Timeline for Are hyperreal numbers isomorphic to formal power series?
Current License: CC BY-SA 3.0
23 events
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| Apr 10, 2017 at 1:39 | history | edited | Todd Trimble | CC BY-SA 3.0 |
added a word to ensure countability
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| Jan 25, 2015 at 2:48 | comment | added | Noah Schweber | Again, "$No$" is from the surreals, not the hyperreals. | |
| Jan 25, 2015 at 2:30 | comment | added | Anixx | @Noah S I think in $No(\omega_1)$ there is. But I still would like a clarification. | |
| Jan 24, 2015 at 16:34 | comment | added | Noah Schweber | Anixx, based on your most recent comment I think you might be confusing the hyperreals with the surreals, which are two very different things. In particular, there's no "special" element of 'the' hyperreals which we deem "$\omega$." | |
| Jan 24, 2015 at 14:50 | comment | added | Todd Trimble | @JoelDavidHamkins I was confused it seems; thanks for setting me straight! I'm going to use my mod powers to edit my comment now. | |
| Jan 24, 2015 at 14:47 | comment | added | Joel David Hamkins | Todd, one may refer to $\sin(\omega)$ or $e^\omega$ for any hyperreal $\omega$, including infinite elements, simply by appealing to the transfer principle---every function on the reals has a nonstandard analogue, which satisfies all the same first-order expressible properties in the hyperreals that the original function did in the standard reals. | |
| Jan 24, 2015 at 14:15 | comment | added | Anixx | Actually to be more precise I was referring to Conway's $No$ ohio.edu/people/ehrlich/Unification.pdf | |
| Jan 24, 2015 at 13:41 | comment | added | Todd Trimble | Anixx, I think a problem here is exactly how you are defining the order if you are considering $x$ to be infinite and considering power series expressions such as $\sin x$. Is this going to be positive or negative (or zero)? I think you have to be careful here! | |
| Jan 24, 2015 at 13:31 | comment | added | Anixx | So if we take $\omega$ as infinite element, there are formal power series (like $\sin \omega$) which are not hyperreals apparently, so the power series seems to be a bigger set than hyperreals? | |
| Jan 24, 2015 at 13:24 | history | edited | Anixx | CC BY-SA 3.0 |
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| Jan 24, 2015 at 13:19 | comment | added | Todd Trimble | Oh, I see (and sorry I was confusing). Right, the way that Puiseux series are usually set up, the $x$ will be infinitesimal [I edited a mistake here in my answer; I often get confused which way it should go], and there, there is no such series for that would express an infinite element such as $e^y$ where $y = x^{-1}$. | |
| Jan 24, 2015 at 13:02 | history | edited | Todd Trimble | CC BY-SA 3.0 |
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| Jan 24, 2015 at 13:01 | comment | added | Anixx | So $\sin (1/\omega)=\sin \varepsilon$ is definitely a hyperreal and infinitesimal, but what about $\sin \omega$ then? | |
| Jan 24, 2015 at 12:56 | comment | added | Anixx | Oh, it seems you misunderstood me! In my question I meant $\omega$ to be infinite, as usually in non-standard analysis it is postulated to be equal to the first ordinal $\omega$. | |
| Jan 24, 2015 at 12:55 | comment | added | Todd Trimble | Yes, $\sin \omega$ will be infinitesimal. | |
| Jan 24, 2015 at 12:54 | history | edited | Todd Trimble | CC BY-SA 3.0 |
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| Jan 24, 2015 at 12:54 | comment | added | Anixx | "No, you can form a hypperreal $\sin ω$". - hmm, I am curious, will it be an infinitesimal? | |
| Jan 24, 2015 at 12:51 | comment | added | Anixx | I think the last comment is very valuable, can you please add this to the answer? | |
| Jan 24, 2015 at 12:50 | vote | accept | Anixx | ||
| Jan 24, 2015 at 12:48 | comment | added | Todd Trimble | (1) No, you can form a hypperreal $\sin \omega$. (2) Yes, one can regard the field of formal power series as an ordered subfield of the hyperreals. | |
| Jan 24, 2015 at 12:45 | comment | added | Anixx | If the hyperreals are bigger, can we say that the formal power series (or converging power series or analytic functions) are isomorphic to a subset of hyperreals? | |
| Jan 24, 2015 at 12:37 | comment | added | Anixx | By the way, $\sin \omega$ does belong to the formal power series but does not to hyperreals, is this wrong or correct? | |
| Jan 24, 2015 at 12:31 | history | answered | Todd Trimble | CC BY-SA 3.0 |