Real Analysis, Lecture 26: Ordinal Numbers and Transfinite Induction

[@jimmydu444]

Thumbs up if you watched through all of the lectures! Thank you HMC and Professor Su!

[@newkid9807]

Wizard of Oz su... A good friend of mine described him perfectly, Win at all costs, the tragedy of Professor Su

[@x0cx102]

@@newkid9807 that makes absolutely no sense

[@quantumgaming9180]

@@x0cx102 indeed

[@LaureanoLuna]

25:50 w+1 (omega plus one) is not the supremum of the set of all ordinals less than w+1, as he says. The set of all ordinals less than w+1 is w+1 itself and its supremum is its (great) union, which is just w, not w+1. Note that w+1 contains w while the union of w+1 does not contain w, since no member of w+1 contains w. Successor ordinals are never their own suprema, only limit ordinals are.

[@LaureanoLuna]

Be careful: he makes the same mistake again at 34:11.

[@BareClause]

Yes, in fact, an ordinal is a limit ordinal if and only if it is equal to its union and an ordinal is a successor ordinal if and only if it is equal to the successor of its union.

[@markusklyver6277]

I thought of this too. This property only holds for limit ordinals.

[@joejoejoe3]

this is a phenomenal lecture series! professor su is one of the best math lecturers i've ever seen, bravo!

[@jordanwoltjer2024]

gonna miss you Professor Francis Su 😢

[@jidrit999]

What happened

[@jordanwoltjer2024]

@@jidrit999 haha sorry, I just meant I had finished the last lecture and was sad to not have any more math content from him. To best of my knowledge he's still alive and well!

[@jidrit999]

@@jordanwoltjer2024 ok you make me scare man

[@jdbrown371]

40:00 Transfinite Induction

[@AdamMcLean]

Professor Su is an excellent lecturer. He presents the sometimes dense material in a clear way. I hope he will issue some further lectures on related subjects. There is a lack of a good video course on complex analysis.

[@beback_]

Presenting dense material to dense learners... :p

[@DrThalesAlexandre]

Thank you for the lectures, Professor. It has been a pleasure!

[@laurenbarghout4449]

Love watching this first thing in the morning with a cup of coffee. ~smiles

[@navidnoroozi8361]

Prof. Su's lectures on Analysis I are really great. Hope that he will provide video lectures of Analysis II.

[@newkid9807]

Navid Noroozi his lectures on analysis 2 completely flopped, they were terrible...

[@smalljbug]

@@newkid9807 where did you find them?

[@anasabdel2376]

There is no link for analysis II please

[@brianbutler2481]

Thanks so much Prof Su! I made it all the way through and I'm glad I did. The lectures were very high quality! Brian, HMC class of 1989

[@Liaomiao]

great set of lectures, hoping to see more lectures such as real analysis 2, complex analysis etc

[@LaureanoLuna]

0:41:39 Frédéric Ouimet, one year ago: In defining inductive sets. it's not necessary to say that they must contain 0. It's enough to say that if everything less than n is in the set, then n is also in the set, for all n. Just as he defines. Since there is nothing less than 0 in the ordinals, everything less than 0 is in any set, hence 0 is in any inductive set so defined. Remember that any conditional with false antecedent is true; 'if m is less than 0, then m is in S' is true for any m and any S, because the antecedent is always false.

[@kashnigahbaruda]

Oh my god! Please give us analysis 2! (or anything with Professor F. Su!)

[@2558rkr]

prof. Fransic su thanks for all real analysis lectures,hopefully we will see u in real analysis-2

[@jagwasenibanerjee1424]

Did you find any lectures on topics of analysis-2? From Prof Su or any other professors?

[@biggerthaninfinity7604]

50:00 Is there a set K in R^2 that intersects every line exactly twice?

[@chuchogl8160]

Thanks Dr Francis, thanks to you I got an A in analysis 1. Waiting for the videos in Analysis 2.

[@jagwasenibanerjee1424]

Did you find any lectures on topics of analysis-2? From Prof Su or any other professors?

[@porterhowland8655]

Really fun lectures! I'm hoping Professor Su also posts videos for Analysis II.

[@jagwasenibanerjee1424]

Did you find any lectures on topics of analysis-2? From Prof Su or any other professors?

[@porterhowland8655]

Jagwaseni Banerjee I didn’t. I ended up just taking the second half of the course at university. For self study, I would recommend just going through Rudin on your own (and asking for help on math stack when you get stuck). The solutions for Rudin’s Principles of Mathematical Analysis are available online. Good luck!

[@jagwasenibanerjee1424]

@@porterhowland8655 okay actually I have no mentor or guide or professor at this moment so I don't know if I am capable of learning from Rudin on my own. A little self doubt issues here..😅😅 I will try my best though. Thank for replying. Good luck to you too.😊

[@porterhowland8655]

Jagwaseni Banerjee good luck man! I can sympathize, because I’ve been in similar situations before: wanting to learn a subject, but don’t have anyone I can get help from when I’m stuck. Just try your best and see how far you get. ;)

[@jagwasenibanerjee1424]

@@porterhowland8655 yes trying my best is all I can do. Thank you for your good wishes. Happy learning !😇

[@NoNTr1v1aL]

Absolutely amazing video!

[@Tetraglot]

Oh, it's an optional lecture… I was starting to get worried my real analysis class was skipping stuff xD

[@류성원-p9c]

Great Lectures! Thank you so much Prof Su!

[@biggerthaninfinity7604]

18:50 S(a) = { a U {a} } rather than S(a) = a U {a} ??? Please, someone would help me. Thank you.

[@biggerthaninfinity7604]

[The Joy of Sets] - Keith Devlin, page 24 n={0,1,2,...,n-1} So, n+1={0,1,2,...,n}=nU{n}. Page 23, As a result of our development of ordinals, we obtain, gratis, a neat definition of the natural numbers. Impressive!!! Now we define the integers, finally!!!

[@jakejohnson6135]

On your proof by contradiction of strong induction: "if the least element n not in A is in N then there is an Sn that is in N that is also in A, therefore, n is in A." What if n is equal to 0, or whatever, such that it has no predecessor in N. Then there would be no Sn in A that comes before it.

[@markusklyver6277]

I don't really understand the induction step of the transfinite induction example. Can someone explain how we verify that K_α satisfies the smiley conditions?

[@markusklyver6277]

I struggle with the third condition.

[@HotPepperLala]

MORE ANALYSIS LECTURES PLEASE!!

[@newkid9807]

Hot Pepper Lala he is done when he says so!

[@jagwasenibanerjee1424]

Anyone have suggestions about any lectures on topics of analysis-2? Is there available lectures by Prof Su or any other professors?

[@xy-xe4zl]

thank you！and why there is no analysis II，i cant finish second half of baby rudin😂

[@newkid9807]

叶笑 because you’re an asshat

[@youdontknowme398]

Amazing video，Glad I've make it to the end

[@daviangel]

OMG there is more! Doesn't it just become complex or functional analysis?

[@aftermath4096]

yes he forgot to mention 1 (or 0) has to be in A for it to be inductive

[@valtih1978]

It is a great lecture. It is more instructive than all papers I have seen around but I still do not understand why w*w != w (owing to uncommunativity, like 2*w was exaplained) and how can you count the uncountable?

[@robertwilsoniii2048]

Valentin Tihomirov It’s an ambiguous convention to adopt the meaning that w*w = w^2 which is no less ambiguous. But, the limit definition of w^2 is not ambiguous, and instead of choosing the x*w definition of multiplication it simply chooses the w*x convention. Thus w*w = w*x where x = w. I know it’s silly but that’s it. Notice, this means you have omega groups of omega. So, to visualize the set and it’s ordinal you have to visualize omega many sets of omega, or omega many sets of N, the natural numbers. This is a countable set. I’m not sure what the associated cardinal number for this set is but there is one.

[@jaykazmi]

Prof Su. I think i'm not the first one to say but what the hell happened to Analysis 2 lectures?. dont get me wrong. You did a fine and noble job but this is plain co*& teasing:)! kidding... but seriously man, if you have them recorded, please upload them. Its been 4 years..And i hope the quality is much better ...Finger crossed. Thanks!

[@brandomiranda6703]

did you find out if he released analysis 2?

[@beback_]

I googled a bit for Advanced Analysis lectures and found this series: kzbin.info/www/bejne/Y5moo4CmZ6uEZtE Not yet sure how good they are.

[@ricardosuarez2707]

Thank you so much

[@rshrott]

Please tell me which textbook he teaches from. Thanks

[@brandomiranda6703]

rudin

[@dopplerdog6817]

Ill miss cyclops smiley

[@christophersedlak1147]

thanks

[@2558rkr]

real analysis walter rudin

[@lomertamahon1]

This lecture died right about 28:00.

[@brandomiranda6703]

why?

[@beback_]

RIP

[@thelastcipher9135]

@@brandomiranda6703 because mind blown.

[@-_Nuke_-]

+1 Vsauce brought me here :)

[@newkid9807]

NUKE 😕

[@matthewv5063]

ME! AND I HATE IT