A related rabbit hole that you can jump down in FoM is that Zermelo's ordinals and von Neumann ordinals cannot both be true at the same time. Wikipedia's intro to the topic [0] is a starting place, see also [1] which might be more in depth.
> If you don't know why set theory is important, it is because set theory is the foundation of all of mathematics.
Nitpick maybe but the types and categories people might prefer sets as "a" foundation instead of "the"? These 3 things are the most useful, get the most attention, and have benefited from the most serious efforts. But IMHO one of the cool things about math is that if you're willing to squint and work at it, then many alternative foundations are possible. For example Conway's surreals[2] hint that you can get numbers/sets by starting with even games as a primitive. I can't quickly find refs, but the visualizations here hint that starting with graph theoretic axioms can lead to sets instead of vice-versa and I think people have worked on that too. Who knows whether alien math builds everything else up starting from geometry or probability, etc.
That’s pretty interesting, thanks. It’s related to abstraction (interfaces) vs representation (implementation) in programming. To my eye, there’s no conflict that 1 \in 3 is true in one representation and not in the other. Trying to use \in like that seems like a violation of an abstract interface, somehow expecting that the various implementations of an abstraction must be identical. It also doesn’t seem to violate platonism, since “numbers” are abstract ideas that are not expressible directly in set theory. Set theory can only encode concrete representations of numbers. Much like any physical chair cannot be identical to the platonic Chair.
[0] https://en.m.wikipedia.org/wiki/Benacerraf%27s_identificatio... [1] https://plato.stanford.edu/entries/philosophy-mathematics/#W...
> If you don't know why set theory is important, it is because set theory is the foundation of all of mathematics.
Nitpick maybe but the types and categories people might prefer sets as "a" foundation instead of "the"? These 3 things are the most useful, get the most attention, and have benefited from the most serious efforts. But IMHO one of the cool things about math is that if you're willing to squint and work at it, then many alternative foundations are possible. For example Conway's surreals[2] hint that you can get numbers/sets by starting with even games as a primitive. I can't quickly find refs, but the visualizations here hint that starting with graph theoretic axioms can lead to sets instead of vice-versa and I think people have worked on that too. Who knows whether alien math builds everything else up starting from geometry or probability, etc.
[2] https://en.wikipedia.org/wiki/Surreal_number