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When there is a hierarchy $1 < 2 < ... < \aleph_0 < \aleph_1 < ...$ Can we say anything reasonable and useful about the ordering of the reciprocals of these numbers. To begin: $1/1 > 1/2 > 1/3$ makes sense, but I am not sure about the infinite ones. (For example, how should $1/\aleph_0$ compare with $1/\aleph_1$? I think there is something more to it than defining both of these things equal to 0.) Is there any research on this?

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closed as not a real question by Simon Thomas, Franz Lemmermeyer, Gjergji Zaimi, Pete L. Clark, Robin Chapman Sep 12 '10 at 14:53

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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$1/x$ is defined when $x$ is a real number. But $\aleph_0$ is not a real number. So (unless you provide a definition) there is no meaning to $1/\aleph_0$. –  Gerald Edgar Sep 12 '10 at 13:31
    
I removed the "descriptive set theory" tag. –  Simon Thomas Sep 12 '10 at 13:39
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I'm tempted to reopen this, so I started a meta discussion - tea.mathoverflow.net/discussion/663/infinitely-small-numbers –  François G. Dorais Sep 12 '10 at 17:22
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Dear Willem, after much discussion on meta, we have reached the conclusion that your question is off-topic for MO. That said, I am sure you will get some very interesting answers if you repost the question on our cousin site math.stackexchange.com –  François G. Dorais Sep 13 '10 at 0:44