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If there are different kind of infinities (aleph0 aleph1 and so on) then are there different kind of infinitesimals? Or should I consider zero the "opposite" of infinity if there is such a thing and ask the same questions with different kind of zeros?

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  • $\begingroup$ Aside: $\aleph_0$ and $\aleph_1$ are the same kind of 'infinity': they are both (infinite) cardinal numbers. Cardinal numbers are unrelated to infinitesimals. $\endgroup$ – user14972 Jul 22 '16 at 21:30
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In a nonstandard model of the reals $0$ is not the reciprocal of any element. There are infinities of various magnitudes, and their reciprocal steps are infinitesimals of various magnitudes. If you throw away the infinitely large and small elements and identify all infinitesimals to $0$, then you obtain the classical reals as a quotient. So, different infinities correspond to different non-zero infinitesimals (bijectively) and $0$ is not the reciprocal of anything.

Addendum following comment below: You can view a one-point compactification as adding a point at infinity. But that is a topological intuition that has little to do with any kind of reciprocal of $0$. The closest I can think of to address your intuition can be gathered from "The reals as rational Cauchy filters", New Zealand Journal of Mathematics, to appear, available here. In that article I show how to construct the reals as rational Cauchy filters. Then $0$ is the minimal Cauchy filter containing every rational neighborhood of $0$. Filters are very robust, and in particular you can often extend partial operations on the rationals to a full operations on filters of rationals, point-wise. So, the partial operation $x\mapsto 1/x$ extends to an operation on filters, and thus $1/0$ is defined. In more detail, $1/0$ is a certain filter, but it is, of course, not a minimal Cauchy filter since that would make it a real number, which is impossible. But it is a very concrete filter. That filter can be shown to have certain characteristics making it an infinitely large extended real number. I hope this is in line with what you are looking for.

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    $\begingroup$ What about the real projective line to consider zero the reciprocal of infinity? $\endgroup$ – Masacroso Jun 28 '16 at 21:01
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    $\begingroup$ @Masacroso I extended my answer to address your further question. $\endgroup$ – Ittay Weiss Jun 28 '16 at 21:56
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There are different ways of formalizing infinity in mathematics. Cantorian set theory is only one of them, and not necessarily the most useful one. Cantor himself was virulently against infinitesimals, claiming that he proved their self-contradictoriness (the published proof was incorrect).

A different approach was that proposed by Abraham Robinson around 1960. Here one constructs a proper order field extension denoted ${}^\ast\mathbb{R}$ of the real field $\mathbb{R}$. The extended field contains both infinite numbers and infinitesimals, and they are precisely in the reciprical relationship that you indicated.

For a gentle introduction to these ideas see Elementary Calculus.

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