Classification of sets - Lec 03 - Frederic Schuller

[@michaelliu2961]

0:08 Overview 7:01 Terminology - Domain, Target, Surjective, Injective, Bijective ("structure-preserving map"), Set-isomorphism 12:27 Classification of Sets - Infinite, Finite 17:35 More on Maps - Composition, Space diagram, Inverse, Identity, Preimage, 25:52 Equivalence Relations - Definition, Equivalence Classes, Quotient Set 44:36 Warning on ensuring maps are well-defined when using representatives in definition 57:38 Construction of ℕ - Successor map, Predecessor map, Addition map 1:10:00 Construction of ℤ - By equivalence relation (on ℕ x ℕ), Inheriting operations (on ℕ) 1:22:14 Construction of ℚ - By equivalence relation (on ℤ x ℤ), Inheriting operations (on ℤ) 1:31:38 Construction of ℝ - Brief sketch (by equivalence relation on "almost-homomorphisms" on ℤ)

[@alpistein]

"2+(-3) is -1. Halleluyah." Best Lecturer Ever!

[@mappingtheshit]

Kudos to those ancient guys that had to rediscover ZFC to brings us the niceties of arithmetics

[@xintongbian]

44:40 care must be taken when defining maps on quotient set using representatives 57:41 defining addition operation on natural numbers 1:09:31 construction of integer numbers 1:31:39 times is up for the most difficult one, the reals...

[@YourFriendlyAlan]

1:09:35 “So, obviously the increase in your calculational power due to these, ah, musings is zero - that’s clear” - Frederic Schuller

[@YourFriendlyAlan]

I have studied foundational set theory for several years now and have never heard my efforts summed up so succinctly and precisely.

[@LeopoldBlooom]

The french's movies joke made me laugh aloud. I can't believe nobody laughed.

[@joefagan9335]

Haha - the ménage a trois joke 😀😀

[@jarrodanderson2124]

They're all math majors. 😔

[@kemaruproduction]

p_1: The students are German. p_2: They are fiddling already with the serious mathematics. p_3 Math has no room to humor. p_4: Humor is left for break. p_5: Therefore, Germans cannot laugh during the class (with Modus ponens out of p_1 U p_2 U p_3 U p_4).

[@moe531]

Actually the lecture is so informative and productive and enjoyable though when he said the French joke I just stopped it for laughing hard😂😂

[@clubsandwedge]

Best exposition ever - extremely clear

[@pritambanerjee7172]

please make the problem sets available to normal internet user so that they can practice too.

[@thomasjames1067]

I second this!

[@martinepstein9826]

At 41:30 to show that M/~ is a set you don't need the power set axiom. The map m -> [m] for m in M is a functional relation so its image must be a set by the axiom of replacement.

[@pittyconor2489]

why is it a functional relation

[@martinepstein9826]

@@pittyconor2489 That just means every domain element m relates to exactly one thing, namely [m]. It's two years later and I don't remember what this lecture is about, but I've learned since that the power set axiom is considered more basic than the axiom of replacement. So probably what he did in the video makes the most sense.

[@Stadtpark90]

32:44 d) is actually funny, yet nobody laughs - everybody is busy writing...

[@Stadtpark90]

could be used in "Big Bang Theory"^^

[@michalbotor]

cameraman seemed to be pondering on that one though.. ;)

[@rajarshichatterjee3281]

i found the french cinema part much more funny ......

[@LeopoldBlooom]

@@rajarshichatterjee3281 I laughed both times, but when he talked about transitivity that was it.

[@LeopoldBlooom]

thought of him watching a french movie while thinking about all of this.

[@DavidRuizTijerina]

Does anyone know whether the problem sets are available somewhere?

[@mappingtheshit]

The exercises on set theory won't help you to grasp, for example, the Lie algebra, which is the main subject of this course. I don't understand why put emphasis on set theory?

[@piercingspear2922]

@@mappingtheshit He explained it in lecture 1

[@daoudhadjab1696]

where is the problem sheet!

[@Hythloday71]

Great lecture, I wish we had covered these concepts in set theory from such a foundational level. Equivalent relation as means of making consistent categories, who'd of thought? We just learned it verbatim. Quotient structures / equivalence classes .. we just ran hard into these in Ring theory without any instruction. (well about 'well defined' considerations)

[@KipIngram]

59:48 - I notice that if we cheat a little and index our entries in N, we can note that N[0] is 0, and then N[i+1] = {N[i]} from there on out.

[@mikeCavalle]

This is a fine set of lectures. bravo.

[@whdaffer1]

It would've been nice if in the section about choosing the proper representatives and the proper maps when dealing with equivalents classes, that he had provided the counter example.

[@millerfour2071]

5:10, 13:48, 32:56, 40:15, 44:08, 53:06, 1:02:28, 1:08:10, 1:13:50, 1:15:52, 1:17:43 (unique 0), 1:22:10, 1:25:08, 1:26:55, 1:30:54 ((a’+b)(c+d)=(a+b)(c’+d) not true given ac’=a’c), 1:32:23

[@shreyashbhattacharjee1412]

Can somebody point to the assignments ? Moreover can somebody suggest a book to go along with the course?

[@ansgarschaefer5411]

Just a quick question: Where does the identity sign "=" come from? We have defined equality for sets before writing down the axioms, but not for "symbols" or "anything" in general.

[@tadfadfgadfasfd2437]

I looked pretty carefully throughout this video looking for any examples of something which was not a set that had an equal sign and could not find any examples. Could you point out an example of an equal sign written between two things that are not sets?

[@ansgarschaefer5411]

@@tadfadfgadfasfd2437 For example at kzbin.info/www/bejne/bHasiIWdYpuZqrM, (9:09) in the definition of injectivity, or at kzbin.info/www/bejne/bHasiIWdYpuZqrM (36:18) in the explanation of equivalence classes.

[@tadfadfgadfasfd2437]

@Ansgar Schaefer At 9:09 in this video a1=a2 it is implicit that a1 in A and a2 in A. ​ At 11:35 of lecture 2 we see on the board that x in y is a proposition iff x and y are both sets. This means that a1 and a2 are both sets. Intuitively if you think about it, if a1 = 1 for instance, we have defined 1 in this course as a set. At 36:18 for equivalence classes, we have the equivalence class defined (right where his hand is) and that is guaranteed to be a set by the principle of restricted comprehension given in lecture 2 at 1:08:33. So equivalence classes are sets, specifically they are subsets of the set they are defined on. So the equation at the bottom is the usual definition of equivalence of sets

[@rupayansaha721]

Problem sheets are needed.

[@taraspokalchuk7256]

sorry for a question from the first lecture. 30:05 suppose m~n is false, m~p id true and m~p is true. then the whole predicat is true (as from false anything can follow). is transitivity satisfied?

[@Zygzak191]

Year after you comment, but there was n~p on second place, not m~p. I it wasn't a typo and you meant to write m~n false, n~p true to satisfy transitivity you would get m~p false, not m~p true.

[@franciscoaguero9028]

Is a predicate a relation when defining a map?

[@ShokoDemon]

wuderbar, Herr Professor!

[@joefagan9335]

Another excellent lecture 👏👏

[@fjolsvit]

In a predicate of n variables, what is the collection of variables? That is, given P(v_1,v_2,...,v_n), what is (v_1,v_2,...,v_n)? For that matter, given P(a,b), what is (a,b)?

[@GunsExplosivesnStuff]

Aren't the integers conveniently set-theoretically isomorphic to ((N\{0})x{0,1})U{0} ? Or indeed even to N itself with some annoying renaming? It seems like this would avoid a quotient set, so I am wondering what is wrong with constructing it like this?

[@whdaffer1]

Although I'm quite certain that he could do it, it would've been nice if he had defined multiplication in the integers. Because he skates right over that when defining the rationals.

[@YourFriendlyAlan]

44:28 “So, if you wish, it’s gender theory in a nutshell that there are several types equivalence relations on the same set normally” - Frederic Schuller

[@YourFriendlyAlan]

I came for the mathematics but I stayed for the gender theory.

[@paulelliott9487]

I'm not clear how his non-standard definition of the ordinals extends to ordinals greater than or equal to omega. What is omega for example? 0 ={}, 1 = { {} }, 2 = { {} , { {} } } so the ordinal n always has n elements in it for n < omega. N+1 = N union { N } and limit ordinal = the union of ordinals less than N. His definition won't work for limit ordinals like omega.

[@steffahn]

It doesn't extend, it needn't even, the construction is only meant to work for defining the natural numbers and nothing beyond. Actually, the only advantage of the {}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}, ... approach is that then your natural numbers and ordinal numbers are precisely the same objects. But then again it gets more difficult again, if you want them to be the same objects as the real numbers 0, 1, 2, 3, .... just as well, so, yeah. Most people do not even care about such constructions a lot after they have seen at least one variation that works and do not worry about it anymore afterwards ;-)

[@davidwright5719]

It appears that the predecessor map requires the Axiom of Choice.

[@xavierpuerta]

I wonder where the induction principle is hidden among the set theory axioms, which are used to define natural numbers, instead of using Peano axioms.

[@danideboe]

You are right. The principle of induction relies on the well-ordering principle. And the induction principle is needed to define the addition on the natural numbers. Also note that usually one picks von Neumann ordinals for the natural numbers instead of his definition. He probably did because of the easy way out with the predecessor map but this is not the usual fashion. Usually you assume well-ordering for non-empty subsets of the natural numbers which implies the induction principle. And this can then together with the successor be used to define the addition by first proving the recursion theorem, then the addition on the natural numbers follows directly as a unique map recursively defined

[@jimallysonnevado3973]

Intuitively, I would define a set A to be finite if there is such a natural number N st. A is (set theoretically) isomorphic to {1,2,3,...,N}. I would define infinite set as the complement of this finite definition. My question is how to show that both definitions are equivalent? If there is no such proper subset of A, that is isomorphic to A, how can I show that there is a Natural number N such that A is isomorphic to {1,2,3,...,N}?

[@xavierpuerta]

sometimes mathematicians prefer to be elegant rather than intuitive

[@UnforsakenXII]

1:30:45 Thank you, student. He got rid of my question, lol.

[@LiamJongsuKim]

me either.

[@UnforsakenXII]

@@LiamJongsuKim lol

[@jurgenblick5491]

Are we talking about Algorithms?

[@s00s77]

this man ffs he could pick Jech at once for a lecture series... but no and he does a work just as good. the presence of Choice at choosing representatives for partitions was a worry of mine ever since i had naive set theory

[@jiahao2709]

I don't fully understand Lec01,02, but I can understand the Lec03, anyone give some reference for better understand the Lec01,02?

[@kunjalvara6113]

I also want that

[@TurtleTube123]

57:04 -- Now *that's* the basis of democracy!! :)

[@juliangarcia3416]

IF A is ISOMORPHIC TO B, then B IS ISOMORPHIC TO A? Isomorphism is conmutative??? Please

[@danideboe]

Yes. The reason is that a bijective function is invertible.

[@reinerwilhelms-tricarico344]

1:33:10 The natural numbers where handed down by God ... by means of the unique empty set ∅ . (The only way to get Mathematicians convinced :-)

[@ChaineYTXF]

There is no mention of God in math. That notion is not required

[@thomassinha5301]

An hour into the third 2 hour lecture of an extremely advanced mathematical physics course we finally prove 2+1=1+2=3 lol

[@Michael-xd7sj]

French cinema transitivity joke was hilarious.

[@jimnewton4534]

it's not clear to me how to construct the rest of the reals from the rationals without the idea of convergence???

[@twwc960]

+Jim Newton I'm not familiar with the concept of "almost homomorphisms", so I do wish the problem sets were available, but the method of Dedekind cuts, which is fairly well known, does not require convergence. The other popular technique, namely equivalence classes of Cauchy sequences does depend on a notion of convergence, but it is very easily defined based on the natural metric structure of the rational numbers (one doesn't need the language of topology.)

[@alvinlepik5265]

this is properly done by Dedekind cuts and from that, a complete ordered field is born, (that we call the real numbers)

[@richerzd]

The most common ways of constructing ℝ is via Dedekind sets or Cauchy sequences, but there are other less common ones. "Almost homomorphisms" is another way to construct ℝ. (Of course, these are all equivalent, since all "complete ordered fields" are isomorphic.) An "almost homomorphism" is a map ϕ : ℤ -> ℤ such that {ϕ(m + n) - ϕ(m) - ϕ(n) | m, n ∈ ℤ} is finite. We define ϕ ~ ϕ' if {ϕ(n) - ϕ'(n) | n ∈ ℤ} is finite. Then ℝ is defined as the quotient of almost homomorphisms with ~. Addition is pointwise, and multiplication is function composition. The main idea is to represent a real number with the "slope" of ϕ. Note that even though the graph of ϕ only contains lattice points, it can still have a slope for every real number. Intuitively, - If r is a "real number", then we can associate it with the almost homomorphism ϕ(n) = floor(n*r). - For an almost homomorphism ϕ, it is associated with the real number lim ϕ(n)/n. It is quite fun to prove the properties of complete ordered fields this way, e.g., well-definedness of the operations, commutativity of multiplication, and existence of least upper bounds.

[@santhoshb7583]

beautiful sir....

[@claschxtreme]

I feel like a geeky bilingual professor, i have exactly the same sweater as Schuller ...

[@callmedeno]

I thought they only made one!

[@juliangarcia3416]

I cant found a biyection between Z and N, to prove that Z is infinitely countable

[@FlavioSchneiderCH]

One easy example would be: 1->1,-1->2,2->3,-2->4,3->5,-3->6 and so on.

[@brunnun]

Flavio explained it well, just to summarize: send positive integers to odd naturals and negative integers to even naturals

[@nournote]

You map them like this: N ----> Z n I----> n/2 if n is even (1-n)/2 if n is odd

[@nurlatifahmohdnor8939]

Fred or Eric?

[@nurlatifahmohdnor8939]

22 21 1 20 2 19 3 18 4 17 5 16 6 6 days ago was 16.12.2023 Page 196 On the night of December 16, 1773, there were three tea-laden cargo ships from England at anchor in Boston Harbor.

[@nurlatifahmohdnor8939]

Several hundred Bostonians, disguised as Indians, raided the vessels and dumped 342 cases of tea into the water.

[@JohnBicknell]

Tilde please, not twiddle, great lecturer. Makes me wish I could be young enough to do my degree all over again in Germany.

[@santoshrrrr]

schuller ! i bow !

[@callmedeno]

Is this undergraduate?

[@krishnakumarsah632]

For mathematics students it is undergraduate and for physics ones it is graduate

[@chongshen4280]

好（helpful）

[@ayushpaliwal1880]

Someone fucking arrange for problem sheets

[@Esloquees]

1:17:42 I end up clapping

[@danielroddy2032]

Camera dude, sleeping on the job…

[@whdaffer1]

The comment about 'gender theory in a nutshell' seems .... odd.

[@whdaffer1]

It would've been nice if in the section about choosing the proper representatives and the proper maps when dealing with equivalents classes, that he had provided the counter example.