I agree with that sentiment. In the surreal numbers, we get the dyadic rational numbers at finite stages, and then all at once we get all the rest of the real numbers. I guess it is a way of seeing that one could have taken Dedekind cuts in the dyadic rationals, rather than in the rational field.
A clip from my conversation with Lex Fridman, in which we discuss the big bang of numbers, the surreal genesis. The surreal numbers are a vast number system unifying all our familiar numbers into one collossal system. www.infinitelymore.xyz/p/the-big-ba... #InfinitelyMore @lexfridman.bsky.social
Today I had my students discuss what makes soup soup. www.youtube.com/watch?v=Y1HV...
My daughter is taking an intro-logic course at Caltech and had told me she was searching online for stuff about definability, and found my essays. How boring and embarrassing to be pointed to your father's writing. Nevertheless, she said, this essay was the key to the homework.
Please enjoy my essay on nondefinabilty. #InfinitelyMore #PanoramaOfLogic
substack.com/home/post/p-...
Including The Liar (a logic song).
www.youtube.com/watch?v=VvcP... @HHMusicOfficial
Today's topic for my infinity class: Infinite Liars!
www.infinitelymore.xyz/p/infinite-l... #InfinitelyMore #BookOfInfinity
Last week's guest lecture in my Infinity class was given by Notre Dame philosophy grad student Jacob Harvey, on the genre of Infinitary Hat Puzzles. www.infinitelymore.xyz/p/infinitary...
I will read that for a while. Pen and paper, no LLM cheating…
Close up of the spectacular John S. Coppin mural.
Detroit public library.
The slides are now available for my Owens Lecture talk today. We had a great discussion in the Q & A.
jdh.hamkins.org/truth-and-pa...
Learned today that Gettier was at Wayne State University when he wrote the famous paper on epistemology.
Yay! Tomorrow in my Infinity class: Philosophical Dialogue day.
Students have all paired up and will perform philosophical dialogues exhibiting various philosophical points of contention. Potentialism versus actualism; Cantor-Hume principle versus Euclid's principle; etc.
Read more on #InfinitelyMore.
www.infinitelymore.xyz/p/natural-pr...
This figure shows how to realize the natural product of two ordinals α⊗β as a linearization of the coordinatewise product order. Sort the blocks arising from the Cantor normal form. Each block in effect is a higher-dimensional rectangular product.
Speaking on Tuesday this week for the Notre Dame Logic Seminar. 2pm 125 Hayes-Healey
jdh.hamkins.org/consistency-...
A pile of notes and false starts on the way to my essay on the natural product of ordinals.
The Natural Product of Ordinals.
Enjoy this latest installment in my series of essays on ordinal arithmetic, part of a larger series on the surreal numbers.
www.infinitelymore.xyz/p/natural-pr...
#InfinitelyMore
Glass half full,
or glass 7/8 empty?!
We specifically discussed this in class, that "choice" is a metaphor to help us understand the axiom, which makes a bald existence statement. Although I believe this metaphor is basically helpful, it can also lead to confusion, for example, when you want to choose an element from just one set.
Today's topic in my Infinity class: The axiom of choice.
www.infinitelymore.xyz/p/axiom-of-c... #BookOfInfinity #InfinitelyMore
Makes sense.
If everyone did very well, would you still devote a lot of class time to discussing the solutions? It seems reasonable to spend time talking about problems that students found difficult, and less time on those where they mostly got it.
Take today's quiz in my Infinity class:
Tell the story of the continuum hypothesis. What does it say? What is the logical status? Mention the relevant thinkers and dates. What philosophical positions and controversies arise particularly in connection with it?
Moby Dick also has long digressions on fascinating topics. A whole detailed chapter on the gruesome process of harvesting a whale.
The book has 400 pages with over 200 high-quality color figures of my own design. See a sample here: jdh.hamkins.org/the-book-of-...
To my way of thinking, the extremely large finite numbers often stand as a gateway to the infinite, and part of what it means to understand infinity is to have a deep understanding of the nature of the very large finite numbers.
