Отметься кто смотрить ролик до конца 🐼......

Infinity is contained within the concept of options which are some of the mechanisms that serve as the foundation of Existence and Non Existence. We experience this through Free Will.

18:11 "Between any two real numbers there's a rational number." You guys are killing me! How do I even begin to get that into perspective with the fact that there are more reals than rationals? Between every Real/post, there is a rational/fence, but the difference in total posts and total fences is uncountable.

@@starc. Where can I learn more about it!?

@@Grateful92 "Most of what we are is non physical, though, our lowest form is physical. All life on our planet has the lowest form, the Body. Our Body is an Animal and the other type of Body on our planet is a Plant. Bodies are bound absolutely to Natural waL (spelt backwards) which is the lowest form of true Law. Natural waL (spelt backwards) is a localised form of Law and is derived from the Laws of Nature. Natural waL (spelt backwards) is the finite and specific foundational control structure ordering the actions and interactions of species, members of species, and the material sources of a planet. The lowest non physical form of what we are is the Mind, which is a Process. There are other forms of life on our planet that have both a Body and a Mind, however, so far as we currently know, there are no Plants and only some Animals that have a Body and a Mind. The lowest forms of Mind, Instinct and Emotion, are predominantly bound to Natural Law. The next higher form of Mind is Intellect which is bound predominantly to the Laws of Nature. Intuition, the highest form of Mind, can be bound or not to both Natural Law and the Laws of Nature separately or together, or to higher forms of Law altogether. Intuition is the truest guide for our Selves. The next non physical form of what we are is the Self, which is an Awareness. There are relatively few other forms of life on our planet that have a Self. The Self is not bound to any form of Law other than One's Own Law. It is the only form of Law that cannot be violated. The foundation of what we are is the highest non physical form of what we are. The highest form of what we are is the Being, which is an Existence. The Being is not bound to any form of Law originating within Existence. The Being is bound absolutely to The Law. Existence, and the Laws of Nature which are the finite and specific foundational control structure ordering the actions and interactions of all elements within Existence, cannot Be without The Law being The Law. So, what is The Law? In a word, The Law is options. Definition option: a thing that is or may be chosen. The word 'option' does convey the idea of The Law in its most basic sense but does not clarify all of what The Law is. Free Will does describe how our species experiences The Law but does not convey all of what The Law is. In clarifying what The Law is; The capitalised form of the word 'The' indicates the following noun is a specific thing. Law is the finite and specific foundational control structure ordering the actions and interactions of all elements subordinate. Together, the words 'The' and 'Law' (in that exact order,) is a proper noun indicating; the singular form of Law that all other forms of Law and all other Laws are founded upon, the singular foundation upon which Existence is founded, the singular foundation upon which Non Existence is founded, the singular foundation connecting Existence to Non Existence, the concept of options, and Free Will. However one thinks, believes, guesses, hopes, or "knows", whether by a gnaBgiB (spelt backwards), a creation story, a computer program, an expansion of consciousness, or whatever means by which Existence could have come to Be, the option for Existence to not Be also exists. Existence and Non Existence, the original options connected by the very concept of options, connected by The Law. Outside of space and before time. Extra-Existential. As we experience The Law in our Being, The Law is Free Will. The First Protector of The Law is Freely Given Consent. The First Violation of The Law is Theft of Consent." - Goho-tekina Otoko

I never did understand that quote, but I eventually got used to it.

what a quote. Honestly there is something about the mathematical language that doesn't fit for every different brain and mental wiring out there. In school it was the hardest thing with all teachers except one who had a different way of explaining things that just flowed. The key really is that we're using a poor method of description a poor method of interpretation of the world

"There's a trick you can use in mathematics called not worrying about it." - Matt Parker

@@SellymeYT Sounds more like an Andrew Ng quote.

That strikes me as fleshing out to be: 1. To understand would require truth. But absolute truth is not accessible to us. As such, there exists no definitive handle upon which to mathematically anchor one's self. [In alignment with Godel's incompleteness theorems] 2. Notwithstanding, we must get on with it. Getting on with it comes via routine. Use routine as an anchor. We have a plethora of routines to choose from. Any anxiety about not having a handle to grab ahold of will surely give way to one's commitment to routine. Very much in line with Mermin's Shut up and Calculate quip.

As I sheepishly put away my ruler...

@@backwashjoe7864"Pixels were never meant to be counted"

Not to scale, naturally.

I loved the "obviously" especially

not only is it not to scale, the real numbers are INFINITELY bigger than the natural numbers. there'd be no WAY to show it to scale

Absolutely had the same feeling for years now 😄

pssst, hey buddy.... wanna buy a Numberphile script? ;-)

fundamental attribution error (that's a real thing, google it)

absolutely, and having watched this channel for more than half a decade now, you can actually notice him getting more and more knowledgeable in all fields of math, just like us watching along

Yes. He'd make a great news reporter.

yes

The best of both worlds.

It's a very specific era of old school, the handheld 'camcorder' style is kind of mid-90s to 2000's era old school

this is a numberphile signature and should never be changed

Even the animations remind me of the CGI visualizations of 90s/00s math educational films (like 'Outside In') in the best way

Yeah, that was like something out of Catch-22.

If your model of set theory has an inaccessible cardinal, you can define the universe of sets up to that cardinality (in the von Neumann hierarchy). That universe doesn't contain its own cardinality, and there is no set in the universe as large as the universe itself, so you do essentially "run out of sets." Or if you use the whole universe V, you can discuss in a philosophical sense the "size" of V, and that can't possibly be the size of a set (because that would have to be a universal set). Rather, it's the size of a proper class. It's consistent that all proper classes have the same size, but it's also consistent that they have different sizes. But even if they all have the same size, that size is not a cardinal, because you can't form an equivalence class of proper classes.

Classic Numberphile moment for sure

@@EebstertheGreat If your model of set theory has an inaccessible cardinal, the universe contains much more then just cardinals up to that inaccessible cardinal.

@@Tian-wi6qr If κ is an inaccessible cardinal, then V_κ is a model of ZFC. It contains everything in the cumulative hierarchy before κ.

It's like the gigachad meme

"...you know, it's an everyday thing one encounters in their life. Nothing too crazy."

Infinity is an occult symbol.

Did you just timeshift infinity?

Wow, so creative. You're definitely the first person to think of that. It definitely applies to infinite numbers.

@@overestimatedforesightI'm sure it was just intended as a silly joke. :)

oh yeah, well, your number +0.1. i win.

8:12

EXACTLY. Thank you. I don't know why power sets weren't mentioned anywhere here when they are key to understanding these concepts on a more than "I just said so" level.

I wish Dr Karagila explicitly mentioned that. It would have been a perfect conclusion. Still a great video

the weird thing is even though you can't count the real numbers, you can come arbitrarily close...

@@michaelsmith4904no you can't, far from it. even after counting off 1 real number every nanosecond forever you'd have counted aleph 0 numbers, while there are aleph 1 ahead

@@michaelsmith4904 what do you mean?

​@@michaelsmith4904not really, as even if you counted them all, you'd still be able to make an entirely new unique real number to add to it. So you can always add another number to the set so you can never have an entire set to count, hence the uncountable.

Not to scale, naturally.

Nuh uh

Can you proof this? Even if numberphile has a very positive effect on all living humans and on all humans that will ever live till the end of the universe this number will be quite small compared to infinity...

@@red.aries1444 my source is dude trust me bro

​@@red.aries1444*resurrects Ernest Zermello*

There are no upper bounds on opinions 😋

The question is super interesting, but the answer is somewhat misleading. While it is true that 2^aleph = 3^aleph = aleph^aleph, 2 isn't a random choice. It represents the power set, which is the set of every subset of aleph

@@shmendusel I'm surprised he didn't remark this!

And the reason why it's notated that way is because the power set is isomorphic to the set of all functions from the set to a set with two elements.

@@galoomba5559True, I forgot to mention that!

​@@galoomba5559Wait isn't it the other way around? Shouldn't it be the functions of X to {0,1}? In that case the isomorphism is very simple. Given a subset Y, a function f_Y and an input x, return 0 if x is not in Y and 1 if x is in Y. Now clearly the functions from X to {0,1} and the subsets of X are 1-to-1.

@@SolMasterzzz Of course, my bad

Yes, all of this... this video seemed lazy and inaccessible to people... not up to the usually quality of Numberphile. There are so many ways to think if the class of infinite cardinalities and how to show that the cardinals do indeed get larger, which was just kind of presented axiomatically here without any constructive proof.

That was my thought too… isn’t he just straight up assuming the continuum hypothesis there?

@@WaffleAbuser No, he isn't, you are thinking about 2^(aleph_0). Aleph_1 is literally defined as the smallest uncountable infinity.

@@Tian-wi6qryou’re just wrong tho lol

@@antoniocortijo-rodgers75 For some reason, there is a _widespread_ misconception about the definition of Aleph numbers. Tian is correct. And I assume you're one of the people who have heard someone define Aleph numbers improperly. So I suggest you look up "aleph numbers" and "beth numbers", and in particular, their relation to the Continuum Hypothesis.

@@antoniocortijo-rodgers75 What am I wrong about?

I think we did get to the proof that the size of the rationals and the Natural numbers were the same, shown by writing then rationals in a certain order and drawing snaking diagonals lines...? I was hoping they might show that in the video. I also hadn't realised that the Reals and the Complex numbers were the same size - would like to have seen that explained.

It's never too late to go for your dream! It's harder, but you can still do it. :)

Sadly, ieee 754 really lacks imagination when it comes to infinity :(

I had worse that that. I remember trying to figure if the number of Xs was infinite, and the number of Ys was infinite, but X was bigger than Y, if you could have one infinity bigger than another- essentially exploring set theory without knowing it. My teacher, who I now know had a degree in teaching NOT maths, told me that infinity was infinity and you *couldn't* have a bigger infinity. I believed that until I was in my 20s when I picked up a popular science book that mentioned sets.

A gift from me to you: while(true) By the way, we have plenty of set theory in computer science so I dont really understand where your sadness comes from

I also would be surprised, if he had forgotten the explanation of the other professors. I remember Dr. Grimes calling the Aleph_0 size sets „listable numbers“. Hence you can list natural, whole and rational numbers, they are all the same size. At real numbers you don‘t even know the next number in the list after 0.

his answer is incredibly honest, whether he remembers all the videos he did on this subject or not... he really is building an intuition for it!

@@SmashXano What do you mean by "not knowing the next number in the list"? You can pick any number to be the next number in the list. The point is that any list you make in this way will not contain all the real numbers.

@@galoomba5559 I could be wrong, but it sounds like another way of stating that, no matter which 2 real numbers you choose, you will always be able to find a real number that lies between them in value.

@@yudasgoat2000 That's also true for the rational numbers, and they are countable.

Although (I believe) there are only countably many numbers that are uncomputable for that specific reason. Since there are only countably many "definable" numbers at all, and the remaining ones are both uncomputable and undefinable.

@@MrCheeze Take the reals and remove any set size aleph-0. Say, the rationals. You are still left with an uncountable amount of numbers, since Cantor's diagonal argument still works with a sequence of irrational numbers, like (π, 2π, 3π, ...) And we can actually define an uncomputable number The sum of 1/TREE(n) or the sum of 1/BB(n) are easy examples of definable but uncomputable numbers By proving there is a bijection between the naturals and a set S of uncomputable numbers, and by defining at least 1 uncomputable number ∉ S, we show that there is a uncountable amount of uncomputable numbers Let S = { [sum(1/BB(n))]^1, [sum(1/BB(n))]^2, [sum(1/BB(n))]^3, ... } Enumerate the elements of S by using the naturals Remember the sum of 1/TREE(n), well this element ∉ S and we already used all the naturals to enumerate S, so there is an uncountable amount of uncomputable numbers ;)

@@MrCheeze Just for clarity, when you write "countable", you presumably mean "countably infinite"?

​​@@landsgevaer countable cardinalities are finite natural numbers and aleph-null

@@alexc4924 Yeah, I know. But you didn't mean to allow for the possibility that there could be finitely many uncomputable numbers? Actually, reading your comment again: "countably many uncomputable numbers"? That cannot be right. The computable ones are countably infinite, so the remaining uncomputable ones must be uncountably many in number.

Pete McPartlan did the animations. 👍🏻

It really is. In some sense it is the second most trivial thing next to nothing.

“Lazy 8” actually comes from branding livestock. Which is, if you squint, a kind of heraldry.

'Lazy 8' is when the symbol is used for cattle branding. The mathematical symbol for infinity is called a lemniscate (Latin for 'decorated with ribbons').

Infinity isn't 8 on the side. 8 is infinity standing on end! - Piet Hein.

@@PilpelAvital 'Losing one glove is certainly painful, but nothing compared to the pain, of losing one, throwing away the other, and finding the first one again.' My favourite Piet Hein quote 😁

Wouldn't it be more fun to mention how the algebraic numbers are also countable? Not just every rational number, but every single solution to any polynomial with integer coefficients. Every strange thing you can make with addiction, multiplication and integer roots. They're countable.

@@xinpingdonohoe3978 Huh, that _is_ fun and surprising! Thanks!

@@xinpingdonohoe3978 the algebraic numbers - the solutions to any polynomial of any degree - are a cool set as well, for sure. I believe that the computable numbers, which includes e, pi etc, are also the same size. In fact I think there's a Matt Parker video about it on this channel from a few years ago.

It's as if there was an infinitely thin silk textile that can let things go through and yet block everything

Between every two irrational numbers there is a rational number and between every two rational numbers there is an irrational number.

Two Klein bottles in fact, the coke bottle is one as well! Well, 3 dimensional analogs of Klein bottles at least

I see the light saber and the (canonical) Klein bottle...where is the Adams reference?

you missed the gameboy

@@seanbirtwistle649 The Ultimate Tetris Machine

Nice pfp

Your 1st grade was wild, man

Hmm. First thing I learned was zero. "There are no more cookies."

Lol, you guys made me realise how funny this comment was. I read it and assumed it meant one of the first things WHEN YOU GET INTO THE MATH COMMUNITY

There are mathematicians who are finitists, who insist that any math done with infinity is not legitimate. And there are even ultrafinitists who insist that very large finite numbers (far bigger than anything that would come up in a physical context) are not "real" or legitimate in some sense. It's a minority position though.

The empty set is empty, but the set of the empty set is not, it contains the empty set. Put another way: There is nothing in the empty set, but the set itself is not nothing. (And from that, everything else follows.)

@@davidwuhrer6704 shut up, nerd

It's not true that the surreals contain every number, in part because "number" is not a well-defined term. The complex numbers aren't contained in the surreals, for example. What is contained in the surreals is every ordered field.

@@galoomba5559 Thank you for clarifying that. I did mean to exclude (hyper)complex numbers from the surreals myself, when I said specifically "number *line*", and then went on to talk about "numbers that aren't on lines".

If number line means an ordered field, the surreals are the most complete number line in that for any other (set-sized) number line, you can find a subset of the surreals that is isomorphic to it. The well-ordered transfinite hierarchy of the ordinals (and cardinals) has to come before the surreal numbers (which are non-well-ordered) can be defined though, because the surreals are defined by induction along the ordinals, and the induction requires well-ordering.

😂

What?

​@@chaman9537 multiplying stuff by i is sometimes understood as a 90° rotation. If you rotate 8 by 90° you get ∞

😦😦😦🤯NO WAYYY 8i`=∞!!!!¡

My thoughts exactly. A lot of "maths" involving infinity becomes absurd and/or quite subjective, I always think twice about the conclusions they postulate

You need to construct sets from other sets. This is the limitation sets to prevent Russell's paradox. Higher infinities some of them cannot be constructed from below so they are no longer sets but they still have describable properties so they are Classes.

Behind the light saber

fish =

no

\{°◇°}/ -[~_~]-

how!? it came out 15 seconds ago >:(

Patrons got an early notification 😂 but I always wonder that myself when I see three comments on a video that just released @@aguyontheinternet8436

Math is great😅

I have caught a glimpse of the shadow of infinity three times in my life, thus far. It is an overwhelming experience which one does not quickly forget.

@@curtiswfranks Maybe it's better I don't ask what that was about.

It isn't. Still interesting despite it's lack of purpose, like most things in life.

Free plastilin. GTFO!

From whom? The terrorists hiding amongst the civilians?

There is no place called palestine... it's just a fake slogan

@@eem19584 lol, keep telling yourself that

"There is no war in ba sign se"​@@eem19584