.

Question: Is there or is there not that $\displaystyle |S|>c$?