About the order of infinity of $\Re^n$

Question

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!

Answer 1

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.