I answered a question on MathOverflow concerning how we are think about running a Turing machine program inside a nonstandard model of arithmetic. mathoverflow.net/a/503573/1946

Midterm exam tomorrow. Wish my students success!

My latest installment in a series of essays on the surreal numbers. This week, we look at several failures of the analogy between the Omnific integers and the integers—Oz is not so very like ℤ after all. www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore

I am truly honoted to be giving the DePrima lecture at Caltech in a few weeks. This distinguished annual lecture series aims to bring mathematical researchers to Caltech to give a talk for the undergraduate mathematical community.

I realized today that the natural sum and product in the ordinals does not play nicely with the concept of even. The every-other pattern breaks down, and we do not have the Euclidean algorithm, because ω cannot be divided by 2 with remainder 0 or 1.

Here are some images I created for a forthcoming article by Barbara Montero and John Toner on the role of effort in expert action. Victory at all costs--the role of effort in expert action.

A paper of mine figures in this MathOverflow question concerning the topic of...wait for it...a strengthening of the weakly superstrong cardinals. mathoverflow.net/q/421788/1946

I have extended my post on the Omnific integers to discuss how they relate to ⟨Ord⟩, the subring generated in No by the ordinal numbers. After all, ⟨Ord⟩ is a discretely ordered subring of the surreals. Is it an integer part? www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore

This week's reading for my undergrad PhilSciMath core seminar.

I answered a question of Elliot Glazer's on MathOverflow which turned on the set-theoretic analogue of the universal algorithm. mathoverflow.net/a/502069/1946

In contrast to Conway, in my new essay I give a purely order-theoretic account of the omnific integers. Also a sign-sequence account, a gap-diameter account, and more, but prove them all equivalent, including Conway's. www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore

The omnific integers are an integer part of the surreal numbers The latest installment in my series of essays on the surreal numbers. www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore

Hit 8000. Thank you subscribers! www.infinitelymore.xyz #InfinitelyMore Weekly content on the mathematics and philosophy of infinity.

Please enjoy my series of essays on the subtle distinction between the game-theoretic concepts of tactics and strategies. www.infinitelymore.xyz/t/tactic #InfinitelyMore

I edited my answer on MO on the fundamental theorem of finite games. Take a look at mathoverflow.net/a/260471/1946

Take a ride on the transfinite subway. See my recent series of essays on the infinite subway paradox. (Start at the oldest essay, at bottom, and move up.) www.infinitelymore.xyz/t/infinite-s... #InfinitelyMore

This Friday, lunch seminar for the History and Philosophy of Science seminar.

How many unit lengths span the interval [0,ω] in the surreal numbers? Find out at www.infinitelymore.xyz/p/surreal-ch.... #InfinitelyMore

I am scheduled to be offering my course on Infinity next semester, and I have been a little worried that previously the class may have been a bit too hard, too demanding of the students. To correct this, my wife Barbara has suggested that I lecture instead on the half infinite.

This week's reading for my undergrad PhilSciMath core seminar. Looking forward to the discussion.

The surreal ω × ω chessboard is bigger—and stranger—than you think! Enjoy my latest post on #InfinitelyMore. www.infinitelymore.xyz/p/surreal-ch...

Asked my pal from Somerset if he knew anyone capable of taking the square root of -𝑟². "I are!" he said. That's a proof by country diction.

I answered a question on MathOverflow about the fruitfulness of considering the closeness-to-falsity of a given statement in mathematics. mathoverflow.net/a/501465/1946

New post-doctoral position at Notre Dame Core analytic/logic/phil math. philjobs.org/job/show/29622 2 year position, $75K annual salary plus $15K expense account. Please apply! (I am chairing this search.)

I am deeply honored to be invited to give the De Prima lecture at Caltech this fall, a distinguished annual math lecture aimed primarily at undergraduates. Help me decide on a topic! What would you request, drawing from topics on which you've seen me write or speak?

I answered a question on MathOverflow about the Boolean ultrapower, regarding the algebraic approach to the ultrapower versus the names-based approach. mathoverflow.net/a/501257/1946

This week's reading for my undergrad PhilSciMath core seminar. Looking forward to the discussion.

If I were a musician... youtu.be/fjesu41f1mY?...

Detail from student quiz on Lakatos, Proofs and Refutations.

Overheard: the wife says "You're always wrong" and the husband replies "You're right".

I have made a post on Substack Notes substack.com/@joeldavidha... to the effect that we should have a Mathematics category on Substack. If you support math on Substack, please like and restack there.

When I was a graduate student in Berkeley, I remember a talk for the Logic Colloquium by Leon Henkin, long a member of the math department amongst the logic group started by Tarski. The title of his talk was "The other two chapters."

The infinite subway paradox, extending to uncountable ordinals. www.infinitelymore.xyz/p/the-uncoun... #InfinitelyMore #FrivolitiesOfTheGods

I answered a question about the first repetition of theories in the cumulative hierarchy. mathoverflow.net/a/500938/1946

Next week's reading for the core PhilSciMath seminar, two sessions.

Today's reading for the Core seminar on Philosophy, Science, & Mathematics. Lockhart's lament.

An addendum about the regrettably common practice of placing quantifiers at the end of a mathematical assertion. math.stackexchange.com/a/5097851/413

Today's reading for the PhilSciMath seminar. Looking forward!

I love lift problems...! #MathsToday #ALevelMaths

What is the probability that 4 points chosen uniformly at random on surface of a sphere form a tetrahedron whose four faces are each acute? Asked on MathOverflow (mathoverflow.net/q/498296/6094) with evidence that the answer is 1/12. But not yet resolved. #MathSky #Mathematics #Geometry #Probability

I realized that it has been 15 years now that I have been the top user on MathOverflow. What a great time its been! So much great mathematics learned and shared. Here is the screen shot from 5 July 2010.

On MathOverflow: a questioner seeks examples of buttons that are not pure buttons in the modal logic of forcing. These are statements that you can turn on and off as you like, but ultimately, if you like, make them necessary, fully on. See my answer at mathoverflow.net/a/500698/1946

I answered a question on MathOverflow about the chromatic number of the antichain hypergraph on the natural numbers. mathoverflow.net/a/500697/1946

Did you know that we can visually see why sqrt(40)=2sqrt(10)? Let's see how we can visualize rewriting square roots!

Although I happily use an abundance of em dashes in my writing—like this, often to set off a comment midstream—I would like to assure everyone that I am in fact human. I do not use AI in my writing.

To which ordinal station can you reach on the transfinite subway line, with a disembarkment at every station along the way, and with a train capacity of n passengers? www.infinitelymore.xyz/p/infinite-s...

Reading Lockhart's Lament in preparation for seminar next week and finding myself underlining nearly every single sentence. What good is that?

Infinite subway paradox. Suppose the train accommodates 25 passengers at most. To which ordinal can you reach, with some schedule of passenger itineraries, while still having at least one disembarkment at every station along the way? www.infinitelymore.xyz/p/infinite-s... #InfinitelyMore

Surprise ending! youtube.com/shorts/y3VDF...

Recently I've had the pleasure of being interviewed on the blog of the American Philosophical Association. You can read the whole thing here: blog.apaonline.org/2025/09/12/a... You don't have to read the whole thing of course, but please enjoy the two snippets that I'm pretty proud of ;)