How can I succinctly but correctly say that a set is finite? If I want to say that a set $A$ is numerable but infinite, I can do so like this: $$|A| = \aleph_0$$
What should I use instead to say that a set is finite? $|A|\in\mathbb{N}$? $|A|< \infty$? $|A|< \aleph_0$? Something else entirely?
 A: In light of your comment below the question (in addition to "What should I use instead to say that a set is finite?"), I suggest using $|A| < |\mathbb{N}|$ (or $|A| < \aleph _0$); see Wikipedia's definition here and Theorem (5.4) here. Note that this allows $A$ to be empty (the empty set is finite, and has a cardinality of zero).
EDIT: Exact quotations from the above links: 1) "Any set $X$ with cardinality less than that of the natural numbers, or $|X| < |\mathbf{N}|$, is said to be a finite set" (where $\mathbf{N}=\lbrace 0,1,2,3,\ldots\rbrace $); 2) "A set $X$ is finite if and only if $|X| < |\mathbb{N}^+|$" (where $\mathbb{N}^+ = \lbrace 1,2,3, \ldots \rbrace$).
A: Why not simply, "$A$ is finite" or $|A| = n$.
A: You should say "The set $A$ is finite." There is nothing wrong with using sentences in mathematics; they often are easier for the reader to understand than a sequence of symbols.
A: You could say,
"There does not exist an injection from $\mathbb{N}$ to $A$."
Personally, however, I would just go with either "$A$ is finite" or "$|A| <\infty$". It does depend on the context though.
A: As a matter of style I would say, "$A$ is finite."  It is best to avoid having a bristling obstacle course of symbols for your reader to penetrate.  Which is easier to read here?
Every nonvoid subset of the positive integers has a least element.
$\forall \emptyset \subset S\subseteq {\Bbb N}$, $ \exists m\in S$ such that $m\le s$ $\forall s\in S.$
The choice is clear to me.  Use notation and symbols to simplify and clarify.
