the infinite weirdness of infinities
I've long been fascinated with infinities, and the infinitesimal, but 'infinite' wasn't always a thing. Maybe the Greeks were the first to take the idea seriously, but infinity was still just a synonym of not quantifiable; e.g. the distance to the stars.
The first formal use of it AFAIK was in describing "lines" in the abstract. In Euclidian geometry, a line has infinite length. The ∞ symbol more recently came to represent growth without bounds, as in the lack of an upper bound for a limit in summations, like in calculus. Finding the area under a curve, taking the derivative of a function, is conceptualized as taking the sum of a number of rectangular slices. The approximation grows more accurate as the size of the slices diminish, better fitting the curve. More precision results from increasing the number of slices. You could say that the best answer is the result of taking that limit to infinity.
People realized there were infinite series pretty early on as well, like Pi. The number of digits you produce for the value Pi depends only on the precision you ask from your computer.
The idea really takes off in the late 19th and early 20th century however, as we get infinite sets and infinite madness beyond that. We get the idea of "countably infinite" which just means that you can enumerate the entire set. Consider Natural numbers, I can literally define the entire set by rule, N={1, 2, 3,... } For any x in the set N, I can give you the next set member, x+1. So we can enumerate as many elements as you'd like, but never all of them.
The size of infinite sets is defined by cardinality, instead of ordinal measure. For example, you'd likely say that the set of natural numbers N has twice as many elements as the set of even numbers E. Specifically, the infinite set of odd numbers {1, 3, 5,... } is wholly contained with the set N of natural numbers, and entirely distinct from the set E of even numbers. That's where cardinality comes in. Georg Cantor proved in the most intuitive way that there exists a one-to-one correspondence between those 2 sets. That just means that there exists a mapping function that takes each member of one set to the other, and that every member of that other set is covered by the process. A one to one, and onto mapping function, namely for every n in N, and e in E, e = 2n. Take each n, multiply by 2 and you get the next unique e. Hence these sets are exactly the same cardinality.
There are larger infinities however. Infinities are ranked using this Aleph notation. Infinite sets of this cardinality are said to be Aleph-0, the 'smallest' infinities. This group also includes the set of all computable numbers, the set of all computable functions, the set of all binary strings of finite length, and more.
What is above Aleph Zero? Consider Real numbers, or to put it simply - consider that I can find a number between any 2 numbers you give me. So, between 2 consecutive integers like 2 and 3, I can use a formula to get a new number n, such that 2 < n < 3. My formula is just n=(a+b)/2 for whatever a and b you give me, including the numbers my formula produces. In fact there are an infinite number of numbers I can find between any two integers by repeating this process recursively with the results. The Real numbers are in fact infinitely larger than the infinite sets we saw earlier, like Natural numbers.
Another way to think about infinitely larger infinities is to consider power sets. A power set is the set of all subsets of a given set. Let's take our natural numbers again, but this time grouped in subsets. {1},{2},{3},...{1,1},{1,2},{1,3},...,{2,1},{2,2},...,...,{1,2,3},... It turns out that collectively a set of all subsets of natural numbers is infinitely larger than the original set of natural numbers. A superset of an infinite set has higher cardinality than the original set.
Lest you think these are the largest infinities, let's think more about power sets. Instead of natural numbers, consider the set of all sets. The set containing all sets ought to be the largest, right?
Hang on, a set containing all sets is a contradiction. If it really contains all sets, it must contain itself, since it is a set. But if it includes itself plus all other sets, it's a superset of itself and must be larger than itself.
Beyond this point, madness ensues. it seems there might be a well ordered hierarchy of infinities, but the details are infinitely weird. If you think there is more to this story, you're right, but you'll need to consult someone who has some understanding of it to get a lucid explanation.