Let $\aleph_{0}$ be the cardinality of the set of all natural numbers, $\aleph_{1}$ be the next larger cardinality of a well-ordered set, and so on. Let $\aleph_{\omega}=\operatorname{sup}\{ \aleph_{n}| n\in \{0,1,2,.. \} \}$. I am not familiar with set theory. So the question may be trivial.
According to the wikipedia https://en.wikipedia.org/wiki/Aleph_number, $2^{\aleph_{0}}\not = \aleph_{\omega}$ under the ZF set theory. My first question can you give the reference for this? I can not find the reference on wikipedia.
On the wikipedia, there is a sentence: "For any natural number n ≥ 1, we can consistently assume that $2^{\aleph_{0}}=\aleph_{n}$, and moreover it is possible to assume that $2^{\aleph_{0}}$ is as least as large as any cardinal number we like. "
I think the first part of the sentence means that if we assume $2^{\aleph_{0}}=\aleph_{n}$ for $n\geq 1$, there is no contradiction under ZF set theory, am I right? However, I am very confused about the second part of the sentence. Whether this means that if we assume $2^{\aleph_{0}}>\aleph_{\omega}$, there is no contradiction under ZF set theory.
In other word, we only know $2^{\aleph_{0}}\not = \aleph_{\omega}$, but we can not show $2^{\aleph_{0}}< \aleph_{\omega}$ or $2^{\aleph_{0}}>\aleph_{\omega}$. Am I right?
