Thanks, this is now fixed.

]]>Ha! We should make it a pizza party!

]]>I am so glad to hear that it is appreciated!

]]>Yes, so if there is no such game, then player 1 cannot win. But if player 2 has a winning strategy, then choice holds for sets of reals, and so there is a nondetermined game after all.

]]>Yes! Thank you, now fixed.

]]>https://x.com/JDHamkins/status/1438225439751905285

However, I’m confused about the following step:

https://x.com/JDHamkins/status/1438225455136526345

> And then player 1 must play a non-determined game on the natural numbers

What is such a game? Isn’t the existence of a non-determined game what we’re trying to prove in the first place?

(In the comments under the tweet, others are also confused by this.)

]]>I don’t know of any simple formula for this case. But in trees, antichains are somewhat simpler and can be recursively generated by replacing any node with an antichain in that upper cone.

]]>I see, thanks. I agree that youâ€™ve successfully argued that the scenario you described *could have* played out that way. But my impression was that the claim was a bit stronger, that if the early analysts had been blessed with rigor about number systems, the hyperreals would have been the most natural thing to use, leading to CH as a way to secure categoricity. I think examples like Levi-Civita, given their simplicity (in hindsight, admittedly) mitigate the sense of inevitability of the alternate history.

I think it could be plausibly argued that the notion of countable saturation may have been too abstract for early analysts, and maybe the concreteness of Levi-Civita would have won more favor, if there had been a contest between them.

]]>Thanks for the comment, and I’m glad you like the paper. That field is concrete enough that one can imagine it coming very early, and perhaps giving substance to the infinitesimal ideas. For my argument, however, I don’t think I am obligated to show that the solution I describe is the only possible thing that could have happened (since it didn’t in fact happen that way in any case), but rather my task was only to argue that it *could* have happened in the way I describe in my thought experiment. And I think I have done that much, so have explain how it could have been that CH is viewed as a fundamental axiom, necessary for mathematics.

]]>Simply: What about the Levi-Civita field? https://en.wikipedia.org/wiki/Levi-Civita_field

This is defined as the field of finitely-supported formal power series $\sum_{q \in \mathbb{Q}} a_q \varepsilon^q$ with real coefficients $a_q$. It is *not* countably saturated, and of course $\omega$-cofinal. However, it has several virtues, including being real-closed and (astonishingly, to me) useful in computational analysis (see the wikipedia article and links inside).

Additionally, it is characterized by a second-order categorical theory. This just piggybacks on the categoricity of $\mathbb R$ and $\mathbb N$.

Now, I donâ€™t know if it has all the advantages of a hyperreal field. But being real-closed, it at least is elementarily equivalent to $\mathbb R$. One can imagine that this relatively concrete field, if it were conceived by early analysts, might have sufficed for the infinitesimal applications they were after.

Levi-Civita is only one among many fields that extend the reals but are not given by an ultrapower. But it seems like a particularly elegant one.

]]>https://www.dijkstrascry.com/lecture3

Purely technically, I would like to suggest that there is (actually) more than one way to go from Turing’s a-machines to halting machines. Davis only gave us one way, and there is (again) no reason to believe that Turing was not more open minded about the issue at hand. Please consider perusing these results: https://www.dijkstrascry.com/RBM

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