tag:blogger.com,1999:blog-8148573551417578681.post6178330725229663774..comments2022-01-24T10:08:02.112-08:00Comments on Dark Buzz: Contest winner misunderstands infinitesimalsRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-8148573551417578681.post-18504659209394460282015-06-17T16:59:09.436-07:002015-06-17T16:59:09.436-07:00Sylvia, you are right that the hyperreals do not h...Sylvia, you are right that the hyperreals do not have the least upper bound property if you include non-internal sets. That is, the infinitesimals are bounded but do not have a least upper bound. But the bounded <a href="http://en.wikipedia.org/wiki/Internal_set" rel="nofollow">internal sets</a> have least upper bounds.<br /><br />I will post more on this subject.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-46619448232958154932015-06-17T13:39:46.162-07:002015-06-17T13:39:46.162-07:00Roger,
I await the second round.
I *enjoy* the ...Roger,<br /><br />I await the second round. <br /><br />I *enjoy* the show.<br /><br />But, I am not a silent spectator. <br /><br />Neither have I read Sylvia's essay. <br /><br />Or yours. <br /><br />Mostly, no need.<br /><br />Over to you. <br /><br />[On the proverbial second thoughts, I might have something to add. Just in case I catch something or the other, from you guys Mathematicians and Physicists and all, you know...]<br /><br />Best,<br /><br />--Ajit <br />(who else? [And, please delete the C language's "[\r]\n" as convenient. (Yes, I am *that* excited now.(See, see, see, who all get excited and for what reason(s)!))])<br />[E&OE]<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-77114845917440760512015-06-16T13:05:39.975-07:002015-06-16T13:05:39.975-07:00Dear Roger,
Thank you for your critical remark...Dear Roger,<br /><br /> Thank you for your critical remarks.<br /><br /> What I mean by an infinitesimal is <b>a number different from zero, of which the absolute value is smaller than any strictly positive real number</b>. There is no such number in the real numbers of standard analysis, whereas there are infinitely many in (any model of) the hyperreal numbers.<br /> It is true that in the context of standard analysis the term 'infinitesimal' is also used (e.g. to refer to epsilon and delta in the definition of the limit) but these are ordinary real numbers and do not satisfy the definition of infinitesimals that I implicitly assumed througout the essay.<br /> The goal of the competition was to write an essay for a general audience, so I decided not to include this definition. A text like this is intended to make people curious and if they look up "non-standard analysis" or "hyperreals", this will pop up soon enough. Your remarks strongly suggest that I <i>should</i> have added this in a footnote.<br /><br /> Doing so might have avoided a lot of confusion, for instance:<br /><br />- What I mean by "loose talk involving infinitesimals" "frowned upon by mathematicians" is that physicists often talk about infinitesimals in a way that is close in spirit to Leibniz's work (and hence to non-standard analysis), which is not compatible with the definition of the classical limit as mathematicians use it in standard analysis.<br /><br />- You wrote: "Infinitesimals were never banned from mathematics. They are completely legitimate if backed up by limits or hyperreals. Maybe physicists never learn that, but mathematicians do."<br /> The idea that infinitesimals (in the sense made precise above) have long been banned from mainstream mathematics is non-controversial amongst historians of mathematics. On the other hand, research on non-Archimedean mathematics never quite stopped: see the work of Ehrlich for details on this topic.<br /><br />- etc. -> I think many of your other remarks also boil down to the same point: I use the term 'infinitesimal' in a more restricted sense than the way you seem to interpret it.<br /><br /> The only remaining point is that you wrote: "I don't know why she says the hyperreals are incomplete. They have the same completeness properties as the real numbers. That is, Cauchy sequences converge, bounded sets have least upper bounds, and odd order polynomials have roots."<br /> This simply is not correct: hyperreals do <i>not</i> obey the least upper bound property! The set of hyperreal numbers are incomplete in the sense that they are not Dedekind complete (as the real numbers are). Non-standard models of the real numbers - by definition - satisfy all the same first-order properties as the standard reals, but Dedekind completeness is a second-order property.<br /><br /> I hope this clears up some of the issues you raised. If there are other points you want me to address, feel free to ask. ;-)<br /><br />Best wishes,<br /> SylviaSylviahttp://www.sylviawenmackers.be/noreply@blogger.com