jamespropp

I am curious whether there are other applications of Cantor's ordinals that I am unaware of.

I'd be curious to know whether anyone can find "God's number" for this puzzle.

I agree (see endnote #5).

Thanks for catching this! I fixed the error.

I added a sentence: “(Warning: This is sort of a trick question, so don’t expect a textbook-style answer!)” I hope that this will prevent others from being similarly frustrated.

I plan to continue using AI tools, but to do so in a mindful way, to avoid becoming addicted to them, and to be prepared to back off if I don’t like the direction they take me in.

Perhaps an additional Moral is, if a calculation can be done in an afternoon, it's probably been done before. (I say this because it's been pointed out in the Comments section of my essay that Jon Lu asked and answered the same question back in 2016.)

I'm not sure I understand your question. Given that it takes an infinite amount of time to write down the decimal expansion of pi, what might it mean to compute pi in finite time?

Thanks for catching this! What was in my mind when I wrote the first of those two sentences was the entirety of my past experience using ChatGPT as a research tool, and all the times it made mistakes. (For a funny example, see my January 17, 2023 essay “Denominators and Doppelgängers” in which I describe ChatGPT's proof that .999... is less than 1.) But you're 100% right that the version of my essay that I posted last week didn't make this clear; I'll update it appropriately. I'll also add a brief description of ChatGPT's bibliographic blunder.

As usual, I'm most interested in comments that will help me improve the article, but feedback of every kind is welcome!

I’ve updated this passage. Let me know if the new version is clearer.

Thanks for pointing this out. I’ll work on this passage tomorrow.

Yes!

Do you disagree with my take or think I’m missing Witt’s point? I’d be happy to hear from people who disagree with me.

Richard Dedekind didn't just invent Dedekind cuts (and rings and ideals); he gave us a new way to think about (or avoid thinking about) the ultimate nature of mathematical reality.

What makes the real number system different from all the smaller number systems it contains? Richard Dedekind found a simple answer: a geometric axiom that Euclid missed.