How the continuum hypothesis could have been a fundamental axiom

Joel David Hamkins, “How the continuum hypothesis could have been a fundamental axiom,” Journal for the Philosophy of Mathematics (2024), arxiv:2407.02463.

Abstract. I describe a simple historical thought experiment showing how we might have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.

The JPM will launch in September 2024. Meanwhile, the preprint pdf is available at arxiv.org/pdf/2407.02463.

See also this talk I gave on the topic at the University of Oslo:

Did Turing prove the undecidability of the halting problem?

Joel David Hamkins and Theodor Nenu, “Did Turing prove the undecidability of the halting problem?”, 18 pages, 2024, Mathematics arXiv:2407.00680.

Abstract. We discuss the accuracy of the attribution commonly given to Turing (1936) for the computable undecidability of the halting problem, eventually coming to a nuanced conclusion.

The halting problem is the decision problem of determining whether a given computer program halts on a given input, a problem famously known to be computably undecidable. In the computability theory literature, one quite commonly finds attribution for this result given to Alan Turing (1936), and we should like to consider the extent to which these attributions are accurate. After all, the term halting problem, the modern formulation of the problem, as well as the common self-referential proof of its undecidability, are all—strictly speaking—absent from Turing’s work. However, Turing does introduce the concept of an undecidable decision problem, proving that what he calls the circle-free problem is undecidable and subsequently also that what we call the symbol-printing problem, to decide if a given program will ever print a given symbol, is undecidable. This latter problem is easily seen to be computably equivalent to the halting problem and can arguably serve in diverse contexts and applications in place of the halting problem—they are easily translated to one another. Furthermore, Turing laid down an extensive framework of ideas sufficient for the contemporary analysis of the halting problem, including: the definition of Turing machines; the labeling of programs by numbers in a way that enables programs to be enumerated and also for them to be given as input to other programs; the existence of a universal computer; the undecidability of several problems that, like the halting problem, take other programs as input, including the circle-free problem, the symbol-printing problem, and the infinite-symbol-printing problem, as well as the Hilbert-Ackermann Entscheidungsproblem. In light of these facts, and considering some general cultural observations, by which mathematical attributions are often made not strictly for the exact content of original work, but also generously in many cases for the further aggregative insights to which those ideas directly gave rise, ultimately we do not find it unreasonable to offer qualified attribution to Turing for the undecidability of the halting problem. That said, we also find it incorrect to suggest that one will find a discussion of the halting problem or a proof of its undecidability in Turing (1936).

Read the full paper at the mathematics arxiv:2407.00680. (pdf download available)

Bibliography

Mathematics, Philosophy of Set Theory and Infinity, Back to the Stone Age interview, May 2024

I was interviewed by Francesco Cavina for the Back to the Stone Age series on May 17, 2024, with a sweeping discussion of the philosophy of set theory, infinity, the continuum hypothesis, beauty in mathematics, and much more.

Daniel Solow Author’s Award 2024

My book, Proof and the Art of Mathematics (MIT Press 2020), has been awarded the 2024 Daniel Solow Author’s Award by the Mathematical Association of America.

Proof and the Art of Mathematics, MIT Press, 2020

The MAA asked me to write a brief response to receiving the award…

One of the great pleasures for any mathematician is to share the fascination and wonder of mathematics with those who are eager to learn it—to teach aspiring mathematical minds the art of mathematics, watching as they bend the logical universe to their purpose for the first time. They bring one idea into reactive contact with another, and we observe a carefully controlled explosion of insight, an Aha! moment, the natural consequence of clear and correct mathematical proof. What a joy it has been for me to experience these moments with my students using my book Proof and the Art of Mathematics, and I am truly honored by the recognition of the Daniel Solow award for this book. I am so glad to learn that others have understood so well what I was trying to do with the book and that they also have benefitted from it.

The book is filled with theorems, good solid theorems, theorems which even experienced mathematicians find compelling, but all of them are amenable to elementary proof. I find it an ideal context for teaching the craft of proof writing, showcasing a range of proof methods and styles. Many theorems are proved several times in completely different ways, using different argument methods that engage the problem from totally different perspectives. One thus realizes how a mathematician’s mind expands.

Every chapter ends with a discussion of various mathematical habits of mind, tidbits of wisdom on how to be a mathematician. State claims explicitly, not only for the benefit of your readers but for the clarity of your own conceptions. In your mathematical thinking and analysis, Use metaphor, which can provide a scaffolding of thought for otherwise difficult or abstract mathematical ideas. For mastery and insight, Express key ideas several times in different ways, thereby exploring your concepts more thoroughly. In every mathematical context, Have favorite examples, for they provide a playground of test cases to deepen understanding.

For a taste of the book, let me ask you: If two polygons have the same area, can you cut the first with a scissors into finitely many pieces that can be rearranged exactly to form the second? You’ll find out in chapter 10. What about a square and circle of the same area, allowing cuts along curves? What about higher dimensions?


The book has a supplementary text with many further examples and extensions of the ideas and discussion, including answers to all the odd-numbered exercises and more.

See also the nice announcement at Notre Dame.

Life Story of Mathematician & Philosopher of Infinity, interviewed by The Human Podcast, May 2024

I was interviewed by The Human Podcast on 17 May 2024. Please enjoy our sweeping conversation about nature of infinity, the nature of abstract mathematical existence, the applicability of mathematical abstractions to physical reality, and more. At the end, you will see that I am caught completely at a loss in answer to the question, “What is it to live a good life?”.

Forcing is simply the iterative conception undertaken with multivalued logic, ForcingFest, Oslo, June 2024

I shall be speaking at the ForcingFest meeting at the University of Oslo, 21 June 2024.

Abstract. I will explain how the forcing construction can be seen as a direct implementation of the iterative conception, giving rise to the cumulative hierarchy, but undertaken in the context of multivalued logic. The shape of the logic that is available in effect enables a certain constructive interference of the truth values in such a way that can affect the truth judgements. The core utility of forcing arises from the fact that we can often control these consequences by making a careful choice of the logic to be used, thereby controlling the truth values even of natural set-theoretic statements such as the continuum hypothesis.

The continuum hypothesis could have been a fundamental axiom, CFORS Grad Conference, Oslo, June 2024

I shall be giving a keynote lecture for the CFORS Grad Conference at the University of Oslo, 19-20 June 2024.

Abstract. I shall describe a simple historical thought experiment showing how our attitude toward the continuum hypothesis could easily have been very different than it is. If our mathematical history had been just a little different, I claim, if certain mathematical discoveries had been made in a slightly different order, then we would naturally view the continuum hypothesis as a fundamental axiom of set theory, one furthermore necessary for mathematics and indeed, indispensable for calculus.

The paper is now available at How the continuum hypothesis could have been a fundamental axiom.

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.

How the continuum hypothesis could have been a fundamental axiom, UC Irvine Logic & Philosoph of Science Colloquium, March 2024

This will be a talk for the Logic and Philosophy of Science Colloquium at the University of California at Irvine, 15 March 2024.

Abstract. With a simple historical thought experiment, I should like to describe how we might easily have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.

The paper is now available at How the continuum hypothesis could have been a fundamental axiom.

What if your potentialism is implicitly actualist? Oxford conference, March 2024

This will be a talk at the conference Challenging the Infinite, March 11-12 at Oxford University. (Please register now to book a place.)

Abstract Many commonly considered forms of potentialism, I argue, are implicitly actualist in the sense that a corresponding actualist ontology and theory is interpretable within the potentialist framework using only the resources of the potentialist ontology and theory. And vice versa. For these forms of potentialism, therefore, there seems to be little at stake in the debate between potentialism and actualism—the two perspectives are bi-interpretable accounts of the same underlying semantic content. Meanwhile, more radical forms of potentialism, lacking convergence and amalgamation, do not admit such a bi-interpretation with actualism. In light of this, the central dichotomy in potentialism, to my way of thinking, is not concerned with any issue of height or width, but rather with convergent versus divergent possibility.

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.

The Gödel incompleteness phenomenon, interview with Rahul Sam

Please enjoy my conversation with Rahul Sam for his podcast, a sweeping discussion of topics in the philosophy of mathematics—potentialism, pluralism, Gödel incompleteness, philosophy of set theory, large cardinals, and much more.

Pluralism in the foundations of mathematics, ASL invited address, joint APA/ASL meeting, New York, January 2024

This will be an invited ASL address at the joint meeting of the ASL with the APA Eastern Division conference, held in New York 15-18 January 2024. My talk will be 16 January 2024 11:00 am.

Note the plurality of Empire State Buildings...

Abstract. I shall give an account of the debate on set-theoretic pluralism and pluralism generally in the foundations of mathematics, including arithmetic. Is there ultimately just
one mathematical universe, the final background context, in which every mathematical
question has an absolute, determinate answer? Or do we have rather a multiverse of
mathematical foundations? Some mathematicians and philosophers favor a hybrid notion, with pluralism at the higher realms of set theory, but absoluteness for arithmetic.
What grounds are there for these various positions? How are we to adjudicate between
them? What ultimately is the purpose of a foundation of mathematics?

See the related paper on the mathematical thought experiment: How the continuum hypothesis could have been a fundamental axiom.

The computable model theory of forcing, Rutgers Logic Seminar, December 2023

This will be a talk for the Rutgers University Logic Seminar, December 4, 2023.

Abstract. I shall discuss the computable model theory of forcing. To what extent can we view forcing as a computational process on the models of set theory? Given an oracle for the atomic or elementary diagram of a model (M,∈M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of the corresponding forcing extensions M[G]. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.

The paper is available on the arxiv at https://arxiv.org/abs/2007.00418.