There are infinitely many sizes of infinity! Let's explore this idea in five levels, ranging from tangible examples with hotel infinity to proving that there are infinitely many sizes of infinity.
It's crazy how we can have a hotel of infinite rooms of integers and natural numbers but if we try and have a hotel of all the real numbers between 0 and 1, it's not possible. Like it's weird how one set of infinity can be countable and another can't but yet it somehow makes sense.
I feel you...or intuition would say that infinity is just infinity, and there is none "bigger" than the other. But at the same time, you can't really argue against what is shown in the video because it indeed makes sense lol. I guess it really depends on your definition of "bigger" when talking about infinite sets? Infinity is fascinating and confusing.
isn't this just adding an infinite amount of guests to the infinite hotel which is doable. Just move everyone to odd rooms again and you have infinite empty rooms for infinite guests
The diagonal argument I've seen included an additional constraint - for example, always choosing the digit 5 if the digit on the diagonal is not 5, and 3 if the diagonal digit is 5. Otherwise, you can run into counterexamples such as 1.00000000.. and 0.99999..., which have no matching digits but are the same number. It's therefore possible for the constructed number to be part of the bijection even though it doesn't match its counterpart in a given digit. I think disallowing 0 and 9 in the constructed number avoids any such case, although there may be a stricter requirement that I don't immediately see.
I agree it would have been better if he’d given a more explicit algorithm for constructing a real that can’t be in the list, but there’s no problem with his argument, at least, insofar as it demonstrates that there are more real numbers than natural numbers. What you’re pointing out is that (countably) many reals between 0 and 1 have two decimal expansions, one ending in an infinite string of 0s and another ending in an infinite string of 9s. If you don’t rule out one representation or the other, the list will have duplicates. In that case, the purported listing of the reals represents a _surjection_ from ℕ onto [0,1] rather than a _bijection_ so in that case your assumption is simply that ℕ is _at least_ as large as [0,1] rather than the _same size_ as [0,1]. And the diagonal construction shows that that assumption is false and, hence, that ℕ is strictly smaller than [0,1].
I often have to cringe my way through RU-vid videos purporting to teach us about logical paradoxes, the infinite, etc, but this was really first-rate. Super clear and not a bit of irrelevant information.
Great video. When I saw the title I thought you were just going to talk about the cardinalities of natural numbers vs real numbers; most people who talk about this topic stop there. I'm happy to see you went a bit further!
Another great video by Dr Sean about a maybe well-known, but often enough misunderstood topic. I personally like the expression 'hotel infinity', which I have never heard before for 'Hilberts Hotel'. In general: your explanations are always very clear, straight forward, simple and easy to understand, and so are the supporting graphics and animations. Very helpful, especially for 'aliens' like me, who are not native English speakers (sorry for any linguistic oddities, this may have caused 😉). I hope the range of your channel will grow as well as the number of followers. I'll stay tuned! Good luck and keep going 👍! 🙂👻
Love it! I’m going through infinite series with my calc BC students, and this will be a fun little diversion. I can’t believe I just called a cardinality discussion a little diversion 😂
A way that helped me understand real number's infinity, was to think about all the numbers between any integer, say 0 to 1. If we were to count all numbers between 0 and 1, and assume we could count a finite amount of aome sort, it will clash with the fact that 1 exists. The only way for 1 to exist after 0, is if there is no existing end between them, otherwise imagine a "void" of some sort a little before 1, making all of math fall apart. Its a good thinking exercise.
I was going to ask about the power set of the empty set before I realised that the power set of the empty set *is* the empty set and *not* the content of the empty set itself, and therefore has one element, which is more than no elements. So it's true in the trivial case, which is always good to check.
Well... It seems the Hilbert Hotel example is indeed a paradox. That is, to show that two infinite sets can be paired up, it suffice to identify them as countably infinite, and we know they are the same size. The paradox comes from that you can add or subtract a finite number of objects in both sets, and they still remain equal in size. This is due the rule to identify infinite sizes with Aleph numbers, so if you have two sets with the same Aleph, you can "pair them up" so to speak, and there will be nothing left. But that means that size of an infinite set doesn't have directly to do with exactly how many objects there are in a set. If the number of objects are |N|, the amount of the natural numbers, and you add 55 objects to it, it can still be "paired up", one object at the time, with itself without having those extra 55 objects. It doesn't matter if you add an infinity of new objects to it, it will still have the property of being able to "pair up" all the elements with the original |N| objects. Only if you try to "pair up" the elements to a different Aleph set, you can't do it. So, is there a more precise way of comparing infinite sizes than by Cantor's cardinal numbers (Alephs)? Yes, there is. It is called Numerosities of labeled sets. So to solve the paradox of Hilbert's Hotel, you could use Numerosities of labeled sets instead, which makes more sense if you work with physical entities. It simply use labels for the elements of the sets, 1, 2, 3, 4, ... and calculates if there is a way to pair up 1 with 1, 2 with 2, and so on, for all the elements in the two sets. This leads to a more precise way of dealing with infinite sets, so that in the Hilbert example we simply don't push the problem "out to infinity"; if you add one more guest, you can't assign that to any of the natural numbers that the rooms are labeled with, and the match is not possible. Which method to use to compare infinite sets depend on the situation. Sometimes it suffice to just consider infinity as all sets that are not finite, sometimes you separate them into positive and negative infinity, and sometimes when considering elements that can be labeled in an obvious way, you may use Numerosities of labeled sets, and then you can just pair up the same labels in both sets to see if there are elements over in one of them.
Are there more numbers between 0 and 1 or above 1? There are exactly as many. You can create a bijection where every number from the first set corresponds to its reciprocal in the second set. 1/33, 9√21/(9√2), you can find a pair for every single number, one will be greater than 1, the other will be between 0 and 1.
For the last prove, what if R is already paired with some number? There is no way to include a real number that is not in R, so how do we prove that the new set that we've constructed cannot be R itself?
So what's the cardinality of the infinity that counts infinities? Is it countable, since we're seemingly going through the natural numbers of orders of infinities?
I’m beginning my masters degree in maths this fall, so I pretty much knew everything you were going to say up until level 5. Analysis has been the hardest but by far the most beautiful subject I’ve ever learned. Looking forward to Lebesgue integration and measure tbeory. I see you study advanced probability theory and stochastic processes, that’s actually where I’m planning to concentrate my studies when I get my PhD. I’m a dual major in psychology and mathematics with a minor in stats, I start an applied math MS this fall but plan to supplement my graduate electives with pure math courses in real analysis. I would like to go into further graduate study particularly in probability theory after my masters. I love the connection between real analysis and probability theory.
Good explanation. Here ist the real deal: Look for "Ridiculously huge numbers (part 1)" from David Metzler. This goes all (almost) all the way up.... and it is humongous in the end of the lecture series.
Power set allows us to get higher and higher in cardinality, but is there any cardinality that we can't reach this way? Some "Level 6" that is not achievable by creating the power set, over and over, countably many times? Can one replace the nesting of "taking the power set" operation with other type of operation, that will now result in uncountable amount of times we made a power set? What about the arbitrarily big amount of operations? Etc., as it feels like there should be more than just 5 levels 🙂
What you’re describing is more or less the idea of an inaccessible cardinal. The obvious next step but it described in the same simple and straightforward way as the levels in the video.
@@TXLogic You will not get to an inaccessible cardinal by starting from the smallest infinite cardinalt and iterating the power set operation countably many times. You will get to a cardinal called ב_ω.
I think that this is a sound explanation. I have read or seen these examples many times and for me it is not easily grasped. I think that I understand better what is going on. So, we need to define a mapping or a rule if you like that does the job. If we can then we are good.
The cardinality of the complex numbers is equal to the cardinality of the Real numbers. This is because complex numbers are R^2. And R^N has the same cardinality as R
I've never understood or liked the Hotel Infinity argument. To me, it starts with a lot of assumptions. Not saying it's wrong, but it's never convinced me. I feel like it more explains our mathematical conventions than an objective reality. When I say assumptions, here's some of the ones I see: 1. That we can represent all the numbers with "..." all the way to infinity. 2. That it's possible for all the rooms to be "full" (in fact, the next step defuses that) 3. That you can just move every number over. Why is that assumed to be possible?
Perhaps you should check out the Zermello Fraenkel Axioms. They are the assumptions we make in here. If you don’t agree with those assumptions that’s fine. But if you do, then this video is a consequence of them
A = {0,1}, P(A) = { }, {0}, {1}, {0,1}. P(P(A)) will now use these four sets as four new values to construct the next power set: P(P(A)) = {{ 1 blank set: [ ], 4 sets of 1: [{ }], [{0}], [{1}], [{0,1}], 6 sets of 2: [{ }, {0}], [{ }, {1}], [{ }, {0,1}], [{0},{1}], [{0}, {0,1}], [{1},{0,1}], 4 sets of 3: [{ }, {0}, {1}], [{0}, {1}, {0,1}], [{ }, {0}, {0,1}], [{ }, {1}, {0,1}], 1 set of 4: [{ }, {0}, {1}, {0,1}] }} This is a total of 16 sets in P(P(A). Normally we could drop some parenthesis for simplicity sake but it's already hard enough to explain this via a RU-vid comment so I thought I would be as accurate as possible. Hopefully this helps.
It's another set of sets. It helps my intuition to relabel things. Consider his example that started with the set of two fruits {apple, orange}, and then generated the power set { {}, {apple}, {orange}, {apple, orange} }. That power set had four elements, so for simplicity let's relabel them like so: {A, B, C, D}, meaning that A={}, B={apple}, C={orange}, D={apple, orange}. Now, what's the power set of the power set? It's all possible subsets of {A, B, C, D}, so it will include {A, B, C} and {B, D} and {A} and...so on and so forth.
It's a shit load of things. If a set has n element, it's power set has 2^n element, meaning the power set of the power set has 2^(2^n) element. For example, the power set of the power set of a set with 10 elements is gonna have around 10^308 elements . If you consider the power set of the power set of the naturals, any list of different subsets of N is an element. For exemple {N, the even numbers, the primes numbers, {1,2,3}} is an element of this set. You can see there are a lot of possible choices.
How can the powerset of the reals not be a bijection when there are uncountably many values on both sides? How would you prove that there are more subsets that haven't been listed than there are real numbers? Couldn't you use the proof from 4 alongside the proof from 5 and say both sets are just as incomplete as each other?
The prove you are proposing would show that both R and P(R) are bigger than N, but that doesn't mean they are equal in size. For example it is true that 3 < 4 and 3 < 5 but that does not mean 4 = 5. The existence of a set with bigger cardinality than real numbers is just a consequence of the definition of a set's cardinality that we chose. It is not intuitive
My thoughts are thus: The Power Set of the Reals has the same cardinality as the set of all possible graphs in all possible dimensionalities. What comprehensible set has the same cardinality as the power set of all possible graphs in all possible dimensionalities?
The power set P(S) of a set S is just equivalent to the set 2^S = {0,1}^S = S->{0;1} of all maps from S to the set {0;1}. If the set {0;1} is too simple for your intuition, you may use the set R of reals, if #S >= #R. Then, you have the set R^S = S->R of all maps from S to R, for S a set of any cardinality #S >= #R. In your special example, look at the set of all graphs with the set R->R of all possible graphs from R to R as the domain, and R as codomain. This can be written R^(R^R) = (R^R)->R = (R->R)->R. More generally, use (...((R->R)->R)->...R)->R, for example. With 2 := {0;1}, N->2, (N->2)->2, ((N->2)->2)->2, (...((N->2)->2)->...2)->2 would be the minimalistic sequence of power sets expressed as sets of maps. Note that P(N) = 2^N = N->2, with N the natural numbers.
What an excellent question! It turns out it has a very interesting answer! The answer is that there is no "set of all infinite cardinalities". There are technical logical reasons why this collection is "too big to be a set". So, from a technical standpoint, there isn't a "cardinality of the amount of different infinities". But as I mentioned above, it's "too big" to be a set, so in some sense, the amount of different infinities is bigger than any of the infinities themselves.
But: f(1) = () f(2) = (1) f(3) = (2) f(4) = (1, 2) f(5) = (3) f(6) = (1, 3) f(7) = (2, 3) f(8) = (1, 2, 3) By f(2^n), we have included all subsets of (1, 2, …n). This means that the set will eventually become the set of natural numbers at 2^(cardinality of natural numbers). For this to have the same cardinality as natural numbers, we need to have a 1to1 correspondence between the natural numbers and natural powers of 2, which exist. Therefore, the power set should have the same cardinality as the set, at least for natural numbers.
First and foremost, your map clearly does not work for any infinite subsets of N: eg. the even natural numbers {2, 4, 6, 8....}. Rather than directly proving the |P(N)| > |N|, you might find it easier and more enlightening to instead create a bijection between P(N) and R. Hint: To simplify things, think of the binary representation of every real number. Could we do something with the digits?
The answer is truly magical - those set's cardinalities are equal. The proof of that fact is too complicated to be written in a comment but I can link you a good one if you are interested. Also, fun fact, if you are wondering if there is a set which has bigger cardinality than N but smaller than R, there is no answer. It was proven that you can neither confirm nor denie this hypothesis
@@feliksporeba5851 The set of transcendental numbers has the same cardinality as R. The set of non-transcendental numbers is countable I believe, but I'm not sure if that has been proven.
@@ozzymandius666 Yes, the set of algebraic numbers has been proven countable. You can use a similar argument to Cantor's proof that the rational numbers are countable to show that the set of all polynomials with integer coefficients is countable. So you can create a "countable list" of polynomials with integer coefficients. And then each one of these polynomials has only finitely many roots. So, you can go through each polynomial, one at a time, and go through all roots of that polynomial one-by-one, and that will enumerate all algebraic numbers.
P(N) = 2^N = N->2 = N->{0;1}. Write each r in R as a binary number. Then, the z-th digit of r is either 0 or 1, and you get a correspondence between R and the set of maps Z->2. With #N = #Z, the sketch of a proof shall be completed for now. You actually get some duplicate representations of the same number r, and infinte non-zero digits in positive and negative z-th digits don't correspond to real numbers. So, strictly speaking, we have only shown #P(N) >= #R, and you also have to prove that #R
Weirdly when I took the introduction to higher math class, most students got this concept easier than they did simple truth table problems. For whatever reason, people couldn't wrap their head around how ridiculous, nonsensical statements could be "true" so long as they fit the proper form in the truth table.
Seems to be beginner's level. Church numbers are missing, still within set theory. Next, any set is small. Large is an ensemble that is larger than any set, for instance the class of all sets. Then, of course we can define the ensemble of all classes, which is not a class, and so forth for a finite hierarchy. Then we unify all those finite hierarchies to the omega-th hierarchy level. We continue with the (omega + 1) th level, and continue with the transfinite ordinal-th ensemble levels. This is beyond any infinity of set theory.
1:46 There's also a bijection between the whole numbers and the positive rationals! And no, not the way you show at 3:13 (I mean the Calkin-Wilf sequence- I just didn't specify)
This ignores the Axiom of Choice. Isn't this like saying "That given a straight line on a plane and a point not on the line, there is always one exactly one line through that line that does not intersect the original line.", without mentioning that this an axiom and therefore an assumption, and there could be zero lines or infinitely many lines depending on your assumptions? There could be an infinite number of sizes of Infinities as you said or there could be 1 size depending on whether you choose the to assume the axiom of choice or not. I realize that this would not be possible to explain in a 15 minute video but I think it should be mentioned. I could be wrong it has been a few decades since I studied this but after a quick look at the Axiom of Choice, I think it is true.
The Axiom of Choice isn't used to prove Cantor's Theorem, so there are infinitely many transfinite cardinal numbers regardless of whether you accept or reject the Axiom of Choice. The Axiom of Choice makes cardinalities _cleaner_ by forcing transfinite cardinalities into a total order. Without the Axiom of Choice, some transfinite cardinalities may be incomparable.
That’s literally the point of Cantor: he showed without a doubt that assuming you can list all real numbers implies that you can create a real number that turns out to be distinct from every real number on the list, which is absurd. Therefore, the assumption that you can list them all must be wrong
Probably gonna lose an infinite amount of money from all the guests having to change room and wanting their money back. Good thing they make an infinite amount of money tho 🤷🏻♂️
I'm not convinced infinity is real - in that, it doesn't seem to occur in nature. I always wonder how a mathematical construct with no basis in the natural world (infinity, imaginary numbers etc.) can nonetheless be used to model it so effectively. However, I also wonder if all of the counterintuitive or paradoxical aspects of working with infinite sets is only possible *because* infinity is an invented construct that doesn't exist in the natural world.
Infinity is an axiom in set theory. It doesn't result in paradoxes. You can also do math without infinity. Similar with the various continuum hypotheses.
I wonder about things like this too. I find it helpful to remember that mathematics is a branch of philosophy, so there's a whole mathematical universe of concepts that don't care about what happens in the natural "real world". Since mathematics only models the real world, I'm reminded of the sentiment that all models are wrong, but some are useful.
The hotel of infinity can host infinity^infinity (countable of course), that is to say, each one with an arbituarily long integer sequence and each integer of them is arbituarily big. The way to host them is make each (x1, x2, x3...) into room number (2^x1 *3^x2 *5^x3 *... *pn^xn *...)
this only works for finite sequences as otherwise the room number mustn't be a natural number. if you allow infinite sequences which is usually meant by IN^IN (the set of functions from IN to IN, precisely the sequences in IN), then it allows you to do a diagonalization argument to show that this set is uncountable
@@zhihuangxu6551 “but omega^omega has the cardinality of aleph_0” - This is true, all "small" ordinals are countable, including all ω^...^ω. Your original statement is unclear wrt the meaning of "infinity^infinity (countable of course)". It seems from your argument that you're considering a map from the set of all subsets of ℤ⁺ onto ℤ⁺. If so, the argument fails because this function may not be a bijection: the set of all subsets is precisely the definition of power set (then follow the argument in the part 5 of the video). Transfinite ordinals aren't relevant to it.
@@cykkm My idea about the infinite hotel is as follows. (Natural language, and then the corresponding mathematical description) 1 new guest = 0 (Guest No.0). Infinity hotel can host (when already full, of course. The same below). 69 nice new guests = 68 (Guests No.0 through 68). Infinity hotel can host. An "Infinite Vehicle" = omega (where each guests has a finite seat number, e.g. Guest No.419) = the first dimension of infinity. Infinity hotel can host. (All instances of "infinity" are countable unless specified.) An "Infinite Fleet" of infinite "Infinite Vehicles" = omega^2 (where each guest has their unique finite car number and finite seat number, e.g. Guest No.419 of Car No.69) = the second dimension of infinity. Infinity hotel can host. An infinite number of "Infinite Fleets" = omega^3 (where each guest is assigned with a finite fleet number, a finite car number and a finite seat number, e.g. Guest No.42 of Car No.419 of Fleet No.69) = the third dimention of infinity. Infinity hotel can host. ... The 69th nice dimension of infinity = omega^69 (where each guest is assigned with a sequence of no more than 69 nice finite positive integers). Infinity hotel can host. ... Infinite dimensions of infinity = omega^omega (where each guest is assigned with an arbituarily long sequence of arbituarily big finite positive integers, removing the restriction of "no more than"). Now, the end of the infinite hotel story introduces the diagonalization argument and tells us that infinite hotel cannot host all real numbers, or all irrational numbers. My opinion is that the infinite hotel cannot host an infinite set of guests if and only if the description of the set mensions one or more uncountably infinite sets (and the uncountability does not in some way cancel out). My construction of "infinity^infinity" above does not encounter any uncountably infinite set. Having nothing to do with uncountably infinite sets in any manner, I assume it manageable for the infinite hotel and not fot the diagonalization argument.
It seems you're using different axioms that support your opinion. If you know anything about axioms, I'm sure you know that you can't call this madness just because it's based on different axioms.
This argument is too glib, because of the existence of Skolem models. While it is true that in a normal set theory, you can always find an infinite hierarchy of infinite sizes, it isn't so great to view the higher cardinalities are objectively larger than the reals, because nearly all the objective questions you can ask about sizes of powersets are undecidable due to forcing. This is a very important point, it's related to what made Cantor's argument controversial at the time it was proposed. It is no longer controversial only because familiarity breeds contempt, it really is a nifty, but optional, point of view. You can treat the higher cardinalities in a different way than that developed by Cantor, and keep the power of the higher cardinalities, perhaps even gain more power, but this is not yet certain.
I have a problemi with the idea of applying a finite procedure to an infinite set. This seems contradictory to me. Any procedure that matches or changes elements of sets is finite (in actuality you need to physically match or change each element of the set, so you need finite steps). Sure, the reason for doing this is applying the /same/ procedure to the whole set. So in theory we can think we can do it, because we understand "sameness" and we think we understand "the whole". But this reason just moves the goalpost: "sameness" can be determined of finite things by comparison. "The whole" is undefined when it is an infinity, because infinity by definition lacks a limit. We just assumed we can apply a finite idea to infinite things again. There is a formal contradiction between the concepts we implicitly assume when we use Cantor's procedure. So of course the demonstration of higher cardinalities can be done only via negation. All it reveals is a contradiction between two ways of applying finite procedures to an infinity. But this contradiction derives from the contradictory assumptions: we are trying to do two opposite things at once, so we think we get opposite results. All we have demonstrated is a contradiction in our way of thinking infinities.
@@kazedcat See I'm ignorant of higher level maths, a friend had mentiones the axiom of choice but didn't explain it. He just mentioned you can use it to get two spheres out of one sphere, all with the same surface. Thanks.for telling me that. I'll look it up. As for the consequence of rejecting useful maths like calculus: shouldn't logical consistency be more fundamental than pragmatic functions? If there is a logical error, you end up carrying that error along to unexpected paradoxes and categorical errors.
@@andreab380 axiom of choice is consistent either way. You can build consistent mathematics by accepting it or by rejecting it the only issue is usefulness. You can build a mathematical system where the only number is zero. It is very consistent but also very useless.
@@kazedcat Please remember I'm stating this in ignorance so I do it humbly. I'm not sure about consistency. (EDIT: Talking about consistency of the foundations rather than their consequence on the systems they found). It seems to me that, as in the example of Cantor, the results of an inconsistency can always be interpreted as proving something via reductio as absurdum, then giving a new name to unintelligible* concepts (such as transfinite sets) that result from said inconsistenxies, and then using these newly named absurdities* as regular mathematical objects. (Unintelligible and absurd only with reference to the non-adherence of finite procedures to infinities.) This still definitely produces intellectual results, I can accept that. Working on paradoxes can provide loads of intellectual instruments. So I won't deny utility, of course. It would be arrogant of me. Absurdities and paradoxes can give us technical instruments that adhere to reality after going the long way round (like when in physics they accept infinities as long as they cancel out in the end). But direct adherence to reality (and its logical conditions) is a different thing to me.
@@andreab380 If we are talking about reality then there is no problem of deconstructing a sphere and assembling them back into to more spheres after all this is already reality with quantum mechanics. If you break up quarks you end up with more quarks. The problem is you think common sense is reality when the real reality agrees more with weird mathematics that is in conflict with your common sense. Reality is weirder than mathematics which in turn weirder than your common sense.
I would claim sets are built from images. But first I will show that numbers are built from images Example , 4 always represents 4 images, like 4 squares for instance. To be specific numbers are "labels" for groups of images 1. The main idea here is that maths is built from images (a) example , geometry is clearly made of images b) example 2, We claim numbers are built from images too, as say 4 , always represents 4 images, like 4 squares for instance. C) imaginary numbers are connected to images too , which is why they have applications in physics D) In general any mathematical symbol that comes to mind is connected to images too.
I think we have a conceptual problem if we assume that there is something larger than infinity. Like saying that every natural number moves in hotel to another room, it is a contradiction, because you have to think limit there, finiteness, but it is infinite. It is impossible for *all of them* to move, because when you think that all of them have moved, there is always someone who says that I didn't move. Thinking of limit seems to lead conceptions like Numberphile's -1/12, which according to them is the sum of all positive integers. It is nonsense.
This kind of thinking does not lead to the sum of the natural numbers being -1/12. This concept is perfectly well-defined. If you have a problem with moving all of the numbers, then you should have a problem with having all of the numbers in the first place. Most infinite things don't seem to exist in reality. If you limit yourself to only finite collections, then you get a much weaker form of mathematics.
@@mikahamari6420 I think you’re focusing too much on language. When we say some infinity is “bigger” than another, we’re just saying that some infinite sets can’t have bijections to other infinite sets. For finite sets, that does have the intuitive interpretation of some set being “bigger” than another. So that’s why we started using that word in general, as a shorthand way of saying “one of the sets can’t have a bijection to the other”.
Oh, and btw, 1+2+3+ … = -1/12 makes sense when you’re talking about the analytical continuation of the Riemann Zeta Function. It’s not nonsense, but we have to make clear what our context is, which numberphile did not make clear
@@AndresFirte Yes, the set of positive integers is infinite and also the set of negative integers is infinite. Also the set combining both of those is infinite, but not larger than any of them.
Studying infinity in this way is actually very useful. Not ridiculous at all, it revolutionized math and has been very useful. Here are some examples of it **Example A:** It’s the basis for other branches: The study of infinity is fundamental for Measure Theory, which is itself crucial for developing other branches like Analysis, Calculus and Probability, *which have tons of applications in real life.* So infinity helps us to be able to study these topics in a *precise and rigorous* way **Example B:** It helps to verify that the axioms we use to study mathematics are indeed able to be used to study mathematics: we study math by assuming a set of axioms (rules) and studying the consequences of those axioms. However, our axioms could be contradictory, and we could get things like 1=2 if we aren’t careful. If that happens, it means that our axioms are contradictory and can’t give us any meaningful information about the mathematics that interest us. One of the sets of axioms we use is the Peano axioms. And we can use infinity (ZFC axioms in particular) to prove that the Peano axioms aren’t contradictory. We can even use infinity again to prove that the ZFC axioms are not contradictory. So we are pretty sure that the math rules we are currently using aren’t going to suddenly break one day. **Example C:** Infinity can even solve questions about *finite* numbers: there’s some questions about finite numbers that can’t be solved unless we start using infinity. For example, Goodstein's theorem can only be proven if we use transfinite ordinal numbers. And if I recall correctly, Goodstein’s theorem has applications in Computer Science. **Example D:** Cantor's work can be used to prove Gödel's incompleteness theorems, which answers very important questions about mathematics, philosophy and even engineering. These theorems helped answer some questions related to the goals of the Hilbert's program (which is very related to example B) that talk about the essence and limitations of mathematics. Gödel's incompleteness theorems can also be used to give an answer to the Halting Problem, which has many implications in real life, particularly in Computer science and Software engineering. For example it has implications in *cybersecurity* (I don't know the specifics of it) and lets us know that *some* algorithms don't exist and it's better to focus effort in finding heuristics solutions to *some* problems instead of wasting time searching for a perfect algorithm that literally doesn’t exist. (As a fun fact, it was later found out that one of Gödel's incompleteness theorems is equivalent to the Halting problem)
Hi! You posted a very interesting comment, so I left you a long and detailed reply. However, youtube sometimes thinks that long replies are spam and hides them So *please* let me know if you can see my reply. If you can’t, then I can try posting it in parts so you can read it. Thanks.
@@AndresFirte this is the one and only comment of you that I see up to now... I guess your lengthened response didn't make it through -.- yet another one of youtube's quirks... that or I'm grossly misjudging the length of your reply... makes me all the more curious ^...^ though I could imagine what some of your argument could be, I'd only suggest that numbers should be seen more like a placeholder that helps us grasp the concept of infinity better... like a label, so despite the kind of set of numbers you decide to use... integers, fractions, decimals of any kind... they all still describe one and the same thing, infinity... a continual expansion of value with no means to end... so it's kind of unnecessary to put one infinity in comparison to another when we could as well just change their "labels" and they would still be valid
@@OLBICHL Part 1 it’s not ridiculous, Studying infinity in this way revolutionized math and is actually very useful. Here are some examples of it **Example A:** It’s the basis for other branches: The study of infinity is fundamental for Measure Theory, which is itself crucial for developing other branches like Analysis, Calculus and Probability, *which have tons of applications in real life.* So infinity helps us to be able to study these topics in a *precise and rigorous* way
RU-vid already has plenty of videos on the same subject, and you're doesn't add anything new or explain things in a particularly better way. Not much of a point to this video
Cantor's 1891 paper is almost never taught correctly. As an example of this, in it he states _explicitly_ that he is not applying it to the real numbers. It can be used with them, but Cantor didn't. This is not very significant. The important one difference, between what is taught and what Cantor did, is the contradiction. That part is wrong. And it is easiest to demonstrate in the abstract. Let's say that you want to prove that the statement A&B cannot be true. So you say "I assume that A&B is true," and then you prove that A-->not(B). This is not a proof by contradiction; it is a direct proof of a lemma that, with an almost immediate next step, proves that A&B cannot be true. In Cantor's Diagonal Argument, statement A is "We have an infinite list of real numbers in [0,1]" and statement B is "This list contains all the real numbers in [0,1]." If you read that 1891 paper (there is a translation in a wiki at the logicmuseum site, but youtube doesn't allow links), he never assumes that all elements of his set M are in his list. He just proves that if you have a list, there is an element E0 of M that is not in that list. That is, A-->not(B). The completion of the proof is by contradiction. Quoting from that site, "From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing E0 would be both an element of M, but also not an element of M."
I disagree with the main part of this. The definition of infinite is limitless. The meaning of infinite as a value is there can be no value greater. If you either limit infinite or make something greater, you have violated the definition of infinite. This isn't a math argument, its a logic argument. Cantors argument is a logical fallacy because it requires the decimal real numbers to have a fixed or limited length. It also requires the grid to be a square. Consider 3 digit decimals like .000, .001, .002 and so on. The grid is not square, its 3 across by 1000 long with .999 being the last one. Only by imposing fallacious rules, like decimals have limited length and that the grid of all reals is a square dos this argument work. In an actual list of all decimals, each item has an infinite length and the list of rows is 10x that size in length; there is no diagonal and all values are listable. In a logical perspective, there is no limit to the list of whole numbers. It has infinite and uncountable size, just like every other infinite set. The Power Set argument is just as illogical. In fact, you showed the opposite of it with your first statement: the whole numbers are an infinite set (we agree). Adding 0 to it does not increase its set size (we agree). Adding negative integers to it doesn't increase its size (agree). Adding increases its size. I say that's false, as the set is already infinite which means limitless. To say set A > set B in cardinality requires there to be a limit of set A, which means its not infinite, which contradicts the conditions of the statement. I realize that math majors need these constructs (countable, uncountable, etc) for certain areas of math, but it doesn't make it logically accurate. I expect someone to prove all infinite sets are the same size and this logical fallacy can be put to rest.
“The definition of infinite is limitless” That may be the way it’s used in everyday life. But in mathematics (particularly in Set Theory), the word “infinity” is used to refer to a very precise and rigorously defined concept: Infinite Sets. What is an infinite set? Again: this is a technical concept. So don’t try to deduce it’s meaning from its name. The definition is this: an infinite set is a set that can have a bijection to a proper subset of itself. That definition is probably too technical and confusing. Which is precisely why we chose the name “infinite set” for it. Because it signals that, in SIMPLE words, it is a set that has an infinite amount of elements in it. But if you want to understand it in depth, and not just with simple words, you would have to fully understand the technical definition: understand what is a set, a proper subset, and a bijection. My point is: the video is correct, because we have to keep in mind that we’re in a technical context, where concepts like “infinity” are used in a different way from what you may be familiar with. So don’t guide yourself too much by the names, and try to understand the concepts instead.
“Cantor’s argument is a fallacy because it requires the decimal real numbers to have a fixed or limited length” Actually, it’s the opposite. Cantor’s argument works with the Real numbers precisely because he took advantage of the fact that they have an infinite amount of digits each.
Cantor was correct, and his work revolutionized math, and has been very useful. **Example A:** It’s the basis for other branches: The study of infinity is fundamental for Measure Theory, which is itself crucial for developing other branches like Analysis, Calculus and Probability, *which have tons of applications in real life.* So infinity helps us to be able to study these topics in a *precise and rigorous* way **Example B:** It helps to verify that the axioms we use to study mathematics are indeed able to be used to study mathematics: we study math by assuming a set of axioms (rules) and studying the consequences of those axioms. However, our axioms could be contradictory, and we could get things like 1=2 if we aren’t careful. If that happens, it means that our axioms are contradictory and can’t give us any meaningful information about the mathematics that interest us. One of the sets of axioms we use is the Peano axioms. And we can use infinity (ZFC axioms in particular) to prove that the Peano axioms aren’t contradictory. We can even use infinity again to prove that the ZFC axioms are not contradictory. So we are pretty sure that the math rules we are currently using aren’t going to suddenly break one day. **Example C:** Infinity can even solve questions about *finite* numbers: there’s some questions about finite numbers that can’t be solved unless we start using infinity. For example, Goodstein's theorem can only be proven if we use transfinite ordinal numbers. And if I recall correctly, Goodstein’s theorem has applications in Computer Science. **Example D:** Cantor's work can be used to prove Gödel's incompleteness theorems, which answers very important questions about mathematics, philosophy and even engineering. These theorems helped answer some questions related to the goals of the Hilbert's program (which is very related to example B) that talk about the essence and limitations of mathematics. Gödel's incompleteness theorems can also be used to give an answer to the Halting Problem, which has many implications in real life, particularly in Computer science and Software engineering. For example it has implications in *cybersecurity* (I don't know the specifics of it) and lets us know that *some* algorithms don't exist and it's better to focus effort in finding heuristics solutions to *some* problems instead of wasting time searching for a perfect algorithm that literally doesn’t exist. (As a fun fact, it was later found out that one of Gödel's incompleteness theorems is equivalent to the Halting problem)
@@AndresFirte I do understand the bijection concept, and surjection, I just disagree with them. My belief is that a better definition will come along and bijection and surjection will be tossed.
well, math is not about belief really. you can definitely try to create your own system and axioms for dealing with whatever you want to define infinity to be. But in standard (set theory) mathematics, these are facts. It's not up to you to deny them.
