Defining $\mathbb{R}^{1/2}$?

Question

If this is the wrong sort of question to ask here, apologies!

Not much background, I have no experience (except idle curiosity) with cardinality, cartesian products, and the like, but I was curious about the idea of some set $\mathbb{R}^{1/2}$ such that $\mathbb{R}^{1/2}\times \mathbb{R}^{1/2}=\mathbb{R}.$ I do know already that cardinality is not very helpful here because $|\mathbb{R}\times\mathbb{R}|=|\mathbb{R}|$. I was wondering if there would be any possible way to define this, and maybe generally define the notion of non-integer "$x$-tuples". Any ideas would be greatly appreciated.

Answer 1

I will answer the easier half of your question.

While there won’t exist a canonical $\mathbb{R}^{\frac{1}{2}}$ there is a notion that generalizes “non integer” tuples.

I’ll assume some familiarity with fractals here (if not just make a comment and I can add an introduction).

We consider 3 fractals: the regular real line $\mathbb{R}$, the $\log_{3}(5)$ dimensional vicsek fractal and then the plane $\mathbb{R}^2$ although for convenience we will consider this to be the same as $\mathbb{C}$.

Elements on the real line have a trinary representation where the elements can be written as $\sum a_n 3^n$ where the $a_n \in \lbrace -1, 0, 1 \rbrace$. Notice the base of the powers is 3, and there are 3 symbols (-1,0,1) to pick from at each digit and $\log_3(3) = 1$. This will be important later.

We might desire a similar representation on the complex plane but it’s not so obvious since we have a pair of real numbers and we want a single base-3 expansion. With a little cleverness we might observe that if we let our symbols range from $S = \lbrace -1-i,-1,-1+i,-i,0,i,1-i, 1, 1+i \rbrace$ suddenly it becomes possible to represent every point on the plane as $\sum a_n 3^n$ where $a_n \in S$ and here we have 9 symbols to pick from at each digit and $\log_3(9)=2$

With this established it becomes clear that “n-tuples of real numbers” become “3^n characters in a single base 3 expansion” and so we can finally try to ask “what does a $\log_3(5)$ tuple look like?” turning our attention to the vicsek fractal.

Here our set of characters are $S = {-1,-i,0,i,1}$ and our base-3 representation of each element is the usual $\sum a_n 3^n$.

So what have we accomplished? We turned integer sized tuple into a base $3$ representation and then found a new base $3$ representation that doesn’t have an integer sized tuple analogue.

This suggests that instead of thinking in terms of tuples we should instead think in terms of base-N representations and suddenly fractional dimensions and integer dimensions are all treated equally nicely on the same footing.

We also get a hint about what $\mathbb{R}^n$ really is. It is those base-$B$ representations for which there exists a natural addition, multiplication, and “carry” rule.

If you relax the notion of “natural” enough you might then find many candidates for a $\mathbb{R}^{\frac{1}{2}}$ though I suspect none will have ALL the properties you want

Answer 2

You write:

"I do know already that cardinality is not very helpful here because |R×R|=|R| ."

There is a kind of category error involved here, in the following sense. One usually speaks of two types of "infinite number": the Cantorian (infinite) cardinals and ordinals. But there is a third type of "infinite number" where extracting roots does not pose a problem. This type has been called ringinal in some recent publications. Thus, instead of using ordinals or cardinals, one should use ringinals.

Ringinals are the "infinite numbers" of nonstandard analysis, where root extraction works just as any operation of classical analysis. In particular, given a ringinal $H$, its square root $\sqrt{H}$ is entirely unproblematic. One only runs into problems if one looks for radicals of the wrong type of infinite number.