This low-quality video proves how great the teacher is. At first, I could not see anything on the blackboard, but after listening to his words, I feel like I saw every single word we wrote. Very clear and well-worded lecture.
In literally every one of my high school math classes, my teacher has shown the proof that the reals are not countable, just for fun. It's like a go-to trick for a magician, but for a mathematician.
Ok one that’s awesome what kind of math classes did you have none of mine ever did that. And yeah diagonalization is a pretty useful technique that shows up frequently in math
Correction at 36:35, it should say "If Jn ~ N then Jn *minus* 1 ~ N". He corrects this at 43:00. Thanks Jenny! Also, there is some disagreement over the definition of 'countable'. Countability is generally taken to mean that one can find an injective function f: A -> N, whereas the textbook's author Rudin requires that it is a bijective function. Despite his esteem, Rudin seems to be in the minority here. Just be aware if you're reading other texts.
At 52:18, Prof. Su says that real numbers bounded below dont have an infimum, that is incorrect. Just wanted to ensure that this is clarified. Also it does not apply to the problem at hand, because those are natural numbers not real numbers. @HarveyMuddCollegeEDU
This is the most difficult lecture so far. I had to play it twice as there is a lot of counterintuitive material in here. Having written that, it is still a superb lecture.
Imagine you have many worlds. One of them is Number World, and the rest are many other worlds. When you want to count objects in another world, which means you assign numbers of Number World to objects.
“We're mapping the the items the people in the first row on to the natural numbers.” Translate physical objects into mathematical objects. Or using anther language to describe physical things.
Proof that Q+ is countable : f: Q+ -->N : m/n -->(p^m)*(q^n) With p,q 2 prime numbers. f is injective, and its domain is an infinite subset of N, so that latter is countable, and so is Q+
you are right thanks. suppose we have Definition: No=not Yes then Axiom opposes law of excluded middle (either P or not P, but not both). notice that this law (which is an axiom itself) is used for proofs of contradiction (in this case, Yes is false because [Yes and not Yes] is false, only by assuming the law of excluded middle). So we cannot use the law of the excluded middle because it would imply that [Law and Axiom] is true, which is false by the same law. so the proof is not valid.
In the example where he shows that the natural numbers and the even natural numbers have the same cardinality, something interesting happens. Or at least I feel that it is very interesting. Some student comments on his example and basically states that: "Well... In a sense there are more natural numbers than even natural numbers because for every even number there is a corresponding odd number which is not contained in the set of all even natural numbers." After this comment it feels as if the professor does not really answer to this comment, merely stating that "by the definition of one-to-one correspondence, these sets must have the same cardinality."He also mentions that you could do a two-to-one mapping and so on (maybe it is here that I am getting confused). Now... My point is that it feels as if the definition of how two finite sets have the same cardinality might be intuitive and hold true, but the definition does not quite extend to infinite sets because we have this apparent discrepancy; that from one perspective (definition) two infinite sets might not have the same cardinality and from another perspective it might have. Its a real struggle for me to accept this notion of two infinite sets having the same size so I would really appreciate if someone could explain why I am wrong in feeling that there is some kind of condradiction lurking behind the scenes here. Hehe.
if you think about the notion of infinite sets growing at the same rate you can accept how the size is the same. for every 1,2,3,4.... there is another 2,4,6,8.... very informal but perhaps helpful
another way of thinking about this is considering a one ot one correspondance. If f(n) = 2n you may list each set at once and understand the cardinality.
The unintuitive notion is the result of the "existence of infinite sets" axiom of ZFC. This gets even weirder when you realize that Q is dense in R (meaning there's a rational number between any two different reals) yet there are somehow more reals.
@@HabibuMukhandi between every two even numbers is an odd number, and vice versa so it makes intuitive sense the cardinality of the two sets are equal. Even though this is also true for rational and reals, the cardinality of the two aren’t the same. Please tell me how this isn’t weird.
yeah he said it failed too. it 'was a mishap. easy on the drama. the rest of the lecture was crystal clear. you seem very eager and energetic though, so please, go ahead and upload an alternative proof.
This low-quality video proves how great the teacher is. At first, I could not see anything on the blackboard, but after listening to his words, I feel like I saw every single word we wrote. Very clear and well-worded lecture.
In literally every one of my high school math classes, my teacher has shown the proof that the reals are not countable, just for fun. It's like a go-to trick for a magician, but for a mathematician.
Ok one that’s awesome what kind of math classes did you have none of mine ever did that. And yeah diagonalization is a pretty useful technique that shows up frequently in math
"fish, fish fish... fish fish fish." LOVE THIS GUY!
if all professors were like him ... i would never miss a class ... i always think that i should not be studying maths in my uni ..
Correction at 36:35, it should say "If Jn ~ N then Jn *minus* 1 ~ N". He corrects this at 43:00. Thanks Jenny!
Also, there is some disagreement over the definition of 'countable'. Countability is generally taken to mean that one can find an injective function f: A -> N, whereas the textbook's author Rudin requires that it is a bijective function. Despite his esteem, Rudin seems to be in the minority here. Just be aware if you're reading other texts.
Ig rudin didn’t want to consider finite sets
I think the function
Goes from N to the set so it has to be bijective … I think that’s how rudin has defined it
At 52:18, Prof. Su says that real numbers bounded below dont have an infimum, that is incorrect. Just wanted to ensure that this is clarified. Also it does not apply to the problem at hand, because those are natural numbers not real numbers. @HarveyMuddCollegeEDU
This is the most difficult lecture so far. I had to play it twice as there is a lot of counterintuitive material in here. Having written that, it is still a superb lecture.
Proof.
Axiom:
Yes=Yes and No
Suppose Yes. Then we have a contradiction.
Therefore No
Best teacher for real ananlysis
Imagine you have many worlds. One of them is Number World, and the rest are many other worlds. When you want to count objects in another world, which means you assign numbers of Number World to objects.
“We're mapping the the items the people in the first row on to the natural numbers.” Translate physical objects into mathematical objects. Or using anther language to describe physical things.
An incredible teacher.
Proof that Q+ is countable :
f: Q+ -->N
: m/n -->(p^m)*(q^n)
With p,q 2 prime numbers.
f is injective, and its domain is an infinite subset of N, so that latter is countable, and so is Q+
One rational number can be mapped to infinitely many N: 1/2 == 2/4, but p^1*q^2 != p^2*q^4
@Andrey : what if we make gcd(m,n)=1 ? will work I think
I wish the videos were at least 360p. Nonetheless, it is still acceptable. Great work by Professor Francis.
The times when 240p and 360p were distinct.
@@onattanriover true
Cool always wondered what the Continuum Hypothesis was about!
Thanks for the tremendous work
How does he choose what to underline when writing on the board?
you are right thanks. suppose we have
Definition:
No=not Yes
then Axiom opposes law of excluded middle (either P or not P, but not both). notice that this law (which is an axiom itself) is used for proofs of contradiction (in this case, Yes is false because [Yes and not Yes] is false, only by assuming the law of excluded middle).
So we cannot use the law of the excluded middle because it would imply that [Law and Axiom] is true, which is false by the same law. so the proof is not valid.
improve the video print...please.
In the example where he shows that the natural numbers and the even natural numbers have the same cardinality, something interesting happens. Or at least I feel that it is very interesting. Some student comments on his example and basically states that: "Well... In a sense there are more natural numbers than even natural numbers because for every even number there is a corresponding odd number which is not contained in the set of all even natural numbers." After this comment it feels as if the professor does not really answer to this comment, merely stating that "by the definition of one-to-one correspondence, these sets must have the same cardinality."He also mentions that you could do a two-to-one mapping and so on (maybe it is here that I am getting confused).
Now... My point is that it feels as if the definition of how two finite sets have the same cardinality might be intuitive and hold true, but the definition does not quite extend to infinite sets because we have this apparent discrepancy; that from one perspective (definition) two infinite sets might not have the same cardinality and from another perspective it might have.
Its a real struggle for me to accept this notion of two infinite sets having the same size so I would really appreciate if someone could explain why I am wrong in feeling that there is some kind of condradiction lurking behind the scenes here. Hehe.
if you think about the notion of infinite sets growing at the same rate you can accept how the size is the same. for every 1,2,3,4.... there is another 2,4,6,8.... very informal but perhaps helpful
another way of thinking about this is considering a one ot one correspondance. If f(n) = 2n you may list each set at once and understand the cardinality.
The unintuitive notion is the result of the "existence of infinite sets" axiom of ZFC. This gets even weirder when you realize that Q is dense in R (meaning there's a rational number between any two different reals) yet there are somehow more reals.
@@EastBurningRed Not weird, Q is a proper subset of R (reals). What do you think reals are?
@@HabibuMukhandi between every two even numbers is an odd number, and vice versa so it makes intuitive sense the cardinality of the two sets are equal. Even though this is also true for rational and reals, the cardinality of the two aren’t the same. Please tell me how this isn’t weird.
Awesome lecture
yeah he said it failed too. it 'was a mishap. easy on the drama. the rest of the lecture was crystal clear. you seem very eager and energetic though, so please, go ahead and upload an alternative proof.
There are no study guides or handouts at the link given above, and the video quality doesn't help either.
@bluestarrfall I know!! me too..
32:28
Real analysis will be the death of me
Are you still alive?
I taking this course LoL,
Are you happy with that 😏?
this guy is awesome :D
Thanks prof.
p,q 2 different prime numbers.
''okay?''
lol how many times he uses this word :P
240p? Impossible to lock at!
Happy with that?
r u with me？
Nice
happy with that ! :D
Theorem: NO. Too funny
Very bad cameraman 👀,but nice teacher 🌚
@Dartme18 with an Ouija board
nice
No because the same applies to Q but Q is countable...
@licensetokill29
wtf