According to the axiom of replacement, the von Neumann hierarchy of sets just keeps going.
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Replying to @ModelOfTheory
In ZFC without Replacement, the number of functions from an N-element set to an M-element set is M * (M - 1) * ... * (M - (N - 1)).
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Replying to @RadishHarmers
You mean injective functions, for finite M,N? True, but does it relate to the von Neumann hierarchy just keeping going?
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Replying to @ModelOfTheory
I mean just a dumb joke about permutations vs. combinations (which may be the same joke you were making for all I know, but I couldn't quite figure it out.)
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Oh! I did not get it at first. I wasn't really making a joke, just noting that the axiom of replacement is what prevents the ordinals from stopping at some point that can be described from below.
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