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Is it acceptable to write "$X$ is an infinite set" as "$|X| = \infty$"?

By "acceptable" I mean that I can use it in a research paper or in a textbook, and a reasonable person won't be against it. Possible arguments against it:

  1. $\infty$ is not actually a value, so it doesn't fit into a standard set-size notation.
  2. $|\mathbb R| = \infty$ and $|\mathbb N| = \infty$, so one may think that $|\mathbb R| = |\mathbb N|$.

(I looked for duplicates with some variations of the title, and found none)

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    $\begingroup$ I would strongly recommend against it. Either write it in words, or write $|X|\ge\aleph_0$, $|X|\ge\omega$, or the like. $\endgroup$ Feb 14, 2021 at 23:01
  • $\begingroup$ While I haven't published any papers or books, I would think it depends on the context. If you're writing about, say, set theory, where precise cardinalities are often a subject of study, using this notation would be very jarring. But, if you're writing about some field where this sort of question doesn't come up, I wouldn't have a problem with it. $\endgroup$
    – Sambo
    Feb 14, 2021 at 23:37
  • $\begingroup$ "$X$ is an infinite set" is unambiguous. Your other option raises questions ("What does "$|\cdot$|" mean for an arbitrary set?" for one.) Why raise unnecessary questions? $\endgroup$
    – JonathanZ
    Feb 14, 2021 at 23:51
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    $\begingroup$ @JonathanZsupportsMonicaC "$\vert\cdot\vert$" is meaningful for arbitrary sets, I don't think that's an issue here. $\endgroup$ Feb 15, 2021 at 0:33
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    $\begingroup$ @JonathanZsupportsMonicaC By definition "$\vert A\vert$" is the smallest ordinal in bijection with $A$. If we want to think of $\vert\cdot\vert$ as a function-like thing, it's an example of a class function: technically not a function but rather a formula with two free variables $\varphi(x,y)$ such that $\mathsf{ZFC}$ proves "For all $x$ there is exactly one $y$ with $\varphi(x,y)$" (or "Every set has a unique cardinality"); in this context its domain and codomain are each $V$, the universe of all sets. (That said, if we drop choice there is a more technical definition of cardinality.) $\endgroup$ Feb 15, 2021 at 2:35

2 Answers 2

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$\infty$ is a bit ambiguous/misplaced in the context of set cardinality, something like an Aleph Number (https://en.wikipedia.org/wiki/Aleph_number) might be more appropriate, i.e. $|X| \ge \aleph_0$, which states "X has the cardinality of at least the smallest possible infinite set".

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  • $\begingroup$ I consider myself reasonable :-), and I'm going to say "Don't do this". If all you really want to do is distinguish finite from infinite, then all you'll do with the $|X| \ge \aleph_0$ notation is show that you've taken some set theory. My guideline is "If you're not going to use $\aleph_1$, then why mention $\aleph_0$?". $\endgroup$
    – JonathanZ
    Feb 15, 2021 at 6:14
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I'm personally strongly against this - we have more exact notation for describing infinite sets, so why not use it?

That said, I have seen this notation used when describing objects which are subsets of some fixed countable set (e.g. sets of natural numbers). Here there's really no possible confusion. However, I still consider this bad practice.

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