Timeline for In surreal numbers, what exactly is $\omega_1$?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Oct 28 at 20:51 | comment | added | Joel David Hamkins | I'd be happy to take a look. But I recognize that you and I have rather different approaches to studying the surreals. | |
| Oct 28 at 20:24 | comment | added | Anixx | Hello, Joel! Would you be interested to take a look at my work-in-progress regarding surreal integration? It is nearly complete, I only need to add some bibliography. | |
| May 7 at 23:52 | comment | added | Joel David Hamkins | @MichaelHardy I have nothing to do with anything actually useful, if that is what you mean. I just meant useful for understanding the surreals themselves. | |
| May 7 at 23:49 | comment | added | Joel David Hamkins | @PeterLeFanuLumsdaine Yes, I agree with that, and I'm sorry this view evidently didn't come through. Part of my point is that one will miss a lot of the intended structure of the surreals by taking it only as a field. But also, one must recognize that the underlying homogeneity of the field means that this extra structure is not inherent. There are necessarily different ways to define exp and log etc, in light of all the automorphism, and one shouldn't presume that these things have a unique determinate interpretation. | |
| May 7 at 22:23 | comment | added | Michael Hardy | "Almost all of my own understanding of [...] is not present in the mere algebraic structure of the surreals." Etc. But my question was about the nature of the usefulness that you referred to. | |
| May 7 at 21:46 | comment | added | Peter LeFanu Lumsdaine | Absolutely agreed with the technical part of this answer, but I’d draw a slightly different conclusion from it, or at least a slightly different emphasis — not so much “the games are just one way of constructing the surreal field” (though that’s certainly true), but more “the field structure is not capturing the full structure we implictly consider on the surreals”. @SamHopkins’ analogy with the complexes is apt: the automorphisms show us that the embedding of the reals and the choice of $i$ are part of our implicit structure on $\mathbb{C}$. | |
| May 7 at 20:45 | comment | added | Mike Shulman | Ah, of course, when you have quantifiers you can do that. | |
| May 7 at 16:12 | comment | added | Joel David Hamkins | @MikeShulman The order is definable from the algebra, since $x\leq y$ if and only if $\exists z\ (x+z^2=y)$. | |
| May 7 at 16:09 | comment | added | Mike Shulman | What about if you can use the ordered field structure of the surreals? | |
| May 7 at 15:54 | comment | added | Joel David Hamkins | @SamHopkins I agree very much with that. We generally understand the complex numbers as having the form $a+bi$, but this form is not deducible from the pure algebraic structure of the complex field. There are many different copies of $\mathbb{R}$ in $\mathbb{C}$, over which $\mathbb{C}$ arises as the algebraic closure, and one cannot define real-part, imaginary-part in $\mathbb{C}$ using only the field structure. This situation is something like the $\{L\mid R\}$ representation of surreal numbers being similarly not respected by the algebraic structure. | |
| May 7 at 15:21 | comment | added | Anixx | @SamHopkins well, in dual numbers it is worse than in complex because of the automorfisms between $aε$ and $ε$ where $a$ is any real. Yet the choice between $ε$ and $-ε$ can be solved analytically mathoverflow.net/questions/423796/…. I believe that in surreals we not only can define $ω$ analytically as a germ, but also can define $\omega_1$ using the approach linked in the question. | |
| May 7 at 14:27 | comment | added | Sam Hopkins | There are wild automorphisms of the complex numbers. For that matter, there is an automorphism exchanging $i$ and $-i$. Yet, the working mathematician still treats each complex number, including $i$, as a definite thing; probably the mental picture of the “complex plane” helps here. Maybe things are not so different with the surreal numbers, where the “game” picture is part of what each surreal number is (even if it cannot be seen purely from the field structure). | |
| May 7 at 14:21 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| May 7 at 13:32 | comment | added | Joel David Hamkins | @MichaelHardy Almost all of my own understanding of the nature of the surreals flows from the recursive construction of it, where one fills gaps at successive ordinal birthdays, and so forth. All of that structure constitutes the framework of my understanding. But that extra structure is not present in the mere algebraic structure of the surreals. | |
| May 7 at 13:30 | comment | added | Joel David Hamkins | @Anixx No, because the 1-type of $\omega_1$ with parameter $\omega$ is still the same as all the other large enough surreals. Anything above all the polynomials in $\omega$ will still be automorphic, because of the limited power to define things in real-closed fields. | |
| May 7 at 13:29 | comment | added | Michael Hardy | "extremely useful" : Does that mean useful for understanding surreal numbers, or useful for using surreal numbers to answer some question asked by someone who never heard of surreal numbers, or something else? | |
| May 7 at 13:27 | comment | added | Anixx | Okay, but if we have already identified $\omega$ with the germ of $x$, can we exactly identify $\omega_1$ then? | |
| May 7 at 13:25 | vote | accept | Anixx | ||
| May 7 at 12:22 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| May 7 at 12:10 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |