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We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended numbers that may include higher order infinities like $\aleph_3$ for example, does that adds any additional structure/properties?

P.S: I'm not a mathematician, but a physicist, so please don't be so abstract or technical.

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    I hope that [this](http://math.stackexchange.com/questions/90758/the-aleph-numbers-and-infinity-in-calculus) will answer your question, but I will be happy to add more in here if it doesn't.2012-10-28
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    Thx Asaf but it just explains what is $\aleph$, which I already have an idea about, what I want to understand, is the connection (if any) between them and geometry, the real link between abstract math and physics.2012-10-29

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