Hyperreal Numbers: An Introduction to Infinitesimals and Nonstandard Analysis

An algebraic construction of the hyperreal number system, which extends the real number system with infinitely small and infinitely large numbers, and an illustration of how the system can be used for calculus.

[Yulenka-]

Thank you so much for this video! Even though I haven't touched algebra in years, it managed to pick me up right where I was, brush up my knowledge just minimally, and walk me past the edge, teaching me what I didn't know without leaving me behind. This is how all educational videos should be structured, paced, and produced! hats off

[spin]

Excelent video, veru concise, just what I needed. Thanks!

[decare]

I especially liked how well you explained the transfer principle. I hadn't quite understood it previously, but now it is clear as day.

[andyw]

Even though this is probably way above what I've learned, you made this comprehendible. Thank you.

[RSLT]

Very good video. I enjoyed watching it. I like the simple and straightforward layout and construction of it.

[agranero]

Thanks: you are one of the few videos about non-standard analysis that was not made by a crackpot claiming to be a genius that calls Cantor an idiot, was not made by an AI that puts images of hands that seem to be from an alien transmorph, with elephants with two trunks and a picture person that does not exist presented as Abraham Robinson. That few other remaining are from sets of filmed classes or in other languages as Turkish and Italian (that seem interesting but unfortunately I can't understand).

[abhijeetmulgund]

Thank you so much! This video was perfect!

[willostrand]

I think I understand. Is it essentially like modular arithmetic where you’re saying “25 = 1 + 2*12 but I’m ignoring 2*12 in Z/Z12 because multiples of 12 are identified as 0 so 25=1” but instead what you identify with zero is a really big or really small thing? Also great video it’s been a little over a year since I’ve seen anything near this level of algebra so I have cobwebs but this honestly helped me understand things I never understood before like ideals never really stuck with me very well but somehow the way you talked about them made them make much more sense to me

[lcfrod]

Amazing explanation!!. Thank you so much.

[diegobellani]

Great video!
I would love to see more videos like this.

[markusklyver]

I have never understood why S cap T belonged to to the ultrafilter. It makes sense to re-write it as S cup T minus the symmetric difference, and of course the symmetric difference ought to be a small set.

[AbuSayed-ervs]

Awesome math videos!!!Plz plz continue to make more...🙂

[wyboo]

i think i prefer the surreal numbers to the hyperreals. maybe it's just conway, but the construction of the surreals seems a lot more intuitive but also contains all of the hyperreals within it

[scollyer.tuition]

Pretty nice. I almost got the feeling that I could understand infinitesimals properly if I watched the video a few times. But I can't say I have much intuition for the ultafilter defn - why should the intersection of two "very large" sets be "very large" - it would be easy to come up with two subsets of R, say, that differ from R by a set of measure zero, but whose intersection is empty, for example.

One question though: you use the expression "almost all" several times, but I don't think that you defined it - in what sense is it being used here? In some measure-theoretic way, or what?

[alannrosas]

Shouldn’t the definition of an infinitesimal number, given at 19:26, include the restriction |a| > 0? Without this restriction, it seems like we can assert that 0 is infinitesimal, which seems unintuitive.

Awesome video by the way :)

[tim]

Should physicists therefore use hyperreals so they actually are saying something when they want to imagine an infinitesimal number dx? It always bothered me when they talk about real numbers that don't exist.

Can the dirac delta function be defined on the hyperreals so it has integral 1 like it supposed to? Pick an infinite number w, once, and define delta(x)=w if abs(x)<w/2, otherwise delta(x)=0. If this works, then the nonsense around the dirac delta is resolved without needing measure theory.

Edit: Wikipedia's Dirac Delta page has a section "Infinitesimal delta functions" that covers this.

[timpani]

Nice presentation! I've been interested in hyperreal numbers for quite a while, particularly for how it makes statements in calculus much more intuitive than the standard formulation due to Cauchy and Weierstrass, etc. Do you think it would make pedagogical sense to switch to a "nonstandard" formulation when teaching calculus to students, or are there some drawbacks to this approach that makes the standard way to teach calculus more desireable? Personally I'm leaning towards the nonstandard analysis approach as the superior one for teaching purposes, but it would be nice to see someone else's perspective on the matter.

[vectorshift]

Are infinitesimals related to the undecidability of the continuum hypothesis? Does it represent a set in-between countable and continuum?

[azizx]

"In other words, Leibniz has been vindicated!"

[AlbornozVEVO]

it's incredibly Kantian. making time "untimely".