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fix Leśniewski (and other typography)
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Emil Jeřábek
Emil Jeřábek
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It seems worthwhile to point out that Steve'sSteve’s answer also essentially answers Carl Mummert'sMummert’s question (in a comment) about why one can'tcan’t get set theory as a definitional extension of mereology by defining points (as things with no proper parts) and then using "point“point $x$ is a part of object $y$"” as the mereological interpretation of $x\in y$. You can indeed handle sets of points this way, but there'sthere’s no good way to handle sets of sets. Mereology (at least in Lesniewski'sLeśniewski’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 LesniewskiLeśniewski did) mereology with ontology, but even then I don'tdon’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'sFrege’s notion of Wertverlauf (which would probably be anathema to LesniewskiLeśniewski).

It seems worthwhile to point out that Steve's answer also essentially answers Carl Mummert's question (in a comment) about why one 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 $y$" as the mereological interpretation of $x\in y$. You can indeed handle sets of points this way, but there's no good way to handle sets of sets. Mereology (at least in Lesniewski's version --- 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 did) mereology with ontology, but even then I 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 notion of Wertverlauf (which would probably be anathema to Lesniewski).

It seems worthwhile to point out that Steve’s answer also essentially answers Carl Mummert’s question (in a comment) about why one 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 $y$” as the mereological interpretation of $x\in y$. You can indeed handle sets of points this way, but there’s no good way to handle sets of sets. Mereology (at least in Leśniewski’s version — 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 Leśniewski did) mereology with ontology, but even then I 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 notion of Wertverlauf (which would probably be anathema to Leśniewski).

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Andreas Blass
Andreas Blass
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It seems worthwhile to point out that Steve's answer also essentially answers Carl Mummert's question (in a comment) about why one 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 $y$" as the mereological interpretation of $x\in y$. You can indeed handle sets of points this way, but there's no good way to handle sets of sets. Mereology (at least in Lesniewski's version --- 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 did) mereology with ontology, but even then I 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 notion of Wertverlauf (which would probably be anathema to Lesniewski).