Ohhh, which of these three options is correct? Given my focus on philosophy of math it's mildly embarrassing not to have a fixed position on this, but I keep going back and forth...
1. Just say the hierarchy of sets goes "all the way up"
2. Say the hierarchy of sets goes "all the way up" in the sense that it contains ordinals corresponding to every distinct combinatorially possible way for some objects to be well ordered *except for the one that it, itself is an instance of*. (this would be appealing but i think it may be impossible to spell out in a consistent way)
3. Say that the hierarchy of sets goes up at least far enough to satisfy the axiom of infinity+ the rest of ZF, and leaves it vague what there is beyond the inacessables - much as our concept mountain leaves it vague how many really tiny mountains there are given that there is such and such a bit of lumpy terrain.
Naive comment: given the still-unknown truth or falsity of the continuum hypothesis, is it even safe to say "up" in there being a single direction in which the hierarchy of sets goes? This seems to force #3 as the only safe option, no?
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