Confusion about a specific notation In the following symbolic mathematical statement $n \in \omega $, what does $\omega$ stand for? Does it have something to do with the continuum, or is it just another way to denote the set of natural numbers?

The notation of $\omega$ is coming from ordinals, and it denotes the least ordinal number which is not finite.
The von Neumann ordinals are transitive sets which are well ordered by $\in$. We can define these sets by induction:


*

*$0=\varnothing$; 

*$\alpha+1 = \alpha\cup\{\alpha\}$;

*If $\beta$ is limit and all $\alpha<\beta$ were defined, then $\displaystyle\beta=\bigcup_{\alpha<\beta}\alpha$.


That is to say that after we have defined all the natural numbers, we define $\omega=\{0,1,2,3,\ldots\}$, then we can continue if so and define $\omega+1 = \omega\cup\{\omega\}$ and so on.

In set theory it is usual to use $\omega$ to denote the least infinite ordinal, as well the set of finite ordinals. It still relates to the continuum since $\mathcal P(\omega)$ is of cardinality continuum, since $\omega$ is countable.
