To Infinity + 1 and beyond!

Not all infinities are the same size, surprisingly enough! This counterintuitive result has been the source of much consternation. Georg Cantor first discovered this in the 1870s, and the results have influenced many aspects of advanced mathematics ever since. Join me as we explore this interesting fact, and why it's true!
Link to 1st video referenced: • The Basic Sets - Where...
Link to 2nd video referenced: • The Complex Numbers: A...
More stuff to look up: Set Theory, Axiom of Choice, Continuum Hypothesis, Hilbert Problems, Aleph Numbers, Beth Numbers
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The Taylor Series aims help people love math. Whether you're a student in a math class or someone who has ever asked the question, "What was the big deal about Calculus?" there's something here for you.
I am aiming to have one new video up every month. I hope you enjoy them!
Also, I named this the Taylor Series after the mathematical construct created by James Gregory and formally introduced by Brook Taylor ... and I picked this one in particular because my name is Derrick Taylor. :)
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Patreon:
/ taylorseries
Facebook:
/ thetaylorseries
Twitter:
@TheTaylorSeries
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Music Credits (from AudioBlocks):
Intro music: Thinking by Patrick Smith
Main background music: Island Fun by Neil Cross
Outro music: Midnight City Lights by Neil Cross

[@ikkuhishikawa7982]

Your sub count should match PewDiePie's

[@GuilhermeDiGiorgi]

Oh, I see you are trying to get some Norway vs Sweden rivalry out here. Lets make it better and say this channel should reach 100M before Pewdiepie.

[@nuklearboysymbiote]

Can u come up with a bijection between this channel's subs and Pewds

[@Qbe_Root]

Broke: Researching your subject to win the debate with solid arguments Woke: Researching set theory to win the “you’re wrong” fight with bigger and bigger infinities

[@TheTaylorSeries]

You're right x uncountable infinity!

[@Patrik-bc2ih]

@@TheTaylorSeries You're right x 2^(uncountable infinity)!

[@hafizajiaziz8773]

Perhaps you should talk about those ZFC Axioms, it should be fun. Axiom of Choice in particular

[@TheTaylorSeries]

Mmm. I feel like I want to understand them better before I go into them; it's been a while. But yes, this is a thing that I hope to do. :)

[@TheTaylorSeries]

@William Boyle Truth. :) And thank you!

[@SledgerFromTDS.]

I honestly think that The Taylor Series Channel should get over 20 Million Subscribers by the end of 2021

[@RalphInRalphWorld]

Hmm, how big is the cardinality of all bijections between countably infinite sets? My guess is that it's uncountable... We could probably construct a proof like Cantor's theorem, which is for power sets.

[@sulfurx777]

If you really enjoy this topic, vsause expands on it more in his video “ How to count past infinity.”

[@Psychomaniac14]

no u x aleph fixed point

[@TheTaylorSeries]

Aleph to look it up.

[@Psychomaniac14]

@@TheTaylorSeries nice joke

[@emuhast]

I love the way your videos makes every topic interesting and not too hard to understand. Keep it up

[@KohuGaly]

Well, Cantor's diagonal technically doesn't prove that the sets are different size. It proves that they are either different size or that at least one of the sets can't be ordered into a list. That is in essence the problem with the axiom of choice, mentioned in the end. The axiom indirectly postulates that all sets can be ordered into a list somehow.

[@purplepenguin8452]

Huh, I have never heard that pointed out. Thanks for sharing. So how do mathematicians define the concept of size when working without the axiom of choice? Or does it just break down and you have to accept that the concept of size may not be comparable for arbitrary sets? For instance, can one still say that the size of the set of transcendentals is the size of the reals?

[@KohuGaly]

​@@purplepenguin8452 axiom of choice only matters for infinite sets. For finite sets, you can define its equivalent from the rest of the axioms, so defining finite sizes is fine. AC basically guarantees that all sizes are comparable. Without it, there is just no way to compare some sizes, because you can't assume there is a bijection between cardinals and ordinals.

[@AlcyonEldara]

You can define a way to compare sizes without the axiom of choice. I'll use , = for the sizes of sets. A >= B is there exist an injection from B to A. You can also prove without the axiom of choice that A >= B and B >=A IF AND ONLY IF there is a bijection between A and B. So we showed that there are more real numbers than natural numbers. And we use the following definitions A is bigger than B if there is an injection from B to A and no injection from A to B A is equal to B is there is a bijection between A and B We need the axiom of choice to be able to compare any two sets, but in the current situation we proved that there is no bijection between the natural and real numbers. Since the natural numbers are a subset of the real numbers, we can conclude that the size real numbers is bigger than the size of the natural number.

[@rowenagarcia6088]

2D > 1D

[@moiquiregardevideo]

In computer science where we go backward : looking at an existing hardware or library and analyze the numerical limits, going from natural to integer, which is replacing unsigned by signed integer, we loose half the range. For example, the maximum value using 32 bit is 4 Gig, but the largest positive value using signed integer become +2 Gig. The bijection view would be a rotation left by one bit. It is a multiplication by two (shift left by one binary bit) and inserting as least significant bit the value that was in the most significant bit before shifting. Since the least significant bit is zero after shifting left, the result is guaranteed to be even, the rotation operation can overwrite this bit. The discrete coding in an emulator would still need to use "OR' or "ADD" to overwrite a single bit in an integer.

[@kacpertobiasz8312]

It's nice seeing more complicated things on this channel, keep it up :)

[@khmer06]

Nothing is complicated on this channel; he just explains everything so perfectly.

[@JonathanMandrake]

I think while it is good to treat infinity not just as any other number, it is a number. For example the alexandroff extension of the real numbers or of finitely dimensional vectorspaces over the reals can use infinity in their direct representation. Yes, I know that those can also be projected onto spheres, however you can still use infinity

[@johnwalker1058]

Your channel deserves an uncountable infinite number of likes and subscribers!

[@sulfurx777]

Wait, if each number can match with itself, (as you show at 6:39), doesn’t that mean a set of all the cardinal numbers would have a cardinal amount of elements?

[@purplepenguin8452]

there is another approach that does not involve counting Why don't they use the natural notion of a subset? the "size" of the set of even numbers is less than the "size" of the set of integers, because it is a proper subset of the integers the "size" of the set of integers is less than the "size" of the set of rationals, because it is a proper subset of the rationals

[@TheTaylorSeries]

Because that is not true. The connection between a proper subset and seize difference does not extend to sets whose cardinality is infinite, though you are correct when both are finite.

[@purplepenguin8452]

​@@TheTaylorSeries Whether a set is a subset of another is still well defined for infinite sets. It is a _different_ concept of "size" (hence the scare quotes) that can also be used without needing counting. Yes, with finite sets, these two concepts of size coincide. With infinite sets, these two concepts are no longer equivalent. Although they are not fully independent. For instance a proper subset will never be of a larger cardinality than the original set. So when thinking of "size" of infinite sets, some intuition built from finite sets may go along with the subset notion, while other bits of intuition may go along with the cardinality. For example with the subset concept of size, you can very much have a set with exactly one more element than the natural numbers. This begs the question I initially asked: Why don't mathematicians use the notion of a subset as the default concept for size? Well, the concept of proper subsets is well defined, so mathematicians are free to use these properties in proofs.However, the notions of size provided by proper subsets and bijection are no longer equivalent for infinite sets, and so the answer ultimately is that it was a choice. And, as with many math definitions, the reasoning behind the choice comes down to "utility". The subset concept is just not as "useful" a choice for exploring the problems mathematicians were interested in. So if a student is struggling with some concept they expect infinite sets to have regarding their "size" it may well be that internally they are expecting something like the subset concept of size ... which coincides the the cardinality for finite sets, but not with infinite sets. Realizing this can be extremely helpful. I had a high school math teacher that would introduce concepts by posing a puzzle, or an intuitive concept, and asking us to try to figure out its properties and formalize it. It was always a fun group discussion, but ultimately it left the choices mathematicians made as simultaneously "obvious" and "clever" when we finally heard them (and wow, all the holes the teacher poked in our attempts at defining a 'limit' left the definition mathematicians settled on looking amazing). We better understood what the choices were and often had a much better appreciation for why definitions were eventually chosen the way they were. (One obvious answer to the "why" here, is how do you compare the size of sets with different types of elements? A little bit of Socratic leading, and you realize the bijection concept of size is more "useful". And with a lot of leading, eventually see with counter-examples that furthermore, these concepts of size are not equivalent with infinite sets.)

[@TheTaylorSeries]

@@purplepenguin8452 Well, there's more than just utility at stake here. you are correct in much of what you say -- cardinality and size mean *slightly* different things. But the problem you mentioned -- that of a bijection -- makes things more challenging. If a bijection exists between two sets, then I would consider any logic -- including the logic of subsets, which as you say is still well defined for infinite sets -- to be invalid were it to suggest that the two sets in question have different cardinality -- or size, in even a colloquial sense. That, at least, is what I consider to be the specific problems alluded to in your final paragraph. :) I love the analysis you're going through, and it's making me think. Also, the teacher you mentioned sounds awesome. :) Anyway, i'm sorry I can't reply with more; I've got the kids and had like 5 minutes to reply, but a post like this deserves more.

[@vinitchauhan2928]

You are one of the fastest growing youtubers I have personally seen, when I first found your channel you had something like 1k subs. Keep up the good work.

[@gegiojonjongegio7998]

Then as you said, the cardinality of the natural numbers is infinity, but in the minute 15:12 it says that the cardinality is aleph null, which i suppose is just a definition for infinity, so infinity = aleph null. But then the cardinality of the real numbers is 2^aleph null, which with that definition would be equal to 2^infinity, and since you proved that the cardinality of the real numbers is bigger than the natural numbers (2^aleph null > aleph null), then 2^infinity > infinity, which i think is a contradiction... Did I mess up?

[@TheTaylorSeries]

That's a great question. :) I might have messed up the notation. But, from what my understanding is, the cardinality of the Continuum -- another name for the set of Real Numbers -- is written out as 2^(aleph_null), because it has the same cardinality as the power set of the Natural Numbers. :)

[@seba8115]

What s next level of ordinals?

[@zixuan1630]

Count in diagonals!

[@therealsachin]

For the problem of matching the cardinality of natural numbers and ratios of integers ... The way you arranged the ratios helped me get the answer instantly! 😄✌️

[@TheTaylorSeries]

Awesome! What is your strategy?

[@PedanticPlanner]

Love the Videos! I'm a major in Math at the United States Military Academy. I'd love to shoot you an email sometime.

[@TheTaylorSeries]

Shoot me a message on FB :) Can share there.

[@EpicMathTime]

I made a similar video that goes into what I suspect you'll talk about next. Spell: Power Set Casting this spell on any set yields a new set with a greater cardinality. Requirements: Cantor's Theorem

[@TheTaylorSeries]

You are on to something!

[@SOBIESKI_freedom]

This probably explains why Cantor went mad figuring out all this stuff about infinities.

[@TheTaylorSeries]

Perhaps. Mostly, he got ridiculed by the mathematical community (a few folks in particular) and was in and out of asylums for the rest of his life. A sad tale.

[@SOBIESKI_freedom]

@@TheTaylorSeries Indeed. Often the old guard needs to die until new(er) and often revolutionary ideas can be understood and accepted. But if one cares not for external validation, then actual insanity (rather than subjectivly perceived insanity) would less likely occur. Kudo to Herr Cantor for his work nonetheless.

[@purplepenguin8452]

What is the best way to fix up the presented argument to handle stuff like: The numbers 1.9999... = 2.0000... even though every single digit differs. So saying you can construct a sequence of digits that differs from every digit sequence on the list by at least one digit is not the same as saying the number you constructed is not in the list.

[@TheTaylorSeries]

I don't see the connection between that and cardinality comparison. Can you clarify?

[@purplepenguin8452]

@@TheTaylorSeries The video argues that no bijection between the natural numbers and reals is possible in principle (and thus the cardinality is different), by saying that for any proposed bijection you could construct a real number by going down the diagonal, choosing a digit different than the digit in the list. By construction, this digit sequence is not in the list. You then assume that means it represents a number not in the list. And thus there is no bijection. However the decimal representation of reals numbers is not unique. And thus, as in the 1.999... = 2.000... case, it is possible for two strings of digits to not match and yet be the same number. Therefore we cannot conclude that because a sequence of digits is not in the list, that the list is missing that number. Almost every youtube video I've seen on this topic makes this same mistake. I realize it is just a detail to fix up, and so I was curious if there is a really simple way to fix the argument without even needing to bring up the side issue (but then why do most youtube videos not just use this method) or if there is some reason that there is no easy fix?

[@purplepenguin8452]

alright, found another discussion of this that inspires the following tweak: Instead of just saying change each digit, specifically change any 2 to 1, and any non-two to 2. The resulting number now, by construction, has a unique representation as a string of decimal digits, and thus doesn't have this problem.

[@TheTaylorSeries]

@@purplepenguin8452 Ah, now I see. It's funny you mention that; I cut a short bit for time constraints (and because I hadn't prooved it and so it felt more like a conjecture), where I said that there are an infinite number of elements that you can construct, because there are infinite other shapes other than the diagonal that can use this logic, not to mention infinite bases in which you can build different elements with different choices. So, I think if you consider all of the constructed elements possible, that would shore up the hole you found, because at least one will not be merely a rewriting of another number. But ... I haven't proven it so who knows! :)

[@otakuribo]

Is 2:04 the biggest finite number ever represented on a single screen?

[@TheTaylorSeries]

So far as I'm aware, but then again, there's still Rayo's number. However, I still consider Rayo's number to be a total cop out. :)

[@otakuribo]

@@TheTaylorSeries I just looked up Rayo's number. Wow. Not sure if brilliant or petty or both. "Smallest number larger than ..." is such a mathematically rigorous way of saying "plus one!"

[@Michael-cg7yz]

You're wrong infinitation (the last operation) of infinity to infinity! ;p

[@TheTaylorSeries]

That would be rather a thing!

[@Michael-cg7yz]

@@TheTaylorSeries Damn, I hoped you would one-up me.

[@TheTaylorSeries]

@@Michael-cg7yz Hm. You are wrong uncountable infinitation? :)

[@Michael-cg7yz]

@@TheTaylorSeries Yay, thank you.

[@TheTaylorSeries]

@@Michael-cg7yz I do what I can. :)

[@xteishi]

How is the attempt to construct a new number not a denial of the assumption of the completeness of the set? If it is a complete set, does that not entail that a number outside of that set that belongs to that set is imposible to be found? Why not just say that the supposedly new number was always in the complete set since, after all, it is complete? It seems to me that either the set is complete and the number was always there, or else the set was incomplete and was lacking the generated number... What am I missing?

[@TheTaylorSeries]

I think your confusion is coming from the use of the word 'complete.' To see what I mean, consider the following question: does the fact that the set of naturals doesn't include -1 make it incomplete? I don't know if we have a good definition here of complete, which would cause the issue. But, that's just my 2c. :)

[@xteishi]

​@@TheTaylorSeries , thanks for answering!. But the way I understand it is that a complete set of naturals by definition is not supposed to include -1, so even if it does not include -1, is not incomplete; but the complete set of reals by definition is supposed to include every possible real number, so if it doesn't include even one real number, then is not the complete set of reals.

[@TheTaylorSeries]

@@xteishi Ok, I better understand what you mean. I think the point to remmeber is that the constructed real number isn't being built in a way that excludes it from the reals -- quite the opposite. It's most certainly a real. It's just one that hadn't been paired with a *natural* as they had already been paired with numbers from the reals.

[@xteishi]

I guess my problem is this: 1. If the real set is complete, then the built number is there. 2. If each natural number is paired with each number of the real set, then some natural number is bound to be paired with the built number. So, to me, saying that the built number was not considered when pairing each natural, is like saying that the real set was not complete. So the way I see it, either the real set was complete and the built number was paired at first with a natural or else the real set was incomplete because it was missing the built number. What am I missing?

[@TheTaylorSeries]

@@xteishi Well, #1 is a bit off, though this comes down to something called uncomputable numbers. I'm not quite sure I can explain them well enough in this medium, so I am afraid I have to do the intellectually lazy thing and ask you to read up on it. However, I can tell you that it's absolutely *fascinating* and leads to something called Gödel's Theorem, which was life changing for me to learn about myself. :) #2 is the problem, though. I never claim to have caught all of the reals in the list of real numbers you're referencing. I don't know how to construct such a thing. My only point was that it was *a* list, and that this fact I illustrate applies to *any* list -- even if you do claim to have figured out how to enumerate every real number. So, when I construct the new number, I'm not saying "this was a complete list, and I've constructed one that isn't there!", rather I'm saying "this is *any* list, complete or not, and I've constructed one that isn't there!" :)

[@RickWhitechest]

To think I've liked to imagine intensely big numbers since I was in kindergarten, hehe. I remember filling this one sheet of paper with the numbers from 0 to 9, over and over and over, even turning it to the other side and thinking to myself "I'm gonna need more paper!".

[@thephysicistcuber175]

3:26 I'm triggered by the absence of 0

[@TheTaylorSeries]

I know, it's always a thing I think about. :)

[@datdudeoj7673]

Great video!

[@connorconnor2421]

14:12 OONK a typo

[@TheTaylorSeries]

I am confused! :)

[@connorconnor2421]

@@TheTaylorSeries cardinalty

[@TheTaylorSeries]

@@connorconnor2421 oh! I see it now. Good eye :)

[@Xnoob545]

@@connorconnor2421 HI CONNOR

[@connorconnor2421]

@@Xnoob545 the what

[@fuseteam]

came to be convinced of Infinite cardinalities Got convinced of Infinite cardinalities plus 'decimals' being a better name for the set of reals Which means the imaginaries should have a better name as well I wonder if we can proof that the cardinality of the imaginaries and the decimals are equal

[@TheTaylorSeries]

I think the reals and complex numbers share a cardinality! But I forget the proof. :)

[@fuseteam]

@@TheTaylorSeries i think so too, but I want to find a proof hehe, it may help understand cardinality q bit better hehe

[@TheTaylorSeries]

@@fuseteam I will look into it :)

[@fuseteam]

@@TheTaylorSeries thanks :)