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
Please enjoy my latest installment in a series of essays on the ordinals—the Cantor Normal Form. An ordinal denotation system, as useful as the decimal number system, but for arbitrary ordinals and grounded in base ω rather than base ten.
www.infinitelymore.xyz/p/cantor-nor... #InfinitelyMore
www.infinitelymore.xyz/p/cantor-nor... #InfinitelyMore
Cantor Normal Form
Cantor proved a remarkable fact about ordinals, providing an ordinal notation system in which every ordinal admits a unique canonical representation by what we now call its Cantor normal form. The not...
www.infinitelymore.xyz
February 12, 2026 at 12:55 PM
Please enjoy my latest installment in a series of essays on the ordinals—the Cantor Normal Form. An ordinal denotation system, as useful as the decimal number system, but for arbitrary ordinals and grounded in base ω rather than base ten.
www.infinitelymore.xyz/p/cantor-nor... #InfinitelyMore
www.infinitelymore.xyz/p/cantor-nor... #InfinitelyMore
I shall be honored to give the Owens Memorial lecture at Wayne State University in April, joining a distinguished list of previous speakers, including many Fields medalists.
"Truth and Paradox in the theory of finite and infinite games."
jdh.hamkins.org/truth-and-pa...
"Truth and Paradox in the theory of finite and infinite games."
jdh.hamkins.org/truth-and-pa...
Truth and paradox in the theory of finite and infinite games, Owens Memorial Lecture, Wayne State University, April 2026
I am honored to be invited to give the Owen G. Owens Memorial Lecture at Wayne State University on 16 April 2026, joining a distinguished list of luminaries giving previous Owens lectures, includin…
jdh.hamkins.org
February 12, 2026 at 2:48 AM
I shall be honored to give the Owens Memorial lecture at Wayne State University in April, joining a distinguished list of previous speakers, including many Fields medalists.
"Truth and Paradox in the theory of finite and infinite games."
jdh.hamkins.org/truth-and-pa...
"Truth and Paradox in the theory of finite and infinite games."
jdh.hamkins.org/truth-and-pa...
Join me next week for the ASL talks at the Central APA Meeting in Chicago, at the elegant Palmer House hotel.
www.apaonline.org/mpage/2026ce...
ASL Association of Symbolic Logic
APA American Philosophical Association
www.apaonline.org/mpage/2026ce...
ASL Association of Symbolic Logic
APA American Philosophical Association
2026 Central Division Meeting | American Philosophical Association
www.apaonline.org
February 11, 2026 at 9:07 PM
Join me next week for the ASL talks at the Central APA Meeting in Chicago, at the elegant Palmer House hotel.
www.apaonline.org/mpage/2026ce...
ASL Association of Symbolic Logic
APA American Philosophical Association
www.apaonline.org/mpage/2026ce...
ASL Association of Symbolic Logic
APA American Philosophical Association
Woo hoo! Ten thousand subscribers on Infinitely More.
Growth, however, has not been fully differentiable.
www.infinitelymore.xyz #InfinitelyMore
Growth, however, has not been fully differentiable.
www.infinitelymore.xyz #InfinitelyMore
February 11, 2026 at 4:06 PM
Woo hoo! Ten thousand subscribers on Infinitely More.
Growth, however, has not been fully differentiable.
www.infinitelymore.xyz #InfinitelyMore
Growth, however, has not been fully differentiable.
www.infinitelymore.xyz #InfinitelyMore
Today's lecture for my Infinity class is on equinumerosity and the notion of size. You are welcome to follow along with the readings for our class.
www.infinitelymore.xyz/p/equinumero... #InfinitelyMore #BookOfInfinity
www.infinitelymore.xyz/p/equinumero... #InfinitelyMore #BookOfInfinity
Equinumerosity and comparison of size
What does it mean for two sets to have the same size? Galileo's paradox identifies tension between notions of equinumerosity and Euclid's principle that the whole is greater than any proper part.
www.infinitelymore.xyz
February 10, 2026 at 4:14 PM
Today's lecture for my Infinity class is on equinumerosity and the notion of size. You are welcome to follow along with the readings for our class.
www.infinitelymore.xyz/p/equinumero... #InfinitelyMore #BookOfInfinity
www.infinitelymore.xyz/p/equinumero... #InfinitelyMore #BookOfInfinity
Today's lecture for my infinity class is on
The Largest Tweetable Number
Follow along on Infinitely More.
www.infinitelymore.xyz/p/the-larges... #InfinitelyMore #BookOfInfinity
The Largest Tweetable Number
Follow along on Infinitely More.
www.infinitelymore.xyz/p/the-larges... #InfinitelyMore #BookOfInfinity
The largest tweetable number
What is the largest tweetable number?
www.infinitelymore.xyz
February 5, 2026 at 4:34 PM
Today's lecture for my infinity class is on
The Largest Tweetable Number
Follow along on Infinitely More.
www.infinitelymore.xyz/p/the-larges... #InfinitelyMore #BookOfInfinity
The Largest Tweetable Number
Follow along on Infinitely More.
www.infinitelymore.xyz/p/the-larges... #InfinitelyMore #BookOfInfinity
"Surreal arithmetic is bi-interpretable with set theory"
I shall be speaking (in person) at the CUNY Logic Workshop on 13 March.
jdh.hamkins.org/surreal-arit...
I shall be speaking (in person) at the CUNY Logic Workshop on 13 March.
jdh.hamkins.org/surreal-arit...
Surreal arithmetic is bi-interpretable with set theory, CUNY Logic Workshop, March 2026
This will be a talk at the CUNY Logic Workshop on 13 March 2026, held at the CUNY Graduate Center. Abstract. I shall introduce the elementary theory of surreal arithmetic (SA), a first-order theory…
jdh.hamkins.org
February 4, 2026 at 6:10 PM
"Surreal arithmetic is bi-interpretable with set theory"
I shall be speaking (in person) at the CUNY Logic Workshop on 13 March.
jdh.hamkins.org/surreal-arit...
I shall be speaking (in person) at the CUNY Logic Workshop on 13 March.
jdh.hamkins.org/surreal-arit...
Can you answer today's quiz in my infinity class?
Consider the balls-in-a-sack paradox. You hold an empty sack, with nearby balls numbered 1, 2, 3, and so on. At each step, you will add the next two balls to the sack, and then remove one ball from the sack and discard it.
Consider the balls-in-a-sack paradox. You hold an empty sack, with nearby balls numbered 1, 2, 3, and so on. At each step, you will add the next two balls to the sack, and then remove one ball from the sack and discard it.
February 4, 2026 at 2:21 AM
Can you answer today's quiz in my infinity class?
Consider the balls-in-a-sack paradox. You hold an empty sack, with nearby balls numbered 1, 2, 3, and so on. At each step, you will add the next two balls to the sack, and then remove one ball from the sack and discard it.
Consider the balls-in-a-sack paradox. You hold an empty sack, with nearby balls numbered 1, 2, 3, and so on. At each step, you will add the next two balls to the sack, and then remove one ball from the sack and discard it.
Today's lecture for my Infinity class is on
The paradox of giants.
You are kindly welcome to follow along with us in the reading—post your comments, and try the exercise questions!
www.infinitelymore.xyz/p/the-parado... #InfinitelyMore #BookOfInfinity
The paradox of giants.
You are kindly welcome to follow along with us in the reading—post your comments, and try the exercise questions!
www.infinitelymore.xyz/p/the-parado... #InfinitelyMore #BookOfInfinity
www.infinitelymore.xyz
February 3, 2026 at 2:58 PM
Today's lecture for my Infinity class is on
The paradox of giants.
You are kindly welcome to follow along with us in the reading—post your comments, and try the exercise questions!
www.infinitelymore.xyz/p/the-parado... #InfinitelyMore #BookOfInfinity
The paradox of giants.
You are kindly welcome to follow along with us in the reading—post your comments, and try the exercise questions!
www.infinitelymore.xyz/p/the-parado... #InfinitelyMore #BookOfInfinity
Indecomposable ordinals, those ordinals that cannot be generated from smaller ordinals.
www.infinitelymore.xyz/p/indecompos... #InfinitelyMore
www.infinitelymore.xyz/p/indecompos... #InfinitelyMore
Indecomposable Ordinals
Which ordinals are closed under addition? Which are closed under multiplication? Let us try to identify them exactly.
www.infinitelymore.xyz
February 2, 2026 at 12:15 AM
Indecomposable ordinals, those ordinals that cannot be generated from smaller ordinals.
www.infinitelymore.xyz/p/indecompos... #InfinitelyMore
www.infinitelymore.xyz/p/indecompos... #InfinitelyMore
Classes will be held in the igloo this week.
wsbt.com/news/local/n...
wsbt.com/news/local/n...
Notre Dame student gets creative with the cold
Some may be sick of the snow by now, but one Notre Dame freshman turned the cold into cool.The student built an igloo on campus that has since turned into a stu
wsbt.com
February 1, 2026 at 8:59 PM
Classes will be held in the igloo this week.
wsbt.com/news/local/n...
wsbt.com/news/local/n...
Reposted by Joel David Hamkins
One superpower #math gives me is the ability to picture an interesting object in my head, spark up #Mathematica, and within a few minutes figure out how to make it "real." It's intoxicating!
(code in ALT)
(code in ALT)
January 31, 2026 at 3:50 PM
One superpower #math gives me is the ability to picture an interesting object in my head, spark up #Mathematica, and within a few minutes figure out how to make it "real." It's intoxicating!
(code in ALT)
(code in ALT)
Today's lecture for my Infinity class is on the infinite coastline paradox.
www.infinitelymore.xyz/p/the-infini... #InfinitelyMore #BookOfInfinity
www.infinitelymore.xyz/p/the-infini... #InfinitelyMore #BookOfInfinity
The infinite coastline paradox
How unexpected difficulties in a practical measurement problem challenged fundamental conceptions of length and led to a new theory of dimension, while revealing monsters lurking at small scales.
www.infinitelymore.xyz
January 29, 2026 at 5:04 PM
Today's lecture for my Infinity class is on the infinite coastline paradox.
www.infinitelymore.xyz/p/the-infini... #InfinitelyMore #BookOfInfinity
www.infinitelymore.xyz/p/the-infini... #InfinitelyMore #BookOfInfinity
A while back I had suggested a certain problem in modal graph theory to my Oxford student Wojciech Wołoszyn. He has now solved the problem, with the help of AI. He explains both the result and his AI process on his new substack.
woloszyn.substack.com/p/can-ai-wri...
woloszyn.substack.com/p/can-ai-wri...
Can AI write serious mathematics?
A worked example in graph minors and modal logic
woloszyn.substack.com
January 29, 2026 at 1:06 AM
A while back I had suggested a certain problem in modal graph theory to my Oxford student Wojciech Wołoszyn. He has now solved the problem, with the help of AI. He explains both the result and his AI process on his new substack.
woloszyn.substack.com/p/can-ai-wri...
woloszyn.substack.com/p/can-ai-wri...
Reposted by Joel David Hamkins
The reverse mathematics of a theorem I proved over 30 years ago is asked on MathOverflow...
Regarding infinite time computation.
mathoverflow.net/q/507499/1946
Regarding infinite time computation.
mathoverflow.net/q/507499/1946
Reverse mathematics of infinite time Turing machines
This question is an attempt to narrow down and make more precise some discussion which occured in the answers and comments to this question.
Recall that infinite time Turing machines (ITTMs) are an
mathoverflow.net
January 28, 2026 at 4:08 AM
The reverse mathematics of a theorem I proved over 30 years ago is asked on MathOverflow...
Regarding infinite time computation.
mathoverflow.net/q/507499/1946
Regarding infinite time computation.
mathoverflow.net/q/507499/1946
Kelly Truelove's AI take on my idea about anthropomorphizing the Russell paradox. open.substack.com/pub/truescip...
Anthropomorphic Thinking
Cogitation in human terms
open.substack.com
January 28, 2026 at 3:46 AM
Kelly Truelove's AI take on my idea about anthropomorphizing the Russell paradox. open.substack.com/pub/truescip...
Today's quiz in my Infinity class.
Consider the Apollonian gasket, obtained by inscribing three congruent circles in a larger circle, and then successively placing the largest possible circles in the regions that remain. en.wikipedia.org/wiki/Apollon...
Consider the Apollonian gasket, obtained by inscribing three congruent circles in a larger circle, and then successively placing the largest possible circles in the regions that remain. en.wikipedia.org/wiki/Apollon...
Apollonian gasket - Wikipedia
en.wikipedia.org
January 28, 2026 at 2:28 AM
Today's quiz in my Infinity class.
Consider the Apollonian gasket, obtained by inscribing three congruent circles in a larger circle, and then successively placing the largest possible circles in the regions that remain. en.wikipedia.org/wiki/Apollon...
Consider the Apollonian gasket, obtained by inscribing three congruent circles in a larger circle, and then successively placing the largest possible circles in the regions that remain. en.wikipedia.org/wiki/Apollon...
Today in my Infinity class we discussed supertasks. Feel free to follow along with the readings...
www.infinitelymore.xyz/p/supertasks #InfinitelyMore #BookOfInfinity
www.infinitelymore.xyz/p/supertasks #InfinitelyMore #BookOfInfinity
Supertasks
Is it perfectly sensible or fundamentally incoherent to speak of completing a task with infinitely many steps?
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January 27, 2026 at 11:37 PM
Today in my Infinity class we discussed supertasks. Feel free to follow along with the readings...
www.infinitelymore.xyz/p/supertasks #InfinitelyMore #BookOfInfinity
www.infinitelymore.xyz/p/supertasks #InfinitelyMore #BookOfInfinity
Ordinal arithmetic. As a part of my series of essays on the surreal numbers, I am making a brief review of ordinal arithmetic.
www.infinitelymore.xyz/p/ordinal-ar... #InfinitelyMore
www.infinitelymore.xyz/p/ordinal-ar... #InfinitelyMore
Ordinal arithmetic
Let's review the basics of ordinal arithmetic, addition, multiplication, and exponentiation, providing both the order-theoretic semantic definitions as well as the recursive definitions.
www.infinitelymore.xyz
January 22, 2026 at 2:04 PM
Ordinal arithmetic. As a part of my series of essays on the surreal numbers, I am making a brief review of ordinal arithmetic.
www.infinitelymore.xyz/p/ordinal-ar... #InfinitelyMore
www.infinitelymore.xyz/p/ordinal-ar... #InfinitelyMore
Today's pleasant task: looking over the proposed book covers from the publisher for The Book of Infinity.
January 21, 2026 at 10:11 PM
Today's pleasant task: looking over the proposed book covers from the publisher for The Book of Infinity.
An update on China. I have been appointed as Guest Chair Professor at Peking University, with planned visits there during the next 3 summers.
January 21, 2026 at 5:15 PM
An update on China. I have been appointed as Guest Chair Professor at Peking University, with planned visits there during the next 3 summers.
I answered another MathOverflow question, concerning the possibility of a well-ordering on classes (so third-order well-order principle) and the effects on large cardinals. mathoverflow.net/a/507207/1946
Are there large cardinal axioms compatible with choice yet not with class well ordering principle?
Add a new primitive binary relation $ \prec$ to the language of Morse-Kelley class theory "$\sf MK$", which is meant to be a strict well order on all classes. So, it obeys the following a...
mathoverflow.net
January 19, 2026 at 9:28 PM
I answered another MathOverflow question, concerning the possibility of a well-ordering on classes (so third-order well-order principle) and the effects on large cardinals. mathoverflow.net/a/507207/1946
I answered a question on MathOverflow about large cardinals in the theory ZFC- without power set. mathoverflow.net/a/507203/1946
What are the known large cardinal axioms for which weaker and stronger set theories "catch up"?
I will clarify what I mean by the title in the following four ways:
For which cardinals $\kappa$ do we have that ZFC-(Powerset axiom)+$\exists\kappa$ is equiconsistent with ZFC? If that is not poss...
mathoverflow.net
January 19, 2026 at 8:05 PM
I answered a question on MathOverflow about large cardinals in the theory ZFC- without power set. mathoverflow.net/a/507203/1946
