Timeline for Surreal numbers vs. non-standard analysis
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 17 at 12:06 | comment | added | Mikhail Katz | Ehrlich has a recent paper where he defines the sine on suitably chosen subfields of the surreals. @EmberEdison | |
| Jan 17 at 11:51 | comment | added | Ember Edison | I would like to know without considering the size question can we define the sin function of surreal and more real analytic functions in surreal counterparts. the HoTT implementation of surreal does not care about the size question. | |
| Jul 30, 2018 at 8:29 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
added 30 characters in body
|
| Jul 29, 2018 at 15:48 | comment | added | Mikhail Katz | @JoelDavidHamkins, I find your claim that "the common limit of all these [hyperreal] fields is: the surreal numbers" a bit of an obfuscation. Namely, the "is" should be replaced by "is isomorphic to" by an isomorphism claimed in Theorem 20 of Philip's paper (note that no proof is provided though it does exist following Keisler, and is certainly non-trivial). | |
| Jul 29, 2018 at 15:34 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
added 255 characters in body
|
| Jul 29, 2018 at 15:14 | comment | added | Mikhail Katz | The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine function. As far as I know the answer is negative. | |
| Jul 29, 2018 at 15:12 | comment | added | Philip Ehrlich | @Mikhail (Continued): Since Oz is not a nonstandard model of arithmetic, this would not be an example of a sin function one would use if one were doing nonstandard analysis, but it is a perfectly fine sin function that is very useful for various model theoretic purposes. Moreover, there is a vast array of initial subfields of No that have sin functions based on portions of this construction. I am doing work on what I call \emp{trigonometric ordered fields} in which this and related matters are discussed. | |
| Jul 29, 2018 at 15:10 | comment | added | Philip Ehrlich | @Mikhail. In addition to agreeing with Joel's comment about the sin function, I'd like to add the following. You say there is no sin function that can be defined on the surreals or, at least, there is no sin function other than one that can be induced from a nonstandard model of analysis. This is false, and has been known to be false for decades. In fact, as Martin Kruskal pointed out, there is a canonical sign function: $\sin(\theta +2\pi n)=sin\theta$, where $n$ ranges over the system Oz of omnific integers of No. | |
| Jul 29, 2018 at 15:07 | comment | added | Joel David Hamkins | Let me add that I find talk about "the" hyperreals to be an inaccurate conflation of the diverse, non-isomorphic models that serve as instances. The common limit of all these fields is: the surreal numbers. | |
| Jul 29, 2018 at 15:04 | comment | added | Joel David Hamkins | My view is that the sine function extends to the hyperreals by means of the ultrapower construction, and the same is true for the surreals, except that it is an iterated ultrapower. This is simply a class-length analogue of the same iterated ultrapower construction that one uses when providing set-sized models of the hyperreals with a specified hierarchy of infinitesimality. | |
| Jul 29, 2018 at 14:34 | comment | added | Mikhail Katz | If I understand the latest comments by Philip under his answer correctly, he agrees that my comments here concerning transfer are on target. @JoelDavidHamkins | |
| Jul 29, 2018 at 14:29 | comment | added | Mikhail Katz | @JoelDavidHamkins, Which construction are you referring to exactly? I am not familiar with results that would identify specific set-size surreals with specific set-size hyperreals. If you have in mind a definition of the sine function over the surreals without passing via the hyperreals, that would certainly be interesting. Again I am not sure what you are claiming precisely. As far as "maximality" is concerned: that was the content of theorem 20 in Ehrlich's paper, which is why I mentioned it. | |
| Jul 29, 2018 at 14:26 | comment | added | Joel David Hamkins | From my perspective, the construction is basically analogous to how one constructs the sine function in a given model of the hyperreals. (And I think maximality has nothing to do with it.) | |
| Jul 29, 2018 at 14:17 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
added 227 characters in body
|
| Jul 29, 2018 at 14:01 | comment | added | Mikhail Katz | @JoelDavidHamkins, the sine function only extends via the identification of class-size surreals with class-size hyperreals, and not by any independent construction over the surreals. Also I would guess that there is no set-size surreal field where the sine function would extend because the isomorphism in question depends crucially on the fact that the models being identified are maximal. I agree that ultrafilters are involved in the construction of the hyperreals, but I am not sure what you are trying to say. | |
| Jul 29, 2018 at 13:52 | comment | added | Joel David Hamkins | So it seems that the sine function does extend to the surreals. But if I understand you, you seem to object to the proof of this, if one should prove it using a limit ultrapower construction? The hyperreals also are usually defined via ultrapowers, and indeed, ultrafilters are inherent in the transfer principle for the hyperreals, as I explain in mathoverflow.net/a/57108/1946. | |
| Jul 29, 2018 at 13:40 | comment | added | Mikhail Katz | True, but such an extension passes via a class-size model which is being identified with a limit ultrapower model for the hyperreals. @JoelDavidHamkins | |
| Jul 29, 2018 at 13:38 | comment | added | Joel David Hamkins | He says that the surreal numbers admit a relational extension to a model of nonstandard analysis, which I take to mean that we can relations on the surreals corresponding to the transfer of any given relation on the reals, such as the graph of the sine function. | |
| Jul 29, 2018 at 13:35 | comment | added | Mikhail Katz | Where is the claim that the sine function extends? Philip is mainly focusing on the properties of a real closed field, rather than a language suitable for analysis that would naturally include the sine function as well. Also, as the answer indicates, they were recently successful in extending the exponential function, but the sine function is still a challenge as far as I know. @JoelDavidHamkins | |
| Jul 29, 2018 at 13:34 | comment | added | Joel David Hamkins | What I am confused about is the final sentence of your answer, where you seem to say that the sine function does not extend to the surreals, whereas the answer of Philip seems to say that it does. Are you claiming that there is no sine function on the surreals (with the expected first-order properties)? | |
| Jul 29, 2018 at 13:05 | comment | added | Mikhail Katz | I addressed that in my answer. Whatever transfer is available over the surreals comes from the identification of the maximal surreal class with a maximal hyperreal class, as asserted in theorem 20 in Philip's paper, and also in his answer. Do you interpret his answer as saying that there is a proof of the transfer principle over the surreals that's independent of the hyperreals? I don't think such an interpretation is correct. @JoelDavidHamkins | |
| Jul 29, 2018 at 13:01 | comment | added | Joel David Hamkins | You say that the $\sin$ function does not extend to the surreals, but Philip seems to say that there is a transfer principle, which would provide a $\sin$ function on the surreals, which whichever desired properties of it we would want. | |
| Jul 29, 2018 at 12:53 | comment | added | Mikhail Katz | @JoelDavidHamkins, I don't detect any contradiction between my answer and my friend Philip Ehrlich's. Could you be more specific? | |
| Jul 29, 2018 at 10:43 | comment | added | Joel David Hamkins | How are we to reconcile your answer with Philip's? | |
| May 3, 2016 at 9:03 | history | answered | Mikhail Katz | CC BY-SA 3.0 |