Start Learning Numbers - Part 1 - Natural Numbers (in Set Theory)

Thanks to all supporters! They are mentioned in the credits of the video :)

This is my video series about Start Learning Numbers. I hope that it will help everyone who wants to learn about the construction of numbers. This is something one needs as a good foundation for all advanced mathematics.

#StartLearningMathematics
#Mathematics
#LearnMath
#calculus
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I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[gnikola]

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.

[one_logic]

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.

[danjwheatley]

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!

[angelmendez-rivera]

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.

[usernamewatcher]

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

[ahmedamr]

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)?

[SimchaWaldman]

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

[dhn]

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.

[raheem]

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.

[murilopalomosebilla]

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

[GabrielMirandaLima-hvoe]

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

[HaCkeMatician]

Thanks!
Really appreciate that 😊 ..!

Do you recommend some textbooks as a companion to the entire topics of the course?

Thanks, ,

[Lockout_]

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

[AleXander-eoiz]

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?

[douglas]

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

[harshitrajput]

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?

[Lockout_]

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

[NAbadi-zgsh]

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.

[medwards]

I think there is an error on the corresponding quiz?

[iunie]

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?