@theboombody it depends how you define size! the convention for sets is if there exists a continuous bijection between the two, then they have the same "size". Clearly [0,2] is twice as long as [0,1] but that's not what we mean by the word "size" in this context. How do you not understand that yours is a problem of terminology?
@Rokker815 Trust me, I'm not the only one with terminology problems. Cantor invented the word cardinality to distinguish it from the word size, but I see them used interchangably on a constant basis. The only reason you can come up with a bijection between positive numbers and integers is because both have unending terms and you can do some shifting. But the integers is still a bigger set. It's just not an "unmatchably" bigger set like the reals are.
@Rokker815 Cantor said, "Well, if I can create a bijection between two finite sets, that clearly proves they are the same size. I'll just try that with infinite sets and say it does the something similar. But I better be careful, and invent a new word called cardinality in case what works in the finite world doesn't work in the infinite." Turns out cardinality is more like "size class" than "size." Integers are the same size class as the positive whole numbers.
@theboombody Read my reply to the other video. Yes, you're right, "size" and "cardinality" are used interchangeably sometimes, but why is that a problem? We know what we mean by the context. Clearly [0,2] is a closed interval with twice the LENGTH of [0,1], but has the same CARDINALITY as [0,1] since there exists a BIJECTION between the two. That's all that matters, I promise you. You don't need to invent a new phrase "size class".
@Rokker815 I think it's a problem because it leads to things like the Banach-Tarski paradox. People blame the axiom of choice for those paradoxes, but I blame the substitution of cardinality for size as the cause of the paradoxes. In fact, wasn't the Banach-Tarski paradox constructed in order to show that the axiom of choice is wrong? I thought I read that somewhere. But instead of disbeleiving the axiom of choice, the paradox was accepted as true.
@Rokker815 Also, I should add that I don't believe all paradoxes contain a false statement. Some are probably legit. Others I think contain a false statement somewhere. And I'm aware that I'm reaching quite a bit with regards to blaming cardinality for paradoxes instead of the axiom of choice. I'm treading into waters that I know very little about. I'm quite unfamiliar with the axiom of choice.
I think always hate the way that textbooks etc. present the bijection as proof of equal cardinality and close the discussion there too, without considering counterarguments. A bijection is used to prove that the set of integers and the set of even numbers have the same cardinality. But surely one set being a subset of another set should be considered sufficient proof that the two sets have DIFFERENT cardinalities? I have never seen counterarguments addressed in a discussion of bijection.
@johnsmithbsc Man, you state my thoughts very well. That's the whole reason why I made this video. I think Cantor invented the word cardinality because he KNEW the word size was inaccurate. Two sets having the same size is a much stronger statement than saying they have the same cardinality. Cardinality seems to mean "size class" more than size. But I see people tossing around cardinality, size, and bijection like they're all the exact same thing so often that I couldn't remain silent.
Continuity is not a easy notion to handle. In reality, I am not sure if one can come up with a continuity example. Hence the intuition from reality won't be helpful to understand "size" of two different sets which has continuum of elements.
@theboombody. "Size" doesn't make much sense or have any use in areas like Topology. To a topologist, [0,1] and [a,b], a<b, are basically the same thing since you can continuously shrink or stretch it from one to the other. Hence, in terms of number of points, though you can't see it but both sets have same matching of elements with one more tightly packed together. This gives an intuition that, in context of uncountability, "size", per se, doesn't make much sense.
@TKKTism I agree. But I have seen in the past lots of people saying that infinite sets were the same size when a bijection existed between them. That's kind of why I made this video, to demonstrate that there's a clear difference between size and cardinality. Things that have the same size have the same cardinality, but things that have the same cardinality may not have the same size. That's my argument. Cardinality deals with bijections, size does not.
The concept "cardinality" is simply a certain _defined_ property of two sets. It's not a good thing to call such sets "the same size", because these words bring a bunch of irrelevant connotations from everyday life which only confuse the issue. It's just a definition. OTOH if you define things differently (by assigning a measure or using some other construct), the segment [0,2] will be twice as large than [0,1]. It's a matter of context and deciding what's useful for a given type of problem.
@JanPBtest I think my goal now is to get math people to differentiate cardinality from size. Some already do, but I know some don't. Most people who know it's different tend to use measure to describe size. I've never studied measure myself. Just cardinality. Same size implies same cardinality, but same cardinality does not imply same size. I haven't proven the first half of the statement, but it's my working hypothesis.
TITOR002 was not expressing a factorial. The notation "!=" is internet speak for "does not equal" or "is not equal to" for anyone who does not know the ascii for a slash-equal symbol.
@DornierPfeil Thanks for the correction. If I were a bit brighter, I probably would have almost sort of learned something. Now I know that I agree with TITOR002, even though my video makes it look like I don't.
x/x = 1 holds for an infinite number of x, that's right. So you can't say x=1 is THE solution, that's perfectly right. But you can say that the set of x's such that x/x = 1 is THE (only) set of solutions.
So is cardinality. For any real number n, one can match (0, 2] with (0, n]. So there is not a single solution, and because of that, one cannot define a single "size". But one can define the set of solutions, giving it a name other than "size", so that's called cardinality
Cool, this is nice:) Good work! And you know this shows that size can only be defined by the size of other :) Don't let the negative envy votes take you down :) This is what math is all about!
Well, we are all struggling to understand this world. The only way to explain it is to find the missing link between infinity and finity. If one would succeed, then the same one might be able to explian how we came to be. We might even put the debate about god/s on the side once and for all.
@steverock85 I believe certain things can't be figured out, particularly some things about God. But fortunately, many things can be figured out, including possibly this link between the finite and infinite that you speak of. There are some infinite series that converge, and some that diverge, and it would be extremely interesting to kind of find the line where divergence and convergence sort of meet. The harmonic series diverges so slowly it almost looks like it will converge.
Superlike on how you think! :D I never even thought of that.
Yeah, I am agnostic about this god stuff but I have been shearching to understand for years and I get this feeling of a infinity wall that can not be crossed for understanding.
I have my hopes for the link between infinity and finity, but it is only a hope.
@steverock85 I'm a fundamentalist myself. I understand why other people aren't though. The Bible has a lot of stuff in it that is certainly superstitious. But I believe it anyway.
Calculus is a pretty good link between infinity and the finite. Every time you use an anti-derivative to find the area under a curve, you're using a formula that was discovered by summing up the area of a infinite amount of infinitely small rectangles. It's truly a wonder.
Yes that is something that we can use infinity for application. But infinity itself is still a little unknown like the gravity. It is there and we can use it, but we do not understand it really as it is not a real finite number but rather an idea.
@Kreadus005 Yeah. Cantor, the mathematician who first examined this stuff, knew that the word size wouldn't fly too well with the infinite, so he used the word cardinality instead. But I get surprised at how often size and cardinality are used interchangeably. I believe if two things have the same size, they have the same cardinality, but I don't believe the opposite. At least not currently anyway.
@ZhangRed Fair argument. Uncountable infinities are bigger than countable infinities though. I had a thought tonight that perhaps two times infinity is not bigger than infinity and is also not the same as infinity. Maybe it's something inbetween. Maybe there's a measure of quantity somewhere that's between the terms bigger and equal. I can't imagine what it would be. Might shed some light on the continuum hypothesis though.
@integralmath Well said regarding the second statement.
However, I insist using element matching to compare sizes of infinite sets can yield indeterminate results though. If I match it one way, I get a two-to-one ratio, and if I match another way, I get a one-to-one ratio. Sort of like if I take a limit of 1/x at zero, I get a positive infinity on the right side, and negative infinity on the left. Implying indeterminism. I conclude cardinality and size are not equal.
@theboombody yeah, okay. The solution is indeterminate - and? This doesn't admit of an answer; it only says we're not capable of finding a particular solution to a problem. In fact, you've gone from the proposition that a given system is indeterminate to a therefore x is true. Given that the proposition isn't even intelligible, there is no such therefore which can follow. Your conclusion isn't defensible given your process.
@integralmath Sure, I'm not giving an answer, but I am saying there are flaws with the method that should be addressed and eventually corrected. And my argument was that if you use cardinality to determine set size relation between the particular example sets I gave, you get multiple answers. Multiple conflicting answers imply cardinality cannot accurately measure size because of certain indeterminate results. If cardinality cannot measure size, then cardinality is not size.
@integralmath Well, to solve x/x=1 for x would give you infinitely many solutions. That's not a problem. What is a problem is to say x=1 is THE solution when there are many, many others. So when someone says (0,2] is the same size as (0,1] through the element matching argument, they ALSO should say it's twice as big, half as big, three times as big, etc., because element matching could be used to show those results as well.
1) First provide a definition, before saying "there's nothing wrong with my definition"
2) "if I match the same two sets in different ways, I get a different result" Yes, of course, it will ALWAYS work like that. For each infinite set, there's a bijunction from that set to a subset of its, and you can choose from an infinite number of different subsets of that set. This does not happen with a finite set, but it's not so weird, because in-finite sets are not finite and do not work the same... -.-
This video has helped me in my lifelong ambition to discover the diameter of an elliptical circle, infinite in size, and squared from pi. Darn if it isn't 666. I thank you sir!
@gmdinformation At the very least this video exposed that there are more people with brains on youtube than we could have imagined. It's like every week someone who obviously has a higher education in mathematics posts a comment here. Where do these guys come from? I figured I'd only get one such comment a year. One a week is extraordinary.
(...) You could give your own definition of set size, replacing #1, in order to make #2 valid even for infinite sets. But that would be your personal definition, which would not be Cantor's one, and textbooks & co. provide the definition by Cantor, which is widely accepted because its results are widely found to be useful.
@GiorgioDeRa If that's the case, I see nothing wrong with my definition. In fact, I speculate in another video that seeming contradictions that arise from the axiom of choice might actually stem from Cantor's cardinality definitions being incorrect. I have no proof of this of course. It's just speculation.
(...) Double the set, and you'll get a different integer, i.e. size. Because they are different, it makes sense to distinguish between them, and say "size of A is twice the size of B".
The "size" of an infinite set is defined in terms of bijections between INfinite sets. Thus, it's NOT the same, and does not have to work the same. It's NOT a positive integer, and if you double the set, you get again the initial "size" as a result. This kind of size is what the world calls cardinality.
@GiorgioDeRa I'm just saying if I match the same two sets in different ways, I get a different result. So maybe I should measure with something besides matching to get an analysis of size as well. This is why I think the integers are larger than the even numbers, even though they have the same cardinality. Different cardinalities automatically imply different sizes, but the same cardinality may not imply the same size.
2) "If a 1-to-1 bijection exists between finite sets, they're equal. If a 2-to-1 bijection exists between finite sets, one is twice as big as the other. Why would one extrapolation work and the other fail?"
Because #1 is a definition, while #2 is a consequence of that definition; more precisely, #2 is the fact that if you double a set, you get one with a different size.
The size of a finite set is defined in terms of bijections between finite sets; it's essentially a positive integer. (...)
1) While it is clear to me that by "size of a finite set" you mean "the number of points I see when I draw them at 1:13", I can't get what do you mean by "size of an infinite set", provided that it's not cardinality. The sentence "[0; 2] is twice the size of [0; 1]" does not make sense until you give a precise definition of what you mean.
The case where (0,2] is partitioned into the subintervals (0,1] and (1,2] to create two separate bijections between (0,1] and the subintervals is not unique. You could partition the interval (0,2] into any number of subintervals of equal distance (and perhaps of unequal distance) and still be able to make bijections between each of the subintervals and (0,1]. It should then follow that (0,1] is not only same size and twice the size of (0,2], but every whole number ratio.
@theboombody I remember one of the things that shocked me most when studying that was proving that the set with all the finite subsets of N has the same cardinality as N. However it is consistent and in a certain way it makes sense, however it may be counter intuitive, but maybe it's only because we haven't trained our intuition xP
@theboombody Still, talking about "twice" the size doesn't make sense when talking about infinite cardinality, it only makes sense when talking about finite sets because every finite set can have a biyection to X_n ={x in N, x<n}, and thus we can treat their cardinality as that n. However with infinte sense that notion makes no sense
@rjmantilla If a one-to-one bijection exists between finite sets, they're equal. Using that logic, Cantor extrapolated it to infinite sets. If a TWO-to-one bijection exists between finite sets, one is twice as big as the other. If I extrapolate like Cantor did, it would follow for infinite sets as well. Why would one extrapolation work and the other fail? I think it's silly to say they both work, so I say they both fail. The only way it wouldn't fail is if cardinality isn't size at all.
@theboombody Well stablishing a two to one biyection inmediatly generates trouble...... Natural numbers have a two to one biyection with themselves 0->0,1 ... 1->2,3.... etc... two to one biyection says that a set is twice its own size.... so the notion of double size is contradictory with infinte sets, the notion of same size isn't
@rjmantilla Well, I think we've both argued our points to a standstill here. One of us believes establishing a two-to-one and a one-to-one bijection between sets follows the same rules regarding what's allowed, and the other believes that one of those bijections is allowed and the other is not. I don't see either one of us changing the other's mind no matter how much further into our points we go.
@theboombody The size of a set does not depen on it's ordering nd that was what I was trying to show you, what I mean is that if we start saying stuff like "every proper subset of a set is smaller than the set itself", we will reach contradictions, which is different from a paradox. Math is filled with paradoxes, but that doesn't make it invali, however we must avoid contradictions. (Well Gödel's theorem states that math is bound to contradictions....)
When talking about sizes of infinite sets there are a lot of things that are counter intuitive, but are true. The first thing that is really hard to understand is the fact that there are some sets which have proper subsets of the same size. There are equal amount of : primes, even numbers, natural numbers and rational numbers. What's even more strange is that you can make an infinite number of sets of infinite size, which all together have the same size as natural numbers.
@rjmantilla It's true that this is what math classes teach. But I never thought this made sense. I would instead prefer to say that the number of naturals is larger than the number of primes, just not unmatchably larger because you can create a bijection between the two. Both are still in the countable category. This would make the mathematics not quite so counterintuitive, and I think it would actually bring it closer to the truth.
@theboombody there is no such thing as truth when talking about mathematics, there's stuff that's consistent and stuff that not, and we basically test to see if what we say is consistent to the basic axioms in math (which normally would be ZFC axioms). To set a definition of size we need to do something that works for both infinte and finite sets, and biyections work our deal, and it makes a lot of sense, as long as qe can match them we have the same amount of elements
@rjmantilla I've always considered it inconsistent for a proper subset of a set to have the same number of elements as the original set. Whether it's infinite or not. I am unable to continue the established formal study of mathematics as long as this inconsistency (in my opinion) is accepted, because I can't follow any lines of logic that stem from that root. Right or wrong, my brain doesn't work that way. How can I move to step 5 when step 4 makes no sense to me?
@theboombody If it weren't true that the some proper subsets have thes same number of elements of the set they are contained we would have a bigger paradox: take N...... take out the 0, that is a proper subset of N, and thus it must have less elements...... substract one from each element, that set will be N, and we already said that it had less elements than N, which makes less sense......
@rjmantilla Well either way, we seem to be comfronted with something that defies intuition. In my opinion, mathematics should always be intuitive, since it almost always starts with intuitive principles that are refined into proper axioms. I agree that my suggestion doesn't exactly make sense, but at least I'm raising awareness in the potential of inconsistency being present in even the most consistent of the sciences. I'm inconsistent and you're inconsistent. Or we can throw out set theory.
but construting math the way you propose gets us to a bigger paradox, let's say you take the natural numbers, and add a element at the end, lets call it X..... you say that N U {X} has to be bigger than N as N is contained in N U {X} however if take each natural number and add one to them, and then put X at the start, calling it 0, we would have the same size as N, and thus the size of the same set N U {X} would b different depending on its ordering, which would make no sense
@rjmantilla I guess it's possible that mathematics itself is paradoxical and inconsistent, which kind of saddens me. But at least if that's true, it makes rare supernatural events more possible to accept. I mean, if logic isn't always right, nonsense might actually be more correct than logic on certain rare occassions.
@rjmantilla This is probably the best comment I've ever seen to any of my videos. {0,2,3,4,5,6,7...} would be the same size as N. Maybe even {0}U{0,2,3,4,5,6,7...} would be. I don't see how one paradox can be bigger than another paradox though. A paradox is a paradox. I mean, the way we got it now, I can say (0,2) is the same size as (0,1) or twice as big as (0,1) depending on how I choose to match it. Is that any worse than saying a size of a set can vary depending on its ordering?
@rjmantilla I was thinking more about this N U {X} example today, and wondered if the size of infinite sets could be measured another way. Let's say you have an infinite number of points evenly spaced along a number line, and let's say each point is a vertex of an equilateral triangle with sides less than the distance between points on the number line. The vertices of the triangles will always be three times the points on the number line, because both are increasing at constant RATES.
However those kinds of thing you¿re thinking are also valid, and thus you'll see further on in math two important concepts, ordinals and measure. With ordinals you'll that in ordered structures it's different to take N and Z because of their order. And with measure you'll see that [0,1] has measure 1 but [0,2] has measure 2....
@miguelcastano It's more contradictory than trivial. I believe there are certain flaws within recently accepted mathematics that need to be corrected. Because if they're not corrected, they begin to lead to ridiculous conclusions (like something being twice as big as itself). To my surprise, most unbiased people tend to accept the ridiculous conclusion rather than question the math that leads up to it. There's also the possibility that logic is inherently broken and can't be corrected.
Does the requirement of equal cardinality--the words size and big have too many imprecise laymen connotations that render them useless in the study of infinities--between sets depend on the _existence_ of a one-to-one and onto mapping or does it require _all_ mappings to be one-to-one and onto? My suspicion from the memory of my training is that the former is the case. Also, 'twice as big' doesn't make sense in infinity math. 2 * "infinity" == "infinity" and 2 * "continuum" == "continuum"
@jasonw137 If twice as big doesn't make sense in infinity math, why should countability? In the finite world, if you have two sets, and you can set up a two-to-one and onto ratio between them, one set would be twice as big as the other. So if I can do that in the finite world, why couldn't I do it in the infinite? Don't Cantor's methods compare the size of infinite sets in the same way we compare the size of finite sets? I thought that's what the one-to-one stuff was all about.
For one thing, countable implies a mapping to the cardinal numbers. So if one relies on the notion of countable, then sets 'bigger' than the cardinal numbers could not be 'sized'.
see wikipedia: Cardinality#Infinite_sets
see also amazon: Gödel, Escher, Bach: An Eternal Golden Braid
@hmiller0 So my video says an infinite set can be twice as big as itself, and you're saying a more or less empty set can be twice as big as itself. Interesting. These two ideas seem to be outside the bounds of quantification.
@hmiller0 This implies that an empty set is twice as big as itself. But since an empty set consists of nothing, the implication is that nothing is twice as big as itself (although I'm mixing semantic with pseudo math here :p)
@theboombody Yup. Simply put: "A pea can be chopped up and reassembled into the sun.". Would be nice if this could be applied in physics. I can see the replicators now :D
@rerere284 Say you take the set of numbers (14, 6, 5, 12). There are four numbers in this set. Multiply every number in this set by two. You get the numbers (28, 12, 10, 24). There are four numbers in this set also. The two sets of numbers are not the same. They just have the same amount of numbers in them (four numbers). So the two sets have the same size (cardinality four). This works well for sizing up finite sets, but what about infinite sets?
I don't understand the practicality or use it would have to follow this way of thinking. it's not that 0-2 was the same size as 0-1, it's just that it had the same quantity of points depicted in the same relative locations. This has no effect on size/scaling but rather a logical comparison of how many points one has in common being displayed with the other. So it is not twice as big as it's self either, it's just you can select different placement of points. Nice video :)
@Shakeitupyes Sometimes things seem to be useless facts at first, but a use is found for them later. As far as I know, cardinality of infinite sets has no practical applications at this point.
I love your comment, and I agree with it more than you realize. In fact, I'm planning on uploading sort of a follow-up to this soon that somewhat addresses what you point out.
Do you have a mathematical background? I'd be amazed if you didn't.
Does that make it more clear @theboombody (Mr Gator).
Similarly .5*Infinty = 10*Infinity, finally Infinity is never less than Infinity (or some "multiple of infinity"). The logical problem with your statements is that you think Infinity (the number of numbers within (0,1) or (0,2) ) is somehow a number that you can count to.... you start counting.... when you get there, let me know :P
I seem to contradict myself when I say infinity + Infinity is not 2 infinity.... and then say that .5infinty = 10 infinity....
the only point is that any addition of infinity and any sum of infinities is still infinity, so after you do all the sums and multiplications, just drop the coefficients and realize its still just infinity........ Hope that helps, although I agree that this argument should be presented in text-books just to illustrate how counter-intuitive cardinal numbers can be! :)
@brydust I think the main focus of the matter should be, what's the difference between something that's counterintuitive and something that's false? Especially in an abstract science like mathematics where you can't do a lab test to verify a hypothesis?
@brydust one cannot add or subtract from infinity, for infinity is infinity, there is nothing to add to it for infinity has already encapsulated all that there is. There is nothing to take away from it because infinity encapsulates all (where would the subtracted number go? what is the subtracted number? how can that be subtracted at all from that which has no number) the whole idea of attempting to use mathematical formula with infinity is chaos. infinity is not a number it is a concept. :)
@brydust If one infinity can be bigger than another, as Cantor suggests, then I don't see why two times infinity can't be bigger than infinity. It sounds like you think all infinities are the same size. But it's pretty clear to everybody in the mathematical community that the set of real numbers is far more numerous than the set of integers. You do have a point though... the mathematical community also accepts that two times infinity is the same infinity.
@HolymackerelProd Agree with what? That something can be twice as big as itself? I don't agree with that idea really. Cantor came up with the idea and I'm just trying to show how ridiculous it can be.
well this aleast sounds more reasonable that 1+1 = 1.9999999 which is a lie but according to instein evrything is relative so every1 is some how right as long as they are not wrong lol well {:-)
But by definition, this has nothing to do with cardinality.
If we were to again take 10 of each, and say divide them into individuals, then we would only have 10 in each case.
the natural numbers are like 10 elephants, and the reals are like one apple. If we must respect the elephants as individuals (like integers) and we are permitted to go infinitesimally small with the apple, clearly we would get more pieces of apple than elephants.
@100qwerth Actually, I'm not sure if your argument is valid. If I take a point from the set of [0,1), [0,2), and the whole real line, each of those points have the same "size." It's not like one point is bigger than another. So a piece from each of these sets is the same "size."
They have the same quantity of elements in the set, but their physical size is different. If you had a set of 10 apples and a set of 10 elephants, they both have the same cardinality, but different physical sizes.
If we were to take apple and (godforbid) an elephant and cut them up into as many pieces as we can, you could cut an apple and an elephant into infinitely many pieces. You could make an argument that if you choose x many pieces, the size of a piece of elephant will be much larger.
Well put into infinity they have the same cardinality but not put into comparison with infinity they are not? Or am I wrong, I just assumed that if you have some numbers in a set that you don't necessarily have to show them into infinity?
@Dolphidood Well, if you count the numbers, 1, 2, 3, 4, and so forth forever, that's a smaller infinity than if you count those whole numbers and all the decimal points inbetween them, including the decimals that have digits that never end, like pi and so forth. That's the clearest example.
@theboombody Okay, I see so the different infinities are really defined by the sets of numbers? I heard that if there is any chance of something happening within infinity then it will happen... how does that tie in exactly with infinite individuals leading different lives on other planets? Thanks
@Dolphidood I think you're going by the idea that time is infinite, and even an event with the smallest probability of happening will eventually happen given so much time for it to happen. This I certainly agree with. I usually disagree with the idea of beings on other planets though because I think the odds of intelligent life flourishing are so bad that even trillions upon trillions of planets aren't enough to beat those odds. I'm amazed that we even exist.
@theboombody Yeah, sorry, I should of been clear. I think its amazing too when you consider the complexity of the human body and mind and then calculate the chances of such a composition taking place in the universe under pure external factors. I wouldn't disagree with that thought because no matter how small the chances are if we have an infinite amount of planets and time then we will have that result EVENTUALLY. How can we have other infinite universes when ours infinite and leaves no space?
@Dolphidood You bring up some very good points. We could go into a deep philosophical discussion on whether or not something will happen. For me though, probability is not a very factual science since I don't really accept that 0!=1, and it shows up a lot in probability. Probability is elegant and impressive, but definitely not flawless in my opinion. In fact, I don't think mathematics itself is flawless. I think there's a lot of contradictory ideas within it, and that's why I made the video.
@theboombody I think if there are continuous circumstances with room for infinity then it would not be possible to make a flawless probability. I too don't think that mathematics or science are necessarily definite knowledge for everything... I believe they are an attempt to make sense of the world but far more useful in its short term application... also, apparently its been proven by Godel that for mathematics to be totally pure the axioms of which it is formed off must contradict each other.
@theboombody ... that being said, it must be by that also impure... i.e. "This sentence is a lie"... I think its fair to say that people make many assumptions that they rely on heavily to fulfil their survival however when it comes to when it comes to the 'truth' of reality, I very strongly think that there is very little truth when I consider what it has to be to be truth... it HAS TO not leave possibility for any other 'truth' or otherwise its only a potential 'truth'.
@theboombody Lastly, are we not simply fulfilling an aspect of the anatomy of the mind that the universe has given us... this aspect being logical thought which could simply be nothing more but a survival mechanism of which gives us a sense of satisfaction and drive that which I/we have made a concept of in the first place? Any how, do you have any role in science and maths?... I would love to be a research mathematician but I'm just seeking guidance on experiences of people..thanks for replying
@Dolphidood Very interesting ideas here which I pretty much agree with. I stopped my formal mathematics study at a bachelor's degree for three reasons, First, the math was pretty hard. Second, I started to disgree with the math that I was presented. Third, I didn't want to invest the money to continue the studies. So now I'm studying some basic accounting in my spare time hoping to take some classes and make more money. You talk about an artificially complex field, that's it right there.
@theboombody Artificially complex strikes me as theory based if that is the case then I think I've met my match. Although I agree with the uncertainty that logic itself brings with endless questions I would still really like to pursue maths in order to simply know more and equip myself. I look forward to watching more of your videos. I have to admit, as I had no interest when I was younger, I didn't get past GCSE level and I'm only now retaking for A* lol but it doesn't matter. Thanks again.
@Dolphidood Well, some things are complex because they won't work otherwise, like computers. Other things like taxes work fine without complexity, but have complexity pumped into them to confuse people on purpose. I think this often happens to disguise criminal activity. I admire natural complexity even though I hate it, but I really can't stand artificial complexity.
I probably won't make any more math videos. I think I've said everything I wanted to with that topic.
Your video misses the point. By definition, an infinite set is a set which forms a bijection with a proper subset of itself, so your drawings are misleading, as they instead depict finite sets.
@ja524309 Since when did intervals depict finite sets? All intervals are infinite. The sets in my video are intervals. No other commenter disputes that they're infinite.
@theboombody Two sets have the same cardinality if there exists a function which is both one to one and onto between the two sets, and is thus invertible. Infinite sets have the unusual property that such functions exist for both the set onto itself (the identity mapping) and onto a proper subset of itself. For example, the natural numbers can be mapped onto both themselves and the positive even numbers. Again, your visuals are misleading because they depict finite sets.
@ja524309 By that argument every visual representation of an infinite set is misleading. Nobody can draw an infinite number of points on an interval. You have to use your imagination.
Why are infinite sets allowed to have the property that they can be mapped to their subsets and finite sets are not allowed to have that property? I don't think such a property should be allowed to any set because it makes no logical sense for a piece of something to be the whole thing.
@theboombody Ah, now I think we've arrived at the crux of the matter. What do you mean by "logical sense"? Do you mean that they entail a contradiction? This is not the case. Attempt to prove me wrong if you wish. Moving on, yes, of course every finite depiction of an infinite set is misleading, unless, say, one use the usually "..." to denote it extends forever in the same fashion. Finally, I don't know what you mean by being "allowed" to do something. You need to clarify this.
@ja524309 I do think something being twice as big as itself is a contradiction, yes. If A is equal to B, A is not twice as big as B. I know I'm simplifying things a bit here, but that really is the crux of the matter. Tell me how this simplication is so different from the scenario we're facing with the cardinality of intervals. For this reason I think it's a bad idea to study how infinite sets vary in size. It's straying away from actual truth. Are you okay with allowing such straying?
@theboombody Having the same cardinality does not make two sets equal. In fact, it is a stronger notion than equinumerousity. Perhaps you meant sets having equal cardinality is a contradiction, when one is a proper subset of the other? Again, I ask you to show how this is so. You remind me of those people who reject imaginary and irrational numbers in history past, because they didn't feel they were "real" enough.
@ja524309 Imaginary numbers have a terrible name. I'd rather call them semi-positive numbers. Since two positives multiplied give a positive, and two negatives multiplied give a positive, perhaps the square root of a negative number isn't negative or positive, but something inbetween. So I favor exploration. I'm just suggesting cardinality shouldn't be equated to size. Bigger cardinality means a bigger set, same cardinality doesn't mean the same size set.
@ja524309 It's funny how we do agree on certain things and not others. Inconsistency is what it is. And truth is more inconsistent than lies. That's a big part of my G-rated Shock Value philosophy.
@ja524309 Good luck to you sir. You seem to be quite diligent in your studies. I'm sure you'll be a very positive influence in the mathematical world. Just remember that being a negative influence is much more fun.
Essentially you're saying that 2*Aleph = Aleph. This is a specific case of the more general case - for any 2 infinite cardinals a,b we have that a*b = max(a,b).
This also means that for any natural number k, the unit interval is k-times bigger than itself. In my opinion, this is why this fact is not mentioned in textbooks - it is rather meaningless.
Also, the definition tenchoosethree was talking about is indeed an equivalent, rigorous definition of an infinite set (where same size=bijection)
@MyBrrr I just don't see why their "size" relationship is classified as the same simply because a bijection exists when so many other non-bijections can be used as well. I would classify their size relationship as indeterminate since in one instance you can make one set twice as big as the other, and in another you can make it three times as small.
is that a potato or a bean wearing a tuxedo?
Kendrakefull 6 days ago
@Kendrakefull A bean, but it's hard to tell in this picture.
theboombody 5 days ago
[0,1] and [0,2] are homeomorphic mate
Rokker815 2 weeks ago
@Rokker815 So I'm guessing you're going to say they're the same size.
theboombody 1 week ago
@theboombody it depends how you define size! the convention for sets is if there exists a continuous bijection between the two, then they have the same "size". Clearly [0,2] is twice as long as [0,1] but that's not what we mean by the word "size" in this context. How do you not understand that yours is a problem of terminology?
Rokker815 1 week ago in playlist Uploaded videos
@Rokker815 Trust me, I'm not the only one with terminology problems. Cantor invented the word cardinality to distinguish it from the word size, but I see them used interchangably on a constant basis. The only reason you can come up with a bijection between positive numbers and integers is because both have unending terms and you can do some shifting. But the integers is still a bigger set. It's just not an "unmatchably" bigger set like the reals are.
theboombody 1 week ago
@Rokker815 Cantor said, "Well, if I can create a bijection between two finite sets, that clearly proves they are the same size. I'll just try that with infinite sets and say it does the something similar. But I better be careful, and invent a new word called cardinality in case what works in the finite world doesn't work in the infinite." Turns out cardinality is more like "size class" than "size." Integers are the same size class as the positive whole numbers.
theboombody 1 week ago
@theboombody Read my reply to the other video. Yes, you're right, "size" and "cardinality" are used interchangeably sometimes, but why is that a problem? We know what we mean by the context. Clearly [0,2] is a closed interval with twice the LENGTH of [0,1], but has the same CARDINALITY as [0,1] since there exists a BIJECTION between the two. That's all that matters, I promise you. You don't need to invent a new phrase "size class".
Rokker815 1 week ago in playlist Uploaded videos
@Rokker815 I think it's a problem because it leads to things like the Banach-Tarski paradox. People blame the axiom of choice for those paradoxes, but I blame the substitution of cardinality for size as the cause of the paradoxes. In fact, wasn't the Banach-Tarski paradox constructed in order to show that the axiom of choice is wrong? I thought I read that somewhere. But instead of disbeleiving the axiom of choice, the paradox was accepted as true.
theboombody 1 week ago
@Rokker815 Also, I should add that I don't believe all paradoxes contain a false statement. Some are probably legit. Others I think contain a false statement somewhere. And I'm aware that I'm reaching quite a bit with regards to blaming cardinality for paradoxes instead of the axiom of choice. I'm treading into waters that I know very little about. I'm quite unfamiliar with the axiom of choice.
theboombody 1 week ago
I think always hate the way that textbooks etc. present the bijection as proof of equal cardinality and close the discussion there too, without considering counterarguments. A bijection is used to prove that the set of integers and the set of even numbers have the same cardinality. But surely one set being a subset of another set should be considered sufficient proof that the two sets have DIFFERENT cardinalities? I have never seen counterarguments addressed in a discussion of bijection.
johnsmithbsc 2 weeks ago
@johnsmithbsc Man, you state my thoughts very well. That's the whole reason why I made this video. I think Cantor invented the word cardinality because he KNEW the word size was inaccurate. Two sets having the same size is a much stronger statement than saying they have the same cardinality. Cardinality seems to mean "size class" more than size. But I see people tossing around cardinality, size, and bijection like they're all the exact same thing so often that I couldn't remain silent.
theboombody 1 week ago
Continuity is not a easy notion to handle. In reality, I am not sure if one can come up with a continuity example. Hence the intuition from reality won't be helpful to understand "size" of two different sets which has continuum of elements.
TKKTism 3 weeks ago
@theboombody. "Size" doesn't make much sense or have any use in areas like Topology. To a topologist, [0,1] and [a,b], a<b, are basically the same thing since you can continuously shrink or stretch it from one to the other. Hence, in terms of number of points, though you can't see it but both sets have same matching of elements with one more tightly packed together. This gives an intuition that, in context of uncountability, "size", per se, doesn't make much sense.
TKKTism 3 weeks ago
@TKKTism I agree. But I have seen in the past lots of people saying that infinite sets were the same size when a bijection existed between them. That's kind of why I made this video, to demonstrate that there's a clear difference between size and cardinality. Things that have the same size have the same cardinality, but things that have the same cardinality may not have the same size. That's my argument. Cardinality deals with bijections, size does not.
theboombody 3 weeks ago
The concept "cardinality" is simply a certain _defined_ property of two sets. It's not a good thing to call such sets "the same size", because these words bring a bunch of irrelevant connotations from everyday life which only confuse the issue. It's just a definition. OTOH if you define things differently (by assigning a measure or using some other construct), the segment [0,2] will be twice as large than [0,1]. It's a matter of context and deciding what's useful for a given type of problem.
JanPBtest 2 months ago
@JanPBtest I think my goal now is to get math people to differentiate cardinality from size. Some already do, but I know some don't. Most people who know it's different tend to use measure to describe size. I've never studied measure myself. Just cardinality. Same size implies same cardinality, but same cardinality does not imply same size. I haven't proven the first half of the statement, but it's my working hypothesis.
theboombody 2 months ago
x != x times 2
fail.
1 != 2
TITOR002 2 months ago
@TITOR002 I've always been suspicious of factorials... At least zero factorial.
theboombody 2 months ago
@theboombody::
TITOR002 was not expressing a factorial. The notation "!=" is internet speak for "does not equal" or "is not equal to" for anyone who does not know the ascii for a slash-equal symbol.
DornierPfeil 3 weeks ago
@DornierPfeil Thanks for the correction. If I were a bit brighter, I probably would have almost sort of learned something. Now I know that I agree with TITOR002, even though my video makes it look like I don't.
theboombody 2 weeks ago
This has been flagged as spam show
x/x = 1 holds for an infinite number of x, that's right. So you can't say x=1 is THE solution, that's perfectly right. But you can say that the set of x's such that x/x = 1 is THE (only) set of solutions.
So is cardinality. For any real number n, one can match (0, 2] with (0, n]. So there is not a single solution, and because of that, one cannot define a single "size". But one can define the set of solutions, giving it a name other than "size", so that's called cardinality
GiorgioDeRa 2 months ago
@GiorgioDeRa I agree.
theboombody 2 months ago
Comment removed
GiorgioDeRa 2 months ago
Cool, this is nice:) Good work! And you know this shows that size can only be defined by the size of other :) Don't let the negative envy votes take you down :) This is what math is all about!
steverock85 2 months ago
@steverock85 The debate has been interesting.
theboombody 2 months ago
@theboombody
Well, we are all struggling to understand this world. The only way to explain it is to find the missing link between infinity and finity. If one would succeed, then the same one might be able to explian how we came to be. We might even put the debate about god/s on the side once and for all.
steverock85 2 months ago
@steverock85 I believe certain things can't be figured out, particularly some things about God. But fortunately, many things can be figured out, including possibly this link between the finite and infinite that you speak of. There are some infinite series that converge, and some that diverge, and it would be extremely interesting to kind of find the line where divergence and convergence sort of meet. The harmonic series diverges so slowly it almost looks like it will converge.
theboombody 2 months ago
@theboombody
Superlike on how you think! :D I never even thought of that.
Yeah, I am agnostic about this god stuff but I have been shearching to understand for years and I get this feeling of a infinity wall that can not be crossed for understanding.
I have my hopes for the link between infinity and finity, but it is only a hope.
steverock85 2 months ago
@steverock85 I'm a fundamentalist myself. I understand why other people aren't though. The Bible has a lot of stuff in it that is certainly superstitious. But I believe it anyway.
Calculus is a pretty good link between infinity and the finite. Every time you use an anti-derivative to find the area under a curve, you're using a formula that was discovered by summing up the area of a infinite amount of infinitely small rectangles. It's truly a wonder.
theboombody 2 months ago
@theboombody
Yes that is something that we can use infinity for application. But infinity itself is still a little unknown like the gravity. It is there and we can use it, but we do not understand it really as it is not a real finite number but rather an idea.
steverock85 2 months ago
@steverock85 No argument here.
theboombody 2 months ago
I think the problem is that the word "size" is implicitly a finite term and has no place being used describing infinities.
Kreadus005 2 months ago
@Kreadus005 Yeah. Cantor, the mathematician who first examined this stuff, knew that the word size wouldn't fly too well with the infinite, so he used the word cardinality instead. But I get surprised at how often size and cardinality are used interchangeably. I believe if two things have the same size, they have the same cardinality, but I don't believe the opposite. At least not currently anyway.
theboombody 2 months ago
2 * Infinite is not > Infinite.
ZhangRed 2 months ago
@ZhangRed Fair argument. Uncountable infinities are bigger than countable infinities though. I had a thought tonight that perhaps two times infinity is not bigger than infinity and is also not the same as infinity. Maybe it's something inbetween. Maybe there's a measure of quantity somewhere that's between the terms bigger and equal. I can't imagine what it would be. Might shed some light on the continuum hypothesis though.
theboombody 2 months ago
It all depends on the size and shape of space itself.
AdamLore 2 months ago
A Tarski sphere is a perfectly elegant mathematical construct. That it doesn't make intuitive sense to you is of no moment.
Many people pick up where the textbooks leave off. In fact, far too many people do as much, and are almost always wrong.
integralmath 2 months ago
@integralmath Well said regarding the second statement.
However, I insist using element matching to compare sizes of infinite sets can yield indeterminate results though. If I match it one way, I get a two-to-one ratio, and if I match another way, I get a one-to-one ratio. Sort of like if I take a limit of 1/x at zero, I get a positive infinity on the right side, and negative infinity on the left. Implying indeterminism. I conclude cardinality and size are not equal.
theboombody 2 months ago
@theboombody yeah, okay. The solution is indeterminate - and? This doesn't admit of an answer; it only says we're not capable of finding a particular solution to a problem. In fact, you've gone from the proposition that a given system is indeterminate to a therefore x is true. Given that the proposition isn't even intelligible, there is no such therefore which can follow. Your conclusion isn't defensible given your process.
integralmath 2 months ago
@integralmath Sure, I'm not giving an answer, but I am saying there are flaws with the method that should be addressed and eventually corrected. And my argument was that if you use cardinality to determine set size relation between the particular example sets I gave, you get multiple answers. Multiple conflicting answers imply cardinality cannot accurately measure size because of certain indeterminate results. If cardinality cannot measure size, then cardinality is not size.
theboombody 2 months ago
@theboombody how many solutions are there to inverse sine of root 2 over 2? Oh, infinitely many. I guess there is a problem there too, huh?
integralmath 2 months ago
@integralmath Well, to solve x/x=1 for x would give you infinitely many solutions. That's not a problem. What is a problem is to say x=1 is THE solution when there are many, many others. So when someone says (0,2] is the same size as (0,1] through the element matching argument, they ALSO should say it's twice as big, half as big, three times as big, etc., because element matching could be used to show those results as well.
theboombody 2 months ago
bull shit.
henrythesuper 2 months ago
@henrythesuper Why do you think I discontinued math? Seriously, they really showed me stuff like this in the classroom.
theboombody 2 months ago
This has been flagged as spam show
1) First provide a definition, before saying "there's nothing wrong with my definition"
2) "if I match the same two sets in different ways, I get a different result" Yes, of course, it will ALWAYS work like that. For each infinite set, there's a bijunction from that set to a subset of its, and you can choose from an infinite number of different subsets of that set. This does not happen with a finite set, but it's not so weird, because in-finite sets are not finite and do not work the same... -.-
GiorgioDeRa 3 months ago
This video has helped me in my lifelong ambition to discover the diameter of an elliptical circle, infinite in size, and squared from pi. Darn if it isn't 666. I thank you sir!
gmdinformation 3 months ago
@gmdinformation At the very least this video exposed that there are more people with brains on youtube than we could have imagined. It's like every week someone who obviously has a higher education in mathematics posts a comment here. Where do these guys come from? I figured I'd only get one such comment a year. One a week is extraordinary.
theboombody 2 months ago
(...) You could give your own definition of set size, replacing #1, in order to make #2 valid even for infinite sets. But that would be your personal definition, which would not be Cantor's one, and textbooks & co. provide the definition by Cantor, which is widely accepted because its results are widely found to be useful.
GiorgioDeRa 3 months ago
@GiorgioDeRa If that's the case, I see nothing wrong with my definition. In fact, I speculate in another video that seeming contradictions that arise from the axiom of choice might actually stem from Cantor's cardinality definitions being incorrect. I have no proof of this of course. It's just speculation.
theboombody 3 months ago
(...) Double the set, and you'll get a different integer, i.e. size. Because they are different, it makes sense to distinguish between them, and say "size of A is twice the size of B".
The "size" of an infinite set is defined in terms of bijections between INfinite sets. Thus, it's NOT the same, and does not have to work the same. It's NOT a positive integer, and if you double the set, you get again the initial "size" as a result. This kind of size is what the world calls cardinality.
GiorgioDeRa 3 months ago
@GiorgioDeRa I'm just saying if I match the same two sets in different ways, I get a different result. So maybe I should measure with something besides matching to get an analysis of size as well. This is why I think the integers are larger than the even numbers, even though they have the same cardinality. Different cardinalities automatically imply different sizes, but the same cardinality may not imply the same size.
theboombody 3 months ago
2) "If a 1-to-1 bijection exists between finite sets, they're equal. If a 2-to-1 bijection exists between finite sets, one is twice as big as the other. Why would one extrapolation work and the other fail?"
Because #1 is a definition, while #2 is a consequence of that definition; more precisely, #2 is the fact that if you double a set, you get one with a different size.
The size of a finite set is defined in terms of bijections between finite sets; it's essentially a positive integer. (...)
GiorgioDeRa 3 months ago
1) While it is clear to me that by "size of a finite set" you mean "the number of points I see when I draw them at 1:13", I can't get what do you mean by "size of an infinite set", provided that it's not cardinality. The sentence "[0; 2] is twice the size of [0; 1]" does not make sense until you give a precise definition of what you mean.
GiorgioDeRa 3 months ago
God i fucking love/hate math
ShaddySoldier 3 months ago
@ShaddySoldier Can't blame you.
theboombody 3 months ago
The case where (0,2] is partitioned into the subintervals (0,1] and (1,2] to create two separate bijections between (0,1] and the subintervals is not unique. You could partition the interval (0,2] into any number of subintervals of equal distance (and perhaps of unequal distance) and still be able to make bijections between each of the subintervals and (0,1]. It should then follow that (0,1] is not only same size and twice the size of (0,2], but every whole number ratio.
asphyxiateDrake00 3 months ago
@asphyxiateDrake00 Absolutely. Glad you pointed that out.
theboombody 3 months ago
ALL THE COMMENTS ARE TL;DR
25hrspastmidnight 3 months ago
@theboombody I remember one of the things that shocked me most when studying that was proving that the set with all the finite subsets of N has the same cardinality as N. However it is consistent and in a certain way it makes sense, however it may be counter intuitive, but maybe it's only because we haven't trained our intuition xP
rjmantilla 3 months ago
@theboombody Still, talking about "twice" the size doesn't make sense when talking about infinite cardinality, it only makes sense when talking about finite sets because every finite set can have a biyection to X_n ={x in N, x<n}, and thus we can treat their cardinality as that n. However with infinte sense that notion makes no sense
rjmantilla 3 months ago
@rjmantilla If a one-to-one bijection exists between finite sets, they're equal. Using that logic, Cantor extrapolated it to infinite sets. If a TWO-to-one bijection exists between finite sets, one is twice as big as the other. If I extrapolate like Cantor did, it would follow for infinite sets as well. Why would one extrapolation work and the other fail? I think it's silly to say they both work, so I say they both fail. The only way it wouldn't fail is if cardinality isn't size at all.
theboombody 3 months ago
@theboombody Well stablishing a two to one biyection inmediatly generates trouble...... Natural numbers have a two to one biyection with themselves 0->0,1 ... 1->2,3.... etc... two to one biyection says that a set is twice its own size.... so the notion of double size is contradictory with infinte sets, the notion of same size isn't
rjmantilla 3 months ago
@rjmantilla Well, I think we've both argued our points to a standstill here. One of us believes establishing a two-to-one and a one-to-one bijection between sets follows the same rules regarding what's allowed, and the other believes that one of those bijections is allowed and the other is not. I don't see either one of us changing the other's mind no matter how much further into our points we go.
theboombody 3 months ago
@theboombody The size of a set does not depen on it's ordering nd that was what I was trying to show you, what I mean is that if we start saying stuff like "every proper subset of a set is smaller than the set itself", we will reach contradictions, which is different from a paradox. Math is filled with paradoxes, but that doesn't make it invali, however we must avoid contradictions. (Well Gödel's theorem states that math is bound to contradictions....)
rjmantilla 3 months ago
When talking about sizes of infinite sets there are a lot of things that are counter intuitive, but are true. The first thing that is really hard to understand is the fact that there are some sets which have proper subsets of the same size. There are equal amount of : primes, even numbers, natural numbers and rational numbers. What's even more strange is that you can make an infinite number of sets of infinite size, which all together have the same size as natural numbers.
rjmantilla 3 months ago
@rjmantilla It simply makes no sense talking about twice as big when talking about infinite sets
rjmantilla 3 months ago
@rjmantilla It's true that this is what math classes teach. But I never thought this made sense. I would instead prefer to say that the number of naturals is larger than the number of primes, just not unmatchably larger because you can create a bijection between the two. Both are still in the countable category. This would make the mathematics not quite so counterintuitive, and I think it would actually bring it closer to the truth.
theboombody 3 months ago
@theboombody there is no such thing as truth when talking about mathematics, there's stuff that's consistent and stuff that not, and we basically test to see if what we say is consistent to the basic axioms in math (which normally would be ZFC axioms). To set a definition of size we need to do something that works for both infinte and finite sets, and biyections work our deal, and it makes a lot of sense, as long as qe can match them we have the same amount of elements
rjmantilla 3 months ago
@rjmantilla I've always considered it inconsistent for a proper subset of a set to have the same number of elements as the original set. Whether it's infinite or not. I am unable to continue the established formal study of mathematics as long as this inconsistency (in my opinion) is accepted, because I can't follow any lines of logic that stem from that root. Right or wrong, my brain doesn't work that way. How can I move to step 5 when step 4 makes no sense to me?
theboombody 3 months ago
@theboombody If it weren't true that the some proper subsets have thes same number of elements of the set they are contained we would have a bigger paradox: take N...... take out the 0, that is a proper subset of N, and thus it must have less elements...... substract one from each element, that set will be N, and we already said that it had less elements than N, which makes less sense......
rjmantilla 3 months ago
@rjmantilla Well either way, we seem to be comfronted with something that defies intuition. In my opinion, mathematics should always be intuitive, since it almost always starts with intuitive principles that are refined into proper axioms. I agree that my suggestion doesn't exactly make sense, but at least I'm raising awareness in the potential of inconsistency being present in even the most consistent of the sciences. I'm inconsistent and you're inconsistent. Or we can throw out set theory.
theboombody 3 months ago
but construting math the way you propose gets us to a bigger paradox, let's say you take the natural numbers, and add a element at the end, lets call it X..... you say that N U {X} has to be bigger than N as N is contained in N U {X} however if take each natural number and add one to them, and then put X at the start, calling it 0, we would have the same size as N, and thus the size of the same set N U {X} would b different depending on its ordering, which would make no sense
rjmantilla 3 months ago
@rjmantilla I guess it's possible that mathematics itself is paradoxical and inconsistent, which kind of saddens me. But at least if that's true, it makes rare supernatural events more possible to accept. I mean, if logic isn't always right, nonsense might actually be more correct than logic on certain rare occassions.
theboombody 3 months ago
@rjmantilla This is probably the best comment I've ever seen to any of my videos. {0,2,3,4,5,6,7...} would be the same size as N. Maybe even {0}U{0,2,3,4,5,6,7...} would be. I don't see how one paradox can be bigger than another paradox though. A paradox is a paradox. I mean, the way we got it now, I can say (0,2) is the same size as (0,1) or twice as big as (0,1) depending on how I choose to match it. Is that any worse than saying a size of a set can vary depending on its ordering?
theboombody 3 months ago
@rjmantilla I was thinking more about this N U {X} example today, and wondered if the size of infinite sets could be measured another way. Let's say you have an infinite number of points evenly spaced along a number line, and let's say each point is a vertex of an equilateral triangle with sides less than the distance between points on the number line. The vertices of the triangles will always be three times the points on the number line, because both are increasing at constant RATES.
theboombody 3 months ago
However those kinds of thing you¿re thinking are also valid, and thus you'll see further on in math two important concepts, ordinals and measure. With ordinals you'll that in ordered structures it's different to take N and Z because of their order. And with measure you'll see that [0,1] has measure 1 but [0,2] has measure 2....
rjmantilla 3 months ago
*gets hot girl*
something in my body grew twice it's size...
mrBorkD 3 months ago
This video sucks. Just trivial math rearranged to confuse people uneducated in math.
miguelcastano 3 months ago
@miguelcastano It's more contradictory than trivial. I believe there are certain flaws within recently accepted mathematics that need to be corrected. Because if they're not corrected, they begin to lead to ridiculous conclusions (like something being twice as big as itself). To my surprise, most unbiased people tend to accept the ridiculous conclusion rather than question the math that leads up to it. There's also the possibility that logic is inherently broken and can't be corrected.
theboombody 3 months ago
* mind blown!
shapein 3 months ago
@shapein Check out the CHistrue channel for similar material. He's much better at this stuff than me.
theboombody 3 months ago
Does the requirement of equal cardinality--the words size and big have too many imprecise laymen connotations that render them useless in the study of infinities--between sets depend on the _existence_ of a one-to-one and onto mapping or does it require _all_ mappings to be one-to-one and onto? My suspicion from the memory of my training is that the former is the case. Also, 'twice as big' doesn't make sense in infinity math. 2 * "infinity" == "infinity" and 2 * "continuum" == "continuum"
jasonw137 4 months ago
@jasonw137 If twice as big doesn't make sense in infinity math, why should countability? In the finite world, if you have two sets, and you can set up a two-to-one and onto ratio between them, one set would be twice as big as the other. So if I can do that in the finite world, why couldn't I do it in the infinite? Don't Cantor's methods compare the size of infinite sets in the same way we compare the size of finite sets? I thought that's what the one-to-one stuff was all about.
theboombody 4 months ago
@theboombody @theboombody see e-mail for complete reply... character limits blow.
For one thing, countable implies a mapping to the cardinal numbers. So if one relies on the notion of countable, then sets 'bigger' than the cardinal numbers could not be 'sized'.
see wikipedia: Cardinality#Infinite_sets
see also amazon: Gödel, Escher, Bach: An Eternal Golden Braid
jasonw137 4 months ago
can something be twice as big as itself?
q x 2 = q
q = 0
0 x 2 = 0
0 is twice as big as it's self, 3 times as big as it's self, 4 times as big as it's self, etc
hmiller0 5 months ago
@hmiller0 So my video says an infinite set can be twice as big as itself, and you're saying a more or less empty set can be twice as big as itself. Interesting. These two ideas seem to be outside the bounds of quantification.
theboombody 5 months ago
@hmiller0 This implies that an empty set is twice as big as itself. But since an empty set consists of nothing, the implication is that nothing is twice as big as itself (although I'm mixing semantic with pseudo math here :p)
FHomeBrew 5 months ago
There's this homologous topic in topology where a sphere can be duplicated.
FHomeBrew 5 months ago
@FHomeBrew Is it that Banach-Tarski theorem thing?
theboombody 5 months ago
@theboombody Yup. Simply put: "A pea can be chopped up and reassembled into the sun.". Would be nice if this could be applied in physics. I can see the replicators now :D
FHomeBrew 5 months ago
@FHomeBrew Tea, very hot.
theboombody 5 months ago
0-1 = 0-2 if you multiply things in 0-1 in 0-2.
BUT YOU MULTIPlIED THEM
rerere284 5 months ago
@rerere284 Say you take the set of numbers (14, 6, 5, 12). There are four numbers in this set. Multiply every number in this set by two. You get the numbers (28, 12, 10, 24). There are four numbers in this set also. The two sets of numbers are not the same. They just have the same amount of numbers in them (four numbers). So the two sets have the same size (cardinality four). This works well for sizing up finite sets, but what about infinite sets?
theboombody 5 months ago
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sythax 5 months ago
wow is eyes are talking to me and the goat behind him on the framed picture,i gotta stop making my own l.s.d
qqaaaqqq 5 months ago
@qqaaaqqq Maybe you had your first drug-free hallucination. Those are the best kind.
theboombody 5 months ago
@theboombody ,i was just trying to be funny,i love the art.
qqaaaqqq 5 months ago
@qqaaaqqq Thanks. I call it G-rated Shock Value. I'm not very good at drawing, so I want other people to make it who can do it more justice.
theboombody 5 months ago
nice try alexander gram dumbbell.
awesomesauceyumyum 6 months ago
What REALLY doesn't make sense is why we are being lectured by an alligator/crocodile professor with angry blue birds' heads for eyes. Lol :)
iParry117 6 months ago
@iParry117 Actually I think they look more like cow udders. But you're right. That doesn't make any sense either.
theboombody 6 months ago
I don't understand the practicality or use it would have to follow this way of thinking. it's not that 0-2 was the same size as 0-1, it's just that it had the same quantity of points depicted in the same relative locations. This has no effect on size/scaling but rather a logical comparison of how many points one has in common being displayed with the other. So it is not twice as big as it's self either, it's just you can select different placement of points. Nice video :)
Shakeitupyes 6 months ago
@Shakeitupyes Sometimes things seem to be useless facts at first, but a use is found for them later. As far as I know, cardinality of infinite sets has no practical applications at this point.
I love your comment, and I agree with it more than you realize. In fact, I'm planning on uploading sort of a follow-up to this soon that somewhat addresses what you point out.
Do you have a mathematical background? I'd be amazed if you didn't.
theboombody 6 months ago
Infinity + Infinity = Infinity; NOT Infinity + Infinity = 2*Infinity.
Does that make it more clear @theboombody (Mr Gator).
Similarly .5*Infinty = 10*Infinity, finally Infinity is never less than Infinity (or some "multiple of infinity"). The logical problem with your statements is that you think Infinity (the number of numbers within (0,1) or (0,2) ) is somehow a number that you can count to.... you start counting.... when you get there, let me know :P
brydust 7 months ago
I seem to contradict myself when I say infinity + Infinity is not 2 infinity.... and then say that .5infinty = 10 infinity....
the only point is that any addition of infinity and any sum of infinities is still infinity, so after you do all the sums and multiplications, just drop the coefficients and realize its still just infinity........ Hope that helps, although I agree that this argument should be presented in text-books just to illustrate how counter-intuitive cardinal numbers can be! :)
brydust 7 months ago
@brydust I think the main focus of the matter should be, what's the difference between something that's counterintuitive and something that's false? Especially in an abstract science like mathematics where you can't do a lab test to verify a hypothesis?
theboombody 7 months ago
@brydust one cannot add or subtract from infinity, for infinity is infinity, there is nothing to add to it for infinity has already encapsulated all that there is. There is nothing to take away from it because infinity encapsulates all (where would the subtracted number go? what is the subtracted number? how can that be subtracted at all from that which has no number) the whole idea of attempting to use mathematical formula with infinity is chaos. infinity is not a number it is a concept. :)
Shakeitupyes 6 months ago
@brydust If one infinity can be bigger than another, as Cantor suggests, then I don't see why two times infinity can't be bigger than infinity. It sounds like you think all infinities are the same size. But it's pretty clear to everybody in the mathematical community that the set of real numbers is far more numerous than the set of integers. You do have a point though... the mathematical community also accepts that two times infinity is the same infinity.
theboombody 7 months ago
@HolymackerelProd Agree with what? That something can be twice as big as itself? I don't agree with that idea really. Cantor came up with the idea and I'm just trying to show how ridiculous it can be.
theboombody 9 months ago
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drrobertoboogie97 9 months ago
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drrobertoboogie97 9 months ago
@drrobertoboogie97 That's implicitly stated in the video. Perhaps not explicitly, but definitely implicitly.
theboombody 9 months ago
yes...... animals
whywhyhk 10 months ago
@whywhyhk Very big animals, or very small animals?
theboombody 10 months ago
@theboombody most animals grow twice its size during the time of the baby and fully grown.
whywhyhk 10 months ago
@whywhyhk Can they be twice as big as themselves at one moment in time?
theboombody 10 months ago
well this aleast sounds more reasonable that 1+1 = 1.9999999 which is a lie but according to instein evrything is relative so every1 is some how right as long as they are not wrong lol well {:-)
KiteFlyingGuy 10 months ago
@KiteFlyingGuy It's like the deeper you dig into reality, the more fake you find it to be.
theboombody 10 months ago
@KiteFlyingGuy wow i meispelled einstein as instein
KiteFlyingGuy 10 months ago
@KiteFlyingGuy Big deal. Content is more important than presentation.
theboombody 10 months ago
(continued)
But by definition, this has nothing to do with cardinality.
If we were to again take 10 of each, and say divide them into individuals, then we would only have 10 in each case.
the natural numbers are like 10 elephants, and the reals are like one apple. If we must respect the elephants as individuals (like integers) and we are permitted to go infinitesimally small with the apple, clearly we would get more pieces of apple than elephants.
100qwerth 11 months ago
@100qwerth Yours is the best explanation I've seen. I love it. If cardinality is not size though, then what is it used for?
theboombody 11 months ago
@100qwerth Actually, I'm not sure if your argument is valid. If I take a point from the set of [0,1), [0,2), and the whole real line, each of those points have the same "size." It's not like one point is bigger than another. So a piece from each of these sets is the same "size."
theboombody 11 months ago
They have the same quantity of elements in the set, but their physical size is different. If you had a set of 10 apples and a set of 10 elephants, they both have the same cardinality, but different physical sizes.
If we were to take apple and (godforbid) an elephant and cut them up into as many pieces as we can, you could cut an apple and an elephant into infinitely many pieces. You could make an argument that if you choose x many pieces, the size of a piece of elephant will be much larger.
100qwerth 11 months ago
Well put into infinity they have the same cardinality but not put into comparison with infinity they are not? Or am I wrong, I just assumed that if you have some numbers in a set that you don't necessarily have to show them into infinity?
Dolphidood 11 months ago
@Dolphidood According to today's accepted mathematics there are many levels of infinity.
theboombody 11 months ago
@theboombody Ooooh could you name some types please? Thanks
Dolphidood 11 months ago
@Dolphidood Well, if you count the numbers, 1, 2, 3, 4, and so forth forever, that's a smaller infinity than if you count those whole numbers and all the decimal points inbetween them, including the decimals that have digits that never end, like pi and so forth. That's the clearest example.
theboombody 11 months ago
@theboombody Okay, I see so the different infinities are really defined by the sets of numbers? I heard that if there is any chance of something happening within infinity then it will happen... how does that tie in exactly with infinite individuals leading different lives on other planets? Thanks
Dolphidood 11 months ago
@Dolphidood I think you're going by the idea that time is infinite, and even an event with the smallest probability of happening will eventually happen given so much time for it to happen. This I certainly agree with. I usually disagree with the idea of beings on other planets though because I think the odds of intelligent life flourishing are so bad that even trillions upon trillions of planets aren't enough to beat those odds. I'm amazed that we even exist.
theboombody 11 months ago
@theboombody Yeah, sorry, I should of been clear. I think its amazing too when you consider the complexity of the human body and mind and then calculate the chances of such a composition taking place in the universe under pure external factors. I wouldn't disagree with that thought because no matter how small the chances are if we have an infinite amount of planets and time then we will have that result EVENTUALLY. How can we have other infinite universes when ours infinite and leaves no space?
Dolphidood 11 months ago
@Dolphidood You bring up some very good points. We could go into a deep philosophical discussion on whether or not something will happen. For me though, probability is not a very factual science since I don't really accept that 0!=1, and it shows up a lot in probability. Probability is elegant and impressive, but definitely not flawless in my opinion. In fact, I don't think mathematics itself is flawless. I think there's a lot of contradictory ideas within it, and that's why I made the video.
theboombody 11 months ago
@theboombody I think if there are continuous circumstances with room for infinity then it would not be possible to make a flawless probability. I too don't think that mathematics or science are necessarily definite knowledge for everything... I believe they are an attempt to make sense of the world but far more useful in its short term application... also, apparently its been proven by Godel that for mathematics to be totally pure the axioms of which it is formed off must contradict each other.
Dolphidood 11 months ago
@theboombody ... that being said, it must be by that also impure... i.e. "This sentence is a lie"... I think its fair to say that people make many assumptions that they rely on heavily to fulfil their survival however when it comes to when it comes to the 'truth' of reality, I very strongly think that there is very little truth when I consider what it has to be to be truth... it HAS TO not leave possibility for any other 'truth' or otherwise its only a potential 'truth'.
Dolphidood 11 months ago
@theboombody Lastly, are we not simply fulfilling an aspect of the anatomy of the mind that the universe has given us... this aspect being logical thought which could simply be nothing more but a survival mechanism of which gives us a sense of satisfaction and drive that which I/we have made a concept of in the first place? Any how, do you have any role in science and maths?... I would love to be a research mathematician but I'm just seeking guidance on experiences of people..thanks for replying
Dolphidood 11 months ago
@Dolphidood Very interesting ideas here which I pretty much agree with. I stopped my formal mathematics study at a bachelor's degree for three reasons, First, the math was pretty hard. Second, I started to disgree with the math that I was presented. Third, I didn't want to invest the money to continue the studies. So now I'm studying some basic accounting in my spare time hoping to take some classes and make more money. You talk about an artificially complex field, that's it right there.
theboombody 11 months ago
@theboombody Artificially complex strikes me as theory based if that is the case then I think I've met my match. Although I agree with the uncertainty that logic itself brings with endless questions I would still really like to pursue maths in order to simply know more and equip myself. I look forward to watching more of your videos. I have to admit, as I had no interest when I was younger, I didn't get past GCSE level and I'm only now retaking for A* lol but it doesn't matter. Thanks again.
Dolphidood 11 months ago
@Dolphidood Well, some things are complex because they won't work otherwise, like computers. Other things like taxes work fine without complexity, but have complexity pumped into them to confuse people on purpose. I think this often happens to disguise criminal activity. I admire natural complexity even though I hate it, but I really can't stand artificial complexity.
I probably won't make any more math videos. I think I've said everything I wanted to with that topic.
theboombody 11 months ago
Your video misses the point. By definition, an infinite set is a set which forms a bijection with a proper subset of itself, so your drawings are misleading, as they instead depict finite sets.
ja524309 1 year ago
@ja524309 Since when did intervals depict finite sets? All intervals are infinite. The sets in my video are intervals. No other commenter disputes that they're infinite.
theboombody 1 year ago
@theboombody Two sets have the same cardinality if there exists a function which is both one to one and onto between the two sets, and is thus invertible. Infinite sets have the unusual property that such functions exist for both the set onto itself (the identity mapping) and onto a proper subset of itself. For example, the natural numbers can be mapped onto both themselves and the positive even numbers. Again, your visuals are misleading because they depict finite sets.
ja524309 1 year ago
@ja524309 By that argument every visual representation of an infinite set is misleading. Nobody can draw an infinite number of points on an interval. You have to use your imagination.
Why are infinite sets allowed to have the property that they can be mapped to their subsets and finite sets are not allowed to have that property? I don't think such a property should be allowed to any set because it makes no logical sense for a piece of something to be the whole thing.
theboombody 1 year ago
@theboombody Ah, now I think we've arrived at the crux of the matter. What do you mean by "logical sense"? Do you mean that they entail a contradiction? This is not the case. Attempt to prove me wrong if you wish. Moving on, yes, of course every finite depiction of an infinite set is misleading, unless, say, one use the usually "..." to denote it extends forever in the same fashion. Finally, I don't know what you mean by being "allowed" to do something. You need to clarify this.
ja524309 1 year ago
@ja524309 I do think something being twice as big as itself is a contradiction, yes. If A is equal to B, A is not twice as big as B. I know I'm simplifying things a bit here, but that really is the crux of the matter. Tell me how this simplication is so different from the scenario we're facing with the cardinality of intervals. For this reason I think it's a bad idea to study how infinite sets vary in size. It's straying away from actual truth. Are you okay with allowing such straying?
theboombody 1 year ago
@theboombody Having the same cardinality does not make two sets equal. In fact, it is a stronger notion than equinumerousity. Perhaps you meant sets having equal cardinality is a contradiction, when one is a proper subset of the other? Again, I ask you to show how this is so. You remind me of those people who reject imaginary and irrational numbers in history past, because they didn't feel they were "real" enough.
ja524309 1 year ago
@ja524309 Imaginary numbers have a terrible name. I'd rather call them semi-positive numbers. Since two positives multiplied give a positive, and two negatives multiplied give a positive, perhaps the square root of a negative number isn't negative or positive, but something inbetween. So I favor exploration. I'm just suggesting cardinality shouldn't be equated to size. Bigger cardinality means a bigger set, same cardinality doesn't mean the same size set.
Merry Christmas by the way.
theboombody 1 year ago
@theboombody Awe, thanks, you too. And yes, I agree: our intuitive notion of size and cardinality are two distinct things.
ja524309 1 year ago
@ja524309 It's funny how we do agree on certain things and not others. Inconsistency is what it is. And truth is more inconsistent than lies. That's a big part of my G-rated Shock Value philosophy.
theboombody 1 year ago
@theboombody So, who are you anyway? Like, where do you come from?
ja524309 1 year ago
@ja524309 I work for a funeral home in Texas. How about you?
theboombody 1 year ago
@theboombody Dude, that's kind of awesome! I'm a sophomore in college actually, majoring in maths! I've never been to Texas before!
ja524309 1 year ago
@ja524309 It's a nice state. Do you know what you want to do after college?
theboombody 1 year ago
@theboombody I would like to teach eventually, if I am able to acquire funding for grad school!
ja524309 1 year ago
@ja524309 Good luck to you sir. You seem to be quite diligent in your studies. I'm sure you'll be a very positive influence in the mathematical world. Just remember that being a negative influence is much more fun.
theboombody 1 year ago
@theboombody Haha so true.
ja524309 1 year ago
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DracOverLordHaton 1 year ago
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DracOverLordHaton 1 year ago
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DracOverLordHaton 1 year ago
Essentially you're saying that 2*Aleph = Aleph. This is a specific case of the more general case - for any 2 infinite cardinals a,b we have that a*b = max(a,b).
This also means that for any natural number k, the unit interval is k-times bigger than itself. In my opinion, this is why this fact is not mentioned in textbooks - it is rather meaningless.
Also, the definition tenchoosethree was talking about is indeed an equivalent, rigorous definition of an infinite set (where same size=bijection)
MyBrrr 1 year ago
@MyBrrr My bad, the infiniteness of both cardinals isn't needed in the first part -- only one of them needs to be infinite.
MyBrrr 1 year ago
@MyBrrr I just don't see why their "size" relationship is classified as the same simply because a bijection exists when so many other non-bijections can be used as well. I would classify their size relationship as indeterminate since in one instance you can make one set twice as big as the other, and in another you can make it three times as small.
theboombody 1 year ago
@theboombody But then you would lose any meaning cardinality has at all, and there would be no point to talk about it.
MyBrrr 1 year ago