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A friend came up with what looks like a very simple and beautiful construction for mapping an uncountable number of dimensions into the plane ($\mathbb{R}^2$).

Specifically, each dimension is indexed by a number $\theta \in [0,2\pi)$. And each location $x$ along dimension $\mathbb{R}_\theta$ gets mapped to the polar coordinate $(\theta, \exp(x)) \in \mathbb{R}^2$.

This construction seems sound and would imply that an uncountable number of uncountable reals can live comfortably inside $\mathbb{R}^2$. Is this correct?

P.S. Okay, after writing this out, I now see that this should definitely be the case, since $\mathbb{R}^2$ is, by construction, an uncountable number of real lines, one for each value of $y$. So we've mainly switched to polar coordinates here.

ted
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  • If I understand the construction, it seems that your friend has mapped a single point in $\mathbb{R}^\mathfrak{c}$ to a set of points in $\mathbb{R}^2$, is that right? – Brian Tung Mar 11 '24 at 20:47
  • Sorry, what is $\mathbb{R}^c$ ? I'm not familiar with that notation. – ted Mar 11 '24 at 20:49
  • It seems you already answered your own question with the last paragraph, no? – freakish Mar 11 '24 at 20:50
  • $\mathfrak{c}$ is simply shorthand notation for the cardinality of the continuum—that is, the cardinality of the reals. – Brian Tung Mar 11 '24 at 20:51
  • If my supposition is correct, I think the construction works, although I think using ordinary Cartesian coordinates is more transparent. You avoid a situation in which multiple dimensions with zero coordinate all get mapped to the origin. In contrast, using Cartesian coordinates is more or less what people are used to with function spaces, for example. – Brian Tung Mar 11 '24 at 20:52
  • After writing it out you saw the answer, so you can delete the question. – Ethan Bolker Mar 11 '24 at 20:54
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    That $\Bbb{R}^2$ comprises an uncountable number of real lines is fine: $c^2 = c$. It is not true that $c^c = c$, so I don't see any sense in which you can map an "uncountable number of dimensions" into the plane. Please rethink your P.S., and don't delete the question just yet, pace @EthanBolker. – Rob Arthan Mar 11 '24 at 21:16
  • @RobArthan The title asks about uncountably many lines, not uncountably many dimensions. Let's see what the OP does. – Ethan Bolker Mar 11 '24 at 21:20
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    @EthanBolker: good point, but I look to the body of the question for the statement, not the title! As you say, over to the OP (who should be careful to make the body of his/her MSE questions self-contained and to make the title consistent with the body). – Rob Arthan Mar 11 '24 at 21:23
  • @RobArthan I agree, and I also fail to see even how this is mapping infinite amount of dimensions, since fundamentally a union of uncountable many copies of $\mathbb{R}$ and $\mathbb{R}^\mathbb{N}$ is a little different. – Jakobian Mar 12 '24 at 21:13
  • Why does your title refine to an uncountable number of lines but you body talks of dimensions? The construction seems to equate a dimension with a line axis but shouldn't it include the entire space. – fleablood Mar 12 '24 at 21:34
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    Doing an uncountable (of the cardinality of $\mathbb R$) number of lines to $\mathbb R^2$ is almost trivial. As $\mathbb R^2$ is already a collection of uncountable many parallel lines (that is to say for any slope $m$, $\mathbb R^2 = \cup_{b\in \mathbb R}{(x,y)|y = mx + b}$) we just map each of the uncountable lines to an appropriate line. (We could do circular polar coordinates as well as your friend did but it doesn't add anything). But that's not an uncountable number of dimensions. – fleablood Mar 12 '24 at 23:03

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If I have understood correctly, this is not right. The function you are describing is $$f\colon [0,2π) \times ℝ → ℝ^2\\ (θ, x) ↦ (θ, e^x) $$ where $(θ, e^x)$ is wiritten in polar coordinates.

First, the domain is not $ℝ^ℝ$.

Pedagogical note on $ℝ^ℝ$: It might be hard to wrap your hand around $ℝ^ℝ$. A point in this space has a continuum many of coordinates $(x_λ)_{λ∈ℝ}$. Hence, a point can be thought of as a real-valued function from $ℝ$ that for each coordinate spits value at that coordinate. Compare this with, say $ℝ^3$, where each point $x= (x_1, x_2, x_3)= (x_i)_{i ∈ \{1,2,3\}}$ can be thought as a function $x\colon \{1,2,3\} → ℝ$ with $x(i) = x_i$.

Second, $f$ is not a surjection because there is no point that is mapped to $(0,0)$ since $e^x > 0$ for all inputs $x$.

Third, as it was mentioned in the comments by Rob Arthan, $$ |ℝ^ℝ| = 2^\mathfrak{c} > |ℝ| = |ℝ^2|. $$ Check out this MSE question for details.

To sum up, I think this might be a nice lesson for you to write down clearly and understand each term you use (I think you were confused with the notion of dimension) to utilize the power of formality! Hope this helps :)

Kamil
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