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Is it possible to prove the uniqueness of a point in space?

Well. I know this might be a silly question, but some response would be nice.

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    Ok. If you take the point (3,4,1) how do you know that it is unique? That there exists no other (3,4,1) ?2011-11-29

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In the comments, the problem has been clarified: how does one know that the point $(3,4,1)$ (say) is unique, that there is no other $(3,4,1)$.

I would say the question is unnecessarily complicated. The problem it raises for space already comes up on the line. So I ask OP, how do we know the number 17 is unique? How do we know there is no other 17?

If you can find an answer to that question, an answer that satisfies you, then you should have no trouble with points in space.

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    Thank you for the clarification.2011-11-29
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It's clear that what is important is not the space itself but the structure and arguments we impose and use on space. For instance, if we model space as R^3 and think of distance as the Euclidean norm then this problem is done.

What you really need to ask is, is it acceptable to assume there is a bijection between R^3 and space (or, to be slightly stronger we want an Isomorphism of sufficient depth)? Once you accept that, uniqueness is trivial (as points in R^3 are unique).