## Infinitely small numbers [closed]

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?

-
See en.wikipedia.org/wiki/Surreal_numbers – Richard Borcherds Sep 12 2010 at 13:24
$1/x$ 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 2010 at 13:31
I removed the "descriptive set theory" tag. – Simon Thomas Sep 12 2010 at 13:39
I'm tempted to reopen this, so I started a meta discussion - meta.mathoverflow.net/discussion/663/… – François G. Dorais Sep 12 2010 at 17:22
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 2010 at 0:44
show 1 more comment