Set theory

During my sixth form, or the junior/senior years for Colonials, I got interested in the nature of infinities and the singularity. What if we treated "Two Divided By Zero" as an actual number, like pi. I raised this to my math teacher / tutor; he recommended I rethink my presuppositions. He recommended Georg Cantor. Not in the original German; in an English-language summary.

Georg Cantor back in 1870s had run a thought-experiment: let's list through all the positive integers in binary-notation. Nerds call these, the natural numbers. First, there's 1. And then 2 - "10". And "11", and "100" (for 3). And so on. Next, pretend you're a Semite and start reading them from top-right.

So, you got this:

0..0001
0..0010
0..0011
0..0100
0..0101

Now: consider the number when you pick digits, diagonally:

0..00001
0..00010
0..00011
0..00100
0..00101

Cantor said: suppose we flip those digits. By definition, it's not 1. It's not 2. It's not 3 ("011") - and moreover, that third digit has flipped boolean to "true", so we know it's greater than 3. As you go down row x, the function of x becomes 4, 12, 28, 60...

By this, Cantor had formally defined (Two to the Power of x, Minus One) Minus Three, with x -> Infinity. He'd defined a new theory: a theory of sets. The natural numbers map one-to-one with these rows of binary digits. And, they can never count to 2^∞-4. Not as in, it would take too long. As in: the unbounded set of the natural numbers does not contain this number.

More: any function you can think of to "hash" the integers, any deterministically-driven list of binary-digit sequences: once you've defined them and listed them, Cantor - that devil - can ruin your day by subsequently doing his diabolical diagonal.

It further turns out that if you stick a decimal in front of these binary digits, that the Continuum of real numbers [0,1] - which include pi-3 - isn't countable, either.

This forced mathematicians to take stock of which sort of numbers can be mapped to countability. They got as far as the algebraic numbers, like 5 and -12 and 3.6 and even root-2 and even even those x solutions in the complex plane for finite polynomials including famously-insoluble x⁵ - x + 1. They also found a way to count two-dimensional space: so, the imaginary / complex algebraic numbers were countable. But they (literally) couldn't square the circle: pi wasn't algebraic. Above all, take e - whose natural logarithm is equal to one. The logarithm function is the inverse of the exponential function.

Infinities and their inverted twins, the infinitesimals, are the bane of mathematics; but this inductive method of Cantor is the same method used in, for a start, calculus. Cantor's incipient set theory did run into "Russell's Paradox" but that paradox was swiftly corrected by the rigourous Zermelo–Fraenkel theory. There's also Cantor's hypothesis on whether there exists some infinity between countable numbers and the uncountable Continuum, which unfortunately Zermelo-Fraenkel's theory couldn't - and can't - address. Or so the Internets tell me. I never got this far.

I got far enough to figure out that no serious mathematician has disproven Zermelo–Fraenkel. So Cantor's discovery of uncountable infinities must stand.