I have seen these three symbols, $\aleph_0$, $\omega$ and $\mathbb{N}$, a lot in my reading (mostly in analysis, I have very limited experience in set theory). I have seen in various places they are used interchangeably, which is confusing for me.
There is no problem that the symbol $\mathbb{N}$ denotes the set of natural numbers. (By convention, the number $0$ may or may not be in the set.) The aleph null $\aleph_0$ is defined as the "cardinality" of the set $\mathbb{N}$. This Wikipedia article says that $\omega$ is the first infinite ordinal. I have seen people use $\mathbb{R}^\omega$ for the set of all real sequences (see, for instance, Munkres's Topology); some people use $\mathbb{R}^{\mathbb{N}}$ instead, which suggests that $\omega$ and $\mathbb{N}$ may be the "same" in some sense. On the other hand, I have never seen $\mathbb{R}^{\aleph_0}$.
The definitions of these three concepts are quite different, yet they seem to be closely related.
So my question is: how exactly are they related to each other and in what sense they are (possibly) the same?
Best Answer
It may be useful to separate the facts that should be true in any reasonable set-theoretic foundation from the facts that are true by convention in the usual foundation.
Generally true: $\aleph_0$ is the cardinal number of a countably infinite set. $\omega=\omega_0$ is the order-type of a simple infinite sequence (an infinite sequence in which each element has only finitely many predecessors). $\mathbb N$ is the set of natural numbers.
Convention 1 (von Neumann): Any ordinal (= order-type of a well-ordered set) is identified with the set of strictly smaller ordinals. Thus, $0$ is the empty set, $1=\{0\}$, $2=\{0,1\}$, etc., and $\omega=\{0,1,2,3,\dots\}$.
Convention 2: A cardinal number is identified with the smallest ordinal of that cardinality. Thus, $\aleph_0=\omega$. (This convention depends on the axiom of choice in general, to ensure that every cardinality is the cardinality of some ordinal. But this is not an issue for $\aleph_0$, which is the cardinality of $\omega$.)
Convention 3: $0$ is a natural number. (A nontrivial number of respectable mathematicians disagree with this and start the natural numbers with $1$.) So $\mathbb N=\omega$.
In the end, if you adopt all these conventions, you have $\aleph_0=\omega_0=\omega=\mathbb N$. If you adopt other conventions (or no conventions), you need to check what they say about these things, but the general facts that I listed first should still be true.