# Pointwise definable and Leibnizian models of arithmetic and set theory, realized in end extensions of a given model, Notre Dame Logic Seminar, October 2022

This will be a talk for the Notre Dame logic seminar, 11 October 2022, 2pm in Hales-Healey Hall.

Abstract.  I shall present very new results on pointwise definable and Leibnizian end-extensions of models of arithmetic and set theory. Using the universal algorithm, I shall present a new flexible method showing that every countable model of PA admits a pointwise definable $\Sigma_n$-elementary end-extension. Also, any model of PA of size at most continuum admits an extension that is Leibnizian, meaning that any two distinct points are separated by some expressible property. Similar results hold in set theory, where one can also achieve V=L in the extension, or indeed any suitable theory holding in an inner model of the original model.

# 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)

See the research papers:

• Set-theoretic mereology
• 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 = {},
}

# O’Hara Professor of Philosophy and Mathematics, University of Notre Dame

I have now taken up a position at the University of Notre Dame as the O’Hara Professor of Philosophy and Mathematics, beginning January 2022.

My appointment is with the Department of Philosophy with an affiliation with the Department of Mathematics. I expect to be teaching and working with students both in philosophy and mathematics.

Notre Dame offers a unique joint PhD degree program between mathematics and philosophy, the program in logic and the foundations of mathematics. For Notre Dame undergraduates of any major, I encourage you to consider the mathematical philosophy minor.

Notre Dame has strong research groups in logic in both philosophy and mathematics. In philosophy, Notre Dame recently came out very well in the speciality PGR rankings in philosophy of mathematics (#2, tied with NYU, Princeton, behind Harvard), mathematical logic (#2 tied with CMU, behind Harvard), and philosophical logic (group 2). In mathematics, Notre Dame has a strong research group in mathematical logic.

# The pluralist perspective on the axiom of constructibility, MidWest PhilMath Workshop, Notre Dame, October 2014

This will be a featured talk at the Midwest PhilMath Workshop 15, held at Notre Dame University October 18-19, 2014.  W. Hugh Woodin and I will each give one-hour talks in a session on Perspectives on the foundations of set theory, followed by a one-hour discussion of our talks.

Abstract. I shall argue that the commonly held $V\neq L$ via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.

Set-theorists often argue against the axiom of constructibility V=L on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible. Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory. In this talk, I shall criticize Maddy’s proposal, pointing out that neither the fairly-interpreted-in relation nor the (strongly) maximizes-over relation is transitive, and furthermore, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s account, contrary to what had been desired. Ultimately, I shall argue that the V≠L via maximize position loses its force on a multiverse conception of set theory with an upwardly extensible concept of set, in light of the classical facts that models of set theory can generally be extended to models of V=L. I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Slides