Joel David Hamkins
@joeldavidhamkins.bsky.social
Mathematics and Philosophy of the Infinite
Professor of Logic, University of Notre Dame
University of Oxford
#InfinitelyMore #BookOfInfinity #PanoramaOfLogic #PhilMaths
https://buymeacoffee.com/joeldavidhamkins
Professor of Logic, University of Notre Dame
University of Oxford
#InfinitelyMore #BookOfInfinity #PanoramaOfLogic #PhilMaths
https://buymeacoffee.com/joeldavidhamkins
Tonight on campus at Notre Dame. I assure you that under that snow, the leaves are the most incredible majestic autumn colors!
November 10, 2025 at 12:48 AM
Tonight on campus at Notre Dame. I assure you that under that snow, the leaves are the most incredible majestic autumn colors!
Barbara Montero on machine consciousness, appearing in the New York Times today: www.nytimes.com/2025/11/08/o...
Opinion | A.I. Is Already Intelligent. This Is How It Becomes Conscious.
www.nytimes.com
November 8, 2025 at 2:13 PM
Barbara Montero on machine consciousness, appearing in the New York Times today: www.nytimes.com/2025/11/08/o...
Take my midterm exam!
This was yesterday's midterm exam for my Core Seminar in Philosophy, Science, and Mathematics, an undergraduate seminar here at Notre Dame. So far, we've read the works in this photo, in truth a fairly demanding load for (half) an undergraduate course.
This was yesterday's midterm exam for my Core Seminar in Philosophy, Science, and Mathematics, an undergraduate seminar here at Notre Dame. So far, we've read the works in this photo, in truth a fairly demanding load for (half) an undergraduate course.
November 8, 2025 at 12:47 AM
Take my midterm exam!
This was yesterday's midterm exam for my Core Seminar in Philosophy, Science, and Mathematics, an undergraduate seminar here at Notre Dame. So far, we've read the works in this photo, in truth a fairly demanding load for (half) an undergraduate course.
This was yesterday's midterm exam for my Core Seminar in Philosophy, Science, and Mathematics, an undergraduate seminar here at Notre Dame. So far, we've read the works in this photo, in truth a fairly demanding load for (half) an undergraduate course.
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
How to understand non-standard halting times?
Take any desired consistent c.e. theory $T$ asserting the basic facts of arithmetic such that $T$ can formalize the operation of Turing machines. There is some Turing machine program $p$ and input ...
mathoverflow.net
November 6, 2025 at 9:01 PM
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!
November 5, 2025 at 7:50 PM
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
www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore
The omnific integers are strange and weird
We shall explore several surprising failures of the analogy between the omnific integers and the integers.
www.infinitelymore.xyz
November 4, 2025 at 6:35 PM
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
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.
November 4, 2025 at 2:37 AM
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.
October 30, 2025 at 3:11 PM
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.
Victory at all costs--the role of effort in expert action.
October 29, 2025 at 3:26 AM
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.
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
mathoverflow.net/q/421788/1946
What's the consistency strength of this strengthening of weakly superstrong cardinals?
Recall that a cardinal $\kappa$ is weakly superstrong if, for every $A \subseteq V_\kappa$, there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \ra...
mathoverflow.net
October 28, 2025 at 10:21 PM
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
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
Is it an integer part?
www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore
The omnific integers are an integer part of the surreal numbers
Can we find a surreal-numbers analogue of the integers? An integer part of the surreal numbers, a discretely ordered subring, for which every surreal number is within 1.
www.infinitelymore.xyz
October 27, 2025 at 3:52 PM
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
Is it an integer part?
www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore
This week's reading for my undergrad PhilSciMath core seminar.
October 27, 2025 at 1:25 AM
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
Can a sentence and its negation both be $\Sigma_2$-conservative over ZFC?
Is there a sentence $\sigma$ such that $\sigma$ and $\neg \sigma$ are both $\Sigma_2$-conservative over ZFC?
Eventual GCH and its negation are both $\Pi_2$-conservative over ZFC. But a $\Sigma_2$
mathoverflow.net
October 25, 2025 at 12:31 PM
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
Can we find a surreal-numbers analogue of the integers? An integer part of the surreal numbers, a discretely ordered subring, for which every surreal number is within 1.
www.infinitelymore.xyz
October 24, 2025 at 1:23 AM
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
The latest installment in my series of essays on the surreal numbers.
www.infinitelymore.xyz/p/omnific-in... #InfinitelyMore
The omnific integers are an integer part of the surreal numbers
Can we find a surreal-numbers analogue of the integers? An integer part of the surreal numbers, a discretely ordered subring, for which every surreal number is within 1.
www.infinitelymore.xyz
October 23, 2025 at 1:27 PM
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
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.
www.infinitelymore.xyz #InfinitelyMore
Weekly content on the mathematics and philosophy of infinity.
October 17, 2025 at 12:48 AM
Hit 8000. Thank you subscribers!
www.infinitelymore.xyz #InfinitelyMore
Weekly content on the mathematics and philosophy of infinity.
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
www.infinitelymore.xyz/t/tactic #InfinitelyMore
October 16, 2025 at 12:44 PM
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
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
Determined, finite games
What is the simplest way to prove that each finite game is also determined? I know that a game is said to be determined if one of the players has a winning strategy. I was hoping to prove by
mathoverflow.net
October 15, 2025 at 10:39 PM
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
(Start at the oldest essay, at bottom, and move up.)
www.infinitelymore.xyz/t/infinite-s... #InfinitelyMore
October 15, 2025 at 10:08 PM
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
(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.
October 14, 2025 at 9:17 PM
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
October 14, 2025 at 12:40 PM
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.
October 14, 2025 at 12:39 AM
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.
October 14, 2025 at 12:07 AM
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...
Enjoy my latest post on #InfinitelyMore.
www.infinitelymore.xyz/p/surreal-ch...
The surreal ω × ω chessboard is bigger—and stranger—than you think
What is the nature of the surreal ω × ω chessboard? How many squares are there? How many chess pieces shall we require to set up the board?
www.infinitelymore.xyz
October 13, 2025 at 5:31 PM
The surreal ω × ω chessboard is bigger—and stranger—than you think!
Enjoy my latest post on #InfinitelyMore.
www.infinitelymore.xyz/p/surreal-ch...
Enjoy my latest post on #InfinitelyMore.
www.infinitelymore.xyz/p/surreal-ch...