This is a really good and concise explanation of Dedekind cuts - well done! For those who are interested, here's a bit of discussion around multiplication of cuts. (I'll let S denote the cut representing the real number s in the following.) Let's use the naive definition of cut multiplication that _S*T_ simply represents all products of rational numbers in _S_ and _T._ Now, if _s_ and _t_ are both strictly positive rational numbers, then in particular, _-2s_ and _-t_ are in _S_ and _T,_ respectively, and so their product is in _S*T_ - however, that product is _2st,_ and so it is larger than _s*t,_ which means our "cut" actually goes beyond the rational number it is meant to represent. In fact, if you've understood this step, it'll be pretty easy for you to show that the product (as we defined it) of two rational cuts greater than the zero cut will actually yield the entirety of the rationals, and so it's not even a cut - which is pretty bad because it violates the field axioms. Since negative numbers really mess up the whole thing, defining the multiplication of two cuts is a bit more subtle. You first start by only considering cuts greater than the zero cut (more precisely, cuts of which the zero cut is a strict subset) and only the rationals greater than 0 in these cuts. Then, the product of _S_ and _T_ may be defined as simply being the union of the zero cut and the set of all products of strictly positive elements of _S_ and _T._ Now, multiplying subsets of the zero cut can simply be defined using established consequences of the field axioms, such as _(-a)*b = -a*b._ Related to that is a technical nuance not discussed in this video - the definition of the additive inverse itself. It's pretty natural to define _-A_ as the cut such that _A + (-A) = 0_ (the zero cut) - but that is not quite it. For instance, if you choose _A_ to be the zero cut, then _-A_ is not a cut because it has a greatest element of _0_ (straightforward exercise to the reader). However, our good friend Weierstrass comes to the rescue - instead defining _-A_ such that, given an element _q_ of _-A,_ there exists a strictly negative real number _ε_ such that, for all _p_ in _A,_ _p + q < ε_ will do the trick. I hope this explanation satisfies those looking for more details - though there are still some things I left out. However, I hope I gave a basic idea of how to deal with these technicalities.

Equivalence classes of Cauchy sequences are bae and you wont convince me otherwise.

I just had a crazy thought? If there was a collab with Dr Peyam and Masahiro Sakurai the universe would implode and projective geometry division by zero would become the new norm.

You have an Interesting style of writting. You always start off with such big letters and somehow manage to write everything without cluttering it.

I think you're a really good teacher for foundational mathematics

I am Brazilian and I study mathematics here. I love watching your videos to improve my English. Kisses dear!

Very helpful, thank you!!

Thanks a lot professor ❤ respect!

You the man Dr Peyam!!!!!!

Thank you so much, finally understood it!!!! Didn’t get that the cuts are the actual reel numbers!!!

this is really well done! could you do the construction of surreal numbers as well? as a tribute to John Conway, who died from covid19 a while ago.

I think a bit more care could be taken in discussing how Dedikind is using the set of everything BELOW the irrational number itself to stand in for the number, and then defining operations on SETS (which don't include the number) to construct a field Q* that we later just swap out for R. It's a subtle point that may be lost on the viewer.

Very helpful thank you

As soon as you start involving infinite sets your mathematics is skating on thin ice.

Real awesome!

Hi dr.Peyam can you make a video if possible on constructing the lebesgue measure?But like without using the "caratheodory approach" i.e. restricting the lebesgue outer measure to the lebesgue sigma algebra,and instead using the riesz representation theorem approach just like in paparudin.

I was looking at my old lecture notes a few months ago it would of been good if these videos were around then , I know the construction were to do with dedekind cuts

From a computational pov, Dedekind cuts and their arithmetic have a problem. We could implement them as a triple, consisting of a member (proof of non-emptiness), a non-member (proof of non-totality) and a characteristic function for set-membership. One can then define the characteristic function of a summation with a bit of approximation trickery, and the member/nonmember come in handy at that stage. Only problem is: this will not work completely when two irrational numbers add up to a rational one. Because in that case the constructed characteristic function will be undefined on one rational number, the actual summation result itself.

Can every algebraic number be written as a cut in a similar fashion to sqrt(2)? If so how would you prove it?

Thanks mate :3

Seems like you changed your thumbnail format. This looks colourful and attractive!

Very intelligent

Can C be constructed by taking the cartesian product of R with itself and then forcing the addition and multiplication properties?

at 4:45 in condition 2, the s that is a rational number less than r, is that the same S as the real number? if not, why use the same variable, and if it is, then I do not understand condition 2.

At 3:03, real numbers are subset of rational numbers? Isn't it the other way around?

You say that any subset of elements greater than a given subset is not a cut. However in my textbook, cuts are defined exactly the other way around, where they have no upper limit and no smallest element. It would make sense that you could define it both ways right?

Here is a definition of π* that doesn't depend on any trigonometric functions: We know π/4=1-1/3+1/5-1/7+.... (the Leibniz formula for π, which is the Maclaurin series for arctan(1)) We can write this alternating series as a sum of positive terms by grouping the terms in pairs: π/4=(1-1/3) + (1/5-1/7) + (1/9-1/11) + .... Or π/4=2/(1×3) + 2/(5×7) + 2/(9×11) +... So π=8/(1×3) + 8/(5×7) + 8/(9×11) +... Now that we have written π as the sum of a series of positive terms, it is the limit and also the supremum of the partial sums of the series. Hence π* is the union of the cuts corresponding to the partial sums of the series. This method can be used to define the cut for any real number expressed as the sum of a series of non-negative terms, for example for e=1+1/2!+1/3!+... .

Why is the cut defined from less than a number? Surely the same logic could be use to create a cut from a number greater than instead of less than to negative infinity of all rational numbers?

Ok. Thank you very much.

Is it true that for all cuts and every rational number eps>0 then there is a number x in X such that x+eps is not in X? I’m pretty sure this is true and have a proof written up but i’m just trying to make sure. So is it true dr peyam. I thank your response if you do respond.

How do you define pi with the cuts?

BRILLIANT, but what does all that mean?

Hi, this "real numbers" playlist, are the videos in the correct order?

So from the first point of dedekind cut -∞ and ∞ are not real numbers. Is my conclusion correct?

I’ve always wondered about dedekind cut math. Any recommendations of online discussions on how to multiply two cuts, divide, square, etc.?

The cut for root 2 seemed like a bit of a hack :(

how do I formally show that pi is a cut? because it is transcendental, so we want use a logic sentence to construct this number

try prooving 0

im a little confused, if we're allowed to define a real number as a set of other numbers, then why couldnt we have done that when defining the rational numbers themselves? instead of inventing an equivalence class, why not just say 1/2 = {(1,2), (2,4), (3,6)...} isnt that basically like defining the number as a set of a bunch of pairs of other numbers? how is that different than the idea of a cut for example?

its pronounced, precisely, “King DeeeeeeDeeeeeDeeeeee”

Can an irrational number times an irrational number ever be an irrational number?

Is this Dedekind Cut?

Looks like surreal numbers

Please don’t get it wrong but this is not a construction of R. You give examples of cuts and defined the dedekind cut but you didn’t construct R from Q. First of all you get sets and not numbers, so if you think about how you construct Q from Z then the new rational numbers look like the numbers we already knew. That’s not the case with the cuts. You actually have sets and not numbers as Elements and that’s why you can picture or visualize them good but are difficult to handle for implementing the idea in other theories. That’s also the reason why the Dedekind cuts can only be used in Dimension 1 and not higher. For calculating or implementation in other theories it makes more sense to use Cantor’s definition via Cauchy sequences. Another notice is that you don’t make a difference between the cuts of Q and R, what I mean by that is for example the rational number 2 is written as the cut r_2={r element of Q | r

umm what ???

17:09 ...smaller then the area of a unit circle.

Love the video, just a bit too quiet

Are you Iranian?

Dr Peyam I love you more than my drumset.

NaMIstAyscr DR

my prediction: this will have to do with the fact that, between any two real numbers, there exists a rational number. So, if you change a real number even a little bit, you will jump over at least one rational number. so the set of rationals associated with the real number will change, and so the rational numbers associated with the real number is what matters

O hi! I am the only recent reply?

I don't know why, but the more I watch this and the more I think about it the more I think this is nonsense and the real number line makes no sense to me. I'll write this for an exam sure, but I'm not believing this made up construction 😅 For example you say we can do addition and multiplication to real numbers. What's e + pi + sqrt 2 then? Oh yeah you gotta use these cuts that make no actual Computational sense. This seems so flawed

Ha King Dededede: Dedekind Cuts - punny. Dr. Peyam if you ever have free time try looking at Kirby lore. It's actually pretty damn crazy. Kirby is a god that fights against eldritch abominations.