Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property

MIT 18.100A Real Analysis, Fall 2020
Instructor: Dr. Casey Rodriguez

Finishing the lecture on Cantor’s notion of cardinality and starting to define the real numbers. Introducing ordered sets, the least upper bound property, and the incompleteness of the rational numbers.

License: Creative Commons BY-NC-SA

[srsykes]

Thank you Dr. Rodriguez. I just started to watching your lectures and I thoroughly enjoy them. I have been a retired electrical engineer since 2013 online lectures such as yours make each day magic. Since back in the 1980's, I had done some part-time teaching at my local county college while going to graduate school and working full time. I was once asked by my employer to conduct classes for some of our younger engineers. I was located on the east coast and all of the students were in the mid-west. I had something called a Smart-Board which reproduced what I wrote in the two remote classrooms. I was in a conference room by myself lecturing. People would pass by and thought I was crazy. I love your humor and good to see you are making it seem as if your students are actually there.

[yuriakahumanity]

With the avant-garde camera angles... film majors 10000%

[deathhunter]

Is there a way to watch this video without getting motion sickness from the camera movements???

[nicolasg.b.]

Hey self-learners! I think there is a minor typo in the lecture notes, at page 9, related to the proof of theorem at @12:47. In the second case, it says "b is not in g(b)", it should says "b is in g(b)".

[aewfawef]

Super educational lectures, really enjoying them so far!

[Luckynina]

“I would not lie to you I would just give you alternative facts” 😂

[Nate-ztrh]

This lecture is phenomenal. The camera is

[DoggARithm]

This camera needs to track just a little higher so when he's writing words high up, they aren't cut off

[lebesgue-integral]

I took measure theory without having taken real analysis, and was hard in the beginning. We had all those terminology: power set, sup, inf, etc.

[rjosephnieto]

I momentarily got stuck at 59:27. Z has no UB, much less LUB, much less LUBP. Maybe Dr Rodriguez may have meant Z- (negative integers) or Z+?

[Kerrosene]

For the Cantor's proof, why do we assume that we can construct such a B? Also, couldn't such a B be constructed for a bijection from A to A and thus we could disprove that |A| = |A|?

[CosmicButterfly]

7:21 Can't you do induction? The base case is the power set of the null set, which has cardinality of 1, which is greater than 0. Then assume it's true for n, and if you add another element to the set, than at least one more element will be added to the Power Set, meaning the power set is strictly larger.

[BalaSujithPotineni]

At 59:31, can someone explain how Z has the least upper-bound property?

[megalo-rex]

50:35 " one and zewo were in the set"

[dtung]

I would love to know the principle on how to construct the wacky (2-x^2)/(2(2x+1)) (Rudin's book used (2-x^2)/(x+2)).

[cooterhead_jones]

Please, I am confused. At 6 minutes, he shows 1-1. f(x) = {x}. Doesn’t that imply that no doubleton set in P(A) is paired and therefore f cannot be surjective?

[abdullahaddous]

For the second case in cantor’s proof, couldn’t you say that since b is not in g(b), then b is not in B, so b is in g(b)? My mind kept going there lol

[KingZero]

i love these lectures... but this episode has LITERALLY the worst cameraman ever...

[animeleepocket]

At 40:46, if b is the greatest of lower bounds, should it say b < b_0, whereas it says b ≤ b_0 for both second cases of least upper bound and greatest lower bound? I checked the notes for the lecture after and the second case; whilst using c and c_0, it says c < c_0. I wondered if it should strictly be less given c_0 or b_0 is considered the greatest lower bound.

[rashpalsingh]

how can we show the transitive property for the set of cartesian product of q * q.