the power of choice.
Infinity is the primordial metaphysical mystery. We, forever constrained by our habits, our routines, our knowledge, our lifespans, find it only natural to wonder about that which transcends it all. That which transcends us all. Our counterpart, unending, unyielding, unstoppable.
So far, it seems only mathematicians have found any success in the struggle against this force. But the path has been littered with false starts, false victories, and false hopes.
Calculus could be considered, in some sense, to be one of mankind's greatest philosophical wins. Through it, the forbidden has landed in our palms, as an object to manipulate through our pens. But of course, all magic is sleight of hand. For we never truly talk about infinity itself, but focus exclusively on eventual behaviors, on tendencies. The lemniscates on our limits are no more genuine than the qualifiers they encode: for every ε greater than zero, there exists δ greater than zero. Through it, we learn little about infinity, other than to avoid it.
Even the applications of analysis to reality have asterisks attached. Two millennia ago, Zeno of Elea first pondered about the nature of motion. And two millennia later, despite immense scientific and philosophical progress, instantaneous velocity remains just as mysterious. The unstoppable crawl of time, its discrete or continuous nature, is yet unknowable. But as long as the epsilons and deltas are small enough, the math checks out, and the illusion is achieved.
Only three centuries later did Georg Cantor first confront the all in its own domain. In doing so, he unraveled a world just as fascinating as it was senseless. For there is not one but many infinities, successively larger and larger. At the bottom of the ladder we find ℵ0, representing the integers, as well as seemingly much larger structures as the rationals, or the roots of integer polynomials. Somewhere later, we find ℶ1, representing the full number line. In between, other infinities lurk, countless or perhaps none.
The arithmetic of this world was twisted. For addition and multiplication were trivial, laughably so. Yet exponentiation was impenetrable.
But most perplexingly of all, for every infinite there is one beyond, for every collection of infinites there is one beyond, even as our indices and symbols run out, so limited in comparison. Every new infinite gives us the possibility to build immensely larger ones, through transfinite nesting and recursion, and we can recurse our recursion processes, and we can recurse these again, and every single process that builds these massive structures can always be repeated, using these very same infinites. And in the end, the total amount of infinities is so much larger than every single one of these; it transcends infinity itself, incapable of fitting even in the idealized mathematical realm. An endless tower had been built, even the first few steps unimaginable, but heaven was so, so much higher.
If there was ever a God in the tangled wires of symbols and numbers, if there was ever a concept intrinsically worthy of reverence, Cantor had found it. And mortals would only ever know the ineffable through circumlocution. Absolute infinity, Ω.
As we explored