doubt in set theory

Question

In the Wikipedia post on the Aleph number https://en.wikipedia.org/wiki/Aleph_number it is mentioned that "it is possible to define a cardinal number for every ordinal number $\alpha$, as described below." but what Cardinal number will be linked to $\omega + 1$? when $\aleph_0$ would have already been used for $\omega$.

Answer 1

If you define each aleph as an ordinal, the association is as follows:

• $\aleph_0$ is the least transfinite ordinal;
• $\aleph_{\alpha+1}$ (often denoted $\omega_{\alpha+1}$) is the least ordinal that can't be injected into $\aleph_\alpha$, so e.g. $\aleph_1$ is the first uncountable ordinal;
• for any limit ordinal $\gamma\ne0$, $\aleph_\gamma:=\bigcup_{\beta\in\gamma}\aleph_\beta$.

(The least ordinal that can't be injected into a given set, never mind ordinal, is always well-defined.)

This associates $0$ to $\aleph_0$, $1$ to $\omega_1$ etc. While every ordinal appears in the subscript of some transfinite cardinal, each such cardinal is the cardinal of multiple ordinals.

Answer 2

There are two different notions here that I think you are confusing.

1. For any ordinal $\alpha,$ there is a unique infinite cardinal $\aleph_\alpha$ that is the $\alpha$-th infinite cardinal. In other words, the class of cardinals is well-ordered, so can be indexed by ordinals. This is what is being referred to in your quote.
2. For any ordinal $\alpha$ (or more generally any (well-orderable) set), there is a unique cardinal $|\alpha|$ such that there is a bijection between $\alpha$ and $|\alpha|.$ This is called the cardinality of $\alpha.$

So for instance, there is the first infinite cardinal $\aleph_0,$ then the least uncountable cardinal $\aleph_1,$ and then the next least $\aleph_2,$ and so on, and then $\aleph_\omega$ is the least cardinal greater than all of $\{\aleph_n:n\in\mathbb N\}.$ Then $\aleph_{\omega+1}$ is the next largest after that. This is the cardinal corresponding to $\omega+1$ in the sense described on wikipedia.

On the other hand, for cardinality, we have, for example, $$|\omega| = |\omega+1| = |\omega+25|=|\omega^{14}+\omega\cdot 3 +1| =|\omega^\omega|= \aleph_0.$$ In other words, there is a whole slew of countable ordinals that all are in one to one correspondence with $\omega$. Then we define the first uncountable ordinal $\omega_1$ as the least ordinal that is too big to have a bijection with $\omega$ and we also identify this with the first uncountable cardinal $\aleph_1$ ($\aleph_1$ and $\omega_1$ are the same set... one notation just contextualizes it as a cardinal and the other as an ordinal).

Then there is a whole slew of ordinals with cardinality $\aleph_1,$ starting with $\omega_1,$ $\omega_1+1,$ $\omega_1+2,$ etc. Then we define $\omega_2=\aleph_2$ as the least ordinal that is too big to be in one-to-one correspondence with $\aleph_1.$ So we have $|\alpha| =\aleph_1$ for all $\alpha$ satisfying $\omega_1\le \alpha < \omega_2.$

And so on...