Before the 1870s, mathematicians mostly treated infinity as a single idea — a thing was either finite or it wasn't, and "infinite" didn't come in sizes. Georg Cantor overturned that with a piece of reasoning simple enough to fit on an index card: a proof that the infinite set of real numbers between 0 and 1 is, in a precise and provable sense, larger than the infinite set of whole numbers, even though both sets go on forever.
Two infinities can be compared by trying to pair them up
Cantor's method for comparing infinite sets was to ask whether their members could be put into a perfect one-to-one correspondence — every whole number matched to exactly one member of the other set, with nothing left over on either side. Sets that can be paired this way are the "same size" of infinity, even if that feels strange: the even numbers can be paired one-to-one with all the whole numbers (1↔2, 2↔4, 3↔6...), so despite feeling like "half" of the whole numbers, they're the same size of infinity. The whole numbers, the integers, and even the fractions can all be paired up this way — they're all the same "countable" infinity.
The diagonal argument: build a number that can't be on any list
Cantor showed the real numbers between 0 and 1 can't be paired with the whole numbers at all, using a proof by contradiction now called the diagonal argument. Suppose you could list every real number between 0 and 1, each paired with a whole number, and write out their infinite decimal expansions. Cantor's trick is to construct a new number by going down the diagonal of that list — taking the first decimal digit of the first number, the second digit of the second number, and so on — and changing every single digit to something different. That new number differs from every number on the list in at least one decimal place, which means it can't actually be on the list, even though the list was supposed to contain every real number between 0 and 1. The list can never be complete, no matter how it's constructed, which means this infinity is too large to be paired one-to-one with the whole numbers — a strictly bigger infinity.
What we're still unsure about
The diagonal argument itself is airtight and universally accepted in modern mathematics, but it opened a question Cantor himself couldn't resolve: is there an infinity strictly between the size of the whole numbers and the size of the real numbers? This is the continuum hypothesis, and in the 20th century mathematicians proved something genuinely strange about it — using the standard axioms of set theory, the continuum hypothesis can be neither proved nor disproved. It isn't that nobody has found the answer yet; it's that the standard rules of mathematics are, provably, not enough to settle the question either way.
This sits inside Set Theory, one of eight topics in Discrete Mathematics, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.