What is the difference between $\omega$ and $\aleph_0$?

Question

The book I'm using says that the cardinality of a set $X$ is the least ordinal $\alpha$ such that $|X| = |\alpha|$. So then $\omega = \aleph_0$, but $\omega + \omega \ne \omega$, while $\aleph_0 + \aleph_0 = \aleph_0$. What am I missing?

Answer 1

You’re talking about two different operations. In $\omega+\omega\ne\omega$ you’re talking about ordinal addition; in $\aleph_0+\aleph_0=\aleph_0$ you’re talking about cardinal addition. When you consider $\omega$ as a cardinal number and perform cardinal arithmetic on it, $\omega+\omega=\omega$. Similarly, the ordinal sum $\omega_1+\omega$ is the ordinal $\omega_1+\omega$, not the ordinal $\omega_1$, but the cardinal sum $\omega_1+\omega$ is the cardinal $\omega_1$.

Those of us who use the $\omega_\alpha$ notation for both ordinals and cardinals must either rely on context to distinguish ordinal and cardinal arithmetic or explicitly specify which one is being used.

Answer 2

$\omega$ is a well ordered set, while $\aleph_0$ is the cardinality of that set. $|\omega|=\aleph_0$. As I've been informed, since $\omega$ is an initial ordinal, it is also considered a cardinal number, so indeed $\omega=\aleph_0$, and $|\omega|=\aleph_0$. However, when we use ordinal arithmetic $|\omega+\omega|=\aleph_0$ as well, even though $\omega+\omega\neq \omega$.

Answer 3

In standard set theory, they are the same set, but seen from different perspectives. The notation $\omega$ emphasizes that it is an ordinal number, whereas the notation $\aleph_0$ emphasizes its role as a cardinal number.

Now whereas $\omega$ and $\aleph_0$ are two different notations for the same set, $+$ and $+$ is the same notation being used for two different operations. The meaning of "$+$" depends on whether we're adding ordinals or cardinals.

Some books use $+_o$ and $+_c$ for ordinal and cardinal addition, but in everyday semi-informal mathematics we just write $+$ for both and let it depend on context which one is meant. (Which is one reason why it is useful to have two different notations for $\mathbb N$ used as an ordinal and cardinal, respectively).