# Gödel incompleteness, graduate course, Notre Dame, Fall 2024

This will be a graduate course at the University of Notre Dame.

Course title: Gödel incompleteness

Course description. We shall explore at length all aspects of the Gödel incompleteness phenomenon, covering Turing’s solution of the Entscheidungsproblem, Gödel’s argument via fixed points, arithmetization, the Hilbert program, Tarski’s theorem, Tarski via Gödel, Tarski via Russell, Tarski via Cantor, the non-collapse of the arithmetic hierarchy, Löb’s theorem, the second incompletenesss theorem via Gödel, via Grelling-Nelson, via Berry’s paradox, Smullyan incompleteness, self-reference, Kleene recursion theorem, Quines, the universal algorithm, and much more. The course will follow the gentle treatment of my book-in-progress, Ten proofs of Gödel incompleteness, with supplemental readings.

# Failing definite descriptions, Notre Dame Food for Thought Seminar, March 2024

I gave a talk for the Food for Thought seminar for the Notre Dame philosophy department.

The topic concerned definite descriptions, particularly the semantics that might be given when one extends first-order logic to include the iota operator, by which $℩x\varphi(x)$ means “the $x$ such that $\varphi(x)$.” There are a variety of natural ways to define the semantics of iota assertions in a model, and we discussed the advantages and disadvantages of each approach. We concentrated on what I call the strong semantics, the weak semantics, and the natural semantics, respectively. Ultimately, I argue for a deflationary perspective on the debate, as each of the semantics is conservative over the base language, with no iota operator, with no new expressive power. In this sense, I argue, the choice of one semantics over another is purely a matter of convenience or ease of expressibility, as all of the notions are expressible without definite descriptions at all.

My lecture notes are below.

# The covering reflection principle, Notre Dame Logic Seminar, February 2024

This will be a talk for the Notre Dame Logic Seminar on 6 February 2024, 2:00 pm.

Abstract. The principle of covering reflection holds of a cardinal $\kappa$ if for every structure $B$ in a countable first-order language there is a structure $A$ of size less than $\kappa$, such that $B$ is covered by elementary images of $A$ in $B$. Is there any such cardinal? Is the principle consistent? This is joint work with myself, Nai-Chung Hou, Andreas Lietz, and Farmer Schlutzenberg.

# Infinity

## Philosophy 20607 01 (32582)

### University of Notre Dame                                                                              Spring 2023

Instructor: Joel David Hamkins, O’Hara Professor of Philosophy and Mathematics
3:30-4:45 Tuesdays + Thursdays, DeBartolo Hall 208

Course Description. This course will be a mathematical and philosophical exploration of infinity, covering a wide selection of topics illustrating this rich, fascinating concept—the mathematics and philosophy of the infinite.

Along the way, we shall find paradox and fun—and all my favorite elementary logic conundrums and puzzles. It will be part of my intention to reveal what I can of the quirky side of mathematics and logic in its connection with infinity, but with a keen eye open for when issues happen to engage with philosophically deeper foundational matters.

The lectures will be based on the chapters of my forthcoming book, The Book of Infinity, currently in preparation, and currently being serialized and made available on the Substack website as I explain below.

Topics. Among the topics we shall aim to discuss will be:

• The Book of Numbers
• Largest number contest
• The googol plex chitty bang stack hierarchy
• Galileo’s Salviati on infinity
• Hilbert’s Grand Hotel
• The uncountable
• How to count (to infinity and beyond!)
• Slaying the Hydra
• Transfinite recursion
• The continuum hypothesis
• The axiom of choice
• Orders of infinity
• The lattice of subsets of ℕ
• Potential versus actual infinity
• Confounding puzzles of infinity
• Infinite liars
• Infinite utilitarianism
• Infinite computation
• Infinite games
• Indescribable numbers
• Extremely remote events of enormous consequence
• The sand reckoner
• The outer limits of reason
• Puzzles of epistemic logic and the problem of common knowledge

Mathematical background. The course will at times involve topics and concepts of a fundamentally mathematical nature, but no particular mathematical background or training will be assumed. Nevertheless, it is expected that students be open to mathematical thinking and ideas, and furthermore it is a core aim of the course to help develop the student’s mastery over various mathematical concepts connected with infinity.

Readings. The lectures will be based on readings from the topic list above that will be made available on my Substack web page, Infinitely More. Readings for the topic list above will be gradually released there during the semester. Each reading will consist of a chapter essay my book-in-progress, The Book of Infinity, which is being serialized on the Substack site specifically for this course. In some weeks, there will be supplemental readings from other sources.

Student access. I will issue subscription invitations to the Substack site for all registered ND students using their ND email, with free access to the site during the semester, so that students can freely access the readings.  Students are free to manage their subscriptions however they see fit. Please inform me of any access issues. There are some excellent free Substack apps available for Apple iOS and Android for reading Substack content on a phone or other device.

Discussion forum. Students are welcome to participate in the discussion forums provided with the readings to discuss the topics, the questions, to post answer ideas, or engage in the discussion there. I shall try to participate myself by posting comments or hints.

Homework essays. Students are expected to engage fully with every topic covered in the class. Every chapter concludes with several Questions for Further Thought, with which the students should engage. It will be expected that students complete approximately half of the Questions for Further thought. Each question that is answered should be answered essay-style with a mini-essay of about half a page or more.

Extended essays. A student may choose at any time to answer one of the Questions for Further Thought more fully with a more extended essay of two or three pages, and in this case, other questions on that particular topic need not be engaged. Every student should plan to exercise this option at least twice during the semester.

Final exam.  There will be a final exam consisting of questions similar to those in the Questions for Further Thought, covering every topic that was covered in the course. The final grade will be based on the final exam and on the submitted homework solutions.

Open Invitation. Students outside of Notre Dame are welcome to follow along with the Infinity course, readings, and online discussion. Simply subscribe at Infinitely More, keep up with the readings and participate in the discussions we shall be having in the forums there.

# 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:

# 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