Is the set of all sequences of positive integers of unlimited length denoted $\Bbb N^\infty$? Is the set of all sequences of positive integers of unlimited length denoted $\Bbb N^\infty$?
I think that's most probably not right as it seems to imply the set of infinitely long sequences of integers but it's all I can come up with.  I want the set $X$ of sequences of integers of unlimited length, satisfying:
$(1,1,1)\in X$
$(1,2,3,6,4)\in X$
but $(1_n:n\in\Bbb N)\notin X$
 A: From what I understand by looking at your examples, you are looking for the set
$$\bigcup_{k=1}^\infty \Bbb N^k$$
A: In set theory, the set of finite strings of natural numbers is denoted $$\omega^{<\omega}.$$ I've also seen "$\mathbb{N}^{<\mathbb{N}}$" used for the set of finite strings of naturals, outside of set theory. (I think variations like "$\mathbb{N}^{<\infty}$" and "$\mathbb{N}^{<\omega}$" would also be understood, but I would prefer the previous two, and I personally cringe at "$\mathbb{N}^{<\infty}$" although that reflects my own set-theoretic biases.)
However, I would certainly understand "$\mathbb{N}^\infty$" to refer to the set of infinite strings of naturals (not even "infinite-or-finite!"), and I think that's not peculiar to me. Ultimately any notation is "permitted" as long as you define it carefully, but I would view this as very confusing.
A: Notation means whatever the author intends.  Most commonly (infinite) sequences are thought of as functions from $\Bbb N$ to the set of possible entries.  A finite sequence on the other hand can be thought of as a function from $\{1,2,\dots,n\}$ to the set of possible entries.
There is precedent for denoting the set of functions from $A$ to $B$ as
$$\{f~:~f~\text{is a function }A\to B\}=B^A$$
Some of the justification behind this notation is for the convenient identity for finite sets then that $|B^A|=|B|^{|A|}$.  You will also as a result commonly see instead of the power set of $A$ notated as $\mathcal{P}(A)$ to instead see it notated as $2^A$ or as $\{0,1\}^A$.
Using this notation as a base and using the interpretation that you want all finite or infinite length sequences of positive integers, the set you describe could be written then as:
$$\Bbb N^\Bbb N\cup \left(\bigcup\limits_{n=1}^\infty \Bbb N^{[n]}\right)$$
where $[n]=\{1,2,3,\dots,n\}$ (or if you prefer $\{0,1,2,\dots,n-1\}$).  If you wish to include the empty-sequence, you may adjust the lowerbound to $n=0$ instead of $n=1$.
