Defining Numbers & Functions Using SET THEORY // Foundations of Mathematics

We are all familiar with numbers and functions....but are these the most basic, most foundational concept in mathematics? Mathematicians use set theory as the basic building blocks of so much of math. In this video we are going to see how we can think of numbers and functions in terms of sets.

0:00 The Definition Problem
1:44 Set Theory
3:50 Numbers in Set Theory
6:48 Functions in Set Theory

COURSE PLAYLISTS:

OTHER PLAYLISTS:
► Learning Math Series
►Cool Math Series:

BECOME A MEMBER:

MATH BOOKS & MERCH I LOVE:

SOCIALS:

[thebreakdown]

Dude u are a f#cking CHAMPION, literally researching Set theory sends you down rabbit holes of completely convoluted jargon! I've studied so much philosophy and math and yet this thing that is almost completely ontological is usually written in the most over complicated way. You explained it perfectly. You really are the definition of a great teacher.

[jawunderwood]

I’m self studying set theory and proof at the moment and you’ve made the process significantly easier. Textbook treatments and exercises can be a little dry but your enthusiasm and exposition towards where it’s all leading is a great motivation.

[HermanToMath]

Hi, Trefor. My name is Herman. I am a Maths tutor from Hong Kong. Your videos are so great! As I am planning to study PhD next year, now I am studying hard to make up for my maths. Thanks for your videos. I have learnt a lot for them!

[Elite]

Man, your analogies work so beautiful.

[utuberaj]

Great job Prof Bazett. I thought you make superb calc videos- my fav subject in math. And my most boring subject in school math- an awkward concept called SET theory due that bloke G. Cantor, was to me a pure math thing, that was pushed down our throats! Until, I saw this video, that is. You have so beautifully explained the concept for nos and functions in terms of SETs that made my day. Here I see a nice exampleof how 'pure math' gets down-to earth business of numbers & functions- that we all know and use all the time in math and other sciences). I would never have known this fact nor cared to read it up in a standard text book, which would make the subject more boring, I guess, but for your superbly presented lecture.
Keep going. I am getting seduced by SETS, when you do it!!

[itsmebenkenobi]

I'm here again in your channel sir with another lesson, since I can't understand my teacher's lectures and this clearly helped me out again. thanks sir!

[Exahax]

You definitely deserve more subscribers.

[Jan_D-vmrk]

Thank you very much for this great video. You answered in a clear way my question whether a set can contain equal elements, for example {3, 3, 3} and made the definition of numbers in terms of sets easier to understand.

[cognitive-carpenter]

I really thought the whole point was to formulate the functions in all empty set format. Your way was easier haha

[Mestil]

I loved it! I learn more in English with you than I learn in Portuguese with my teachers

[jordanlazaro]

This video is superb. This is a very philosophical topic in which is predicated on the basis that Logic is consistent. Indeed, the very existence of value seems to be something inherent in us, something we intrinsically agree, something against-the-nature of ourselves for us to even debate about because it seems that all of the mathematics narrows down to logic, and logic narrows down to existence or absence of something. Very good video.

[saralewis]

That's amazing! I never thought function this way.

[Ferdimitry]

First comment! Love Trefor. I wish I had a teacher like you when I was much younger.

[rdabdao]

Was studying this last week! For our Topology and Geometry for Physics class :)

[JAUNEtheLOCKE]

How does this not have more views? Instantly subscribed!!

[markjosephalfred]

1st teacher i see to make real world exmples 1st teacher to make me think im good at math thx so much:)

[dandelobo]

Really great way to teach math! Thank you!

[SP-qxtc]

Its like standard model or string theory...we invented/ discovered math as counting numbers like we thought basic building blocks were atoms...then we came to know about subatomic particles and build quantum theory like set theory here...this is just amezing .

[alkankondo]

Man, I was CLOSE to guessing how to define functions set-theoretically! I really surprised myself with how close I was, considering how abstract this type of thinking is and how many different ways it seems you can take it. My thinking was to enclose each element of the co-domain in its own additional set of brackets. I.E.:

f(1) = 2 becomes {1, {2}}

This sets apart the domain from the co-domain. This would also enable an easy extension to non-function relations, where some elements of the domain have multiple outputs. What do you think? Does this work, too, or is there a lapse in my logic?

[yoananda]

great video, very pedagogic.
I foresee that set is great for discrete functions, but what about continous functions ? is there a trick to describe a function in R ?