Recently I learned about the infinite cardinal $\aleph_0$, and stumbled upon a seeming contradiction. Here are my assumptions based on what I learned:
- $\aleph_0$ is the cardinality of the natural numbers
- $\aleph_0$ is larger than all finite numbers, and thus cannot be reached simply by counting up from 1.
But then I started wondering: the cardinality of the set $\{1\}$ is $1$, the cardinality of the set $\{1, 2\}$ is $2$, the cardinality of the set $\{1, 2, 3\}$ is 3, and so on. So I drew the conclusion that the cardinality of the set $\{1, 2, \ldots n\}$ is $n$.
Based on this conclusion, if the cardinality of the natural numbers is $\aleph_0$, then the set of natural numbers could be denoted as $\{1, 2, \ldots \aleph_0\}$. But such a set implies that $\aleph_0$ can be reached by counting up from $1$, which contradicts my assumption #2 above.
This question has been bugging me for a while now... I'm not sure where I've made a mistake in my reasoning or if I have even used the correct mathematical terms/question title/tags to describe it, but I'd sure appreciate your help.
This is a good example where intuition about a pattern breaks down; what is true of finite sets is not true of infinite sets in general. The natural numbers $\textit{cannot}$ be denoted by the set $A=\{1,2,...,\aleph_0\}$ as the set $\aleph_0$ is not a natural number. It is true that the cardinality of $A$ is $\aleph_0$ (a good exercise), but it contains more than just natural numbers.
If $\aleph_0$ were a natural number then, as you point out, we would have a contradiction. However $\aleph_0$ is the $\textit{cardinality}$ of the natural numbers, and not a natural number itself. By definition, $\aleph_0$ is the least ordinal number with which the set $\omega$ of natural numbers may be put into bijection.