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Sniffnoy comments on No coinductive datatype of integers - Less Wrong Discussion
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No, it isn't. The only primitive relations in ZFC are set membership and possibly equality (depending on how you prefer it). "x is a subset of y" is defined to mean "for all z, z in x implies z in y".
Can I downvote myself? Somehow my mind switched "subset" and "membership", and by the virtue of ZFC being a one-sorted theory, lo and behold, I wrote the above absurdity. Anyway, to rewrite the sentence and make it less wrong: subsets(x,y) is defined by the means of a first-order formula through the membership relation, which in a one-sorted theory already pertains the idea of 'subsetting'. x E y --> {x} <= y. So subsetting can be seen as a transfinite extension of the membership relation, and in ZFC we get no more clarity or computational intuition from the first than from the second.