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.
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!
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!!
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!
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.
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.
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.
Thank you for this video. Though you mention that a set is unordered at time 8:45, you better explicitly point that out in the definition given for a set at time 1:50. Not many books explicity define a set as "a collection of distinct unordered objects". This makes a difference to a novice especially if he/she doesn't watch the video to the end. As always your good effort is noticable.
Great concepts and video, one question though: In the set formulation of the function x^2+1, the first element of the corresponding set is {0,{0,1}}. But isn't {0,1} equal to two according to the interpretation of natural numbers from before?
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 ?
Well first you'd need to agree that there can be infinite sets. Then there's no difference if the domain is discrete or continuous, a function is just the ordered triplet of a set called Domain (which will be R), another set called Codomain (which will also be R), and the set "F" of (ordered) pairings within DxC such that for each value x in D there exists a unique pair in F that has x in its "first spot".
@Professor: At around 12:00 when you define function: shouldn't that be A->B where there is **at-most** one mapping from A to B (instead of exactly one)?
Nice video.🙏 At first, a decade ago I was astonished by the set theoretical grounding of numbers. I think you are referring to the Barber Paradox of Russell as an exception. I have subscribed to you👍
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 .
Man, I was SOOOOO 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?
Nice! This looks totally reasonable to me two. Basically we just need some method to distinguish which is the element in the domain, and which is the element in the codomain and your method does that too.
@@DrTrefor Cool! Thanks for feedback. This was a well-explained video! Your statement of the goal in this construction of the natural numbers -- that "numbers can be thought of as nothing but sets" -- really helped make sense of the abstractness and weirdness.
Your definition works for the natural numbers, but it might not for other types of numbers/sets. If we have x={a} for any a and b={y}, then we have {x, {y}}={{a}, {y}}={{a}, b}. This can be interpreted as either f(x)=y or f(b)=a, which are two very different statements. Basically, if the input and/or output already have brackets around them, it can be impossible to tell them apart from the "marker" brackets that denote one object as the output. For the definition in the video, {x, {x, y}}, the _only_ way for a number to be the input is if it's redundant in a very specific way. If you wanted to make it ambiguous, you would have to let x={{x, y}, a}, for some a. which would give {x, {x, y}} (this means f(x)=y) ={{{x, y}, a}, {x, y}} (which means f({x, y})=a) However, a set being a member of itself isn't allowed in Zermelo-Fraenkel set theory. So, this definition is never ambiguous, and it can be generalized to be used for any type of sets as inputs and outputs. This is important, especially since things like the rational numbers, the real numbers, negative numbers, complex numbers, etc. would be complicated sets based on the natural numbers that might not all follow the same rules. Hope that made sense!
As a first attempt, this is a very nice definition, but as already pointed out, this definition encounters problems rather immediately, which is why mathematicians do not use a definition like this when defining what a function is. In set theory, the agreed upon definition of an ordered pair that almost every mathematician uses is that (x y) := {{x}, {x, y}}. The definition of a function is now as follows: the set {(x, y) : x in A, y in B} is a function f iff (x, y) is in f for all x in A and if (x, y) and (x, z) are both in f, then y = z.
When kids engage you in their inevitable infinite questions conversation, it seems like Set theory of reality itself. - "Why do you cut the grass every Saturday?" - "Because it'll grow too high if I don't." - "Why does grass grow?" - "Because the Sun shines on it." - "Why does the Sun shine?" - "Well, umm, gravity forces stuff to come together, and all that pressure makes it hot." - "What's gravity?" - "Gravity is a force. It's why your ball falls to the ground when you throw it." - "What's a force?" - "A force is like when you push something." - "So what's pushing my ball down?" - "Gravity." - "But what makes gravity push?" - "Um...hey son, you wanna go get some ice cream when I'm done?" Kids will take you down the rabbit hole to the absolute limits of your explanations, forcing you to distract them!
I am trying to understand if the set theory is based on a different outlook from a field, a group, or a vector space. I find it fascinating that any mathematical object, and I am assuming that sets are just another mathematical object, tries hard to investigate closure under addition, multiplication, and find identity and inverse, as well as compliance with algebraic properties such as commutativity and associativity. Anyway, I know that some that stuff might not be well defined with sets, but I just want to know where are you going with the sets. I know basic set theory and the major operations performed on sets such as unions, intersections, compliments and some laws that ensure the elements in the set are distinct (avoiding repetitive counting), but I am a bit uncertain over the purpose of defining empty sets and sets of empty sets as elements of a bigger set. Your analogy of empty boxes inside a bigger box makes sense, but as this blows out into more empty boxes inside each other, it becomes much like vanity, although theoretically could be just fine to allow such definitions in the set theory. I like the idea of comparing the empty set with the element zero in the natural number set. As for the cardinality of the infinite sets and set paradoxes, I like to see more presentations on the subject.
Great video! How to imagine a set of empty set plus another a set of empty set equal to a set of a set of empty set and a set of a set of empty set? 1+1 = 2
Why do the set of integers require all previous sets and not just contain only the single highest valued set? Like set for "2" only needs the set for "1" in order to be distinct. So 0 = (), 1 = ( () ), 2 = (( () )), 3 = ((( () ))), etc. Where the number is the amount of times the internal set needs to be extracted in order to reach the empty set zero?
Is the writing of functions in this way as a set convention only to avoid ambiguity, or does it allow us to manipulate these sets in the same way we can manipulate the mathematical objects we call functions. Also what is the notion of an operation on sets if we add 1 + 2 the union nor intersection yield 3 so how does that work. Also with functions that take Reals to Reals how would this set notation of a function work, if the Reals are uncountably infinite does it make sense to notate the function in this way as a set. Sorry for all the questions and thanks for the videos.
The goal isn't manipulation. The formalism here is two cumbersome to spend more than a passing note about. When you want to manipulate numbers or functions, do it in the "usual" way (for example union isn't sufficient to get 1+2=3 from the set notation). The goal is more philosophical, saying that these bizarre objects CAN be written in terms of sets, which means that if you and I agree on the concept of a set, then you and I agree on the concept of a function as well.
Functions f:A->B are sets with elements {x, {x,y}} where every element in A is associated with exactly one element in B. So (x,y) = {x, {x,y}} = {x, {y,x}}. Yes?
Would it not be better to encode a function like {{0, {1}}, {1, {2}}, {2, {5}}, ... }? Here we have a set of sets which contain a member of domain, and the member of a set of the codomain. No repetition required.
I've seen some of these definitions put forth in a few different textbooks on set theory, particularly the definition of natural numbers in terms of sets, however, I have struggled to find or devise a simple operation such as ADDITION under this framework. For example, prove that {0} + {0,1} = {0,1,2} i.e., 1+2=3. Does anyone have a guide/resource on this?
Would it be fair to say that the irrational numbers is an x s.t. x exists in the Real number Union with the x such that it doesn't exist in the Rational numbers? I know I could express them R/Q, but I just really wanted to make sure I could define it differently to show I really understand set notation.
Functions as sets of sets containing 2 elements, one of which just contains a number (which is just a set of sets) and the other containing 2 numbers (except where f(x) = x) which of course are both sets of sets. Phew. It's great to have an axiomatic foundation but as an encoding scheme it is both mind bending and verbose. I also predict an international shortage of curly brackets. It's not only an explanation of how set theory provides an underpinning but at a meta level explains exactly why we normally use higher level concepts.
@@DrTrefor i really need to have the rest of the subject i mean what is the the next of the video because the subject is really important i have taken a look into the Cool math series but i couldn't find it . if there is no next video for that please make the another one therefore i can get and understand the wohle point. the Foundations of mathematics is really crucial for us .Thank you so much for your efforts
@@DrTrefor I mean I suggest that you make an independent Playlist in which you explain in you amazing way as usual these things of the foundations of mathematics so that i get not blind when i study math :) Thank you so much
They make sense by not being just a regular set. They are either a sequence (basically, a function like s : N->DU{null} with s(0)=1, s(1)=s(2)=2, s(3)=3, s(4)=s(5)=4, and s(n)=null for n>5) or a set of pairs {(0,1),(1,2),(2,2),(3,3),(4,4),(5,4)}.
How would the identity function work? If f(0) = 0, then presumably we would want our input/output "pair" to be {0, {0, 0}}. But {0, 0} is not correct because sets must have different elements. What about if an element maps to itself then we set the input/output pair to something like {0, {0}}? But wait, isn't {0} identified with 1? It seems like the only way to handle an element mapping to itself would be to encode it as a set with a single element, e.g. {0}. In this way, the identify function becomes {{0}, {1}, {2}, ...}, and a function like f(x) = x^2 would be {{0}, {1}, {2,{2,4}}, {3,{3,9}}, ...} (where the elements which do not map to their self remain as {x, {x,y}} pairs). One other approach would be to just leave out the elements which map to themselves, but then we would lose the concept of a well-defined domain (i.e. when an element is missing, how do we know if that is because the element maps to itself or because it is not in the domain?).
Are you the Christopher Locke I know? If so hi! If not, well hi too! Anyways, I agree that {a,{a,a}} should get replaced with {a,{a}} to avoid repetition, but I then think that is unambiguous. If you see something like {0,1} aka { {}, {{}} } you have to decide which element is in the domain. However, you know that you are looking for an element and a set containing that exact same element so it must be that {} aka 0 is the element in the domain. Thus f(0)=0.
@@DrTrefor Yes, it is I. Thanks, that makes sense. One other quick question, in the set theory formulation of math, is the "type" of a set also uniquely defined? In other words, can I tell unambiguously whether a given set represents a number, function, set, differentiable manifold, etc?
@@project.eutopia No, you are wrong. There is absolutely no requirement in the definition of a set or from the axioms of set theories that says you cannot repeat the elements of a set. If you repeat an element of a set, then the set remains unchanged, so the canonical way to denote a set is without repetition, whenever possible, but this does not mean you cannot repeat an element if you want to. As such, (0, 0) := {{0}, {0, 0}} is a perfectly well-formed definition, and a perfectly valid way of denoting the set {{0}}. There is nothing incorrect here. As for your question on types, the answer is "yes and no." It is complicated. The number of braces is often useful for indicating "the type" of a function, but this also only works with respect to certain types of elementary sets. This is because, for example, the set Z is defined in terms of equivalence classes of ordered pairs of elements of the set N, which results in the elements of Z having many more brackets despite still being numbers. This also means that, while you can identify N with the set of nonnegative elements of Z, both sets are not actually equal, since the elements are not actually equal: 1 is not the same as [(1, 0)]. This is why you need to rigorously make s distinction between 1 the natural number and 1 the whole number. Similarly, you need to make a distinction between 1 the whole number and 1 the rational number, and 1 the real number, and 1 the complex number. In utmost generality, if we are talking in terms of the class of all sets, then no, there is no foolproof method that works for literally every set in this regard, when it comes to indicating type. However, there is a weaker indicator: regardless of how many brackets the elements of a set has, a function on this said will always have two additional layers of brackets, for example. So while the absolute count of bracket layers of the elements is not sufficient to be a type indicator, the count of brackets relative to other structures on the set does, in a unique and well-defined manner, help distinguish between types.
9:24 that still allows for a little bit of nitpicking... At first glance i see 2 quirks with this type of notation: * in that {0,{0,1}} element, the 2nd part is actually what you earlier described as "2"... so that's {0,2}... and we go back to the earlier issue of is that 0->2 or 2->0? Guess we need to include the condition that one of the "2" (see nitpick #2 below) terms needs to be an element of the other: {x,z} where x∈z. * what if f(x) = x? We get {x,"{x,x}"} but that is in fact {x,{x}}... So the way to "retrieve" the value of a function out of this notation will have to be worded in a way that applies to these cases as well. :-B ** special case of that last one... f(1)=0... {1,{1,{0}}} = {1,{1,1}} = {1,{1}} = {{0},{1}}. Okay, not too ambiguous, but what'd really grind my gears would be functions from a domain consisting of sets to a codomain of more sets. No TY. 😈
Addition and Subtraction are both functions with two inputs. Mathematicians define addition as repeated successor function. Successor function is define as S(x)= {x,{x}} this is basically (x+1). Subtraction is a lot more complicated but they are still define as a function.
Why does the domain have to be included in the range set? Surely just the fact that one of the numbers is inside a set is enough to identify that that number is the range
haha, I know I was having fun with the complex series. Don't worry, it isn't going away, just going on pause for a short bit to get some different videos in:D
It does not quite work like that. Yes, you can use simple axioms of set theory to construct sets n+1 := {n} of natural numbers. But you cannot have a function with a domain and a co-domain consisting of all those natural number sets, unless there is a set ℕ that contains exactly those sets as its elements. For that you need to talk about the "axiom of infinity" AI, to ensure that such a set exists in the first place.
The point I'm stuck on is why? You say "0➡️ empty set" but what is gained over saying "0➡️ blueberry". I can intuitively make an association so far between 0 and the null but I don't logically see how it flows from the axioms. You then you associate 1 with the set that contains the null set. Why? I don't know why you make that association rather than associating 1 with a set of "unique blueberry" Is the association chosen specifically because it builds on the first association?
It reduces the number of objects that needs to be mathematically define. This method only need to define the empty set and how to construct the next number from an empty set and the rest follows. You are able to define infinite number of objects with two definition. Very efficient.
It is originally define that way but the new definition is a lot cleaner because the axiom of sets already have the union operation. And the construction is just doing the union operation. Also the new definition have each number a set that contains exactly the amount of elements that the number represent. This is very helpful in defining equivalence and equality. It is also needed to define infinities and other objects.
So if one is trying to get some property from inputs on a machine, the concept might be called a "lambda function" ? E.g.: " x => x + 1 ", which looks kind of like {x, x+1}, but that ordering is handled linguistically by the machine. So now I wonder if the only way to construct multi-dimensional ranges from scalars with only sets is to use a set of sets of functions, which sounds initially like a 2x2 matrix, but the cells actually contain these %spoilers% elements ? Very interesting to think: it's like seeing a 2d plane as one eye is closed ..? So fields can be made with sets ? Makes sense that to build a field you'd need sets of things & to build some functions... 😵
I was interested by the title and the thumbnail so I clicked to learn deep explanations of set theory. But at 01:50 I saw what I learnt in school: a start by a definition of sets. And for me, that’s a problem. The axioms of ZF(C) set theory are precisely the rules to create sets, to manipulate sets and to exclude what are not sets. Nowhere, it gives explicitly what is a set, i.e. it gives no definition of a set. Anyway, nice vid (a little adjustment of sound may be needed). I would llike to see more.
I agree. My intention at that point wasn't actually to define a set in a normal sense, as in to define it in terms of more primitive mathematical objects. Instead it was to give a bit of intuition behind the word by associating it with our mental model of a "collection". Nothing in ZFC has to actual correspond to an object with have intuition about, but this is helpful for students to understand sets.
How is that different than any definition of anything in math? All definitions work by saying what the conditions of being a given thing are. That is what the set theory axioms do.
@@francescaerreia8859 That’s right. There’s no difference. There is no critic of the content which is actually well done. And it is what I said. But, I expected a kind of new insight by seeing the title and the thumbnail while it deals with the standard way to define mathematical objects. It is exactly what Dr. Bazett replied to me. For the mathematical aspects, it means "numbers" are "sets" and "sets" are defined using ZF(C) axioms. Hence, no more insights and we all agree.
@11:00 - is not true. If you gave me the set of some examples of x²+1, I can never know that those values represent that function. Unless your set contained all the examples to infinity. I am uncertain what the other functions would be, but how can I assume they do not exist? ie: a complete function can not be known from limited examples. Am I wrong?
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.
The only way I immediately see around this problem is to change the representation of raw numbers to be the nesting level of sets: 0 = {}, 1 = {{}}, 2 = {{{}}}, etc. Then you have the freedom to encode all "sets of more than one element" as something else. With the representation given in the video, I think you'll always run into problems of ambiguous interpretation. Still an awesome video, btw.
The point is a set is not defined in terms of a previous precise concept. So we just have to agree to agree, basically as it is a foundational concept in this formulation. So talking about boxes is a metaphor to help convey meaning.
@@wernerhartl2069 Do you not agree that a box is a container, and that inside containers, there are collections? There is nothing to disagree about here
Set theory is the king all these years.. But nowdays we have category theory, type theory and homotopy type theory that claim the throne...What should we believe..What theory is the true foundation of maths.
Why can't x²+1 be represented { {0,{1}}, {1,{2}}, {2,{5}}, ... }. I don't understand the purpose of intentionally adding waste to the representations in set theory. It seems to be all over the place and without logic.
Is set theory a meta-mathematics? ... instead of distinguishing the properties of one 'thing' ... a number or numerical proxy such as "x" ... we're talking about properties of ALL things and in this way, a more 'pure' or 'core' symbolic representation of reality?
I suppose it sort of depends what you want to call math and what you want to call "meta-math". I think it might be hard to properly delineate. Regardless, I think of this type of topic as "math foundations" upon which the rest of math gets built up.
I wouldn't think of it as mathematics, but rather as a commonly accepted feverdream, :). This literally solves nothing, and only pushes back the definition of what a number is behind yet another abstraction, not making >anything< any clearer or more well-formed. In contrast, the geometric definition of number is far, far more logical and complete. But I guess some people need this intellectual circle-j**k to feel smart.
Yeah its very simple consept its not like its algebraic topology. You went to far with those empty boxes. Unless it was usefull for you yobhelp you grasp the concept
But now all numbers are the same. They all have the same element and only the same element, the empty set. You have just said that if we repeat the same thing it's still the same thing. So 0=1. Set theory is useless
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.
Glad it's been helping!
me too a self-learner, thanks for ur video
Always amazing to see someone explain in such intuitive topics like numbers in a complete different manner, thank you professor!
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!
I joined your membership! :D
Hey, thank you, I really appreciate that!! Good luck in your studying for your PhD:)
Thank you very much😀
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!!
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!
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.
Man, your analogies work so beautiful.
Thank you kindly!
You definitely deserve more subscribers.
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.
I loved it! I learn more in English with you than I learn in Portuguese with my teachers
First comment! Love Trefor. I wish I had a teacher like you when I was much younger.
Nice one! Thank you so much:)
I have a blue shell
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.
Thank you for this video. Though you mention that a set is unordered at time 8:45, you better explicitly point that out in the definition given for a set at time 1:50. Not many books explicity define a set as "a collection of distinct unordered objects". This makes a difference to a novice especially if he/she doesn't watch the video to the end. As always your good effort is noticable.
or "an unordered collection of distinct objects"
The best video I've found about the topic by far
Great concepts and video, one question though: In the set formulation of the function x^2+1, the first element of the corresponding set is {0,{0,1}}. But isn't {0,1} equal to two according to the interpretation of natural numbers from before?
1st teacher i see to make real world exmples 1st teacher to make me think im good at math thx so much:)
I really thought the whole point was to formulate the functions in all empty set format. Your way was easier haha
Well, you COULD do it solely in terms of the empty set, but... I do not think you actually want to see what it would look like if you did that
He did, hes just using shortcuts as numbers to represent sets of sets of empty sets
@@baci5190_Yes, but even better to say the shortcuts are symbols or names for the sets.
That's amazing! I never thought function this way.
How does this not have more views? Instantly subscribed!!
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 ?
Well first we have to define R itself, no trivial task!
Well first you'd need to agree that there can be infinite sets. Then there's no difference if the domain is discrete or continuous, a function is just the ordered triplet of a set called Domain (which will be R), another set called Codomain (which will also be R), and the set "F" of (ordered) pairings within DxC such that for each value x in D there exists a unique pair in F that has x in its "first spot".
Really great way to teach math! Thank you!
Was studying this last week! For our Topology and Geometry for Physics class :)
That's awesome!
the empty box metaphor is so, so helpful. thank you
@Professor: At around 12:00 when you define function: shouldn't that be A->B where there is **at-most** one mapping from A to B (instead of exactly one)?
Functions need to be well-defined for every element of A.
Nice video.🙏 At first, a decade ago I was astonished by the set theoretical grounding of numbers. I think you are referring to the Barber Paradox of Russell as an exception. I have subscribed to you👍
thank you!!
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 .
Man, I was SOOOOO 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?
Nice! This looks totally reasonable to me two. Basically we just need some method to distinguish which is the element in the domain, and which is the element in the codomain and your method does that too.
@@DrTrefor Cool! Thanks for feedback. This was a well-explained video! Your statement of the goal in this construction of the natural numbers -- that "numbers can be thought of as nothing but sets" -- really helped make sense of the abstractness and weirdness.
Your definition works for the natural numbers, but it might not for other types of numbers/sets. If we have x={a} for any a and b={y}, then we have {x, {y}}={{a}, {y}}={{a}, b}. This can be interpreted as either f(x)=y or f(b)=a, which are two very different statements.
Basically, if the input and/or output already have brackets around them, it can be impossible to tell them apart from the "marker" brackets that denote one object as the output.
For the definition in the video, {x, {x, y}}, the _only_ way for a number to be the input is if it's redundant in a very specific way. If you wanted to make it ambiguous, you would have to let x={{x, y}, a}, for some a. which would give
{x, {x, y}} (this means f(x)=y)
={{{x, y}, a}, {x, y}} (which means f({x, y})=a)
However, a set being a member of itself isn't allowed in Zermelo-Fraenkel set theory. So, this definition is never ambiguous, and it can be generalized to be used for any type of sets as inputs and outputs. This is important, especially since things like the rational numbers, the real numbers, negative numbers, complex numbers, etc. would be complicated sets based on the natural numbers that might not all follow the same rules.
Hope that made sense!
As a first attempt, this is a very nice definition, but as already pointed out, this definition encounters problems rather immediately, which is why mathematicians do not use a definition like this when defining what a function is.
In set theory, the agreed upon definition of an ordered pair that almost every mathematician uses is that (x y) := {{x}, {x, y}}. The definition of a function is now as follows: the set {(x, y) : x in A, y in B} is a function f iff (x, y) is in f for all x in A and if (x, y) and (x, z) are both in f, then y = z.
Great explanation! Thank you!
You're full of tricks, wizard.
Honestly though, love your content =)
I remember this in my first course in abstract algebra, First year polytech Yaounde
When kids engage you in their inevitable infinite questions conversation, it seems like Set theory of reality itself.
- "Why do you cut the grass every Saturday?"
- "Because it'll grow too high if I don't."
- "Why does grass grow?"
- "Because the Sun shines on it."
- "Why does the Sun shine?"
- "Well, umm, gravity forces stuff to come together, and all that pressure makes it hot."
- "What's gravity?"
- "Gravity is a force. It's why your ball falls to the ground when you throw it."
- "What's a force?"
- "A force is like when you push something."
- "So what's pushing my ball down?"
- "Gravity."
- "But what makes gravity push?"
- "Um...hey son, you wanna go get some ice cream when I'm done?"
Kids will take you down the rabbit hole to the absolute limits of your explanations, forcing you to distract them!
Where Mathematics Comes From: George Lakoff/Rafael Nunez. Great read.
Much thoughts following. Thanks!
3:25 ZERMELLON FRA
6:00 Sets, sets, sets, and sets, numbers
You are a great teacher .
Hmm,,, von Neumann Ordinal and Ordered Pair.
I hope you make a video on non well founded set theory. Or even Classes, Types and Categories.
How do you define sum with sets? Do you use union/intersection?
Well done!
Dr Bazett please make videos on Quadratic Reciprocity. Most Math RUclipsrs make it cumbersome.
Thank you for your content. Can you recommend some books for logical math.
Loved this!
Thank you!
I am trying to understand if the set theory is based on a different outlook from a field, a group, or a vector space. I find it fascinating that any mathematical object, and I am assuming that sets are just another mathematical object, tries hard to investigate closure under addition, multiplication, and find identity and inverse, as well as compliance with algebraic properties such as commutativity and associativity. Anyway, I know that some that stuff might not be well defined with sets, but I just want to know where are you going with the sets. I know basic set theory and the major operations performed on sets such as unions, intersections, compliments and some laws that ensure the elements in the set are distinct (avoiding repetitive counting), but I am a bit uncertain over the purpose of defining empty sets and sets of empty sets as elements of a bigger set. Your analogy of empty boxes inside a bigger box makes sense, but as this blows out into more empty boxes inside each other, it becomes much like vanity, although theoretically could be just fine to allow such definitions in the set theory. I like the idea of comparing the empty set with the element zero in the natural number set.
As for the cardinality of the infinite sets and set paradoxes, I like to see more presentations on the subject.
Great video! How to imagine a set of empty set plus another a set of empty set equal to a set of a set of empty set and a set of a set of empty set? 1+1 = 2
can you make please a playlist for the foundation of mathematics so that we can understand this key part of the whole Mathematics .
I really appreciate your video’s thanks 🙏
Why do the set of integers require all previous sets and not just contain only the single highest valued set? Like set for "2" only needs the set for "1" in order to be distinct.
So 0 = (), 1 = ( () ), 2 = (( () )), 3 = ((( () ))), etc. Where the number is the amount of times the internal set needs to be extracted in order to reach the empty set zero?
Is the writing of functions in this way as a set convention only to avoid ambiguity, or does it allow us to manipulate these sets in the same way we can manipulate the mathematical objects we call functions. Also what is the notion of an operation on sets if we add 1 + 2 the union nor intersection yield 3 so how does that work. Also with functions that take Reals to Reals how would this set notation of a function work, if the Reals are uncountably infinite does it make sense to notate the function in this way as a set. Sorry for all the questions and thanks for the videos.
The goal isn't manipulation. The formalism here is two cumbersome to spend more than a passing note about. When you want to manipulate numbers or functions, do it in the "usual" way (for example union isn't sufficient to get 1+2=3 from the set notation). The goal is more philosophical, saying that these bizarre objects CAN be written in terms of sets, which means that if you and I agree on the concept of a set, then you and I agree on the concept of a function as well.
You can define addition in terms of set unions, though. Look at the Von Neumann constructive model of the Peano axioms, as an example.
What would be the problem of defining a function as having members {x,{y}} rather than {x,{x,y}}?
Functions f:A->B are sets with elements {x, {x,y}} where every element in A is associated with exactly one element in B.
So (x,y) = {x, {x,y}} = {x, {y,x}}. Yes?
Exactly
Would it not be better to encode a function like {{0, {1}}, {1, {2}}, {2, {5}}, ... }? Here we have a set of sets which contain a member of domain, and the member of a set of the codomain. No repetition required.
I've seen some of these definitions put forth in a few different textbooks on set theory, particularly the definition of natural numbers in terms of sets, however, I have struggled to find or devise a simple operation such as ADDITION under this framework. For example, prove that {0} + {0,1} = {0,1,2} i.e., 1+2=3. Does anyone have a guide/resource on this?
Would it be fair to say that the irrational numbers is an x s.t. x exists in the Real number Union with the x such that it doesn't exist in the Rational numbers? I know I could express them R/Q, but I just really wanted to make sure I could define it differently to show I really understand set notation.
Functions as sets of sets containing 2 elements, one of which just contains a number (which is just a set of sets) and the other containing 2 numbers (except where f(x) = x) which of course are both sets of sets. Phew.
It's great to have an axiomatic foundation but as an encoding scheme it is both mind bending and verbose.
I also predict an international shortage of curly brackets.
It's not only an explanation of how set theory provides an underpinning but at a meta level explains exactly why we normally use higher level concepts.
What about pi, complex number and negative number ?
could you till the name of the playlist that contains this video .Thank you
This is in the cool math series, not really core to any major course
@@DrTrefor i really need to have the rest of the subject i mean what is the the next of the video because the subject is really important i have taken a look into the Cool math series but i couldn't find it . if there is no next video for that please make the another one therefore i can get and understand the wohle point. the Foundations of mathematics is really crucial for us .Thank you so much for your efforts
@@DrTrefor I mean I suggest that you make an independent Playlist in which you explain in you amazing way as usual these things of the foundations of mathematics so that i get not blind when i study math :) Thank you so much
How does lists containin the same thing many time makes sense in set theory ?
Like [1,2,2,3,4,4]
They make sense by not being just a regular set. They are either a sequence (basically, a function like s : N->DU{null} with s(0)=1, s(1)=s(2)=2, s(3)=3, s(4)=s(5)=4, and s(n)=null for n>5) or a set of pairs {(0,1),(1,2),(2,2),(3,3),(4,4),(5,4)}.
How would the identity function work? If f(0) = 0, then presumably we would want our input/output "pair" to be {0, {0, 0}}. But {0, 0} is not correct because sets must have different elements. What about if an element maps to itself then we set the input/output pair to something like {0, {0}}? But wait, isn't {0} identified with 1? It seems like the only way to handle an element mapping to itself would be to encode it as a set with a single element, e.g. {0}. In this way, the identify function becomes {{0}, {1}, {2}, ...}, and a function like f(x) = x^2 would be {{0}, {1}, {2,{2,4}}, {3,{3,9}}, ...} (where the elements which do not map to their self remain as {x, {x,y}} pairs). One other approach would be to just leave out the elements which map to themselves, but then we would lose the concept of a well-defined domain (i.e. when an element is missing, how do we know if that is because the element maps to itself or because it is not in the domain?).
Are you the Christopher Locke I know? If so hi! If not, well hi too!
Anyways, I agree that {a,{a,a}} should get replaced with {a,{a}} to avoid repetition, but I then think that is unambiguous. If you see something like {0,1} aka { {}, {{}} } you have to decide which element is in the domain. However, you know that you are looking for an element and a set containing that exact same element so it must be that {} aka 0 is the element in the domain. Thus f(0)=0.
@@DrTrefor Yes, it is I. Thanks, that makes sense. One other quick question, in the set theory formulation of math, is the "type" of a set also uniquely defined? In other words, can I tell unambiguously whether a given set represents a number, function, set, differentiable manifold, etc?
@@project.eutopia No, you are wrong. There is absolutely no requirement in the definition of a set or from the axioms of set theories that says you cannot repeat the elements of a set. If you repeat an element of a set, then the set remains unchanged, so the canonical way to denote a set is without repetition, whenever possible, but this does not mean you cannot repeat an element if you want to. As such, (0, 0) := {{0}, {0, 0}} is a perfectly well-formed definition, and a perfectly valid way of denoting the set {{0}}. There is nothing incorrect here.
As for your question on types, the answer is "yes and no." It is complicated. The number of braces is often useful for indicating "the type" of a function, but this also only works with respect to certain types of elementary sets. This is because, for example, the set Z is defined in terms of equivalence classes of ordered pairs of elements of the set N, which results in the elements of Z having many more brackets despite still being numbers. This also means that, while you can identify N with the set of nonnegative elements of Z, both sets are not actually equal, since the elements are not actually equal: 1 is not the same as [(1, 0)]. This is why you need to rigorously make s distinction between 1 the natural number and 1 the whole number. Similarly, you need to make a distinction between 1 the whole number and 1 the rational number, and 1 the real number, and 1 the complex number.
In utmost generality, if we are talking in terms of the class of all sets, then no, there is no foolproof method that works for literally every set in this regard, when it comes to indicating type. However, there is a weaker indicator: regardless of how many brackets the elements of a set has, a function on this said will always have two additional layers of brackets, for example. So while the absolute count of bracket layers of the elements is not sufficient to be a type indicator, the count of brackets relative to other structures on the set does, in a unique and well-defined manner, help distinguish between types.
this definition works well for functions with natural numbers but what about functions with real value inputs? won't that get really messy.
9:24 that still allows for a little bit of nitpicking... At first glance i see 2 quirks with this type of notation:
* in that {0,{0,1}} element, the 2nd part is actually what you earlier described as "2"... so that's {0,2}... and we go back to the earlier issue of is that 0->2 or 2->0? Guess we need to include the condition that one of the "2" (see nitpick #2 below) terms needs to be an element of the other: {x,z} where x∈z.
* what if f(x) = x? We get {x,"{x,x}"} but that is in fact {x,{x}}... So the way to "retrieve" the value of a function out of this notation will have to be worded in a way that applies to these cases as well. :-B
** special case of that last one... f(1)=0... {1,{1,{0}}} = {1,{1,1}} = {1,{1}} = {{0},{1}}. Okay, not too ambiguous, but what'd really grind my gears would be functions from a domain consisting of sets to a codomain of more sets. No TY. 😈
24k views? This golden video only have 24k views …
thanks bro
is there a way to define operations like addition and subtraction like this?
Addition and Subtraction are both functions with two inputs. Mathematicians define addition as repeated successor function. Successor function is define as S(x)= {x,{x}} this is basically (x+1). Subtraction is a lot more complicated but they are still define as a function.
@@kazedcat thank you!
Super
Why does the domain have to be included in the range set? Surely just the fact that one of the numbers is inside a set is enough to identify that that number is the range
what about complex numbers and analysis
haha, I know I was having fun with the complex series. Don't worry, it isn't going away, just going on pause for a short bit to get some different videos in:D
@@DrTrefor Thank U Professor !
Can we write f(0) like: {{0},{0,1}} ??
It does not quite work like that. Yes, you can use simple axioms of set theory to construct sets n+1 := {n} of natural numbers. But you cannot have a function with a domain and a co-domain consisting of all those natural number sets, unless there is a set ℕ that contains exactly those sets as its elements. For that you need to talk about the "axiom of infinity" AI, to ensure that such a set exists in the first place.
The point I'm stuck on is why? You say "0➡️ empty set" but what is gained over saying "0➡️ blueberry". I can intuitively make an association so far between 0 and the null but I don't logically see how it flows from the axioms. You then you associate 1 with the set that contains the null set. Why? I don't know why you make that association rather than associating 1 with a set of "unique blueberry" Is the association chosen specifically because it builds on the first association?
It reduces the number of objects that needs to be mathematically define. This method only need to define the empty set and how to construct the next number from an empty set and the rest follows. You are able to define infinite number of objects with two definition. Very efficient.
Now write PI in set notation.
haha, that would be cumbersome indeed!!
Why not represent numbers by just the inclusion of the previous one? So 0=∅, 1={∅}, 2={{∅}}, and so on?
It is originally define that way but the new definition is a lot cleaner because the axiom of sets already have the union operation. And the construction is just doing the union operation. Also the new definition have each number a set that contains exactly the amount of elements that the number represent. This is very helpful in defining equivalence and equality. It is also needed to define infinities and other objects.
thanks doc
You're most welcome!
So if one is trying to get some property from inputs on a machine, the concept might be called a "lambda function" ? E.g.: " x => x + 1 ", which looks kind of like {x, x+1}, but that ordering is handled linguistically by the machine.
So now I wonder if the only way to construct multi-dimensional ranges from scalars with only sets is to use a set of sets of functions, which sounds initially like a 2x2 matrix, but the cells actually contain these %spoilers% elements ?
Very interesting to think: it's like seeing a 2d plane as one eye is closed ..? So fields can be made with sets ? Makes sense that to build a field you'd need sets of things & to build some functions... 😵
I was interested by the title and the thumbnail so I clicked to learn deep explanations of set theory. But at 01:50 I saw what I learnt in school: a start by a definition of sets. And for me, that’s a problem.
The axioms of ZF(C) set theory are precisely the rules to create sets, to manipulate sets and to exclude what are not sets. Nowhere, it gives explicitly what is a set, i.e. it gives no definition of a set.
Anyway, nice vid (a little adjustment of sound may be needed). I would llike to see more.
I agree. My intention at that point wasn't actually to define a set in a normal sense, as in to define it in terms of more primitive mathematical objects. Instead it was to give a bit of intuition behind the word by associating it with our mental model of a "collection". Nothing in ZFC has to actual correspond to an object with have intuition about, but this is helpful for students to understand sets.
How is that different than any definition of anything in math? All definitions work by saying what the conditions of being a given thing are. That is what the set theory axioms do.
@@francescaerreia8859 That’s right. There’s no difference. There is no critic of the content which is actually well done. And it is what I said. But, I expected a kind of new insight by seeing the title and the thumbnail while it deals with the standard way to define mathematical objects. It is exactly what Dr. Bazett replied to me.
For the mathematical aspects, it means "numbers" are "sets" and "sets" are defined using ZF(C) axioms. Hence, no more insights and we all agree.
orfered pairs: what if (a, (a, a)) = (a, (a)) ???
@11:00 - is not true. If you gave me the set of some examples of x²+1, I can never know that those values represent that function. Unless your set contained all the examples to infinity. I am uncertain what the other functions would be, but how can I assume they do not exist? ie: a complete function can not be known from limited examples. Am I wrong?
yes, you're wrong, the set literally has infinite elements the way it is defined
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.
Don't you run into issues with {0, 1} being exactly the same thing as your set representation of 2?
The only way I immediately see around this problem is to change the representation of raw numbers to be the nesting level of sets: 0 = {}, 1 = {{}}, 2 = {{{}}}, etc. Then you have the freedom to encode all "sets of more than one element" as something else. With the representation given in the video, I think you'll always run into problems of ambiguous interpretation.
Still an awesome video, btw.
The set {{},0} is equal to {0} or {{}} ?
What do you mean by 0?
0 = {}, so {{}, 0} = {{}, {}} = {{}}.
@@angelmendez-rivera351 But, 0 = {}, so {{}} = {0} The answer is both are correct.
@@aslpuppy1026 Yes
Why so less subscribers man!¡!¡!¡!¡
😧😧😧😧😧😧😧😧😧
who would dislike this video?
Crazy people!!
@@DrTrefor Definitely crazy maybe even psychotic.
Me encantó
We don’t agree. You say a set is s collection of objects. Then you instantly say it is a box with objects in it. Which is it?
The point is a set is not defined in terms of a previous precise concept. So we just have to agree to agree, basically as it is a foundational concept in this formulation. So talking about boxes is a metaphor to help convey meaning.
@@DrTrefor I believe this clears it up:
ruclips.net/video/F7nDXd9uMJ8/видео.html
@@wernerhartl2069 Do you not agree that a box is a container, and that inside containers, there are collections? There is nothing to disagree about here
@@angelmendez-rivera351 I agree with you. That’s what I said in the link in my previous.reply.
I don't understand why ƒ(x)=X^2+1 isn't already a set. What's the difference if you write it as {N^2+1, ...} where N is a real number.
f(x) = X^2 + 1 isn't a set, it's simply an equation with not-currently-defined terms.
4:03 No zero is not a natural number...........
the definition of natural numbers isn’t fixed, sometimes people include zero, sometimes not.
It is a natural number, according to the ISO, so you have no real business trying to correct people about this.
Set theory is the king all these years.. But nowdays we have category theory, type theory and homotopy type theory that claim the throne...What should we believe..What theory is the true foundation of maths.
False basis: numbers are derived by dividing infinity, not by multiplying zero.
Why can't x²+1 be represented { {0,{1}}, {1,{2}}, {2,{5}}, ... }. I don't understand the purpose of intentionally adding waste to the representations in set theory. It seems to be all over the place and without logic.
I love you
i have question please
Is set theory a meta-mathematics? ... instead of distinguishing the properties of one 'thing' ... a number or numerical proxy such as "x" ... we're talking about properties of ALL things and in this way, a more 'pure' or 'core' symbolic representation of reality?
I suppose it sort of depends what you want to call math and what you want to call "meta-math". I think it might be hard to properly delineate. Regardless, I think of this type of topic as "math foundations" upon which the rest of math gets built up.
@@DrTrefor Good. Thank you.
Why not {{0, {1}},{1, {2}} ...
אחלה
teaching without enthusiasm should be outlawed .
Does anyone else find the way he says zero kinda crazy 😂
how is this maths
I wouldn't think of it as mathematics, but rather as a commonly accepted feverdream, :). This literally solves nothing, and only pushes back the definition of what a number is behind yet another abstraction, not making >anything< any clearer or more well-formed. In contrast, the geometric definition of number is far, far more logical and complete. But I guess some people need this intellectual circle-j**k to feel smart.
Yeah its very simple consept its not like its algebraic topology. You went to far with those empty boxes. Unless it was usefull for you yobhelp you grasp the concept
But now all numbers are the same. They all have the same element and only the same element, the empty set. You have just said that if we repeat the same thing it's still the same thing. So 0=1. Set theory is useless
Please... get a lavalier.