On defining sequences Can two infinite sequences be "concatenated"? Two examples:
\begin{align*}
S &= (2,4,6,8,10\ldots,1,3,5,7,9\ldots)\\
\\
T &= (2,3,7,13,19,\ldots,5,11,17,23\ldots)
\end{align*}
My hunch is no. The only reason I can provide at the moment is that it would be impossible to establish a bijection between $\mathbb{N}$ and the terms of the sequence.
A related question I have is about this "concatenation operator". Is that defined anywhere? It's reminiscent of union. But the concatenation of two countable sequences as above would be uncountable then, wouldn't it?
A thank you to everyone (except one commentator) who provided me with clarifications, hints, leads, and links to further reading.
 A: As was mentioned in the comments, primes are really irrelevant: what you’re interested in here is the notion of concatenating two ordinary infinite sequences to produce a longer sequence-like object. As several people noted in the comments, this is entirely possible, and the resulting sequence-like objects (and others even longer and more complicated) are very useful in some parts of mathematics; set theory and some areas of general and set-theoretic topology come to mind.
These objects are sometimes called transfinite sequences. Just as ordinary sequences are typically indexed by elements of $\Bbb N$ or $\Bbb Z^+$, these more general sequences are typically indexed by ordinals. Ordinary sequences are indexed by the ordinal $\omega=\{0,1,2,\ldots\}$, which you can identify with $\Bbb N$ if you like. Your concatenated sequences are indexed by the ordinal $\omega+\omega$, which ‘looks like’ two copies of $\omega$ strung end to end:
$$0,1,2,3,\ldots,n,\ldots,\omega,\omega+1,\omega+2,\omega+3,\ldots,\omega+n,\ldots\;,$$
or pictorially $\longrightarrow\longrightarrow$. If you concatenated three ordinary sequences, you’d get something indexed by $\omega+\omega+\omega$; an example is
$$0,3,6,9,\ldots,3n,\ldots,1,4,7,10,\ldots,3n+1,\ldots,2,5,8,11,\ldots,3n+2,\ldots\;,\tag{1}$$
indexed by
$$0,1,2,\ldots,n,\ldots,\omega,\omega+1,\omega+2,\ldots,\omega+n,\ldots,\omega+\omega,\omega+\omega+1,\omega+\omega+2,\ldots\;.$$
All of these generalized sequences are still countable. In the case of the simple concatenation of two ordinary sequences you actually demonstrated that in your original post: the fact that you can take the countably infinite set $\Bbb N$ (or $\omega$) and rearrange it as 
$$0,2,4,6,\ldots,1,3,5,7,\ldots$$
shows that this concatenated ‘sequence’ is still countable. My example $(1)$ shows that the same is true of the generalized sequence obtained by concatenating three ordinary sequences.
You might be interested in a simple example of a generalized sequence that looks like a whole sequence of sequences concatenated together. Define an order $\preceq$ on $\Bbb N\times\Bbb N$ as follows: for $\langle k,\ell\rangle,\langle m,n\rangle\in\Bbb N\times\Bbb N$, $\langle k,\ell\rangle\preceq\langle m,n\rangle$ if and only if either $k<m$, or $k=m$ and $\ell\le n$. (This is the so-called lexicographic order (or dictionary order) on $\Bbb N\times\Bbb N$.) If you play with the order a bit, you’ll see that it looks like this:
$$\begin{align*}
&\langle 0,0\rangle\prec\langle 0,1\rangle\prec\langle 0,2\rangle\prec\langle 0,3\rangle\prec\ldots\\
&\langle 1,0\rangle\prec\langle 1,1\rangle\prec\langle 1,2\rangle\prec\langle 1,3\rangle\prec\ldots\\
&\langle 2,0\rangle\prec\langle 2,1\rangle\prec\langle 2,2\rangle\prec\langle 2,3\rangle\prec\ldots\\
&\qquad\qquad\qquad\quad\vdots
\end{align*}$$
Arranged horizontally, with each ordinary sequence (i.e., each line of the display above) represented by an arrow, that’s $\longrightarrow\longrightarrow\longrightarrow\cdots$. And this is still a countable generalized sequence, since I can rearrange $\Bbb Z^+$ into such a generalized sequence:
$$\begin{align*}
&1,3,5,7,9,11,\ldots\\
&2,6,10,14,18,22,\ldots\\
&4,12,20,28,36,44,\ldots\\
&8,24,40,56,72,88,\ldots\\
&\qquad\qquad\vdots
\end{align*}$$
Here the $n$-th row contains the numbers divisible by $2^{n-1}$ but not by $2^n$.
The Wikipedia articles on well-orders, ordinals, and ordinal arithmetic are a place to start getting some idea of these matters, though if you really get interested, you’ll probably want to look at a serious introductory set theory text; I like Karel Hrbacek & Thomas Jech, Introduction to Set Theory, Third Edition, Revised and Expanded, which is written for advanced undergraduates and first-year graduate students. (The first two editions lack some of the newer material in this edition, but they’re still fine introductions.)
A: You can partition the primes into two subsets if you wish, though you should not write it like that if you want to be clear.
One possibility might be primes not of the form $6n-1$ and the other of primes of the form $6n-1$.  If that is your pattern, then it has been done before: see OEIS A045331 and A007528 
