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Thanks. How I would've imagined this could work (if it could be made to work) is to somehow account for the notion that your "infinity" just got halved (yes I g

Thanks. How I would've imagined this could work (if it could be made to work) is to somehow account for the notion that your "infinity" just got halved (yes I get that infinity is not part of the set, but let me wave my hands here while we're breaking our axioms), and everything "after" it is no longer in the set. Which would imply that "{1, 2, 3, ...}" is no longer a sufficient description for the set of natural numbers; you'd probably need extra information (maybe a "scale factor" for the infinity or something). I imagine you're right that this leads to a contradiction somewhere, but I (obviously) haven't thought it through. I just would love to see someone try to break enough axioms to make it work and see what comes out of it, or show that it contradicts either itself or something we see in the real world if we do that.

dullcrisp 2y ago
You can have the partial order A ≤ B iff A ⊆ B. But if you want size({1}) = size({2}), size({2}) = size({3}), etc., then you’ll find that size({1,2,3,…}) = size({2,3,4,…}) and there’s really nothing you can do about it. But yes, the ordinals might be more to your liking. If you equip your sets with more structure you can say more things about them.