Timeline for Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 9, 2014 at 23:27 | comment | added | Mike Battaglia | I'm pretty sure it's a theorem that r·ω is an omnific integer for any real r, but either way it's definitely true that ω/2 is an omnific integer. | |
| Feb 9, 2014 at 0:26 | comment | added | Joel David Hamkins | Well, that also sounds very likely, so I'm not sure. | |
| Feb 9, 2014 at 0:06 | comment | added | Mike Battaglia | Isn't $\sqrt{\omega}$ in the omnific integers? | |
| Feb 8, 2014 at 23:00 | comment | added | Joel David Hamkins | The question would be whether $\omega$ is prime in the omnific integers. I find it likely that it is. | |
| Feb 8, 2014 at 22:50 | comment | added | Mike Battaglia | Also, I found interesting that even without specifying the ultrafilter, you can still decide a lot of information about the relative ordering of things... For instance, take the Ross-Littlewood paradox. At some step n, the number of balls in the jar is 9n, the lowest-numbered ball is n+1, and the highest-numbered ball is 10n, so after the supertask is done, there will be 9ω balls in the jar, consisting of all balls numbered ω+1 to 10ω. Even though you don't know "which" number ω is, you still know you have less balls than if you added only 2 at each new step, for instance. | |
| Feb 8, 2014 at 18:51 | comment | added | Mike Battaglia | I was quite curious if an analogous situation might exist with the surreals, e.g. the primeness of $\omega$ is simply undecidable and you can choose your own adventure regarding it. Or, alternatively, perhaps those things really are decidable in the surreals, with $\omega$ being decided prime by some undeniably natural way, and then a field embedding of $^*\Bbb R$ into them which maps $[(1,2,3,...)] \to \omega$ would decide some equally natural ultrafilter or set of ultrafilters which makes it all work out. Just a thought. | |
| Feb 8, 2014 at 18:40 | comment | added | Mike Battaglia | You can then add the axiom that $\omega$ is composite if you like, which narrows down the set of ultrafilters you can choose from; the different ultrafilters then become like different "models" of the theory in some sense in which every proposition about $\omega$ has a definitive answer. (cont'd again) | |
| Feb 8, 2014 at 18:36 | comment | added | Mike Battaglia | Thank you for your response! I see what you mean about $\omega$ being undetermined. I don't know whether this makes sense or is just an errant intuition of mine, but I've been thinking of it as though that its primeness vs compositeness as something which is "undecidable" simply given the basic axioms of the hyperreals, e.g. that you have these sequences of reals with "an ultrafilter" on them without specifying anything beyond that. (cont'd) | |
| Feb 8, 2014 at 18:11 | vote | accept | Mike Battaglia | ||
| Feb 8, 2014 at 3:44 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |