Say I use the natural numbers to generate the following sequence/set:
$$A = \{n\bmod 2\pi \mid n \in \mathbb{N}, n > 6\}$$
where I start above 6 to exclude the initial integers. Clearly this must have the same cardinality as $\mathbb{N}$, but while generating a countably infinite number of irrational numbers in $(0, 2\pi)$. It also seems clear that for any number you pick in this interval, you can find a member of $A$ that is arbitrarily close to it. I know because of the cardinality that this must be distinct from generating the actual interval $(0, 2\pi)$, but I'm not sure how to argue this in any other way?
To put it differently, I'm interested in what exactly is contained in $A$. Clearly we're not generating any rational numbers, so all elements come from $\mathbb{R}\setminus\mathbb{Q}$. So pick any element of $(0, 2\pi)\cap\mathbb{R}\setminus\mathbb{Q}$, say $e$. Is $e \in A$? I suspect not, and that I can only get close to it. But how can I say for sure?
I suppose my real question is more about how, if at all, irrational numbers can map to each other, but I lack the vocabulary to search for it.