Timeline for Can rules of set theory be founded by paralleling parts of atomic Mereology?
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| Mar 23, 2018 at 14:42 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins most of what you are doing is to found Mereology in Set theory, or in founding set theory in some extension of Mereology (like by addition of a primitive singleton function on top of mereology). I'm doing neither here, I'm simply having both primitive relations and seeking to interpret rules of set theory by mimicry in Mereology. I didn't see that in ANY of the theories you've mentioned. | |
| Mar 16, 2018 at 11:16 | vote | accept | Zuhair Al-Johar | ||
| Mar 15, 2018 at 20:32 | comment | added | Zuhair Al-Johar | I read Lewis book on Parts of classes, I didn't see any real motivation for set theoretic axioms, he only explains membership using the singleton function on top of Mereology, he explains also Ur-elements, sets, classes, proper classes, etc.. but the rules of set theory he gives no real explanation of that stems from his mereology, actually in his account they are motivated by size principles, that's why he himself calls mathematics as size theory. Here you see a motivation whether direct (like the individual axioms of $\text{ZF}$ mentioned above) or indirect (like the case of replacement). | |
| Mar 15, 2018 at 19:31 | comment | added | Joel David Hamkins | Thanks, Philip, I'm glad you like the results. I really like them, also, in part because they are mathematically interesting as well as philosophically relevant. In new work, with Ruizhi Yang, we are looking at other definable reducts of $\langle V,\in\rangle$, for example, to the unary union operator. This theory also is decidable, but if you have unary union and binary union, you can define singleton and subset and hence $\in$. It follows that binary union is not definable from unary union. | |
| Mar 15, 2018 at 19:01 | comment | added | Philip Ehrlich | Joel, this a a remarkable set of results. Collectively, they seem to suggest that mereology is so close, yet so very far from being able to provide an adequate foundation for mathematics | |
| Mar 15, 2018 at 11:23 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Mar 15, 2018 at 1:09 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |