Physics student here. I knew natural numbers had a construction based on set theory, but I didn't know how it worked. When you showed how to build them as sets, I just started laughing. I don't know why, I just couldn't stop. I love this.

@brightsideofmaths

3 жыл бұрын

Exactly for this reason I am doing this :) Thanks for sharing!

Yay, another resource for set theory. I love learning about set theory and how it can serve as a foundation for many mathematical concepts. I am going to watch these videos soon. I see you still respond to comments, so that has me excited knowing you'll read and possibly respond to this message. I have always tried learning about set theory, but it has been hard self-teaching deep set theory concepts as its hard to find the necessary learning sources. All I've really been searching for is an individual who can help me learn about these math concepts and try answering my possible questions. I'll be sure to watch all of these videos relating to set theory. I am passionate for pure mathematics. I have a few set theory thoughts I would like to share or ask if you are willing. Thank you for making these videos. Thanks for reading.

I love your work. Thank you.

great!! thanks! would be great to also have subtitles

I've just found your channel and am extremely impressed already! considering signing up to support you to get at the quizzes :) I was wondering what software you are using to do all the presentation graphics, I love the style!

@brightsideofmaths

9 ай бұрын

Thanks a lot! :) I have a video about this: tbsom.de/s/tools

this was god damn freakin amazin

@brightsideofmaths

3 жыл бұрын

Thank you very much :)

I suppose one technical detail that could be added, but is not strictly necessary for the purposes of the video, is that one should say that the successor map S is a subset of P(N), (or P(N0), depending on the notation you are choosing, I am choosing the ISO notation), where P(N) denotes the power set of the set N. The unique characterization of N in the model is then that {} is an element of N, and S is a subset of P(N) such that S is an injection from N to N mapping n to union(n, {n}), and its canonical surjection is from N to N\{}, which is a form of characterizing the axiom that there is no natural number n of which 0 = {} is its successor. This maybe a stronger and more rigorous characterization, in my opinion, but the details are a lot more technical, so I acknowledge that this can be seen as completely unnecessary.

@BboyKeny

2 жыл бұрын

I don't know what use it is to know this technical detail yet, but I do know that I actually understand the notation you're using. Which makes me a happy camper.

@davidebizzarri9460

Жыл бұрын

I'm not sure, is it correct to say that the successor map S is a subset of P(N)? As far as I understand S is not a set

@angelmendez-rivera351

Жыл бұрын

@@davidebizzarri9460 S is a set. All maps are sets.

@davidebizzarri9460

Жыл бұрын

@@angelmendez-rivera351 what kind of set is S then? Is it true that it's a subset of P(N)?

@angelmendez-rivera351

Жыл бұрын

@@davidebizzarri9460 S is the set {{{0}, {0, 1}}, {{1}, {1, 2}}, {{2}, {2, 3}}, ..., {{n}, {n, m}}, ...}, where m is the successor of n. I said that S is a subset of P(N), but this is actually wrong. S is a subset of P(P(N))). The whole point is that S(n) = m if and only if {{n}, {n, m}} is in S. Normally, we write (x, y) := {{x}, {x, y}}, so you would instead write S = {(0, 1), (1, 2), (2, 3), ...}, which makes it more obvious that S is a function.

😮 I am starting to question if I am not a set too haha

I'm a bit lost here. so a set with number 3 has only one element or four? If it has four then wouldn't the element with the number three have the same cardinality with another set with the numbers 1,2&3?

@delta3244

2 жыл бұрын

to avoid some brackets, ∅ = {} 3 = {∅, {∅}, {∅, {∅}}} The set which contains only 3 is {3} = {{∅, {∅}, {∅, {∅}}}} which has 1 element, which is {∅, {∅}, {∅, {∅}}} That 3 has three elements does not have any effect on the number of elements in {3}

Great explanation! Two questions: 1) To define N instead of N_0, we just say that 1:={ empty set} and drop (a) from the axioms right? 2) Are there sets other than N_0 whose axioms are only (a) and (b)?

@brightsideofmaths

9 ай бұрын

N and N0 are not really different. So after defining N0, you can just define N as N0 without the first element.

Thanks! Really appreciate that 😊 ..! Do you recommend some textbooks as a companion to the entire topics of the course? Thanks,,

@brightsideofmaths

3 жыл бұрын

Good question! There are a lot of good books for the introduction into mathematics out there. Maybe, just have a look and take the one that you like. Books are a matter of taste, I guess.

@lemigiojames3643

2 жыл бұрын

@@brightsideofmaths Very true, but what is your personal taste?

Thanks! But I wonder if there is a simpler way to define natural numbers. For example, I define 0 := empty set, and the successor map s(x) := {x}, then 1={0}, 2={{0}}, 3={{{0}}} and so on.

@brightsideofmaths

4 ай бұрын

Yes, but why do you think it's simpler?

@dhn6943

4 ай бұрын

@@brightsideofmaths Thanks for responding. I dont have the definition for "simpler" but intuitively, i think s(x) := {x} does look simpler than s(x) := x ∪ {x}. My point is why do mathematicians choose that definition over the others. Is it because that definition has a nice property that it ensures each number n is defined by a set of n elements?

@brightsideofmaths

4 ай бұрын

@@dhn6943 I guess, the next videos will answer your question :)

What would be a example of a superset of N0? I can't think of any

@douglas5260

3 жыл бұрын

with the same 2 properties I mean

@HabibuMukhandi

Ай бұрын

@@douglas5260 A set of integers?

Hi! I have a question about 3:21. How do we know that 1 != 0? I mean, how do we know that {empty} != empty? It seems intuitive but I can't figure out how to prove it.

@brightsideofmaths

2 жыл бұрын

The one set is empty and the other one has one element.

@brightsideofmaths

2 жыл бұрын

@@NAbadi-zg3sh There is only set with one element. This is the empty set.

@NAbadi-zg3sh

2 жыл бұрын

@@brightsideofmaths Yes, I understand that intuitively, but shouldn't you prove that {} != {{}} without the use of integers? After all, if the question is "how do we know that 1 != 0?" independently of set theory, the answer can't be "because 1 has 1 element and 0 has 0", because this would imply that 1 != 0

@brightsideofmaths

2 жыл бұрын

@@NAbadi-zg3sh For {} we know: x \in {} is false, for every x. On the other hand for { {} }, we know: x \in { {} } is true for x = {}. Hence { {} } is not the same set as {}.

Is n + 1 := n ∪ {n} equal to {0 , ... , n - 1 , {0 , ... , n - 1}} ?

@brightsideofmaths

3 жыл бұрын

Yeah, it is :)

Where did you get the whiteboard from?

@brightsideofmaths

Жыл бұрын

I bought it in an electronic store in Germany.

@andreiinthedesktopworld1178

Жыл бұрын

@@brightsideofmaths I can't get it because I'm not in Germany

@brightsideofmaths

Жыл бұрын

@@andreiinthedesktopworld1178 I see the problem ;)

Can {-1,0,2,3...} or {-2,-1,0,1,2,3...} be not a superset of N0 that satisfies the axioms for natural numbers?

@brightsideofmaths

2 жыл бұрын

The question is: "what is -1?"

@1iunie2012

2 жыл бұрын

@@brightsideofmaths At minute 4:47 the 2 properties of the set N0 are stated. -1 could be defined like this: 0 is the successor of -1, and -1 would be the successor of -2, and the resulting set would satisfy both properties, right? Or is this idea wrong, and a better idea would be the following? A "bigger" set as N0 let's call it M (N included in M) that satisfies the two properties we could construct like this: M = N joined with {triangle, square} with the property that the triangle is the successor of the square and the square is the successor of the triangle. Thank you!

@brightsideofmaths

2 жыл бұрын

@@1iunie2012 You need to construct the set using sets :)

@1iunie2012

2 жыл бұрын

@@brightsideofmaths In the end I thought I can use only sets and logic. A "bigger" set could be N0 U M, (the union of set N0 as you build it and set M). Set M is constructed in the following way: triangle (abbreviation T) is an element. Set 1T:= {T} . Set 2T :={T, 1T}. Set 3T:= {T, 1T, 2T}... So there can be an infinite "bigger" sets of N0 complying with the two axioms. I hope I am right in the end.

@brightsideofmaths

2 жыл бұрын

@@1iunie2012 You are going in the right direction. Of course, there are bigger sets that satisfy the axiom. However, N0 will be the smallest one.

Also, if 0 = the empty set and 1 is the set that contains 0, doesnt that mean 1 is also an empty set?

@brightsideofmaths

8 ай бұрын

No, it's a set with one element in it :)

@Lockout_31019

8 ай бұрын

@@brightsideofmaths so the nature of that element doesn't really matter? just so long as it has one element?

@HabibuMukhandi

Ай бұрын

@@Lockout_31019 Yes, you need to remember the definition of a set. A set is a collection of distinct objects. These objects can also be other sets. Except for the empty set, it is a set with no elements in it.

2:20 we mean that '0' (the symbol) will denote the size of the empty set right? And then '1' will denote the size of the set containing the empty set?

@brightsideofmaths

Жыл бұрын

No, we mean literally that 0 denotes the empty set and so on. Of course, in background, you should have in mind how many elements the sets have.

@harshitrajput6865

Жыл бұрын

@@brightsideofmaths Ohhh. Thanks for the clarification

@StaticBlaster

11 ай бұрын

@@brightsideofmaths The cardinality of the empty set is 0. right? and the cardinality of the set containing the empty set is 1? correct?

@HabibuMukhandi

Ай бұрын

@@StaticBlaster The statement that the cardinality of the empty set is 0 is an axiom. However, here the empty set itself is the definition of 0.

@StaticBlaster

Ай бұрын

@@HabibuMukhandi because there's nothing inside the set. right? that's why the cardinality is zero.

I think there is an error on the corresponding quiz?

@brightsideofmaths

10 ай бұрын

Mistakes can happen, of course, but I checked the quiz and it's seems fine. Which question do you mean?

@medwards1086

10 ай бұрын

@@brightsideofmaths question two? I think the answer is: If x is an element of N, then x U {x} is an element of N. But the answer has the intersection not the union.

@brightsideofmaths

10 ай бұрын

But the question is: What is **not** a property of this set? @@medwards1086

@medwards1086

10 ай бұрын

@@brightsideofmaths Reading is not my strong point 😄 That’s why I enjoy mathematics. Thanks you for your patients. Love your work. 👌

4:50

@brightsideofmaths

8 ай бұрын

What is about that?

@Rahul-yu3ro

8 ай бұрын

@@brightsideofmathsoh I was just putting a timestamp , I had to go do something. I enjoyed the whole video it helped me understand much better 👌

@brightsideofmaths

8 ай бұрын

You can definitely use the comments in that way. Please just put something in front like: "personal timestamp". Otherwise, I might thing that there is a mistake or some problem at the given timestamp :)@@Rahul-yu3ro

Cool, but, I think Alonzo Church construction of natural numbers, through lambda calculus and church numerals is better

@brightsideofmaths

10 ай бұрын

I don't know the details there. In the end, the concrete construction does not matter anyway :D

@GabrielMirandaLima-hv7oe

10 ай бұрын

@@brightsideofmaths yes, what really matters is getting a consistent structure which behave as natural numbers should behave, It's simply a personal preference

lol the start learning numbers playlist is a subset of start learning mathematics playlist

@brightsideofmaths

8 ай бұрын

And so is "Start Learning Sets" :)

In one quiz question, why S(0)=0 , it should be s(0)=1.

@brightsideofmaths

Жыл бұрын

Oh yeah. I correct that! Thanks!

I would claim sets are built from images. But first I will show that numbers are built from images Example , 4 always represents 4 images, like 4 squares for instance. To be specific numbers are "labels" for groups of images 1. The main idea here is that maths is built from images (a) example , geometry is clearly made of images b) example 2, We claim numbers are built from images too, as say 4 , always represents 4 images, like 4 squares for instance. C) imaginary numbers are connected to images too , which is why they have applications in physics D) In general any mathematical symbol that comes to mind is connected to images too.

you're so adorable