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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?

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    "The set A is not infinite".2011-07-06

5 Answers 5

1

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.

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    I understand what you are saying, I just don't understand how it can be true if$AC$is independent of ZF. I was meaning, unless I am missing something from this independence...2011-07-07
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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.

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    Unless the "reader" is a computer, in which case a symbolic language is far more precise.2011-07-07
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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 $\mathbb{N}^+ = \lbrace 1,2,3, \ldots \rbrace$).

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    @Asaf: Thanks for this reply.2011-07-06
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Why not simply, "$A$ is finite" or $|A| = n$.

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    While "A is finite" is fine, "$|A|=n$" makes me ask _What is $n$??_ ($n=\aleph_0$?) You'll have to write "$|A|\in\mathbb N$" as in Swlabr's answer.2011-07-06
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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.

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    I think I got it three years ago. :-P2014-08-07