This article on Wikipedia states the following:
Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
I was wondering why is it the case for finite-dimensional spaces, but not for $\mathbb{R}^{\omega}$. Can't a PMI proof of a bijection between $[0;1]$ and $\mathbb{R}^{\omega}$ be established, or would that proof be showing that one can establish a bijection between $[0;1]$ and $\mathbb{R}^n$ for any given $n\in \mathbb{R}/\mathbb{N}$ but not $\mathbb{R}^{\omega}$? Is there something special about infinitely-dimensional $R$ that makes it have more points than $\mathbb{R}$ or any segment on $\mathbb{R}$? Thanks a lot.