tbf this uses a lot of medical jargon that most people who aren’t doctors will never need. put simply, it’s a case of a man who put hydrofluoric acid in his ass (owie) while high from snorting cocaine
edit: wrong acid, it’s actually way worse than i thought
actually you can show that the naturals, integers and rationals all have the the same size.
for example, to show that there are as many naturals as integers (which you do by making a 1-to-1 mapping (more specifically a bijection, i.e. every natural maps to a unique integer and every integer maps to a unique natural) between them), you can say that every natural, n, maps to (n+1)/2 if it is odd and -n/2 if it is even. so 0 and 1 map to themselves, 2 maps to -1, 3 maps to 2, 4 maps to -2, and so on. this maps every natural number to an integer, and vice-versa. therefore, the cardinality (size) of the naturals and the integers are the same.
you can do something similar for the rationals (if you want to try your hand at proving this yourself, it can be made a lot easier by noting that if you can find a function that maps every natural to a unique rational (an injection), and another function that maps every rational to a unique natural, you can use those construct a bijection between the naturals and rationals. this is called the schröder-bernstein theorem).
it turns out that you cannot do this kind of mapping between the naturals (or any other set of that cardinality) and the reals. i won’t recite it here, but cantor’s diagonal argument is a quite elegant proof of this fact.
now, this raises a question: is there anything between the naturals (and friends) and the reals? it turns out that we don’t actually know. this is called the continuum hypothesis
i think the argument here is more that saying “you can use this however you like, no questions asked” is a bad idea because it allows corporations to approriate the work
wait that says hydrofluoric acid that seems way worse???
i think my brain just kinda autocorrected it to HCl because it scares me to think that there’s someone out there who a) has access to HF and b) is dumb enough to put it in their ass