There’s a compelling reason to include 0 as a natural number (20:39), namely: 0 = {} 1 = {0} 2 = {0, 1} 3 = {0, 1, 2} … w = {0, 1, 2, …} w + 1 = {0, 1, 2, …, w} … Where w represents the ordinal number omega, the set of all natural numbers: mathworld.wolfram.com/OrdinalNumber.html (Currently, I haven’t seen your entire series. You may have included this in one of your later videos.)
Well, that's assuming the unrestricted comprehension principle. If you go with the axiom of choice, you have to "choose" if it is a part of your set or not, you could include sets in sets.
Great lecture. Perfect balance of simplicity and complexity, fun and challenge, insight and foresight. I'm looking forward for more lectures. You give symbols not just its meaning but how it connects to other disciplines. This technique makes the learner not only understand the symbol but how it is connected to other disciplines.
Thank you, sir! I'm learning higher mathematics to help me with software engineering, as well as just gaining general math knowledge. These videos are excellent!
Metaphysics as set theory I believe that Leibniz-Plato is not really philosophy, but mathematics. In fact, set theory. Unfortunately I am not a mathematician, and need some help. But here are some initial thoughts. In Leibniz, the physical object in spacetime is represented as a mental point called a monad. Mind is a formalization or form of mathematics in Plato-Leibniz. From set theory we find that a set is a collection of objects that it owns. This ownership I believe is what we call control, perception or creation. That is, a set controls the objects or sets within it hierarchically. Using the notation < to mean is conceptually contained in (a sidewise U), M as plato's Mind (the One), T as the domain of time, and S as the domain of space, p as a particle is spacetime, and m(p) as its monad, we have the sets m(p) < S < T < M The motion of an object in spacetime begins with its birth as a still point in Plato's Mind. That is. as a point in the all-encompassing set of points, Plato's Mind, which is the point containing all other (mental) points. The mental is the mathematical, the formal in terms of set theory. Time and space are quantized, as is experience, so experience is given to us a set of movie frames that we expeience as a movie. -- Dr. Roger B Clough NIST (retired, 2000). See my Leibniz site: rclough@verizon.academia.edu/RogerClough For personal messages use rclough@verizon.net
i generally do not comment on videos but cant stop myself after watchinng this.. great work this is awesome when i study for 30 minutes in class it feels like days but here i enjoyed this 30 minutes like i enjoy watching my fav programm on tv.great great great work pls upload more
Yes, multisets are in fact a generalization of sets. You can kind of "construct" a multiset as a set of ordered pairs, in which the first part is some element and the second part is a "multiplicity", meaning how many times the element is included in the multiset. So for example, the multiset {a, b, b, c} can be represented as {(a,1), (b,2), (c,1)}. Multisets are quite useful in combinatorics. You should also look into fuzzy sets for another generalization! (I'll PM you my email address.)
Nice video as always, but I'd quibble with your decision to disallow sets as members of other sets in every case. Yes, this resolves Russell's paradox, but it goes too far IMO -- e.g. what do you call the power set of S? you can't call it a set?? Also there's the example of certain axiomatic set theories where the *only* possible elements of a set are other sets. (Yes, the word "family" is common in the literature, but it's used for clarity; formally a family is just a set, in most treatments.)
This is why we have to be careful with mathematical terminology! XD If D is a subset of A, then A contains all elements of D. This would be different from saying that D itself was an ELEMENT of A. And as I said in one of the annotations, sets CAN contain other sets (a good example being the Power Set of a set), but only as long as we're careful to avoid Russell's Paradox. Zermelo-Fraenkel set theory, which is the kind we use today, has axioms built in that keep this from happening.
@@BillShillito In the video you were under the impression that sets containing other sets were the source of the paradox. That's actually not quite the way the paradox is solved - neither in the Russell or the ZFC way. If anything, the distinction between family and set doesn't help either, because that alone won't solve the paradox in the Russell style (not to mention that a family is in many cases a set too). The key actually lies in the introduction of a higher collection called "class" - and claiming that the "Russell set" is a class that happens to be not a set. So essentially, a class is what was naively known as a set, but now a set has to be redefined so as to tighten up the rules and the formation processes.
Set theory (a theory of objects) applied to Leibniz and Meinong Mathematical sets are a collection of "things" (objects or numbers, etc). Each member is called an lement of the set. There should be only one of each member (all members are unique).By “things” here are meant distinct mental or physical objects. Leibniz's metaphysics, as described in his Monadology, originates from the concept of partless substances, such a concept being called a monad. A monad is the mental correspondent of a physical object. In the philosophy of materialism, the world consists completely of physical bodies, which is also true of Leibniz, except that in Leibniz, each physical body also has a mental correspondent called a monad. These monads are also Plato's Many, controlled by Plato's One (Mind). I call this latter version of Leibniz the metaphysics of Plato-Leibniz (See link at the bottom) While monads are partless, they may and usually are, formed into a composite body of monads, so that the monad for entire body is a composite monad. Composite monads may be either or both additive monads within a composite body, or be nested. A flock of sheep would then be an additive collection of individual sheep, but each sheep will also be a nested composite of organs, such as brain, liver, etc., and within each organ will be additive or nested collections of cells. Leibniz's monads are not the purely mathematical monads of Haskell language, which have no physical correspondents. Perhaps a set theoretician could better differentitate between the two, as the Haskell monads do appear to me as if they might be something like nested sets. With some psychological background, unlike Russell, the german philosopher Meinong was aware that there are simply mental objects which do not have physical correspondents, which are selected or intended by Mind in Meinong's theory of objects. Thus we might describe mind as consisting of an intender or thinker (Mind or mind, depending on the level of concern) whose intent or thought is focussed on either of two sets: a) Intendeds - sets of intended or though purely mental objects b) Monads-- sets of mental objects (monads) which have correspondwnces to physical objects Then mind would consist of an encompassing set (mind or Mind) containing two sets, Intendeds and Monads. And as described above, these sets may be additive, and also may be nested. And the breakdown may be ad inifitum, as Leibniz seemed to think. -- Dr. Roger B Clough NIST (retired, 2000). See my Leibniz site: rclough@verizon.academia.edu/RogerClough For personal messages use rclough@verizon.net
Bill !! Im thrilled with your videos. I am embarking on the same journey, I love logic and set theory and abstract mathematics. Though I have a question, seems to me that your definition "sets cannot contain other sets" is wrong, because what about the power set of say a set A: P(A). This is the set of every possible subset of A, and given a subset of a set is a set itself it cannot be true, given the power set exists, that a set cannot contain other sets in it.
I modified this statement in an annotation. I was originally going with something akin to Russell's type theory, in which something that contains sets isn't really a "set" but a "family". Nowadays, though, we do still allow sets to contain other sets, and the issue of Russell's paradox is solved in a better way (with an axiom that basically means sets can't refer to themselves.)
Yeah, I noticed the pointer thing too late ... I'm trying to fix it in all the videos I've been making since I started noticing it. Sorry about that! Glad you enjoy the videos though!
Fantastic videos! Thank you for posting. Please post more. My only crit is the pointer being in the capture, its a little distracting. Still, its cream of the crop. Bravo!
It is just a definition, but it helps to use the most current definition, especially if it's an international standard like ISO 31-11! I do want to get my hands on ISO 80000-2 which is actually newer, but the document costs like $300...
Which "larger sets" are you talking about? And I do talk about algebraic and transcendental numbers in a later lecture, the one about constructing the real numbers. :)
Great video(s)... FINALLY! I GET IT! THANK YOU!! I was most impressed by the methodical and easy-to-understand approach with EXCELLENT EXPLANATIONS of the meaning and use of SHORT-HAND MATH SYMBOLS. If you want to LEARN MATH, or LEARN HOW TO TEACH MATH, then I strongly encourage you to WATCH THESE VIDEOS! GREAT JOB BILL SHILLITO (- If Samuel L. Jackson could type ;-) )
It's pretty close! It's been stylized differently (∈), but it very closely resembles the lunate epsilon (ϵ), since the former evolved from the latter. On Wikipedia's article about the letter Epsilon: The lunate epsilon (ϵ) is not to be confused with the set membership symbol (∈); nor should the Latin uppercase epsilon (Ɛ) be confused with the Greek uppercase sigma (Σ). The symbol ∈ , first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") did, however, evolve from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word "est".
the rational numbers are part of the real numbers? in the number line you may try to accommodate them, approximately, because, some of them extend to infinity. So, to put them in the number line, and if we are to do math rigorously, it would be very difficult.
My question was: "is there a real theory on real numbers?" and the second was: where exactly pi, square root of two, and e, are in the number line." Don't take these questions as an offense, please. I am a amateur mathematician, and was reading this very interesting book; "the loss of certainty" by Morris Kline, it is why I posted those questions, thanks.
Why would maths not be rigorous if we accommodate rational numbers? If m and n are integers then (m+n)/2 is also on the number line. So between 0 and 1 you have 1/2 and so on. To see irrationals on the line, draw a unit square resting on the points 0 and 1, with a diagonal line from 0 cutting the square into two right triangles - then apply Pythagoras to find the longest side/diagonal line. This gives sqrt(2). By simple inspection the length of the longest line corresponds to a point between 1 and 2, and that's where sqrt(2) occupies in the line. There are more advanced and beautiful ways of seeing it, though (Dedekind cuts, Cauchy sequences) but they're more advanced.
Yeah, Russel's paradox is cute, but not a real problem. Putting itself inside a set not only creates paradox, but also an endless recursion (if we take, say, a set that contains all real numbers and a set that contains all real numbers and a set...). I mean, if you think about sets as being boxes with thing inside, you obviously can't put a box inside itself, so this operation should just be permitted.
Recursive types are heavily used in programming. For example, List a = Null | Cons a (List a). So the idea is valueable exactly by the reason it allows to express infinities...
25:37 if set cannot contain other sets, how do we define sequences (tuples) and concepts such as the order? Without putting sets inside of sets, in terms of formal definition, we are limited to first order logic concepts.
oh, I'd like to share my opinion on whether zero is a natural number and if it should be included--i have to say yes. 0 is an important point in a graph. if you take away zero then you take away the intersections, the curves, etc. zero in a graph has value. for instance, in an oscillation of a particle following a fourier series formula, the particle crosses from -x to x. it can never cross to positive x without approaching zero. in reality, that zero only means the point of origin or a point that divides negative and positive values like that of a temperature.
Ahh I see. What I was describing, it turns out, is Russell's "Type Theory". I'm going to have to do a bit more reading to understand exactly how ZF gets around the paradox, starting with that axiom of regularity. Thank you for pointing me in that direction! Never let it be said that I think I know everything.
This video (like your entire lecture series) has been incredible to watch! You really poured your heart into this project, sir. I want to comment on your point at 25:32 that sets cannot contain other sets. Sets CAN contain other sets, they just can’t contain themselves, because it generates paradoxes. Most people don’t see why Russell’s paradox is such a big deal, so this is my take on it: From set builder notation, we agree that sets are defined by their elements, and more specifically the properties of their elements. This seems reasonable. For instance, A = {x | x is all natural numbers less than 5} = {1,2,3,4} . Properties define sets. But in Russell’s paradox, the guy defines a set B = {x | x is not a member of itself}. Of course, no set is a member of itself, we think. Cool. But we need to look at the universe of all sets to see if they satisfy this property. All of them do. Still good. But then we notice that B is also a set, and trouble strikes. B doesn’t contain itself, as one would expect of a set. By that logic, however, the set B is then an element of the set B, because it satisfies the property of the elements of the set. But if B is an element of B, it does not belong to the set with property “x is not a member of itself” so we remove it from B This circular logic generates a paradox, and we can’t escape it for two reasons. 1) It turns out that not all properties are fair game in naive set theory. If you try hard enough, you can generate logical properties that “break the game”. This is a big deal because if you can’t use any old properties to construct a set, what properties can you use? This is where ZFC comes in, with all the axioms and such. The axiom of class construction tells you how to generate classes (sets are a specific type of class, it turns out). This escapes Russell’s Paradox because you can generate a class C = {x | the set x is not a member of itself}. x is a set whereas C is a class, so C is not checked for the required property. Thus, no paradox is generated. 2) If someone gives you a set, any set(!), it turns out that they are also giving you the power set of that set for free. The power set is a set containing all the subsets of the set you were given, and we’re certain that our set must have at least the trivial subsets. This is actually a problem in naive set theory because let’s say someone defines a universal set U with all the elements in your mathematical universe. Someone can always find the power set of your Universal set U...it must exist! But that power set of U is not in U, so logically, there is no “universal set”. I know that’s how they teach it in elementary school, but ZFC recognizes that, if you’re careful about how you construct sets, you’ll find that all sets have power sets so there can be no universal sets. But all is not lost, because via the axiom of class construction, you can create a class of all elements that are equal to themselves I.e. M = {x | x=x}. Let the element be a set and your class contains all sets. It is, in fact, the closest you can get to a “universal set” in ZFC Theory. Also, M is a class while x is an arbitrary set, so M cannot contain M ... potential paradox avoided. The class M of all sets has a fancy name but I forgot what it is. I apologize if anything I wrote doesn’t make sense. I’m not formally educated so I don’t always process information in a scholarly way. Make no mistake, I still love your playlist! It’s 4am and I’m binging it to my hearts content #COVID. Math is fun. Best wishes, and I’ll try not to write such a long essay next time haha 😂
@@ozzyfromspace Makes sense. Since making these videos a long time ago I've done a LOT more study of math, and man I wish I could go back and edit videos. (I *used* to have an annotation on this one when it got to the sets-cannot-contain-other-sets thing, but then CZcams got rid of all my annotations.) At some point I'll find the time to make new ones...
I believe (though I could be wrong on WHO did this) that Russell resolved this by basically creating a hierarchy of sets. My terminology's off since I'm just recalling this from memory, but there were level 0 sets which contain base elements (numbers, etc.), level 1 sets which could contain up to level 0 sets, level 2 sets which could contain up to level 1 sets, etc. I'm interested in how other axiomatic set theories deal with this problem though. Do you have any good resources on the matter?
fc0chelsea Take a look at my annotation; it would just be a set. A "family" really just ends up being a word we sometimes use to describe sets of sets, but you can still think of it as a set as long as you're careful to avoid Russell's paradox.
Can there be a subset of only they have only one element? Ex. A = {1,2,3} B = {3,4,5} Is B a subset of A? If yes or no, could you explain? There's very little on the internet about this. Thanks in advance.
no, because in order for a set A to be a subset of another B, EVERY element in A must also be in B. in your example neither A is a subset of B, nor viceversa, in fact I can find elements of A that are not in B, and vice versa
+Sander Suverkropp Except that the null set can't be a proper subset of the null set. It would be more correct to say that the null set is a proper subset of every *nonempty* set.
Very interesting video on set theory. It just feels like one of those days you know... Also, for anyone interested in the principles of higher math learning in general. I've found this visual guide to be rather insightful. mathvault.ca/10-commandments
Why can't you make the topic of subsets clearer by saying there are two types of subsets: a Proper Subset and an Improper Subset. A Proper Subset is a subset which has at least one element less than the Superset. An Improper Subset is a set that can have the same elements as the Superset or it can have at least one element less than the Superset.
Good point! I've actually never heard the phrase "improper subset" used, but that would be good to have a word to describe what you're talking about. I like it! You just have to make sure that it IS in fact a subset. In fact, if you describe it like that, the only improper subset of a given set A will be A itself.
I had a huge annotation that corrected the part about families of sets, but recentlyish CZcams got rid of all annotations. I need to just go through all my old videos and post errata in the descriptions when I have the time. 😵
Actually, zero is more and more prevailingly counted among the natural numbers by mathematicians and logicians as well as computer scientists. There are a number of reasons for this. For one, ZFC's construction of the natural numbers starts at zero being the empty set. Even Bertrand Russell, possibly the first person to use the term "natural numbers" in English, said himself that zero should be the first natural number. You're up against more than just the computer scientists, I'm afraid!
Right, but if I give you an empty box and tell you to count the balls in it, you'll say "zero". Which is probably why ZFC starts with zero as the empty set.
lets put it this way, probably way too pessimist, why would we do with the decimal system if we exclude zero from natural numbers? You may say, "well we use zero within the rational and real numbers," yeah but, zero is need in the natural numbers, and it is becoming more and more familiar with mathematicians.
Popo Sandybanks I'm pretty sure there are mathematicians who study set theory. In particular, I'm always interested to see how people find ways to extend and generalize sets, such as fuzzy sets and multisets. (I've actually been reading recently about multisets in which an element can have negative membership!)
+Bill Shillito "Fellow mathematicians, you don't seem to understand what I meant when I said that 2 is *definitely* not in set A. So I invented negative membership and developed theory around it. Now I can say that 2 has *-2* membership in A. It is *doubly definitely not* in A!"
I don't quite understand why sets cannot contain sets... it's like saying that a russian doll cannot contain another doll, or that we cannot put a box inside a box. And the answer for who shaves the barber, well, another barber... his mom...
The barber is a woman. Done. Lol, really though........ The barber represents an object that is somehow different from a mere man, oh, I mean set! The barber stands for a class. Sets are a special type of class. ZFC Theory clears this up and resolves the paradox with a more careful class construction.
Thank you for breaking set theory down so an old lady with a GED can understand, Your teaching skills is the BOMB!!!!!
It's very elegantly presented. I was only confused once in the last video, and my brain complicates everything.
There’s a compelling reason to include 0 as a natural number (20:39), namely:
0 = {}
1 = {0}
2 = {0, 1}
3 = {0, 1, 2}
…
w = {0, 1, 2, …}
w + 1 = {0, 1, 2, …, w}
…
Where w represents the ordinal number omega, the set of all natural numbers:
mathworld.wolfram.com/OrdinalNumber.html
(Currently, I haven’t seen your entire series. You may have included this in one of your later videos.)
Is there anyway, we can get the mathematical community to vote for the trollface as the accepted symbol for paradoxes?
Well, that's assuming the unrestricted comprehension principle. If you go with the axiom of choice, you have to "choose" if it is a part of your set or not, you could include sets in sets.
Great lecture. Perfect balance of simplicity and complexity, fun and challenge, insight and foresight. I'm looking forward for more lectures. You give symbols not just its meaning but how it connects to other disciplines. This technique makes the learner not only understand the symbol but how it is connected to other disciplines.
Thank you, sir! I'm learning higher mathematics to help me with software engineering, as well as just gaining general math knowledge. These videos are excellent!
Great lesson: Deep, simple and intuitive!
Metaphysics as set theory
I believe that Leibniz-Plato is not really philosophy, but mathematics. In fact, set theory. Unfortunately I am not a mathematician, and need some help. But here are some initial thoughts.
In Leibniz, the physical object in spacetime is represented as a mental point called a monad. Mind is a formalization or form of mathematics in Plato-Leibniz. From set theory we find that a set is a collection of objects that it owns. This ownership I believe is what we call control, perception or creation. That is, a set controls the objects or sets within it hierarchically.
Using the notation < to mean is conceptually contained in (a sidewise U), M as plato's Mind (the One), T as the domain of time, and S as the domain of space, p as a particle is spacetime, and m(p) as its monad, we have the sets
m(p) < S < T < M
The motion of an object in spacetime begins with its birth as a still point in Plato's Mind. That is. as a point in the all-encompassing set of points, Plato's Mind, which is the point containing all other (mental) points. The mental is the mathematical, the formal in terms of set theory. Time and space are quantized, as is experience, so experience is given to us a set of movie frames that we expeience as a movie.
--
Dr. Roger B Clough NIST (retired, 2000).
See my Leibniz site: rclough@verizon.academia.edu/RogerClough
For personal messages use rclough@verizon.net
i generally do not comment on videos but cant stop myself after watchinng this.. great work this is awesome when i study for 30 minutes in class it feels like days but here i enjoyed this 30 minutes like i enjoy watching my fav programm on tv.great great great work pls upload more
your lectures are fantastic....you are the personification of lucid didactic communication
Yes, multisets are in fact a generalization of sets. You can kind of "construct" a multiset as a set of ordered pairs, in which the first part is some element and the second part is a "multiplicity", meaning how many times the element is included in the multiset. So for example, the multiset {a, b, b, c} can be represented as {(a,1), (b,2), (c,1)}.
Multisets are quite useful in combinatorics. You should also look into fuzzy sets for another generalization!
(I'll PM you my email address.)
just watched this one and I´m hooked. thank you very much.
Nice video as always, but I'd quibble with your decision to disallow sets as members of other sets in every case. Yes, this resolves Russell's paradox, but it goes too far IMO -- e.g. what do you call the power set of S? you can't call it a set?? Also there's the example of certain axiomatic set theories where the *only* possible elements of a set are other sets. (Yes, the word "family" is common in the literature, but it's used for clarity; formally a family is just a set, in most treatments.)
i dont understand at the end of the video you said a set cannont contain other set
but earlier we said A contain D , D is a subset of A ?????
This is why we have to be careful with mathematical terminology! XD
If D is a subset of A, then A contains all elements of D. This would be different from saying that D itself was an ELEMENT of A.
And as I said in one of the annotations, sets CAN contain other sets (a good example being the Power Set of a set), but only as long as we're careful to avoid Russell's Paradox. Zermelo-Fraenkel set theory, which is the kind we use today, has axioms built in that keep this from happening.
thanks for the explication and the quick answer i appreciate it
@@BillShillito In the video you were under the impression that sets containing other sets were the source of the paradox. That's actually not quite the way the paradox is solved - neither in the Russell or the ZFC way.
If anything, the distinction between family and set doesn't help either, because that alone won't solve the paradox in the Russell style (not to mention that a family is in many cases a set too). The key actually lies in the introduction of a higher collection called "class" - and claiming that the "Russell set" is a class that happens to be not a set.
So essentially, a class is what was naively known as a set, but now a set has to be redefined so as to tighten up the rules and the formation processes.
Set theory (a theory of objects) applied to Leibniz and Meinong
Mathematical sets are a collection of "things" (objects or numbers, etc). Each member is called an lement of the set. There should be only one of each member (all members are unique).By “things” here are meant distinct mental or physical objects. Leibniz's metaphysics, as described in his Monadology, originates from the concept of partless substances, such a concept being called a monad. A monad is the mental correspondent of a physical object. In the philosophy of materialism, the world consists completely of physical bodies, which is also true of Leibniz, except that in Leibniz, each physical body
also has a mental correspondent called a monad. These monads are also Plato's Many, controlled by Plato's One (Mind). I call this latter version of Leibniz the metaphysics of Plato-Leibniz (See link at the bottom)
While monads are partless, they may and usually are, formed into a composite body of monads, so that the monad for entire body is a composite monad. Composite monads may be either or both additive monads within a composite body, or be nested. A flock of sheep would then be an additive collection of individual sheep, but each sheep will also be a nested composite of organs, such as brain, liver, etc., and within each organ will be additive or nested collections of cells.
Leibniz's monads are not the purely mathematical monads of Haskell language, which have no physical correspondents. Perhaps a set theoretician could better differentitate between the two, as the Haskell monads do appear to me as if they might be something like nested sets. With some psychological background, unlike Russell, the german philosopher Meinong was aware that there are simply mental objects which do not have physical correspondents, which are selected or intended by Mind in Meinong's theory of objects.
Thus we might describe mind as consisting of an intender or thinker (Mind or mind, depending on the
level of concern) whose intent or thought is focussed on either of two sets:
a) Intendeds - sets of intended or though purely mental objects
b) Monads-- sets of mental objects (monads) which have correspondwnces to physical objects
Then mind would consist of an encompassing set (mind or Mind) containing two sets, Intendeds and Monads. And as described above, these sets may be additive, and also may be nested. And the breakdown may be ad inifitum, as Leibniz seemed to think.
--
Dr. Roger B Clough NIST (retired, 2000).
See my Leibniz site: rclough@verizon.academia.edu/RogerClough
For personal messages use rclough@verizon.net
Bill !! Im thrilled with your videos. I am embarking on the same journey, I love logic and set theory and abstract mathematics. Though I have a question, seems to me that your definition "sets cannot contain other sets" is wrong, because what about the power set of say a set A: P(A). This is the set of every possible subset of A, and given a subset of a set is a set itself it cannot be true, given the power set exists, that a set cannot contain other sets in it.
I modified this statement in an annotation. I was originally going with something akin to Russell's type theory, in which something that contains sets isn't really a "set" but a "family". Nowadays, though, we do still allow sets to contain other sets, and the issue of Russell's paradox is solved in a better way (with an axiom that basically means sets can't refer to themselves.)
Yeah, I noticed the pointer thing too late ... I'm trying to fix it in all the videos I've been making since I started noticing it. Sorry about that! Glad you enjoy the videos though!
thanks so much! you cannot imagine how grateful I am for having such a nice course available :)
Thanks Mr. Shilito, I think I have learned more from watching your videos then I have from listening to my university lecture...
Fantastic videos! Thank you for posting. Please post more. My only crit is the pointer being in the capture, its a little distracting. Still, its cream of the crop. Bravo!
I just wanted to express how much I love your videos.
Bill this is a very great lecture. You are simply a wonderful teacher
Amazing!! So clear and concise! They never explained the inclusive or in algebra, that drove me crazy. Who needs a teacher?
It is just a definition, but it helps to use the most current definition, especially if it's an international standard like ISO 31-11!
I do want to get my hands on ISO 80000-2 which is actually newer, but the document costs like $300...
YOU ARE DIVINE. Your videos are exactly what I needed!! Thank you very much; you explain very well!
For me u r divine
2:27 I lost The Game. :(
Which "larger sets" are you talking about?
And I do talk about algebraic and transcendental numbers in a later lecture, the one about constructing the real numbers. :)
thank you so much Mr. Shilito :)
Damn thanks for these lessons man! Currently going to start post grad studies in chemistry and this is excellent for recapping stuff.
An excellent explanation apart from the book.
Thank you for your lecture, it helps me making my tests at school!
Great video(s)... FINALLY! I GET IT! THANK YOU!! I was most impressed by the methodical and easy-to-understand approach with EXCELLENT EXPLANATIONS of the meaning and use of SHORT-HAND MATH SYMBOLS. If you want to LEARN MATH, or LEARN HOW TO TEACH MATH, then I strongly encourage you to WATCH THESE VIDEOS! GREAT JOB BILL SHILLITO (- If Samuel L. Jackson could type ;-) )
you completed the internet holy shit this was such a good tutorial
The element of comparator is not the greek letter epsilon
It's pretty close! It's been stylized differently (∈), but it very closely resembles the lunate epsilon (ϵ), since the former evolved from the latter.
On Wikipedia's article about the letter Epsilon:
The lunate epsilon (ϵ) is not to be confused with the set membership symbol (∈); nor should the Latin uppercase epsilon (Ɛ) be confused with the Greek uppercase sigma (Σ). The symbol ∈ , first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") did, however, evolve from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word "est".
@@BillShillito Wow! I didn't expect you to reply. I was just being pedantic. I love this series so far. Thanks for making it!
How is a family defined such that it is not a set?
the rational numbers are part of the real numbers? in the number line you may try to accommodate them, approximately, because, some of them extend to infinity. So, to put them in the number line, and if we are to do math rigorously, it would be very difficult.
My question was: "is there a real theory on real numbers?" and the second was: where exactly pi, square root of two, and e, are in the number line." Don't take these questions as an offense, please. I am a amateur mathematician, and was reading this very interesting book; "the loss of certainty" by Morris Kline, it is why I posted those questions, thanks.
Why would maths not be rigorous if we accommodate rational numbers? If m and n are integers then (m+n)/2 is also on the number line. So between 0 and 1 you have 1/2 and so on. To see irrationals on the line, draw a unit square resting on the points 0 and 1, with a diagonal line from 0 cutting the square into two right triangles - then apply Pythagoras to find the longest side/diagonal line. This gives sqrt(2). By simple inspection the length of the longest line corresponds to a point between 1 and 2, and that's where sqrt(2) occupies in the line. There are more advanced and beautiful ways of seeing it, though (Dedekind cuts, Cauchy sequences) but they're more advanced.
@ 23:29: Did he mean to say rational instead of natural numbers? I guess so.
Ah. I see your point. I blame this caffeine and reading a discrete mathematics book for the last 6 hrs
Great video it helped me review the information I need for my data science program.
22:16, in the Spanish language, the whole numbers are "los numeros enteros."
"Los número s enteros" are integers, not whole.
Yeah, Russel's paradox is cute, but not a real problem. Putting itself inside a set not only creates paradox, but also an endless recursion (if we take, say, a set that contains all real numbers and a set that contains all real numbers and a set...). I mean, if you think about sets as being boxes with thing inside, you obviously can't put a box inside itself, so this operation should just be permitted.
Recursive types are heavily used in programming. For example, List a = Null | Cons a (List a). So the idea is valueable exactly by the reason it allows to express infinities...
Haha, swiss cheese.. These videos are amazing though, much clearer and well made then any other Math tutorial videos I have ever seen!
Good video Bill
25:37 if set cannot contain other sets, how do we define sequences (tuples) and concepts such as the order? Without putting sets inside of sets, in terms of formal definition, we are limited to first order logic concepts.
I've since revised the whole "sets cannot contain other sets" deal. There should be an annotation saying as much.
oh, I'd like to share my opinion on whether zero is a natural number and if it should be included--i have to say yes. 0 is an important point in a graph. if you take away zero then you take away the intersections, the curves, etc. zero in a graph has value. for instance, in an oscillation of a particle following a fourier series formula, the particle crosses from -x to x. it can never cross to positive x without approaching zero. in reality, that zero only means the point of origin or a point that divides negative and positive values like that of a temperature.
Ahh I see. What I was describing, it turns out, is Russell's "Type Theory". I'm going to have to do a bit more reading to understand exactly how ZF gets around the paradox, starting with that axiom of regularity. Thank you for pointing me in that direction!
Never let it be said that I think I know everything.
This video (like your entire lecture series) has been incredible to watch! You really poured your heart into this project, sir.
I want to comment on your point at 25:32 that sets cannot contain other sets. Sets CAN contain other sets, they just can’t contain themselves, because it generates paradoxes. Most people don’t see why Russell’s paradox is such a big deal, so this is my take on it:
From set builder notation, we agree that sets are defined by their elements, and more specifically the properties of their elements. This seems reasonable. For instance, A = {x | x is all natural numbers less than 5} = {1,2,3,4} . Properties define sets. But in Russell’s paradox, the guy defines a set B = {x | x is not a member of itself}. Of course, no set is a member of itself, we think. Cool. But we need to look at the universe of all sets to see if they satisfy this property. All of them do. Still good. But then we notice that B is also a set, and trouble strikes. B doesn’t contain itself, as one would expect of a set. By that logic, however, the set B is then an element of the set B, because it satisfies the property of the elements of the set. But if B is an element of B, it does not belong to the set with property “x is not a member of itself” so we remove it from B This circular logic generates a paradox, and we can’t escape it for two reasons.
1) It turns out that not all properties are fair game in naive set theory. If you try hard enough, you can generate logical properties that “break the game”. This is a big deal because if you can’t use any old properties to construct a set, what properties can you use? This is where ZFC comes in, with all the axioms and such. The axiom of class construction tells you how to generate classes (sets are a specific type of class, it turns out). This escapes Russell’s Paradox because you can generate a class C = {x | the set x is not a member of itself}. x is a set whereas C is a class, so C is not checked for the required property. Thus, no paradox is generated.
2) If someone gives you a set, any set(!), it turns out that they are also giving you the power set of that set for free. The power set is a set containing all the subsets of the set you were given, and we’re certain that our set must have at least the trivial subsets. This is actually a problem in naive set theory because let’s say someone defines a universal set U with all the elements in your mathematical universe. Someone can always find the power set of your Universal set U...it must exist! But that power set of U is not in U, so logically, there is no “universal set”. I know that’s how they teach it in elementary school, but ZFC recognizes that, if you’re careful about how you construct sets, you’ll find that all sets have power sets so there can be no universal sets. But all is not lost, because via the axiom of class construction, you can create a class of all elements that are equal to themselves I.e. M = {x | x=x}. Let the element be a set and your class contains all sets. It is, in fact, the closest you can get to a “universal set” in ZFC Theory. Also, M is a class while x is an arbitrary set, so M cannot contain M ... potential paradox avoided. The class M of all sets has a fancy name but I forgot what it is.
I apologize if anything I wrote doesn’t make sense. I’m not formally educated so I don’t always process information in a scholarly way.
Make no mistake, I still love your playlist! It’s 4am and I’m binging it to my hearts content #COVID. Math is fun.
Best wishes, and I’ll try not to write such a long essay next time haha 😂
@@ozzyfromspace Makes sense. Since making these videos a long time ago I've done a LOT more study of math, and man I wish I could go back and edit videos. (I *used* to have an annotation on this one when it got to the sets-cannot-contain-other-sets thing, but then CZcams got rid of all my annotations.) At some point I'll find the time to make new ones...
your work is greatly appreciated. thank you!
Am I stupid for not really grasping the concept of the Barber?
haha Thanks for the video. Definitely made me less anxious about my finals
I believe (though I could be wrong on WHO did this) that Russell resolved this by basically creating a hierarchy of sets. My terminology's off since I'm just recalling this from memory, but there were level 0 sets which contain base elements (numbers, etc.), level 1 sets which could contain up to level 0 sets, level 2 sets which could contain up to level 1 sets, etc. I'm interested in how other axiomatic set theories deal with this problem though. Do you have any good resources on the matter?
Nice video on proofs
brilliant lecture Bill. thank you so much!
just wanna say many thanks for these videos :) certainly easier than using wikipedia >_>
Sir plz make more videos like this
So the power set of integers, for example, is a family instead of a set? It sounds weird to me.
fc0chelsea Take a look at my annotation; it would just be a set. A "family" really just ends up being a word we sometimes use to describe sets of sets, but you can still think of it as a set as long as you're careful to avoid Russell's paradox.
Bill Shillito Thank you for the answer.
how can I get the slides?
hey man this is a great series so clearly explained thank you!
Can there be a subset of only they have only one element?
Ex.
A = {1,2,3}
B = {3,4,5}
Is B a subset of A? If yes or no, could you explain? There's very little on the internet about this. Thanks in advance.
no, because in order for a set A to be a subset of another B, EVERY element in A must also be in B. in your example neither A is a subset of B, nor viceversa, in fact I can find elements of A that are not in B, and vice versa
+Sander Suverkropp Except that the null set can't be a proper subset of the null set. It would be more correct to say that the null set is a proper subset of every *nonempty* set.
Could someone please tell me what the double struck O set is called at 24:54 next to the quaternions?
I just found out: they are the Octonions.
Great job! Very thorough!
if a set can't have sets as elements than there is no set of real numbers, according with Dedenkind's cut definition of a real number
+Alkis05 That is correct. This is why I annotated the video to say that I gave an incorrect explanation. Please look at the annotation.
+Bill Shillito sorry I was watching on mobile
Thank you so very much for these videos!!!!
Very interesting video on set theory. It just feels like one of those days you know...
Also, for anyone interested in the principles of higher math learning in general. I've found this visual guide to be rather insightful. mathvault.ca/10-commandments
the link doesn't work
Thank you very much, this was really helpful
What about families of families?
forbidden because of the definition of family
Brother .. you are awesome :)
Thank you!!!!!!!
if there were no restriction on sets, then sets will plagued with infinite regression.
Integer ( Latin ) - "Whole or Complete."
*Entire.
*Integrate.
*Integrity.
*Holy.
Why can't you make the topic of subsets clearer by saying there are two types of subsets: a Proper Subset and an Improper Subset. A Proper Subset is a subset which has at least one element less than the Superset. An Improper Subset is a set that can have the same elements as the Superset or it can have at least one element less than the Superset.
Good point! I've actually never heard the phrase "improper subset" used, but that would be good to have a word to describe what you're talking about. I like it! You just have to make sure that it IS in fact a subset. In fact, if you describe it like that, the only improper subset of a given set A will be A itself.
Bill Shillito You can Google "Improper Subset" to find that the term is used by others.
There's symmetry and beauty in the world of set theory with the Improper Subset-Proper Subset categories.
The last part is wrong.
I had a huge annotation that corrected the part about families of sets, but recentlyish CZcams got rid of all annotations. I need to just go through all my old videos and post errata in the descriptions when I have the time. 😵
@@BillShillito This videos are pretty fun to watch. I will tell you if I found something wrong.
Excuseme for my english.
Actually, zero is more and more prevailingly counted among the natural numbers by mathematicians and logicians as well as computer scientists. There are a number of reasons for this. For one, ZFC's construction of the natural numbers starts at zero being the empty set. Even Bertrand Russell, possibly the first person to use the term "natural numbers" in English, said himself that zero should be the first natural number.
You're up against more than just the computer scientists, I'm afraid!
Right, but if I give you an empty box and tell you to count the balls in it, you'll say "zero".
Which is probably why ZFC starts with zero as the empty set.
Couldn't you argue that the null set contains itself
Then it would no longer be the null set as it would then have a cardinality of 1, the null set has a cardinality of 0 i.e. it contains nothing.
lets put it this way, probably way too pessimist, why would we do with the decimal system if we exclude zero from natural numbers? You may say, "well we use zero within the rational and real numbers," yeah but, zero is need in the natural numbers, and it is becoming more and more familiar with mathematicians.
If you're having trouble grasping the Barber, imagine this ... what would happen if Pinocchio said "My nose will grow now"?
Are there mathematicians who study Set Theory, or is it only used as a tool?
Popo Sandybanks I'm pretty sure there are mathematicians who study set theory. In particular, I'm always interested to see how people find ways to extend and generalize sets, such as fuzzy sets and multisets. (I've actually been reading recently about multisets in which an element can have negative membership!)
+Bill Shillito "Fellow mathematicians, you don't seem to understand what I meant when I said that 2 is *definitely* not in set A. So I invented negative membership and developed theory around it. Now I can say that 2 has *-2* membership in A. It is *doubly definitely not* in A!"
the barber is a woman :3
No problem! Wikipedia is nice ... if you can figure out what you need. :P
I don't quite understand why sets cannot contain sets... it's like saying that a russian doll cannot contain another doll, or that we cannot put a box inside a box.
And the answer for who shaves the barber, well, another barber... his mom...
Please read the note in the description. :)
The barber could be a woman
The barber is a woman. Done.
Lol, really though........
The barber represents an object that is somehow different from a mere man, oh, I mean set! The barber stands for a class. Sets are a special type of class. ZFC Theory clears this up and resolves the paradox with a more careful class construction.
that god damn mouse pointer pisses me off
aha yes
The barber can't have a beard?
squished cheese