Timeline for Why hasn't mereology succeeded as an alternative to set theory?
Current License: CC BY-SA 4.0
17 events
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| Jan 6, 2023 at 10:19 | history | edited | Rafał Gruszczyński | CC BY-SA 4.0 |
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| Jan 6, 2023 at 0:59 | comment | added | user76284 | There's a typo in 𝑧⊥𝑦⟺¬∃𝑢(𝑢⊑𝑧∧𝑢⊑𝑧). | |
| Jan 23, 2017 at 23:44 | comment | added | Rafał Gruszczyński | @godelian No, non-atomicity (which you call non-wellfoundedness) is non inconsistent with antisymmetry. There are good models of various nereological theories with parthood being antisymmetrical (which is standard and natural approach), but with no minimal objects. | |
| Jan 19, 2017 at 7:53 | history | edited | Rafał Gruszczyński | CC BY-SA 3.0 |
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| Aug 18, 2012 at 23:25 | comment | added | godelian | @Steve: Thanks for the reference! I quickly glanced at the article, and got already an answer: non-wellfoundedness is inconsistent with the antisymmetry of the parthood relation (which there is called extensionality). | |
| Aug 18, 2012 at 15:04 | comment | added | Rafał Gruszczyński | The fact that the only structure satisfying axioms for mereology plus the schema in question can be shown directly using the fact that mereology axioms entail existence of the unity $\mathbf{1}$, that is the object $x$ such that $\forall y(y\sqsubseteq x)$. One can now put $\varphi(x)\iff\forall y(y\sqsubseteq x)$. Since for any object $y$ it is the case that $y\sqsubseteq\mathbf{1}=\bigl[x\mid\varphi(x)\bigr]$, the axiom entails $\forall z(z\sqsubseteq y)$, that is $y=\mathbf{1}$. Andreas, thank you very much once again for the comment! | |
| Aug 18, 2012 at 11:36 | comment | added | Rafał Gruszczyński | @godelian: You can find something about non-wellfounded approach to mereology in the paper by A.J. Cotnoir and A. Bacon "Non-wellfounded mereology", Review of Symbolic Logic / Volume 5 / Issue 02 / June 2012, pp. 187-204 . Hope this helps. | |
| Aug 18, 2012 at 11:34 | comment | added | Rafał Gruszczyński | @Andreas: You are right. Unfortunately, the result of linearity is narrowing down the class of models to just one (up to isomorphism) degenerate (i.e. one-element) structure. Thanks for the comment. | |
| Aug 18, 2012 at 11:17 | history | edited | Rafał Gruszczyński | CC BY-SA 3.0 |
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| Aug 18, 2012 at 8:38 | comment | added | godelian | Thanks Steve for your detailed answer. I really like the fact that you cannot distinguish an object and its mereological singleton, and agree that this is an obstacle to the interpretation of ZF. Do you know of the attempts of adding an axiom that forbids atoms? Such an axiom would be appealing to me for, say, ontological reasons, but I'm not sure whether introducing non-welfoundedness would make things easier, as Andreas comment to this answer shows. | |
| Aug 17, 2012 at 20:33 | comment | added | Andreas Blass | You wrote that it may be interesting to consider mereology with the additional axiom that if $x$ is part of the sum of the $\phi$-ers then $x$ is itself a $\phi$-er. This axiom looks very strange to me for the following reason. Consider any two things $a$ and $b$, and let $\phi(z)$ say "$z=a$ or $z=b$". Let $s$ be the sum of the $\phi$-ers, i.e., of $a$ and $b$. Since $s$ is part of itself, your axiom would require $\phi(s)$. So $s$ would be one of $a$ and $b$, say $a$. Since $b$ is part of $s$, we'd get that $b$ is part of $a$. Conclusion: Of any two things, one is part of the other. | |
| Aug 17, 2012 at 19:43 | history | edited | Rafał Gruszczyński | CC BY-SA 3.0 |
Minor corrections.
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| Aug 17, 2012 at 19:37 | history | edited | Rafał Gruszczyński | CC BY-SA 3.0 |
Corrected meaning of $\bot$.
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| Aug 17, 2012 at 19:11 | history | edited | Rafał Gruszczyński | CC BY-SA 3.0 |
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| Aug 17, 2012 at 19:06 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
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| Aug 17, 2012 at 19:03 | history | edited | Rafał Gruszczyński | CC BY-SA 3.0 |
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| Aug 17, 2012 at 18:56 | history | answered | Rafał Gruszczyński | CC BY-SA 3.0 |