Question

It seems worthwhile to point out that Steve's Steve’s answer also essentially answers Carl Mummert's Mummert’s question (in a comment) about why one can't can’t get set theory as a definitional extension of mereology by defining points (as things with no proper parts) and then using "“point $x$ is a part of object $y$" y$” as the mereological interpretation of $x\in y$. You can indeed handle sets of points this way, but there's there’s no good way to handle sets of sets. Mereology (at least in Lesniewski's Leśniewski’s version --- I'm — I’m not familiar with other versions) would make no distinction between a collection of sets and the union of those sets. I think you can get somewhat closer to set theory by combining (as Lesniewski Leśniewski did) mereology with ontology, but even then I don't don’t think you get anywhere near ZF. To really handle something like the cumulative hierarchy of ZF (or even the shorter hierarchy of Russell-style type theory, I believe), mereology would have to be supplemented with some way to treat sets as (new) points, something like Frege's Frege’s notion of Wertverlauf (which would probably be anathema to Lesniewski)Leśniewski).