> cardinals are mathematical games Totally agreed, as with 2+2=4. It's an interesting topic in philosophy that we often gloss over because the naturals are exc

> cardinals are mathematical games Totally agreed, as with 2+2=4. It's an interesting topic in philosophy that we often gloss over because the naturals are exceptionally predictive of so many things in the real world. > cardinals aren't necessarily commonsense Maybe. The two notions being applied are (1) that if you just rename everything in your set you haven't changed it's size, and (2) that if all renaming attempts necessarily leave some elements out then the set containing those elements must be bigger. Any "size" capturing those two ideas is equivalent to a renaming of the cardinals. > naturals vs even naturals Any topological sort of the infinite subsets of the naturals preserving the set inclusion partial order is going to have some weird oddities. In particular, you'll have a lot of incomparable sets (like the evens and odds, or like all the vertex-deleted N-{x} sets) which are artificially bigger/smaller than each other or artificially the same size, despite the order superficially looking like one based on set inclusion. > other more commonsense alternatives to cardinals Even just looking at infinite subsets of the naturals, any ordering you choose must necessarily have the property that there exist infinitely many sizes which each either do not have a next bigger element or do not have a next smaller element. That's a decidedly weird property that doesn't manifest for finite sizes, and it's entirely unavoidable. I'm potentially okay with "size" not being used as the name, but as a counterpoint the motivation is to extend the notion of "size" for finite sets as best as possible. > cardinals aren't useful They're often used to prove things that you do use directly. Sometimes those proofs can get convoluted, but a surprisingly effective technique in many domains is just arguing that two sets have different sizes so they can't be the same (or can't map to each other in the desired way or whatever). Cardinalities agree with our intuition about size on finite sets well enough to directly extend to those sorts of proofs. As one example, the math leading up to the fixed point theorems proving the optimal solution to a GAN yields the desired probability distribution is usually done (and originally done iirc) via infinite counting arguments. I'll be the first to admit that my day-to-day as a programmer doesn't often do much with infinite cardinalities, but given that the mathematical world we've constructed for ourselves happens to often line up nicely with the real world, "mathematical games" that leverage existing intuition to expand our knowledge of that mathematical world still seem valuable.