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We say that two integers have the same parity if they are both even or both odd. Is there a word like this but for infinite/finite instead of odd/even? That is, if a quantity $X$ is finite if and only if a quantity $Y$ is finite, can I express this by saying that $X$ and $Y$ have the same [fill-in-the-blank]? Something sort of like "order of magnitude", but where the only two orders are "infinite" and "finite".

Context: I'm writing a paper in probability, and I want to talk about comparing the expectations of two random variables. One is infinite if and only if the other is infinite. If there isn't a word for what I'm looking for, that's fine, I can say it some other way, but I feel like there should be a word for it.

D-Slo
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    finitude (need to type 15 characters more for stackexchange to accept the comment) – hunter Nov 13 '20 at 20:53
  • Thanks! Can you provide a source using finitude in that way? If I look at Wiktionary, for example, it doesn't list that usage. https://en.wiktionary.org/wiki/finitude – D-Slo Nov 13 '20 at 21:05
  • this isn't quite what you're looking for, and I don't think it's standard, but I've seen the notation $|A|\dot{=} |B|$ or ($\kappa\dot{=}\lambda$ where $\kappa,\lambda$ are cardinals) to mean "either $A$ and $B$ are both infinite or $|A|=|B|$" – Atticus Stonestrom Nov 13 '20 at 21:09
  • also, @hunter's suggestion is a good one; even if it's not commonly used, you could just add a short "notation" section to the start of your paper and say something like "we say that two random variables have the same finitude when the expectation of one is infinite iff the expectation of the other is too" – Atticus Stonestrom Nov 13 '20 at 21:13

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