Timeline for Why is it said that all surreal numbers with birthdate $<\omega_1$ are isomorphic to a Hardy field?
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| Jun 28 at 11:46 | history | edited | Anixx | CC BY-SA 4.0 |
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| Jun 13 at 22:38 | history | edited | Anixx | CC BY-SA 4.0 |
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| Jun 13 at 22:15 | comment | added | Anixx | @JoelDavidHamkins $\omega_1$ being uncountable cannot be represented as a germ (otherwise with the above formula we could buld a sequence with numerosity equal to it). But it can be represented in integral forms: $\omega_1=\frac1\pi \int_0^1 \omega dx=\frac1\pi\int_0^\infty\ln \omega dx=\frac1\pi \int_0^1 \int_0^\infty dt dx=\frac1\pi\int_0^\infty\ln \int_0^\infty dt dx=\frac1\pi \int_0^1 \int_0^\infty \frac 1{t^2}dt dx$ | |
| Jun 13 at 22:00 | comment | added | Anixx | @JoelDavidHamkins regarding uncountable surreals, I just define $\omega_1$ to be the numerosity of $[0,1)$. It is an interesting area and I have some formulas developed but it works different from the countable formula. Here are some formulas I have: mathoverflow.net/questions/472792/… | |
| Jun 13 at 21:57 | comment | added | Joel David Hamkins | I didn't know that Mathematica could accept arbitrary countable oracle input. Do you also have a Mathematica formula that will work with uncountable surreals? | |
| Jun 13 at 21:52 | answer | added | Joel David Hamkins | timeline score: 6 | |
| Jun 13 at 21:48 | comment | added | Anixx |
@JoelDavidHamkins here is the Mathematica formula for converting a countable surreal number to a set: S = Log[\[Omega]]; DifferenceDelta[Integrate[Normal[SolveValues[S == k, \[Omega]]], k], k] /. C[1] -> 0 // Last // FullSimplify // Expand
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| Jun 13 at 21:26 | comment | added | Anixx | @JoelDavidHamkins here I described the opposite process: mathoverflow.net/a/472814/10059 I am really amused how simple the formula turned out to be as opposed to various uses of ultrafilters and the like. | |
| Jun 13 at 21:22 | comment | added | Anixx | @JoelDavidHamkins the definition is basically in this post: we represent surreal as a germ of Hardy field, then as divergent integral, then divide it into segmaents of area 1, and then the centers of mass of each segment form a set, whose numerosity is the original surreal. | |
| Jun 13 at 21:20 | comment | added | Anixx | @JoelDavidHamkins well, in canonical embedding they also introduce derivation operation on surreals, coinciding with derivative of a germ. | |
| Jun 13 at 21:16 | comment | added | Joel David Hamkins | Is that enough to determine an embedding? Also, you refer several times to numerosities as surreals, but I don't see any definition of this. My understanding of numerosity is that this is an axiomatic theory, which might be possible to realize with the surreal numbers (but this goes beyond ZFC I believe since it will involve global choice), but it is certainly not unique, in light of the automorphisms of the surreals. So what do you mean? | |
| Jun 13 at 21:14 | comment | added | Anixx | @JoelDavidHamkins I think, the canonical embedding is where the germ of identity function is considered equal to $\omega$. | |
| Jun 13 at 21:13 | comment | added | Joel David Hamkins | You mention a "canonical" embedding of a Hardy field into the surreals, but is there any such thing? What does this mean? Since the surreal numbers have automorphisms moving the so-called countable surreals to uncountable surreals, this concept is not inherent to the field structure of the surreals. | |
| Jun 13 at 21:12 | comment | added | Anixx | @JoelDavidHamkins my point is, there could be countable surreals, which are greater than any germs, for instance, the numerosity of $\mathbb{Q}$ or any set with infinite number of accumulation points. | |
| Jun 13 at 21:11 | comment | added | Joel David Hamkins | Well, yes, $\omega_1$, by definition, is the first uncountable ordinal. | |
| Jun 13 at 21:10 | comment | added | Anixx | @JoelDavidHamkins exactly, I think $\omega_1$ is usually identified with the first uncountable ordinal. | |
| Jun 13 at 21:10 | comment | added | Joel David Hamkins | I guess it might mean the surreals born on a countable ordinal birthday, and if so, then $\text{No}(\omega_1)$ would be the set of countable surreals. | |
| Jun 13 at 21:09 | comment | added | Anixx | @JoelDavidHamkins actually I have encountered this term in some papers about surreals, I will search where I could have seen it. | |
| Jun 13 at 21:08 | comment | added | Joel David Hamkins | I'm not sure about any downvotes. But what do you mean by a countable surreal? | |
| Jun 13 at 21:05 | comment | added | Anixx | Why is the downvote? I was writing this several hours only t get downvote in seconds. | |
| Jun 13 at 21:04 | history | edited | Anixx | CC BY-SA 4.0 |
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| Jun 13 at 20:58 | history | asked | Anixx | CC BY-SA 4.0 |