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$\begingroup$ I certainly agree that demanding that universe levels fit exactly the natural numbers is a bit much. We need to know what < means to test inhabitation and <= for cumulativity. And we also need to know that that's all we need to know, in order to transport constructions homomorphically between systems of universe levels. Modularity is absolutely the thing to care about! $\endgroup$
– pigworker
Commented Feb 22, 2022 at 19:14

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