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.

[@agranero6]

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).

[@blargoner]

Thanks! I think one of the crackpots was very unhappy with my video. The AI ones sound trippy.

[@tim57243]

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)

[@spin7765]

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

[@DarkestValar]

10:08 That plot twist tho.

[@algebraicoo]

Genious

[@willostrand6555]

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

[@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?

[@blargoner]

In this context, "almost all" n just means for all n lying in some set belonging to the free ultrafilter. I must have done the world's worst job making that clear as it seems several viewers have been confused by this.

[@wargreymon2024]

It is crucial to explain the ultrafilter, which is bizarre. Anyway, i think you have made the topic clearer like no other.

[@markusklyver6277]

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.

[@decare696]

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

[@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

[@aristoclesdialectic]

ghosts of departed quantities

[@davidalexander4505]

This was brilliant, bravo!!

[@aziz0x00]

"In other words, Leibniz has been vindicated!"

[@alannrosas2543]

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 :)

[@blargoner]

Under this definition, zero is infinitesimal. This provides some algebraic benefits. We require an infinitesimal to be nonzero when needed.

[@andyw3683]

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

[@abhijeetmulgund5266]

Thank you so much! This video was perfect!

[@vectorshift401]

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

[@blargoner]

No, there are continuum many infinitesimals. However, there are interesting connections to CH. For example, if you assume CH, then *R is unique up to isomorphism.

[@tomkerruish2982]

You're in good company. According to John Conway (I think it's mentioned in his biography), Kurt Gödel once stated that the "correct" theory of infinitesimals could help decide the Continuum Hypothesis. After Conway discovered the surreals, he visited Gödel, who agreed that they were indeed the correct theory of infinitesimals, but that he (Gödel) had been wrong about their utility in this regard and did not remember making his previous statement.

[@RSLT]

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

[@sectorgamma]

Fantastic video!

[@eonasjohn]

Thank you for the video.

[@AJ-et3vf]

great video! Thank you!

[@AlbornozVEVO]

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

[@1906Farnsworth]

It sounds like the Hyperreals are the same as Surreals. Are they different in some way that my pea brane can't fathom?

[@christiancunzeman1460]

Absolute banger video mate.

[@wyboo2019]

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

[@wargreymon2024]

How many axioms do you need to calculate 2+3? Nonstandard analyst: yes

[@timpani112]

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.

[@blargoner]

Good question. There are some textbooks that have tried this, like Jerome Keisler's. While there's definitely intuitive appeal to the approach, I'm not sure whether there might be drawbacks for beginning students. I'd be curious what others think.

[@friedrichhayek4862]

@@blargoner I think that nonstandard calculus must become the standard calculus because it is more rigurous in my view.

[@willostrand6555]

I mean I studied physics as a major and math as a minor so I have a different perspective. I love algebra but I think it intuitively makes less sense. I struggled so hard to understand it intuitively, but when it clicked I swear everything in math just started to click with me somehow. However I think for high schoolers the level of abstraction needed to understand groups is just too much to ask, while limits and series isn’t terribly hard. I think it would be very interesting though and maybe it would work better, I know plenty of people who just couldn’t comprehend any level of calculus as it is right now

[@lcfrod]

Amazing explanation!!. Thank you so much.

[@roibinia3507]

Why is the construction independent of the choice of M? There might be a few different maximal ideals containing the eventually 0 sequences.

[@blargoner]

It's not. No one said it was.

[@AbuSayed-er9vs]

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

[@garfungled7093]

fantastic video

[@game_in_black9901]

I have a question how can we compare two hyperreals for instance x = (1 , -1 , 1 , -1 , . . .) and 0 = (0, 0, 0, . . .) how do we know if x is positive ? because { n in N | x_n > 0 } is infinite and { n in N | x_n < 0 } is infinite too. I am confused.

[@blargoner]

In general it depends on the free ultrafilter.

[@game_in_black9901]

Si how do we construct thé ultrafilter to know which set is inside ?

[@blargoner]

You don't explicitly "construct" it, you choose one using the axiom of choice (AC). I'd recommend doing some reading in set theory to understand this better.

[@NoNTr1v1aL]

Amazing video!

[@diegobellani]

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

[@allyourcode]

@14:09 I don't understand part of this slide. I thought we identified "eventually zero" sequences with zero (i.e. the sequence of all zeros). That does not seem to be the same thing as "all sequences that are 'almost all' zero" though (as stated in this slide). E.g. the following would be a sequence that is almost all zero, but never eventually becomes all zero: 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 ... Why switch from saying "eventually zero" to "almost all zero"? Did we just grow M to make it a maximal ideal?? I thought "eventually zero" was already maximal?

[@blargoner]

We chose a maximal ideal M which contains the eventually zero sequences, but it contains other sequences as well. Whether or not a particular sequence (like the example you gave) is "almost all" zero depends upon the choice of M. Because the existence of M relies on the axiom of choice, we can't define it explicitly.

[@shoopinc]

Encore!

[@seanskinner1672]

Thanks, terrifically paced.

[@CharlieYoutubing]

Thanks

[@tomkerruish2982]

3:17 To borrow from Douglas Adams, "Ah, this is obviously some strange use of the word love that I wasn't previously aware of."

[@DJTrulin]

Do you ever produce writings that accompany these videos? I would definitely read a white paper on this topic.

[@blargoner]

I haven't, but I'm planning to do some writing on linear algebra in the future. I like the constraints of the slide format in that you can't write a lot of text if you're doing it right.

[@ChaineYTXF]

Stop saying l'Hopital, the guy hired Bernoulli do the math work and then stole his work...😁 Edit: you gained a subscriber by the way😊

[@blargoner]

Bernoulli just made a poor business decision.

[@ChaineYTXF]

@@blargoner ha ha then again, maybe not, I understand that he was payed well and that at some point, the Marquess de L'Hôpital acknowledged that this was not his work