Excellent video Mr.Woo!I always hated the idea of Infinity and thought that it was a wrong idea,because it has many contradictions,but now I'm beginning to understand that it is just that we have to get used to the idea and understand the nature of 'endless',and what it means for infinities to be of different kinds and sizes and that the general notions and literal meanings are the enemies of this idea.And then how intuitive some of the things start to seem when the mind is opened up to new facets of thinking and reasoning and leaves its rigid,conservative space and truly develops.I hope,with time I will understand even more.You are great Mr.Woo!My respect!
Great video! My only problem would be that you kind of sweeped negative numbers under the rug, but I guess it's not hard to expand this for negative numbers. You only really have to change the sign of the numerator, and start the diagonal on the left. You also avoid the denominator = 0 problem.
Sir first of all my English is very weak, Thank you, sir for providing this world huge beneficial knowledge. Dear sir, I wanna say you that we have not basic ideas of sets (like integers, real, rational and irrational etc) please guide us for the basic tool of mathematics like sets subsets etc. please please please please
Mr.woo I can agree with the diagonal method of matching to an extent and although, I typically do agree that the conclusions you come to are usually correct and likely well documented studies within professional mathematical literature, I can not agree to the matching of the natural numbers with the even number example that you showed. This is due to the fact that you matched up all the numbers in an orderly and countable way (1-2, 2-4, 3-6, 4-8....) I believe this proof of natural set of numbers = even set of numbers (or any other set for that matter) only works for depending on where you BEGIN to match the set's and ongoing. Sure you may BEGIN to match all the numbers as it's given in order however if you began to match like 1 2 (connect 2 with 2) 2 4 (connect 4 with 4) 3 6 (connect 6 with 6) 4 8 etc 5 10 etc . . etc . . . . . . connecting them diagonally will still be able to successfully match each even number with one of the natural numbers and STILL have unmatched natural numbers left. I am aware that a counter argument could be "well this is not real matching since the lines within the set must be matched in order" or "that does not undermine the fact that starting at the beginning will still allow you to match up each value". However those justifications for objecting what I have presented just seams to be illusionary and a bit dishonest form of reasoning as to why one argument should still have credence over another. There simply needs to be more careful consideration and logical consistency as to why this proof actually holds rather than some other proof that can utilize a similar method (different in quality of order) and yet breed different results (I apologize if I have straw manned or not brought up an actual counter argument one may have, these were simply the first that came to mind).
Two sets have the same number of elements if there exists AT LEAST ONE one-to-one mapping of all the elements of the first into the second. The fact that there may be additional mappings that don't do this doesn't prove there are different numbers of elements in the two sets. Consider this example. Let A and B both be the set of natural numbers. Both A and B have the same number of elements (because they are essentially the same set). Take a mapping of n --> n - 5. This would not assign the numbers 1, 2, 3, 4, or 5 from A to any number in B. They would be "extra" elements is A. But if they have the same number of elements, how can set A have "extra" elements, if they are the exact same set. So, simply showing that there exists a mapping that doesn't map all the elements doesn't indicate that the two sets don't have the same number of elements. It's not that your argument has "less precedence" than the other, it's just that it doesn't necessarily follow that the sets "don't match". It just means that particular mapping doesn't match up all the elements. Once you have a valid mapping that DOES map all element from one set to all the element of the other set, even if it is the only one that does so, you have demonstrated that the two sets must have the same number of elements. Presenting additional mappings that don't doesn't counter the mapping that does. I hope this make sense.
@@sir_elfman I think you have missed one more property of two sets that have the same size. Not only does there need to be a one-to-one mapping of all the elements of the first set into the second, but the mapping has to be bijective.
@@HabibuMukhandi Yes, that's true. I did forget that because most of my work is in algebra and calculus, so I normally only work with functions and typically, when limiting to functions, a 1 - 1 function implies a bijection, as the range only has numbers that can be generated by the function. I don't usually think about it in terms of two "independent" sets. When that is the case (as in usual set theory), you do need the bijection for equal set size.
You're right that there are repeats on the diagonal. We just skip any fractions that can be reduced. For example 6/4 is skipped because it can be expressed as 3/2.
The notion countability has been disproved. If all positive fractions can be enumerated, then the natural numbers of the first column of the matrix 1/1, 1/2, 1/3, 1/4, ... 2/1, 2/2, 2/3, 2/4, ... 3/1, 3/2, 3/3, 3/4, ... 4/1, 4/2, 4/3, 4/4, ... 5/1, 5/2, 5/3, 5/4, ... ... can be used to index all fractions (including those of the first column). In short, there is a permutation such that the X's of the first column XOOOO... XOOOO... XOOOO... XOOOO... XOOOO... ... after exchanging them with the O's cover all matrix positions. But this is obviously impossible.
Are you saying Countability disproved, or only the countability of one of these two sets (integers and fractions)? Remember, you need only to prove ONE such mapping exists to prove countability. The fact that you can find a mapping to doesn't fit the bill doesn't mean that the sets aren't countable.
While going diagonally, you should not recount the fractions. For example, 1/1 is the same as 2/2, hence it should be skipped. But there would be still a one to one correspondence between rational numbers and integers. Also, you could have proved by including negative fractions also.
The reasoning is flawed and implies that infinites all are the same size. Measure theory handles this type of problem more appropriately and arrives at the intuitive correct answer that there are more unique rational numbers (between any two values, including negative or positive countable infinity) than there are a proper subset such as natural numbers or its proper subsets smaller even or odd.
How is the reasoning "flawed"? It tells us that the set of rational numbers are countable and thus the same size as the set of natural numbers. It does this because it shows a method of arranging all possible rational numbers such that any one we can name would be reached eventually by going through them in order one by one. There is no such ordering that can be found for the full set of real numbers and therefore it cannot be used to argue that they are countable and the same size as the other two.
By counting forever! Put it this way, can you give a natural number that we will not eventually reach? Look into Cantor's diagonal argument for why the rational numbers can not be counted for an example of an uncountable infinity.
Errr no. Clearly the diagonal argument is wrong. There are n integers and nxn fractions. Hence for any n integers there are at least n rationals. This holds for all n including n=infinity. Hence there are infinitely more rationals than positive integers. Ditto primes and integers.
schontasm no,I don't think it applies for n=infinity.Who told you it does?Besides,you just can't go about stating that any argument is wrong without having some solid proof against it that attacks it on its foundation.You think that you contradicted the argument,but the problem is that you are never running out of integers either.Matching up is a very valid technique,and the argument you wanna provide works for real numbers,because their density is probably undefined (idk what to call it) and you have CDA right?I hope I am right and you understand why Mr.Woo was right too,because he explicitly stated in his video that why your argument won't work.
No, the diagonal argument means that you'll eventually reach the fraction you want to if you keep counting, just like with integers. It's still a countable infinity and all countable infinities are the same size.
Infinity is not a number, it's a concept. You can't just multiply infinity times a number and say it's larger or smaller. You can't multiply infinity times infinity and say it's bigger.
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Excellent video Mr.Woo!I always hated the idea of Infinity and thought that it was a wrong idea,because it has many contradictions,but now I'm beginning to understand that it is just that we have to get used to the idea and understand the nature of 'endless',and what it means for infinities to be of different kinds and sizes and that the general notions and literal meanings are the enemies of this idea.And then how intuitive some of the things start to seem when the mind is opened up to new facets of thinking and reasoning and leaves its rigid,conservative space and truly develops.I hope,with time I will understand even more.You are great Mr.Woo!My respect!
your energy is brilliant. first time i've subscribed to anything in ages.
i am lucky to have such a brilliant teacher in the earth..thanks
Omg I was on the verge of crying trying to understand this until I'd finally opened your video. Thanks a lot for uploading!
Great video! My only problem would be that you kind of sweeped negative numbers under the rug, but I guess it's not hard to expand this for negative numbers. You only really have to change the sign of the numerator, and start the diagonal on the left. You also avoid the denominator = 0 problem.
Sir first of all my English is very weak,
Thank you, sir for providing this world huge beneficial knowledge.
Dear sir, I wanna say you that we have not basic ideas of sets (like integers, real, rational and irrational etc) please guide us for the basic tool of mathematics like sets subsets etc.
please please please please
Sir you are an awesome teacher!
Mr.woo I can agree with the diagonal method of matching to an extent and although, I typically do agree that the conclusions you come to are usually correct and likely well documented studies within professional mathematical literature, I can not agree to the matching of the natural numbers with the even number example that you showed. This is due to the fact that you matched up all the numbers in an orderly and countable way (1-2, 2-4, 3-6, 4-8....) I believe this proof of natural set of numbers = even set of numbers (or any other set for that matter) only works for depending on where you BEGIN to match the set's and ongoing. Sure you may BEGIN to match all the numbers as it's given in order however if you began to match like
1 2 (connect 2 with 2)
2 4 (connect 4 with 4)
3 6 (connect 6 with 6)
4 8 etc
5 10 etc
. . etc
. . .
. . .
connecting them diagonally will still be able to successfully match each even number with one of the natural numbers and STILL have unmatched natural numbers left. I am aware that a counter argument could be "well this is not real matching since the lines within the set must be matched in order" or "that does not undermine the fact that starting at the beginning will still allow you to match up each value". However those justifications for objecting what I have presented just seams to be illusionary and a bit dishonest form of reasoning as to why one argument should still have credence over another. There simply needs to be more careful consideration and logical consistency as to why this proof actually holds rather than some other proof that can utilize a similar method (different in quality of order) and yet breed different results (I apologize if I have straw manned or not brought up an actual counter argument one may have, these were simply the first that came to mind).
Two sets have the same number of elements if there exists AT LEAST ONE one-to-one mapping of all the elements of the first into the second. The fact that there may be additional mappings that don't do this doesn't prove there are different numbers of elements in the two sets.
Consider this example. Let A and B both be the set of natural numbers. Both A and B have the same number of elements (because they are essentially the same set). Take a mapping of n --> n - 5. This would not assign the numbers 1, 2, 3, 4, or 5 from A to any number in B. They would be "extra" elements is A. But if they have the same number of elements, how can set A have "extra" elements, if they are the exact same set.
So, simply showing that there exists a mapping that doesn't map all the elements doesn't indicate that the two sets don't have the same number of elements. It's not that your argument has "less precedence" than the other, it's just that it doesn't necessarily follow that the sets "don't match". It just means that particular mapping doesn't match up all the elements. Once you have a valid mapping that DOES map all element from one set to all the element of the other set, even if it is the only one that does so, you have demonstrated that the two sets must have the same number of elements. Presenting additional mappings that don't doesn't counter the mapping that does.
I hope this make sense.
@@sir_elfman I think you have missed one more property of two sets that have the same size. Not only does there need to be a one-to-one mapping of all the elements of the first set into the second, but the mapping has to be bijective.
@@HabibuMukhandi Yes, that's true. I did forget that because most of my work is in algebra and calculus, so I normally only work with functions and typically, when limiting to functions, a 1 - 1 function implies a bijection, as the range only has numbers that can be generated by the function. I don't usually think about it in terms of two "independent" sets. When that is the case (as in usual set theory), you do need the bijection for equal set size.
i have learnt a lot from this lecture
for you many many thanks
If there are repeats in the rationals diagonal, then doesn't that mean they don't pair up 1 to 1?
no because you imagine you would just construct a set that does not contain the repeats and it would have one-to-one correspondence still
You're right that there are repeats on the diagonal. We just skip any fractions that can be reduced. For example 6/4 is skipped because it can be expressed as 3/2.
The notion countability has been disproved.
If all positive fractions can be enumerated, then the natural numbers of the first column of the matrix
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
...
can be used to index all fractions (including those of the first column). In short, there is a permutation such that the X's of the first column
XOOOO...
XOOOO...
XOOOO...
XOOOO...
XOOOO...
...
after exchanging them with the O's cover all matrix positions. But this is obviously impossible.
Are you saying Countability disproved, or only the countability of one of these two sets (integers and fractions)? Remember, you need only to prove ONE such mapping exists to prove countability. The fact that you can find a mapping to doesn't fit the bill doesn't mean that the sets aren't countable.
Love the cheeky Pokemon reference.
What reference?
Why is there a whiteboard atop another whiteboard?
While going diagonally, you should not recount the fractions. For example, 1/1 is the same as 2/2, hence it should be skipped. But there would be still a one to one correspondence between rational numbers and integers.
Also, you could have proved by including negative fractions also.
The reasoning is flawed and implies that infinites all are the same size. Measure theory handles this type of problem more appropriately and arrives at the intuitive correct answer that there are more unique rational numbers (between any two values, including negative or positive countable infinity) than there are a proper subset such as natural numbers or its proper subsets smaller even or odd.
How is the reasoning "flawed"? It tells us that the set of rational numbers are countable and thus the same size as the set of natural numbers. It does this because it shows a method of arranging all possible rational numbers such that any one we can name would be reached eventually by going through them in order one by one. There is no such ordering that can be found for the full set of real numbers and therefore it cannot be used to argue that they are countable and the same size as the other two.
Sir I am preparing for kset, may I get some guidelines from you please please
But why is the argument that set of rational > set of integers on the number line? There is obviously another set of infinities between integers?
If something goes upto infinity, then how could we count them??
Just like the very basic natural numbers.... How they are countable????
By counting forever! Put it this way, can you give a natural number that we will not eventually reach?
Look into Cantor's diagonal argument for why the rational numbers can not be counted for an example of an uncountable infinity.
Maybe "listable" is a better term (credit to Numberphile)
Another way to think of the term countable is that every item on the list is a countable number of spots away from the start.
Errr no.
Clearly the diagonal argument is wrong. There are n integers and nxn fractions. Hence for any n integers there are at least n rationals. This holds for all n including n=infinity.
Hence there are infinitely more rationals than positive integers.
Ditto primes and integers.
schontasm no,I don't think it applies for n=infinity.Who told you it does?Besides,you just can't go about stating that any argument is wrong without having some solid proof against it that attacks it on its foundation.You think that you contradicted the argument,but the problem is that you are never running out of integers either.Matching up is a very valid technique,and the argument you wanna provide works for real numbers,because their density is probably undefined (idk what to call it) and you have CDA right?I hope I am right and you understand why Mr.Woo was right too,because he explicitly stated in his video that why your argument won't work.
@@lifeofphyraprun7601 "Matching up is a very valid techhnique."
It is not the only test.
No, the diagonal argument means that you'll eventually reach the fraction you want to if you keep counting, just like with integers. It's still a countable infinity and all countable infinities are the same size.
Infinity is not a number, it's a concept. You can't just multiply infinity times a number and say it's larger or smaller. You can't multiply infinity times infinity and say it's bigger.