Joel David Hamkins

Do you know a basis for P(omega) as a vector space over Z_2 under the operation of symmetric difference? Or P(kappa) or the set-theoretic universe V, for that matter.

Tactics versus strategies in the theory of games How do tactics differ from strategies? Does the fundamental theorem of finite games hold for tactics? Must every finite game have a winning tactic for one player or drawing tactics for both?

Tactics versus strategies in the theory of games
This week's post on Infinitely More
www.infinitelymore.xyz/p/tactics-ve... #InfinitelyMore #FrivolitiesOfTheGods

But an interval is also just an ordered pair of points, if you think of it as determined by the endpoints.

This is precisely the point of the question, since the answer depends on what you mean by cover. Is it a set of intervals? Or a class of intervals? Different questions, different answers for compactness & connectedness. With a proper class space, one must think differently about what a topology is.

It is a proper class.

6. In what senses is the surreal line connected or disconnected?

China 2025 · Jun 19 – Jul 30 📸 Shared album · Tap to view!

Finished my China trip for this summer!
10 lectures at Peking University
1 conference talk at PKU
7 lectures at Fudan University, Shanghai,
1 mini-conference talk at Fudan
1 conference talk in Changchun
1 conference talk in Lanzhou
photos.app.goo.gl/tVLoQnzVEoeJ...

Tic Tac is a first-player win. Same goes for Tic.

The curvature of space An excerpt from Lectures on the Philosophy of Mathematics

The curvature of Space
A free excerpt from Lectures on the Philosophy of Mathematics #InfinitelyMore #PhilMaths
www.infinitelymore.xyz/p/curvature-...

Different ways of drawing the exact same thing….just a plain 32x32 square grid ☺️

If you mean the surreal numbers of the form ⟨+-+++...⟩, then these are too close together to fit a unit interval. After all, ⟨+-⟩=1/2, and ⟨+-+⟩=3/4, ⟨+-++⟩=7/8, etc. So you can't fit unit intervals (that is, length 1 intervals) into those gaps. Or did you mean something else with RLRRR...?

The surreal numbers All the numbers great and small. The surreal numbers, generated in a recursive process of completion, unify the real numbers, the ordinals, and the infinitesimals in a single stupendous number system

My essay on the surreal numbers.
www.infinitelymore.xyz/p/surreal-nu...

5. How would you define the surreal analogue of the Cantor set?

4. Is the { L | R } representation of a surreal number structural? That is, is it respected by automorphisms of the surreal field? Is the ordinal birthday of a number structural?

3. Are there automorphisms of the surreal field No that move numbers like π and e within their infinitesimality class? Can we move ω to ω+5?

2. Is the surreal unit interval [0,1] compact? That is, does every open cover by intervals admit a finite subcover?

1. How many disjoint unit intervals fit in the interval [0,ω] in the surreal line?

Here is the list of challenge questions I asked the audience (mostly graduate students) at my talk yesterday on the surreal numbers.
Can you answer? Post your solutions in response to each question separately.

I don't find it a legal position to have a pawn on the 8th rank.

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Random chess position How likely is a random arrangement of the chess pieces on the chessboard to be a legal position?

Cast the chess pieces randomly on the board. What are the chances you get a legal position?
This week's installment on #InfinitelyMore.
www.infinitelymore.xyz/p/random-che... #FrivolotiesOfTheGods

"Barbeque. But also another dish—roasted pork with a sweet tangy sauce."

Isn't that the same as BBQ?

"No, it's totally different!"

I guess I know what I shall be eating in Changchun.

"Are you really going to Changchun?" the student asked, having seen my blog post. "That's my home town!"

Yes, we leave tonight from Shanghai.

What should we eat there?

The syntax/semantics dichotomy, however, is usually taken as a sweeping fundamental organization of concepts, while the use/mention distinction is often more narrowly focused on particular instances. In this sense, best to view use/mention (actually mention/use) as an instance of syntax/semantics.

YouTube video by Joel David Hamkins What is Proof?

The syntax/semantics dichotomy is a core distinction in mathematical logic, organizing our thinking on diverse topics, and the dichotomy at bottom is extremely similar to the underlying distinction in use/mention. You can see how I think about it here: www.youtube.com/watch?v=4LS4...

Mention is syntax. You mention the word, not the first-order meaning of the word. You use the meaning of the word.

I don't agree with mention/use = meta/object, although there is a connection, but I would find a stronger connection and similarity between mention/use with syntax/semantics, especially since these terms are used to refer to essentially identical distinctions arising, but in different subject areas.

China 2025 · Jul 3 – 13 📸 Shared album · Tap to view!

Meanwhile, I am enjoying some awesome food. photos.app.goo.gl/tVLoQnzVEoeJ...

Potentialist conceptions of infinity, Peking University, June 2025 This will be a talk for the Conference on Infinity, a collaborative meeting of logicians and specialists in Chinese philosophy here at Peking University, 24 June 2025, in the philosophy department.…

I spoke on potentialist conceptions of infinity for the cross-discipline meeting in Beijing between the logic group and the Chinese philosophy group at PKU. jdh.hamkins.org/potentialist...