The model theory of set-theoretic mereology, Notre Dame Math Logic Seminar, February 2022

This will be a talk for the Mathematical Logic Seminar at the University of Notre Dame on 8 February 2022 at 2 pm in 125 Hayes Healy.

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? How well is a model of set theory ⟨M,∈⟩ captured by its mereological reduct ⟨M,⊆⟩? In short, how much set theory does set-theoretic mereology know? In this talk, I shall present results on the model theory of set-theoretic mereology that lead broadly to negative answers to these questions and explain why mereology has not been successful as a foundation of mathematics. (Joint work with Makoto Kikuchi)

Handwritten lecture notes

See the research papers:

  • Set-theoretic mereology
    • [DOI] J. D. Hamkins and M. Kikuchi, “Set-theoretic mereology,” Logic and Logical Philosophy, Special issue “Mereology and beyond, part II”, vol. 25, iss. 3, p. 285–308, 2016.
      [Bibtex]
      @ARTICLE{HamkinsKikuchi2016:Set-theoreticMereology,
      author = {Joel David Hamkins and Makoto Kikuchi},
      title = {Set-theoretic mereology},
      journal = {Logic and Logical Philosophy, Special issue ``Mereology and beyond, part II''},
      editor = {A.~C.~Varzi and R.~Gruszczy{\'n}ski},
      year = {2016},
      volume = {25},
      number = {3},
      pages = {285--308},
      month = {},
      doi = {10.12775/LLP.2016.007},
      note = {},
      eprint = {1601.06593},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      url = {http://jdh.hamkins.org/set-theoretic-mereology},
      abstract = {},
      keywords = {},
      source = {},
      ISSN = {1425-3305},
      MRCLASS = {03A05 (03E70)},
      MRNUMBER = {3546211},
      }
  • The inclusion relations of the countable models of set theory are all isomorphic
    • J. D. Hamkins and M. Kikuchi, “The inclusion relations of the countable models of set theory are all isomorphic,” ArXiv e-prints, 2017.
      [Bibtex]
      @ARTICLE{HamkinsKikuchi:The-inclusion-relations-of-the-countable-models-of-set-theory-are-all-isomorphic,
      author = {Joel David Hamkins and Makoto Kikuchi},
      title = {The inclusion relations of the countable models of set theory are all isomorphic},
      journal = {ArXiv e-prints},
      editor = {},
      year = {2017},
      volume = {},
      number = {},
      pages = {},
      month = {},
      doi = {},
      note = {Manuscript under review},
      eprint = {1704.04480},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      url = {http://jdh.hamkins.org/inclusion-relations-are-all-isomorphic},
      abstract = {},
      keywords = {under-review},
      source = {},
      }
Notre Dame campus in snow

On set-theoretic mereology as a foundation of mathematics, Oxford Phil Math seminar, October 2018

This will be a talk for the Philosophy of Mathematics Seminar in Oxford, October 29, 2018, 4:30-6:30 in the Ryle Room of the Philosopher Centre.

Abstract. In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, it is natural to wonder whether one might find a similar success for set-theoretic mereology, based upon the set-theoretic inclusion relation $\subseteq$ rather than the element-of relation $\in$.  How well does set-theoretic mereological serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics in terms of the subset relation to the same extent that set theorists have argued (with whatever degree of success) that we may find faithful representations in terms of the membership relation? Basically, can we get by with merely $\subseteq$ in place of $\in$? 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.

This is joint work with Makoto Kikuchi, and the talk is based on our joint articles:

The talk will also mention some related recent work with Ruizhi Yang (Shanghai).

Slides