you are wrong about no matchings having more "credibility" than others. We want a BIJECTION between them which means the mapping is INJECTIVE!!! NOT a 2-to-1 mapping. You can't map 0.5 to 0.5 and to 1.5 that is not a well-defined mapping!!!
@Rokker815 Since I can map the same two sets in an injective way AND a 2-to-1 way and in either case the onto criteria is fulfilled suggests to me that you can no longer say that two infinite sets are the same size because you can match the sets element by element (using a bijection). Because I can use the same argument to say I can match the sets two elements to one element to show one of the sets is twice as big. Just because a bijection exists doesn't mean it's the only thing I can use.
@Rokker815 Perhaps, but I have a way to show that (0,2] has exactly twice as many elements as (0,1], which would imply that it's twice as big. A bijection just shows they have the same number of elements. I can show a bijection too as well as showing one of the sets has two times as many elements. Both use very similar logic. Whether I'm familiar with well-definied mapping or not, it makes no difference.
@theboombody a well-defined map means that an element of the domain can have at most one image in the range. This does not imply injectivity or surjectivity. For instance the map f: R -> R given by square rooting. The element "4" say, has two images, -2 and +2. So this is not a well-defined mapping. In this instance we restrict the range to just the non-negative reals so we have a well-defined mapping. The problem you have is your map is not well-defined. [cont]
@Rokker815 Your right, my map is definitely not well-defined then. Either I have exactly two elements in the range for every element in the domain, or I have exactly two elements in the domain for every element in the range. But I wasn't trying to make a function or anything, so it doesn't worry me.
@theboombody I don't wish to sound rude but you don't know what you're talking about. By your logic, I can take the singleton set {1} and the infinite set of positive integers {1, 2, 3, 4, ...}, and map the number 1 to every element of {1, 2, 3, 4, ...}. Hence, according to you, the singleton set {1} is infinite. If you cannot see the flaw in your logic I can help you no further. "I wasn't trying to make a function or anything, so it doesn't worry me" it SHOULD worry you.
@Rokker815 No, if you do that you actually show that the positive integers are infinitely bigger than 1, because you have a many-to-one pairing, since you have to use all of the elements in the {1} set many times to get all of the elements in the other set. You have to count how many times each element is matched, and you can't match sets partially. Not matching the sets partially fulfills what would be called the onto criteria in a normal bijection. These are like multi-bijections
@theboombody [cont] Since I can also use your logic to say, map the (0,1] to the set (0,2] by mapping the set (0,1/3] to (0, 2/3], (1/3, 2/3] to (2/3, 4/3], and (2/3, 1] to (4/3, 2] we see that actually (0,1] is three times as "big" as (0,2]. And we haven't even started on the fact you've mysteriously dropped the 0 so you're looking at (0,1] instead of [0,1]. It is a pointless exercise since things that work with finite sets stop working with infinite sets. It doesn't matter [cont]
@Rokker815 I have to have the zero excluded, otherwise I'll have one point overlapping when I try the two-to-one matching. And you're right, with my method, I can show any interval is some finite number of times bigger than any other. When I discovered this, I said, "Wow, I wonder why bijections even came to be important in the first place if I can do stuff like this. They mean a lot less." But at least the lack of a bijection shows a definite difference in size.
@theboombody [cont] that we can construct the mappings in this manner. Firstly they are not well-defined so they automatically carry less weight than the well-defined bijection. Secondly there is a big difference between the "size" of an infinite set and the LENGTH of the interval. A big, big difference. Finally, we use the word "cardinality" instead of "size" to AVOID confusion, not cause it!!!
@Rokker815 I'll agree that there's a difference between the size of an infinite set and the length of a continuous interval, but I could have used this same two-to-one matching method with the integers and the positive whole numbers to show that bijections aren't the only way to compare sizes of infinite sets.
These maps you describe are not two-to-one. They are one-to-many, which are not proper maps, they are not well-defined. In an earlier comment of mine where I wrote "NOT a 2-to-1 mapping. You can't map 0.5 to 0.5 and to 1.5 that is not a well-defined mapping!!!" I realised I meant to write "1-to-2". You can't take an element and decide it has more than one image. E.g. take 2 and map it to both -2 and 2.
@theboombody [cont] THE POINT IS THIS. Whichever map you come up with, to accurately relate the relative size of two different sets, must be well-defined in "both directions". So even though your map from (0,2] to (0,1] is well-defined, since each element in the domain has ONE image in the range, the corresponding map viewed "the other way round", i.e. the inverse map from (0,1] to (0,2], is NOT well-defined. Where does 1/2 go, exactly? Does it go to 1/2, or 3/2?
@Rokker815 It goes to both. That's why the other set is twice as big. For every element in the first set, there are TWO elements in the other set that match it. None of this is taught in any math class. That's why it seems wrong, but I believe it to be correct. It just doesn't have good terminology and definitions yet. I always agree with the math textbooks, but on this one issue, I can't bring myself to do it. It's possible in rare instances for the smarter person to be wrong.
@theboombody I am tired of how stubborn you are. You are completely wrong. "For every element in the first set, there are TWO elements in the other set" but you've said yourself it's possible to construct a map that says that for every element in the second set, there are, say, ONE HUNDRED elements in the first set. Does this not give it away that you must be wrong? Your mistake is equivalent to writing, say, infinity = 2*infinity, and "dividing both sides by infinity" gives 1=2.
@Rokker815 Right. This is essentially a proof by contradiction. I'm showing that it's idiotic to pair up elements between infinite sets to prove their equality. When you try to do so, things happen like you say they happen. You get things being twice as big as themselves, which makes no sense. Bijections should not be used to prove infinite sets are the same size. Because bijections are essentially nothing more than pairing up elements. Works well in the finite world, not in the infinite.
@theboombody at what point has someone told you that the sets are equal?? No-one is saying that (0,1) = (0,2), or that the real line is the same as (0,1). But in terms of their cardinality they are the same, since (by definition) there exists a bijection between them. That's all. You are completely misguided. Trust me, I'm an (actual) mathematician. From reading these responses you appear to be so bad at understanding this level of maths that you don't even know that you're bad.
@theboombody I REPEAT. We are not showing that (0,1) and (0,2) are EQUAL. An easy way to do this (much easier than your supposed way) is to show that e.g. 1.5 is not an element of (0,1). This is nothing to do with what you're talking about. Both of these sets contain an INFINITE number of elements, and we need a way of dealing with the "level" of infinity. Since there exists a continuous bijection between the two sets, we conclude that the "level" of infinity is the same for both.
@Rokker815 I called the "level" of infinity a "size class" earlier. And I agree with all you're saying here, but many do not. Go to Wikipedia and see what they say about Dedekind-infinite set. It says a bijection is the same as being equinumerous. It says a subset of infinity is equal to infinity. Do I agree? Heck no, but it seems there are people out there that do. That's why I have to make videos like these.
@theboombody if two sets are infinite, and both are at the same "level" of infinity (i.e. "size class", cardinality) then indeed both sets contain the "same number" of elements. The problem you continue to have is to try to give meaning to this number. There isn't really a "number" of elements, since they are infinite. They are the same level of infinite though, and that's good enough. Finally, if you read that wikipedia entry, almost everything on there is just DEFINITIONS. [cont]
@theboombody [cont] IF a set contains a proper subset which CAN be mapped bijectively back on to the original set, (so we don't always HAVE to choose that map, but if one exists, we're in business for proceeding with what the definition has to say), then we say that original set is Dedekind-infinite. What is the problem here? (0,2) has a proper subset (0,1) which can be mapped bijectively onto (0,2) does it not? So by definition, (0,2) is Dedekind-infinite. (0,2) also [cont]
@theboombody [cont] contains the infinite subset {x: 0<x<2, x is rational}, but this does not map bijectively onto (0,2). It is a different cardinality: countable. This means there is a way of listing the elements such that it takes a finite time to count to any given element (sure, an infinite amount of time to count them all, but given an element of the set, I can be sure I'll eventually reach that particular element). And nowhere in the article does it say anything about [cont]
@theboombody [cont] subsets of infinity equalling infinity. You need to read the article again. There is nothing in there that should be causing you concern. It is not saying that the subset (0,1) is EQUAL to (0,2), but since (0,1) is a proper subset of (0,2) and can be mapped bijectively onto it, (0,1) and (0,2) are equinumerous, which if you bothered to read the article on what that means, it doesn't mean same number of elements, it means same cardinality. You agree that [cont]
@theboombody [cont] (0,1) and (0,2) have the same cardinality, right? Then by DEFINITION, they are equinumerous. At no point has Dedekind or any other mathematician formally said anything about the same "number" of elements since such a statement when dealing with infinity is MEANINGLESS!!!!!! I still don't get your problem! You seem to have FUNDAMENTALLY MISUNDERSTOOD the article to which you referred me. How can you disagree about a DEFINITION??
@Rokker815 Equinumerous is an extremely misleading word in this case, you have to admit. It would be like me defining a tree as "water falling from the sky." Mathematically, I can do that, but it sure leads people to think I'm talking about rain.
I admit a slight obsession working with things like infinity which cannot logically be grasped. I just didn't like someone telling me, "For sure, these two lines of different length have the same number of points exactly."
@theboombody it doesn't matter what equinumerous sounds like it means. Point is, you didn't even bother clicking the hyperlink to check what it really meant. And at no point has anyone EVER told you "For sure, these two lines of different length have the same number of points exactly." This is what YOU, with your limited understanding, have inferred from what you have read. Ever even questionned how a set with an infinite number of elements can possibly have a "number" of elements?
@theboombody and then I decide to give this "level" of infinity a special word: CARDINALITY. And when dealing with infinite sets, we can indeed say things like "this set is twice as big as itself" if we know that, secretly, we just mean that "two times infinity equals infinity" which is a meaningless statement unless we give it a sensible meaning - - -which you have repeatedly failed to do.
We can then match Beta to Alpha, Beta b/w 0-1 and Alpha b/w 0-2 Ironically this property could be used to prove that the size b/w any two intervals is the same.
Which means that Aleph-1 is the same size b/w any two intervals.
@azakareal Well, you're clearly showing that a bijection exists between these two sets. No argument there. But it's my belief that while establishing a bijection between two finite sets shows them to be equal, extending this logic to infinite sets doesn't quite work. Just because a bijection exists between two infinite sets doesn't make them equal. It just makes them have the same cardinality. They're in the same "size class" but aren't the same exact size.
After thinking about it I came up with I think is a far more interesting way of proving that the cardinal set of number b/w 0-1 is the same as 0-2.
Take Cantors diagonal number proof.
We take a set of numbers b/w 0-1 and match them b/w 0-2. So, 01 on the left 02 on the right . We take the right column and as Cantor did generate a new number thats not matched that is b/w 02 call it Alpha.
We take Alpha, we divide by ten, and again use it to generate a new number b/w 0-1 call it beta.
Hi I am no mathematician but I'll try to work a solution for you.
Basically there are at least 2 types of Infinity at play here, countably infinite Aleph-0 and uncountably infinite Aleph-1. Canto Suggested that Aleph 1 is greater than Aleph0 by using his now famous Diagonal Argument.
I guess one could say that any mappings that are not bijections are irrelevant. It is easy enough to create mapping between finite sets that are not bijection but of course that doesn't imply that they are different sizes. Therefore, to show that two sets are not the same size, it is not sufficient to give any amount of mappings (as these do not contradict equal cardinality) but instead it is neccesary show that NO bijection can exist.
That's true, various amounts of mappings do not contradict equal cardinality. And for sets to have different cardinality, no bijection can exist at all. I agree with that. But I'm trying to differentiate cardinality from size. I believe (0,2) is bigger than (0,1) even though they have the same cardinality. Cardinality is an attempt to apply what works in the finite world to the infinite. But I don't trust the extrapolation 100%.
@Desrathedemon Beats me. I know nothing about measure. I believe cardinality has something to do with size though because higher cardinal numbers imply a larger set size. Plus when you type in "cardinal number" at answers.com, you see this:
Cantor identified the fact that one-to-one correspondence is the way to tell that two sets have the same size, called "cardinality", in the case of finite sets. Using this one-to-one correspondence, he applied the concept to infinite sets.
I heard mention of the axiom of choice but little on the problem with it...Also, Surjectivity is, by definition, one to one and onto, so there cannot be a "two to one surjective map".. You need to get a bit more educated.
@POWLIHERE22 Surjectivity is just onto from what I understand and not necessarily one to one. A bijection is both one to one and onto. My point is, I'm showing a set is twice a big as another set by taking two elements from one set, matching them with one element in the other, and the entire range of the mapping function chosen is used. Very rarely do I ever see two to one mapping, so I had to get creative with the vocabulary.
@POWLIHERE22 theboombody is right. Surjective maps are when each element in the image has AT LEAST one element in the pre-image. Injection is when each element has a unique image (1-to-1). A bijection implies both i.e. 1-to-1 and no overlap in image.
@POWLIHERE22 Nope, the reals are not countable. In fact, I once tried to show that the naturals are uncountable as well, but of course that logic failed. However, I did show that the wholes are uncountable if you consider a number with infinite digits and no decimal point to be a whole number.
@theboombody The key to understanding infinities is "A set is infinite if and only if it is equivalent to one of its proper subsets. " That is a theorem.
@POWLIHERE22 I was not aware that such a theorem existed. Then I looked up and found that it originates from a guy named Richard Dedekind. I believe he used the word "similar" instead of equivalent though. Similar I agree with. Equivalent I don't.
@POWLIHERE22 It looks like the proofwiki website uses the word equivalent. The answers website claims Dedekind used the word similar. I believe proofwiki is in error.
@POWLIHERE22 Equivalent is a much stronger claim than similar. In this case, I think similar means they're in the same "class" of size, but perhaps not the same exact size. When you find a bijection exists between two finite sets, they're clearly the exact same size, but when you find a bijection between infinite sets, they're probably only the same "class" of size. I think Cantor invented the word "cardinality" to make this distinction.
@POWLIHERE22 Most of the stuff I learned in school, but I don't think the school system has it all correct. I think you going out and learning this stuff on your own gives you an advantage that you don't realize you have. It's certainly impressive.
...Firstly, that fact that the interval 0-1 is a subset of 0-2 is not inherently a problem for them having the same size since one of the charecteristics of a set being infinite is that it has proper subsets of the same cardinality. However, I accept this is tautologous so I offer another reason. Infinitely large numbers cannot be operated on in the same way as natural numbers. 2*infinity is still infinity. Alas, the proof for this might also fall into the same pitfalls. What do you think?
@DemiDragonSon I agree that clearer definitions are more of a problem than cardinality itself is. The fact that the set of reals is larger than the set of naturals makes it hard for me to say that a doubled standard countable infinity is the same size as the plain standard countable infinity, even though the standard countable infinity and its double have the same cardinality. If one infinity is bigger than another I'm reluctant to say a proper subset of infinity is still infinity
Great video. It's deFINITELY given some food for thought. I agree that the mapping can be used in two different ways but I don't it's the idea of mapping (to express cardinality) itself that is the problem but more it's definition which is often given loosely. I wish someone could show us a more watertight and unambigious way of talking about this. However, I do have some response to your case...
(1) there exist a possible way to match elements in the sets (among possibly many possible ways to match the elements), which gives a 1-to-1 matching.
(2) there exist one and only one way to match elements in the sets (or one and only one "valid" way), which gives such a matching.
It seems like you are discussing the second case. I'm certainly not educated about this (my education is wikipedia), but it seems like at least case 1 could hold for your examples.
@arson86 I would actually diagree with #2. If the sets are the same size, you should be able to match them up on a one-to-one basis in multiple different ways, BUT you should NOT be able assign exactly two elements in one set to exactly one element in the other set. Then instead of saying the sets are the same size, you're saying one is twice as big as the other. Since we can actually do this with intervals, I argue that one-to-one pairing is not a good method to measure set size here.
The issue is one of countability. You can pair [0,1] to [0,2]. Thus, they are countable with respect to each other. That does not mean that they are "equal" to each other. Simply countable. That is what pairing establishes.
You are right in a way, though. Pairing is not enough. There has to be a deeper logic to it. That is what more complex set theory establishes, in my mind.
@CHistrue I think the initial idea behind pairing was to establish an equal number of elements (an equal size) but eventually became used to emphasize countability more than size. And everyone agrees that an uncountable set is bigger than a countable set, and a countably infinite set is bigger than a finite set, and a finite set can be bigger than another finite set, but for some reason one countably infinite set can't be bigger than another countably infinite set. Why?
@Soggysilicon Well, I figure either the world doesn't make sense, or set theory is a little off somewhere. The world not making sense is easier to believe. As annoying as the nonsense of the world can be, it can certainly be beautiful, I agree.
you are wrong about no matchings having more "credibility" than others. We want a BIJECTION between them which means the mapping is INJECTIVE!!! NOT a 2-to-1 mapping. You can't map 0.5 to 0.5 and to 1.5 that is not a well-defined mapping!!!
Rokker815 1 week ago
@Rokker815 Since I can map the same two sets in an injective way AND a 2-to-1 way and in either case the onto criteria is fulfilled suggests to me that you can no longer say that two infinite sets are the same size because you can match the sets element by element (using a bijection). Because I can use the same argument to say I can match the sets two elements to one element to show one of the sets is twice as big. Just because a bijection exists doesn't mean it's the only thing I can use.
theboombody 1 week ago
I think you fundamentally misunderstand what it means for a map to be well-defined.
Rokker815 3 days ago in playlist Uploaded videos
@Rokker815 Perhaps, but I have a way to show that (0,2] has exactly twice as many elements as (0,1], which would imply that it's twice as big. A bijection just shows they have the same number of elements. I can show a bijection too as well as showing one of the sets has two times as many elements. Both use very similar logic. Whether I'm familiar with well-definied mapping or not, it makes no difference.
theboombody 3 days ago
@theboombody a well-defined map means that an element of the domain can have at most one image in the range. This does not imply injectivity or surjectivity. For instance the map f: R -> R given by square rooting. The element "4" say, has two images, -2 and +2. So this is not a well-defined mapping. In this instance we restrict the range to just the non-negative reals so we have a well-defined mapping. The problem you have is your map is not well-defined. [cont]
Rokker815 3 days ago in playlist Uploaded videos
@Rokker815 Your right, my map is definitely not well-defined then. Either I have exactly two elements in the range for every element in the domain, or I have exactly two elements in the domain for every element in the range. But I wasn't trying to make a function or anything, so it doesn't worry me.
theboombody 3 days ago
@theboombody I don't wish to sound rude but you don't know what you're talking about. By your logic, I can take the singleton set {1} and the infinite set of positive integers {1, 2, 3, 4, ...}, and map the number 1 to every element of {1, 2, 3, 4, ...}. Hence, according to you, the singleton set {1} is infinite. If you cannot see the flaw in your logic I can help you no further. "I wasn't trying to make a function or anything, so it doesn't worry me" it SHOULD worry you.
Rokker815 2 days ago
@Rokker815 No, if you do that you actually show that the positive integers are infinitely bigger than 1, because you have a many-to-one pairing, since you have to use all of the elements in the {1} set many times to get all of the elements in the other set. You have to count how many times each element is matched, and you can't match sets partially. Not matching the sets partially fulfills what would be called the onto criteria in a normal bijection. These are like multi-bijections
theboombody 2 days ago
@theboombody [cont] Since I can also use your logic to say, map the (0,1] to the set (0,2] by mapping the set (0,1/3] to (0, 2/3], (1/3, 2/3] to (2/3, 4/3], and (2/3, 1] to (4/3, 2] we see that actually (0,1] is three times as "big" as (0,2]. And we haven't even started on the fact you've mysteriously dropped the 0 so you're looking at (0,1] instead of [0,1]. It is a pointless exercise since things that work with finite sets stop working with infinite sets. It doesn't matter [cont]
Rokker815 3 days ago in playlist Uploaded videos
@Rokker815 I have to have the zero excluded, otherwise I'll have one point overlapping when I try the two-to-one matching. And you're right, with my method, I can show any interval is some finite number of times bigger than any other. When I discovered this, I said, "Wow, I wonder why bijections even came to be important in the first place if I can do stuff like this. They mean a lot less." But at least the lack of a bijection shows a definite difference in size.
theboombody 3 days ago
@theboombody [cont] that we can construct the mappings in this manner. Firstly they are not well-defined so they automatically carry less weight than the well-defined bijection. Secondly there is a big difference between the "size" of an infinite set and the LENGTH of the interval. A big, big difference. Finally, we use the word "cardinality" instead of "size" to AVOID confusion, not cause it!!!
Rokker815 3 days ago in playlist Uploaded videos
@Rokker815 I'll agree that there's a difference between the size of an infinite set and the length of a continuous interval, but I could have used this same two-to-one matching method with the integers and the positive whole numbers to show that bijections aren't the only way to compare sizes of infinite sets.
theboombody 3 days ago
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These maps you describe are not two-to-one. They are one-to-many, which are not proper maps, they are not well-defined. In an earlier comment of mine where I wrote "NOT a 2-to-1 mapping. You can't map 0.5 to 0.5 and to 1.5 that is not a well-defined mapping!!!" I realised I meant to write "1-to-2". You can't take an element and decide it has more than one image. E.g. take 2 and map it to both -2 and 2.
Rokker815 2 days ago
@theboombody [cont] THE POINT IS THIS. Whichever map you come up with, to accurately relate the relative size of two different sets, must be well-defined in "both directions". So even though your map from (0,2] to (0,1] is well-defined, since each element in the domain has ONE image in the range, the corresponding map viewed "the other way round", i.e. the inverse map from (0,1] to (0,2], is NOT well-defined. Where does 1/2 go, exactly? Does it go to 1/2, or 3/2?
Rokker815 2 days ago
@Rokker815 It goes to both. That's why the other set is twice as big. For every element in the first set, there are TWO elements in the other set that match it. None of this is taught in any math class. That's why it seems wrong, but I believe it to be correct. It just doesn't have good terminology and definitions yet. I always agree with the math textbooks, but on this one issue, I can't bring myself to do it. It's possible in rare instances for the smarter person to be wrong.
theboombody 2 days ago
@theboombody I am tired of how stubborn you are. You are completely wrong. "For every element in the first set, there are TWO elements in the other set" but you've said yourself it's possible to construct a map that says that for every element in the second set, there are, say, ONE HUNDRED elements in the first set. Does this not give it away that you must be wrong? Your mistake is equivalent to writing, say, infinity = 2*infinity, and "dividing both sides by infinity" gives 1=2.
Rokker815 2 days ago
@Rokker815 Right. This is essentially a proof by contradiction. I'm showing that it's idiotic to pair up elements between infinite sets to prove their equality. When you try to do so, things happen like you say they happen. You get things being twice as big as themselves, which makes no sense. Bijections should not be used to prove infinite sets are the same size. Because bijections are essentially nothing more than pairing up elements. Works well in the finite world, not in the infinite.
theboombody 2 days ago
@theboombody at what point has someone told you that the sets are equal?? No-one is saying that (0,1) = (0,2), or that the real line is the same as (0,1). But in terms of their cardinality they are the same, since (by definition) there exists a bijection between them. That's all. You are completely misguided. Trust me, I'm an (actual) mathematician. From reading these responses you appear to be so bad at understanding this level of maths that you don't even know that you're bad.
Rokker815 2 days ago
@theboombody I REPEAT. We are not showing that (0,1) and (0,2) are EQUAL. An easy way to do this (much easier than your supposed way) is to show that e.g. 1.5 is not an element of (0,1). This is nothing to do with what you're talking about. Both of these sets contain an INFINITE number of elements, and we need a way of dealing with the "level" of infinity. Since there exists a continuous bijection between the two sets, we conclude that the "level" of infinity is the same for both.
Rokker815 2 days ago
@Rokker815 I called the "level" of infinity a "size class" earlier. And I agree with all you're saying here, but many do not. Go to Wikipedia and see what they say about Dedekind-infinite set. It says a bijection is the same as being equinumerous. It says a subset of infinity is equal to infinity. Do I agree? Heck no, but it seems there are people out there that do. That's why I have to make videos like these.
theboombody 1 day ago
@theboombody if two sets are infinite, and both are at the same "level" of infinity (i.e. "size class", cardinality) then indeed both sets contain the "same number" of elements. The problem you continue to have is to try to give meaning to this number. There isn't really a "number" of elements, since they are infinite. They are the same level of infinite though, and that's good enough. Finally, if you read that wikipedia entry, almost everything on there is just DEFINITIONS. [cont]
Rokker815 1 day ago
@theboombody [cont] IF a set contains a proper subset which CAN be mapped bijectively back on to the original set, (so we don't always HAVE to choose that map, but if one exists, we're in business for proceeding with what the definition has to say), then we say that original set is Dedekind-infinite. What is the problem here? (0,2) has a proper subset (0,1) which can be mapped bijectively onto (0,2) does it not? So by definition, (0,2) is Dedekind-infinite. (0,2) also [cont]
Rokker815 1 day ago
@theboombody [cont] contains the infinite subset {x: 0<x<2, x is rational}, but this does not map bijectively onto (0,2). It is a different cardinality: countable. This means there is a way of listing the elements such that it takes a finite time to count to any given element (sure, an infinite amount of time to count them all, but given an element of the set, I can be sure I'll eventually reach that particular element). And nowhere in the article does it say anything about [cont]
Rokker815 1 day ago
@theboombody [cont] subsets of infinity equalling infinity. You need to read the article again. There is nothing in there that should be causing you concern. It is not saying that the subset (0,1) is EQUAL to (0,2), but since (0,1) is a proper subset of (0,2) and can be mapped bijectively onto it, (0,1) and (0,2) are equinumerous, which if you bothered to read the article on what that means, it doesn't mean same number of elements, it means same cardinality. You agree that [cont]
Rokker815 1 day ago
@theboombody [cont] (0,1) and (0,2) have the same cardinality, right? Then by DEFINITION, they are equinumerous. At no point has Dedekind or any other mathematician formally said anything about the same "number" of elements since such a statement when dealing with infinity is MEANINGLESS!!!!!! I still don't get your problem! You seem to have FUNDAMENTALLY MISUNDERSTOOD the article to which you referred me. How can you disagree about a DEFINITION??
Rokker815 1 day ago
@Rokker815 Equinumerous is an extremely misleading word in this case, you have to admit. It would be like me defining a tree as "water falling from the sky." Mathematically, I can do that, but it sure leads people to think I'm talking about rain.
I admit a slight obsession working with things like infinity which cannot logically be grasped. I just didn't like someone telling me, "For sure, these two lines of different length have the same number of points exactly."
theboombody 15 hours ago
@theboombody it doesn't matter what equinumerous sounds like it means. Point is, you didn't even bother clicking the hyperlink to check what it really meant. And at no point has anyone EVER told you "For sure, these two lines of different length have the same number of points exactly." This is what YOU, with your limited understanding, have inferred from what you have read. Ever even questionned how a set with an infinite number of elements can possibly have a "number" of elements?
Rokker815 11 hours ago
@theboombody and then I decide to give this "level" of infinity a special word: CARDINALITY. And when dealing with infinite sets, we can indeed say things like "this set is twice as big as itself" if we know that, secretly, we just mean that "two times infinity equals infinity" which is a meaningless statement unless we give it a sensible meaning - - -which you have repeatedly failed to do.
Rokker815 2 days ago
We can then match Beta to Alpha, Beta b/w 0-1 and Alpha b/w 0-2 Ironically this property could be used to prove that the size b/w any two intervals is the same.
Which means that Aleph-1 is the same size b/w any two intervals.
azakareal 2 weeks ago
@azakareal Well, you're clearly showing that a bijection exists between these two sets. No argument there. But it's my belief that while establishing a bijection between two finite sets shows them to be equal, extending this logic to infinite sets doesn't quite work. Just because a bijection exists between two infinite sets doesn't make them equal. It just makes them have the same cardinality. They're in the same "size class" but aren't the same exact size.
theboombody 2 weeks ago
After thinking about it I came up with I think is a far more interesting way of proving that the cardinal set of number b/w 0-1 is the same as 0-2.
Take Cantors diagonal number proof.
We take a set of numbers b/w 0-1 and match them b/w 0-2. So, 01 on the left 02 on the right . We take the right column and as Cantor did generate a new number thats not matched that is b/w 02 call it Alpha.
We take Alpha, we divide by ten, and again use it to generate a new number b/w 0-1 call it beta.
azakareal 2 weeks ago
Hi I am no mathematician but I'll try to work a solution for you.
Basically there are at least 2 types of Infinity at play here, countably infinite Aleph-0 and uncountably infinite Aleph-1. Canto Suggested that Aleph 1 is greater than Aleph0 by using his now famous Diagonal Argument.
azakareal 2 weeks ago
I guess one could say that any mappings that are not bijections are irrelevant. It is easy enough to create mapping between finite sets that are not bijection but of course that doesn't imply that they are different sizes. Therefore, to show that two sets are not the same size, it is not sufficient to give any amount of mappings (as these do not contradict equal cardinality) but instead it is neccesary show that NO bijection can exist.
DemiDragonSon 1 month ago
That's true, various amounts of mappings do not contradict equal cardinality. And for sets to have different cardinality, no bijection can exist at all. I agree with that. But I'm trying to differentiate cardinality from size. I believe (0,2) is bigger than (0,1) even though they have the same cardinality. Cardinality is an attempt to apply what works in the finite world to the infinite. But I don't trust the extrapolation 100%.
theboombody 1 month ago
@theboombody From what little I know about measure theory, this notion of size is encapsulated in the measure of a set, not in it's cardinality.
Desrathedemon 1 month ago
@Desrathedemon Beats me. I know nothing about measure. I believe cardinality has something to do with size though because higher cardinal numbers imply a larger set size. Plus when you type in "cardinal number" at answers.com, you see this:
Cantor identified the fact that one-to-one correspondence is the way to tell that two sets have the same size, called "cardinality", in the case of finite sets. Using this one-to-one correspondence, he applied the concept to infinite sets.
theboombody 1 month ago
I heard mention of the axiom of choice but little on the problem with it...Also, Surjectivity is, by definition, one to one and onto, so there cannot be a "two to one surjective map".. You need to get a bit more educated.
POWLIHERE22 1 month ago
@POWLIHERE22 Surjectivity is just onto from what I understand and not necessarily one to one. A bijection is both one to one and onto. My point is, I'm showing a set is twice a big as another set by taking two elements from one set, matching them with one element in the other, and the entire range of the mapping function chosen is used. Very rarely do I ever see two to one mapping, so I had to get creative with the vocabulary.
theboombody 1 month ago
@theboombody You are correct!
POWLIHERE22 1 month ago
@POWLIHERE22 theboombody is right. Surjective maps are when each element in the image has AT LEAST one element in the pre-image. Injection is when each element has a unique image (1-to-1). A bijection implies both i.e. 1-to-1 and no overlap in image.
DemiDragonSon 1 month ago
The reals are not countable.
POWLIHERE22 1 month ago
@POWLIHERE22 Nope, the reals are not countable. In fact, I once tried to show that the naturals are uncountable as well, but of course that logic failed. However, I did show that the wholes are uncountable if you consider a number with infinite digits and no decimal point to be a whole number.
theboombody 1 month ago
@theboombody The key to understanding infinities is "A set is infinite if and only if it is equivalent to one of its proper subsets. " That is a theorem.
POWLIHERE22 1 month ago
@POWLIHERE22 I was not aware that such a theorem existed. Then I looked up and found that it originates from a guy named Richard Dedekind. I believe he used the word "similar" instead of equivalent though. Similar I agree with. Equivalent I don't.
theboombody 1 month ago
@POWLIHERE22 It looks like the proofwiki website uses the word equivalent. The answers website claims Dedekind used the word similar. I believe proofwiki is in error.
theboombody 1 month ago
@theboombody And what is the difference?
POWLIHERE22 1 month ago
@POWLIHERE22 Equivalent is a much stronger claim than similar. In this case, I think similar means they're in the same "class" of size, but perhaps not the same exact size. When you find a bijection exists between two finite sets, they're clearly the exact same size, but when you find a bijection between infinite sets, they're probably only the same "class" of size. I think Cantor invented the word "cardinality" to make this distinction.
theboombody 1 month ago
@POWLIHERE22 By the way, your comments have helped me more than anyone elses' on youtube. I sure do appreciate them.
theboombody 1 month ago
@theboombody Your welcome. This stuff is difficult, I learned of it on my own, so I appreciate the autodidacts out there.
POWLIHERE22 1 month ago
@POWLIHERE22 Most of the stuff I learned in school, but I don't think the school system has it all correct. I think you going out and learning this stuff on your own gives you an advantage that you don't realize you have. It's certainly impressive.
theboombody 1 month ago
@theboombody It is all about the hard work studying and dedication to Truth not Ego. :-)
POWLIHERE22 1 month ago
...Firstly, that fact that the interval 0-1 is a subset of 0-2 is not inherently a problem for them having the same size since one of the charecteristics of a set being infinite is that it has proper subsets of the same cardinality. However, I accept this is tautologous so I offer another reason. Infinitely large numbers cannot be operated on in the same way as natural numbers. 2*infinity is still infinity. Alas, the proof for this might also fall into the same pitfalls. What do you think?
DemiDragonSon 1 month ago
@DemiDragonSon I agree that clearer definitions are more of a problem than cardinality itself is. The fact that the set of reals is larger than the set of naturals makes it hard for me to say that a doubled standard countable infinity is the same size as the plain standard countable infinity, even though the standard countable infinity and its double have the same cardinality. If one infinity is bigger than another I'm reluctant to say a proper subset of infinity is still infinity
theboombody 1 month ago
Great video. It's deFINITELY given some food for thought. I agree that the mapping can be used in two different ways but I don't it's the idea of mapping (to express cardinality) itself that is the problem but more it's definition which is often given loosely. I wish someone could show us a more watertight and unambigious way of talking about this. However, I do have some response to your case...
DemiDragonSon 1 month ago
Are not these two different things:
(1) there exist a possible way to match elements in the sets (among possibly many possible ways to match the elements), which gives a 1-to-1 matching.
(2) there exist one and only one way to match elements in the sets (or one and only one "valid" way), which gives such a matching.
It seems like you are discussing the second case. I'm certainly not educated about this (my education is wikipedia), but it seems like at least case 1 could hold for your examples.
arson86 3 months ago
@arson86 I would actually diagree with #2. If the sets are the same size, you should be able to match them up on a one-to-one basis in multiple different ways, BUT you should NOT be able assign exactly two elements in one set to exactly one element in the other set. Then instead of saying the sets are the same size, you're saying one is twice as big as the other. Since we can actually do this with intervals, I argue that one-to-one pairing is not a good method to measure set size here.
theboombody 3 months ago
The issue is one of countability. You can pair [0,1] to [0,2]. Thus, they are countable with respect to each other. That does not mean that they are "equal" to each other. Simply countable. That is what pairing establishes.
You are right in a way, though. Pairing is not enough. There has to be a deeper logic to it. That is what more complex set theory establishes, in my mind.
CHistrue 3 months ago
@CHistrue I think the initial idea behind pairing was to establish an equal number of elements (an equal size) but eventually became used to emphasize countability more than size. And everyone agrees that an uncountable set is bigger than a countable set, and a countably infinite set is bigger than a finite set, and a finite set can be bigger than another finite set, but for some reason one countably infinite set can't be bigger than another countably infinite set. Why?
theboombody 3 months ago
@theboombody What way is preferable to pairing? Can you explain because pairing is rather much implied in the whole idea of a primitive recursion.
CHistrue 3 months ago
@CHistrue Maybe someone will find a better way in the future. Might take a few decades or centuries though. Who knows. Lots of exploration out there.
theboombody 3 months ago
@Soggysilicon Yeah, stuff starts getting weird when you allow partial units. That's why I think the continuum hypothesis is true at this point.
theboombody 5 months ago
@Soggysilicon Well, I figure either the world doesn't make sense, or set theory is a little off somewhere. The world not making sense is easier to believe. As annoying as the nonsense of the world can be, it can certainly be beautiful, I agree.
theboombody 5 months ago
>not intelligent enough to know
Me, either!
IllPropaganda 5 months ago
♥ your submissions.
vigaroux 5 months ago