Is the number of 3-dimensional slices slices of a 5-dimensional space less than the number of 3-dimensional slices of a 4-dimensional space?

Question

I’m only in high school, so I’m not certain whether I could have used better terminology to describe this.

I’m initially thinking of it using dimensional analogy. I think, tentatively, that the question is similar to asking if the number of points on a line is greater than the number of points on a plane.

I‘m not certain whether this question would have an equivalent answer to that of the question “Is the number of 3-d slices of a 5-d hypercube greater than the number of 3-d slices of a 4-d hypercube?”, so if you could tell me whether they’re the same (and if not, what the answer to that one is), I would also greatly appreciate it.

I’m guessing that the solution may have something to do with the infinite number of real number coordinate possibilities (Aleph-1?) that can be chosen for each given new dimension. If that’s the case, maybe this question is really asking “Is Aleph-1 to the power of 5 greater than Aleph-1 to the power of 4?” I’m not sure whether “Infinity” or “Aleph-1” is the right term to use in this context.

Answer 1

It turns out that the number of points on a line is equal to the number of points on a plane. In other words, we know that there is a function from the line to the plane that touches each point of the plane exactly once. That doesn't mean the function will be continuous or in any way "nice." We just know that such a function exists. So in the same way, there are just as many $3$-dimensional sections of a $5$-dimensional hypercube as there are of a $4$-dimensional hypercube.

It sounds like you're aware that there are many different infinite cardinalities. It turns out (using techniques you probably won't learn until graduate school) that we can't know which of those cardinalities is the cardinality of the real line. That's because (assuming our model of mathematics is consistent at all) for almost any possible cardinality, we can create a model where that is the cardinality of the real line. In other words, using our usual axioms of set theory, the cardinality of the real line is undecidable.

But whatever it is, it's the same cardinality (in whatever model we end up choosing) as all of the other sets we're talking about in this question.

Answer 2

Let us ask an analogous question in a space we can imagine: are there more lines in the plane, or in all of 3D space? This is equivalent to your original question, minus two dimensions.

On the one hand, both of these quantities are (uncountably) infinite, and so in this sense the two amounts are equal. On the other hand, any line in the $xy$-plane can be easily embedded into 3D space, and so there are "more" lines in a volume than in a plane. The most useful way (for this question) to compare these two spaces might be to discuss how many parameters are necessary to parametrize each collection.

A line is uniquely determined by two points, and so in the plane a line is uniquely determined by four numbers (two points $\times$ two coordinates per point). But in 3D space, a line is determined by six numbers (two points $\times$ three coordinates per point), and so in this sense it requires more information to specify a line in 3D space than it does to specify a line in a 2D space. This is intuitive and it gets close to formalizing what we might guess about the original question on first read.

But we should be more careful, if we want a precise answer. You see, two points are enough to specify a line uniquely, but some pairs of points actually specify the same line. Consider the line $y = x$ in the $xy$-plane, which is specified both as "the line passing through $(0,0)$ and $(1,1)$", as well as "the line passing through $(-1,-1)$ and $(12,12)$". We should try to parametrize the collection of all lines in the plane, in such a way that each line is uniquely specified by a given choice of parameters.

In fact, a line in the $xy$-plane can be uniquely described by just two numbers: the angle $\theta \in [0,2\pi)$ that the line makes with the positive $x$-axis, and the point $x_0 \in (-\infty, \infty)$ where the line intersects the $x$-axis (if $\theta = 0$, then interpret the number $x_0$ as the $y$-intercept instead). This is similar to identifying a line in the form $y = mx + b$ with the pair $(m,b)$, but allowing $m$ to take the value $\infty$ and interpreting $b$ as the x-intercept in that case.

Conversely, how to specify a line in 3D space? You can specify the direction that the line points with two numbers, the spherical angles $\theta$ and $\phi$. Then, to specify the placement of the line, you need a point $(x_0,y_0,0)$ where the line intercepts the $xy$-plane. There are edge cases here as well; you need to reinterpret the point as lying in the $yz$ or $xz$ plane(s) in the event that $\theta = 0$ or $\phi = 0$. Edge cases aside, we can see that four real numbers uniquely specify a line in 3D space - two more than is necessary to specify a line in 2D space. This is actually the same answer we got with the naive parametrization, but it's nice to be certain.

I'll leave it to you to wonder about parametrizing hyperplanes in 4D and 5D space, but I anticipate that you will come to a similar-looking answer.