Georg Cantor made many invaluable contributions to mathematics, it is to him that we owe the earliest technical explanation of transcendental numbers, the formulation of the Heine–Cantor theorem, and most famously his development of set theory. The latter is a mainstay in grade school mathematics today, but it remains difficult to explain at its most basic level absent some analogous comparison.

In middle school algebra, most of us learned that-

n {0, 1e-10, 2e-10...1e-9... 1...∞} > n {1, 2, 3, 4, 5...∞}
or rather that is to say there are an infinitely greater number of real numbers than there are integers, even though there are already an infinite number of integers. While we can correlate different sets of integers such as odd numbers and even numbers by offsetting their values, as in--

1 ----> 2
3 ----> 4
5 ----> 6
∞ ----> ∞+1 (yes I realize how ridiculous this notation is)

ratio: 1:1

the number of values we can equate any whole integer with in an set of infinite decimals between integers, is, in itself infinite (ratio 1:∞). So there are far more possible values in a set of real numbers approaching infinity than there are rational numbers, even as a dense set. However, at the same time, in a set approaching infinity there are as many rational numbers as there are integers, because they make use of defined units that have one-to-one correspondence with all integers. For more reading on how infinitely divisible real numbers are an exception to this, you can refer to the Wikipedia article on Cantor's diagonal argument.

So if I have even made any remote sense in explaining the concept of infinite sets and "more infinite" (uncountable) sets, how can we translate this into something that high school students can understand, let alone most adults? I submit one of the best ways this can be explained- sine functions and radio frequencies.


With the above sine function we have a simple graphical representation of correlating numbers in infinite sets; each corresponding hue represents a single whole digit of a set of 10. The gradient transitions between these represent infinitessimally smaller and smaller decimal values between each whole digit. Now, let's say this axis goes on infinitely in either direction to represent the entire spectrum of potential frequencies, not just the visible light spectrum. There will always be units by which we can express frequency, and in a sense we can always tune into these units.

For example, 1000 hertz go into 1 kilohertz.

When we realize that hertz are an arbitrary unit that can be broken down further, approaching an infinite number of divisions, we can see how
-1, graphically represented on an infinite set of real numbers, the spectrum in hZ appears as densely lines instead of a perfect gradient
-2, how non-matching sets of infinity ("more infinity"), are like tuning into a specific frequency lacking units in a broader spectrum of possible numbers.

Is this a proof? Nope. Not in the least.
Does it show some semblance of how you can have infinite and uncountably infinite sets? I hope so. I also hope that this conflation of the electromagnetic spectrum with mathematical numbers does a good job of illustrating this concept for teaching. I'll try to reword this in the future to make it more concise, but until then, I welcome any and all criticism on this lesson design.


Image credits: 1. "Infinite Tangent Circles", Florida Center for Instruction Technology Clipart ETC (Tampa: U of S. Florida, 2007)

2. Self made, using Grapher on OS X

3. Derivative of work from AmateurSpectroscopy.com