I shall be giving a keynote plenary talk for the 16th International Congress of Logic, Methodology and Philosophy of Science and Technology (CLMPST 2019), to be held 5-10 August 2019 at the Institute of Philosophy of the Czech Academy of Sciences in the beautiful city of Prague . The CLMPST congress is held every four years, and the theme of the 2019 meeting is, “Bridging across academic cultures.”
Can set-theoretic mereology serve as a foundation of mathematics?
Abstract: Mereology, the study of the relation of part to whole, is often contrasted with set theory and its membership relation, the relation of element to set. Whereas set theory has found comparative success in the foundation of mathematics, since the time of Cantor, Zermelo and Hilbert, mereology is strangely absent. Can a set-theoretic mereology, based upon the set-theoretic inclusion relation ⊆ rather than the element-of relation ∈, serve as a foundation of mathematics? Can we faithfully interpret arbitrary mathematical structure in terms of the subset relation to the same extent that set theorists have done so in terms of the membership relation? At bottom, the question is: can we get by with merely ⊆ in place of ∈? Ultimately, I shall identify grounds supporting generally negative answers to these questions, concluding that set-theoretic mereology by itself cannot serve adequately as a foundational theory.
Please join me in Prague! See the Call for Papers, requesting contributed papers and contributed symposia on twenty different thematic sections, from mathematical and philosophical logic to the philosophy of science, philosophy of computing and many other areas. I am given to understand that this will be a large meeting, with about 800 participants expected.
- J. D. Hamkins and M. Kikuchi, “Set-theoretic mereology,” Logic and Logical Philosophy, special issue “Mereology and beyond, part II”, vol. 25, iss. 3, pp. 285-308, 2016.
- J. D. Hamkins and M. Kikuchi, “The inclusion relations of the countable models of set theory are all isomorphic,” ArXiv e-prints, 2017.
The formula for |s \cup t| \geq n on slide 30 (or 71 of 134) has a typo: It should read |s \cap t| \leq j instead of |s \cap t| \geq j. The same typo is also present in the corresponding paper from 2016. The link to the blog entry for that paper above also has a typo: the jdh.hamkins.org prefix is missing.
Thanks for the slides. So far (have not read part slide 30 yet), I really enjoyed reading them.
Could you explain your reasons? I think my slides and paper are correct, and that your proposed change would be wrong. The point is that (s union t) has size at least n, just in case there is a way of allocating the three parts of the Venn diagram with i,j,k so as to realize the three clauses. Certainly, if the right-hand side holds as I’ve described it, then of course, (s union t) will have at least size n, and conversely, if s union t has size at least n, then there will be some i+j+k=n realizing the desired pattern.
So I don’t regard this as a “typo”. But perhaps I’m missing something?
(Meanwhile, I have corrected the link; thanks very much for that.)
But what happens in your original formula, if s = t = s \cap t and |s| = n-1? In this case, we have s = s \cup t, hence |s \cup t| = n-1, so the condition on the left side is not satisfied. But with i = 1, j = n-2 and k = 1, the corresponding term of the disjunction on the right side is satisfied.
My correction of the typo above was still insufficient. In addition, the case |s \cap t| \geq n must also be added to the disjunction over the cases, since it is no longer included in the other cases after fixing the typo.