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I would like to ask a question about the order of infinity of the $n$-dimensional space $\mathbb{R}^n$. I am not sure whether I use the appropriate notation/mathematical language or not - please correct me, if necessary. If I am not confused, $\mathbb{N}$ is countably infinite, but $\mathbb{R}$ is not; it's uncountably infinite. So, $\mathbb{R}$ is one order of infinity greater than $\mathbb{N}$? What's true about the $\mathbb{R}^n$? What I want more, by asking this question, is to emphasize the right notation and language that I have to use in order to describe a(n) (infinite) set of constraints for the $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$. What exactly should I say about those constraints?

I am not adequatelly familiarized with this issue, as you can see. By the way, could you suggest to me some enlightening stuff (notes/tutorials/books)?

Thanks in advance!

nullgeppetto
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  • This question may be useful: http://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr – tylerc0816 Nov 14 '13 at 12:27
  • Thanks @tylerc0816! I'll have a look! If someone else has something to contribute, please so be it! – nullgeppetto Nov 14 '13 at 12:31
  • Hmm.. The previous article do not exaclty help.. Anyone else? It's rather an issue of notation. @tylerc0816, thanks anyway! – nullgeppetto Nov 14 '13 at 12:43
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    Both $\mathbf{R}$ and $\mathbf{R^n}$ are uncountable. Functions proving this get a little perverse, but are interesting. You may want to read these articles: http://en.wikipedia.org/wiki/Space-filling_curve, http://en.wikipedia.org/wiki/Continuum_hypothesis – doppz Nov 14 '13 at 12:51
  • Thanks a lot @doppz! This is something! But can we say that $\mathbb{R}^n$ is "multiple infinite" in comparison with $\mathbb{R}$? Or, maybe, could we say that $\mathbb{R}^n$ is $n$ orders greater than $\mathbb{R}$? Could it be expressed using the aleph-null notation? Thanks a lot again! – nullgeppetto Nov 14 '13 at 12:55
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    gepetto, this is where your intuition may lead you astray. It is natural to think "since $\mathbf{R^n}$ contains all those copies of $\mathbf{R}$ it surely must be larger!" The result is much like how $\mathbf{N}$ and $\mathbf{Q}$ are both countable, despite what your intuition may want you to think. Of course, there are other ways of showing certain sets are larger than others on a more refined notion than just cardinality. For instance, measure in $\mathbf{R^n}$, for each $k<n$, $\mathbf{R^k}$ has measure zero when viewed as a subset of $\mathbf{R^n}$. – doppz Nov 14 '13 at 13:08
  • That was enlighting, @doppz! Appretiate your help! – nullgeppetto Nov 14 '13 at 13:10

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It all depends on what you'd like to mean with "order of infinity".

If you mean "cardinality", then $\mathbb R$ and $\mathbb R^n$ are indistinguishable. That is, there is a bijection from $\mathbb R$ to $\mathbb R^n$.

Of course, this answer could be somewhat unsatisfactory. Heuristically, an element of $\mathbb R$ has one "degree of freedom", whereas an element of $\mathbb R^n$ has $n$ degrees of freedom. This idea is formalised by the notion of dimension of $\mathbb R^n$ as a $\mathbb R$-vector space. The dimension of $\mathbb R =\mathbb R^1$ as a $\mathbb R$-vector space is $1$, the dimension of $\mathbb R^{10}$ is $10$, and so on. Again speaking heuristically, the dimension gives you a way to distinguish the "order of infinity" of the different $\mathbb R^n$s, in the sense of their degrees of freedom. I would like to stress that it's all up to you to choose the way to distinguish things. If you choose cardinality, then every $\mathbb R^n$ is the same; if you choose dimension, then they are all different, and you could also say (heuristically) that "$\mathbb R^n$ is $n$ orders greater than $\mathbb R$", with the precise meaning that I tried to outline above.

Francesco Genovese
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  • Ok, I see! I am not interested in expressing things heuristically, although I get exactly what you state about the dimension of $\mathbb{R}^n$. The second line of your answer is what I wanted! So, thank you very much! – nullgeppetto Nov 14 '13 at 13:57