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See how we develop even more concepts from this mathematical foundation.
Ever wondered how numbers are actually defined? In this video, you'll learn the most common way it's done by mathematicians.
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#SoME2
00:00 - Intro
02:19 - Motivation
04:12 - Physical Units: An Analogy
08:26 - First Attempt
10:39 - The Perils of Intuition
12:40 - Definition in Principle
15:38 - Sets and Russell's Paradox
23:44 - Rules of Sets
31:46 - Constructing the Numbers
38:25 - Cake
38:29 - Closing Remarks
*Galileo's Paradox: In his final book, which bears the catchy title, "...," Galileo discusses the size of the set of natural numbers versus that of the set of square numbers (not even numbers, as in my video, but the point is the same). It's written as a dialogue, not as a letter as I erroneously claimed.
**Formulations of ZF: There are several equivalent ways of formulating ZF set theory. Some have that the empty set exists as an axiom, but not all. Others prove the existence of the empty set by using the axiom of infinity then the axiom of subsets, and others argue similarly to how I argue in that in first-order logic, something exists. The point is, I shouldn't have claimed that the axiom of the empty set is always part of ZF.
Images
IPK:
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Prototype metre bars:
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All music by Danijel Zambo.
Thank you for watching! I recently hit 10K subscribers and planning a Q&A video. Head over to the Another Roof subreddit to ask your questions. If I get enough questions, I'll make the video -- should be a fun, less scripted one. www.reddit.com/r/anotherroof/comments/wj8hhn/10k_subscriber_qa/ While I'm here, let me respond to some of the common questions related to this video: 1. Doesn't "pairing" use the definition of 2 in its statement? This is just sloppy language on my part, and thanks to those who called me out on it. We can properly state pairing as such: "If x and y exist, then {x,y} exists." That way, we don't appeal to the numbers. 2. But "red" is a matter of perception / maybe there isn't a "noun" red! Yeah, maybe. I'll concede that the analogy falls apart if you push it to breaking point, but I'd encourage everyone to keep in mind the purpose of the analogy. It isn't to say that it's easy to come up with one, objective, be-all-and-end-all definition of "red" -- it's just there to draw the distinction between the adjective/noun forms of words. 3. Why are we allowed to assume that duplicates exist? One way to think about it: Anything we can construct, we can construct again, thereby creating a duplicate. I justified the existence of the empty set by saying that a nebulous something, x, exists, therefore {x} exists by singletons, therefore {} exists. Follow the same steps again and you'll have another empty set to play with. Another way to think about it: this is all happening in abstract thought-space so there's really no harm in considering copies of abstract objects at this level! 4. But what about other numbers? Hit that subscribe button and get ready to find out! 5. You should submit this for #SoME2! Thanks, I already have! 6. Is this the only way of defining the natural numbers? No -- see my closing remarks. There are other systems of axioms, I just outline a bit of ZF here. 7. How about a video on [insert proposed topic here]? I love these comments -- I've got loads of ideas already but I endeavour to read all comments and emails; suggestions like these often end up on my list of topics!
What is 2x*2x its 4x^2 What is a number *faints*
Great video really extremly well and intuitively explained! I just subscribed. Just one small thing that got me a tiny but confused, shouldn't one of the sticks at the end have been on the purple box. 🤔
Fuck no
A greater objection that I had is that the things you called red appear to be better-described as orange; they're close to Reddit's "orangered"; more substantively, a good scientific definition of "red" would mention tristimulus values, although whichever set of such values is used is heavily culturally mediated, considering that although the vast majority of the world's languages have a basic color term centered around red, some of them have a much more expansive concept than others (which you hinted at by pointing out that the pink thing may well be considered red to some). (More specifically, all languages have basic color terms for light and dark colors, close analogues to "white" and "black", but there are some that don't even have a separate basic color category for red and similar colors; in those languages, red colors would be a type of or , the same way as in some languages, pink and orange are types of red rather than distinct categories, and until Middle English, our own language did not have orange as a separate category from red and yellow. Even now, a few languages have two different basic color terms for blue, while English has just one; an example is Italian, where "blu" refers to the darker blues and the ones closer to purple, while "azzuro" refers to the lighter blues and the ones closer to green, and although that word is cognate to "azure", we consider azure to be a type of blue, while Italian does not consider azzuro to be a type of blu or vice-versa.)
@@JamesLewis2 Because you're color blind?
"You're not going to need 90% of the maths you learn at school, but some of it you will need like the other 20%." That's hilarious. Well-played.
To be fair, in this part of the series we don't know how to add or that fractions exist.
@@cashkurtz5780 lmao
i fail to parse this
@@donaastor Not to worry. It's within the 90% you won't need.
@@masterbaraman9372 i am not sure if you understand what i mean. i fail to parse his sentence. his second sentence to be precise. how do we parse it into nouns adjective and adverb clauses... i am not sure what relates to what
This guy just explained one of the simplest possible concepts in the most complicated way possible in the most understandable way possible
Hey natural numbers aren’t that simple! That we usually (and historically) start doing math from gripping with them, is not much an indicator that they should be simple. Or, well, they are pretty simple as things stand, yep, but that’s only when you consider just addition*. When we add other useful structure like ordering, multiplication, exponentiation… it becomes complicated. Even if they look (and are) natural to define for natural numbers, it requires elaborations or additional notions (depending in which way you go at it). * That thing are complicated** is illustrated by that this simple structure can be expressed as “a free monoid over one element” (or maybe better to the spirit, “the simplest nontrivial free monoid”). This is quite a few terms but all they are here for is to say that we’re talking about quantities of just one type of object A (where a single A is a generator of the free monoid). It’d also be fitting to say “free commutative monoid” and not just “free monoid” because we aren’t concerned with order of As-but in this very case it’s irrelevant to add because this free monoid is commutative (and this is the only one except the trivial monoid which isn’t interesting by itself; all other free monoids are noncommutative). ** But unfortunately we can’t _define_ naturals in this way, and not because it might seem weird to define monoids (and what is a free one) before anything else, but because we’ll need some notion of natural numbers already to prove that such a free monoid then indeed does exist in some sense and is not just us wanting too much from a monoid. Though we can always postulate that, like we postulate Peano axioms or axioms of various set theories.
Has anyone else noticed the random hex numbers that appeared throughout the video? I think I found them all: 68 45 4C 70 Translating these to ASCII it reads "hELp" That aside, really great content. I am a Software Engineering student on my first year and I actually studied calculus I in my first semester and they talked about this subject for the first few lessons. I had already watched other youtube content explaining it, but I must say this is by far the most well done and informative. Hope to see more content soon!
noticed them, but I was hoping someone else would take the time to note those numbers and translate them, because 43 minutes it quite a long time for a video and I didn't want to go through the video again, even if the video was great :) So thank you for taking that time.
@@comical_rushing can't wait for the set theory ARG
There is a help needed section on his website with a password field :) Have fun :)
@@comical_rushing I did this and put the video into a spectograph which seems to reveal numbers but I cant read all of them, maybe I am seeing things.
I didn't, but I did notice the man in the gorilla suit.
0:25 "when you point at red things" *points at orange things* "[the definition of red] doesn't care how you think about it" Brilliant. A master class in the fallibility of definitions and their application.
I thought it was an accident... Color correction can be a bit annoying sometimes 😅
Yeah this is 100% my bad. It's my first video so it was a lot of "firsts" for me (lighting, filming, recording audio, editing, sound mixing, colour correction and colour grading etc etc). I learnt a lot by doing it, but a big mistake was focusing on my skin tone for my whole colour correcting/grading process. I thought that was the most important to get right... forgetting that I explicitly talk about the "redness" of things in the first minute >_< Hopefully you can forgive this and generously interpret my overall message!
@@AnotherRoof So I was right. Color correction isn't something I usually have to do (since my higher quality videos doesn't include real life) I do know color correction can be a pain. Amazing video, despite your small mistakes. Amazing job. Better than I could do 😅
@@AnotherRoof glad to know this too. Thought there was a gag to it later or you might have been colour blind!
When he uses a word, it means just what he chooses it to mean - neither more nor less.
The "Do mathematicians always think numbers like this" question reminds me of computer science. When writing code, you don't need to know the exact assembly language instructions to know what a function does, or even the exact workings of the processor to do that instruction, the level of abstraction given by the name is enough and most people use that instead to go faster
Yeah, abstraction is not a creation of computer scientists. It's everywhere.
Yes -- absolutely love this insight, thanks for sharing! My PhD was very computational but we had no idea how the software package actually worked, and we often said "...and we computed this through black magic." But as you say, you can unpack it to find its inner workings, or just use it as part of your own program.
@@AnotherRoof which is why things like Free Software/Open Source and right to repair, etc. are so important. So we can check what the inner workings, especially when things happen we don't expect.
@@AnotherRoof I mean, admitedly, writing code also involves a lot of black magic, especially the lower a level you're programming at
On the other hand, if you know the assembly part, you are able to write same logic more effectively. Maybe to good mathematicians, this allow to do maths more efficiently? :-D
The presentation, the jokes, the effort, the education... Everything about this video is perfect!! I can't wait to see more of you in the future! :)
Can't agree more! The presentation is what caught my attention, sticked to the end
Totally agree
We've measured the earth, theres no curve anywhere. We see mountains from 300 miles away, thats not possible on NASA's globe. Theres no proof the earth is moving. The 2nd law of thermodynamics says outerspace isnt real. Cannot have gas pressure next to a vacuum. NASA brainwashes children with globe propaganda from birth. NASA steals $60 million a day from you to shoot helium balloon rockets and satellites into the ocean. Air bubbles in "space", green screens, hair spray in hair to fake zero G, actornots on wires and harnesses. All government and military design documents assume a flat and non rotating earth. Pilots admit its flat. "Flat Earth" is openly censored by government. Real flat earth youtube channels are deleted and anti-flat earth channels are promoted (corporate welfare). NASA means "to deceive" in hebrew. NASA has 666 in their math everywhere. Every picture of space is a literal cartoon image NASA admits is fake. You could collect $20,000 if you prove earth spins. You could collect $200,000 if you prove earth curve. Mockery, slander, extortion, blackmail, subversion, character assassination and lies wont make the earth a ball. 1
@@xvhkgreen6297 bro go touch some grass or something, don't bother us with this crap
I hated every minute of it. He tries too hard. His arguments goes for too long. Trash video.
I’ve studied a lot of set theory, so I didn’t really learn anything new, but I still watched through the whole video. The video is really well put together, and your delivery is on point. I really hope you’ll keep making videos, this channel has great potential.
so you came to brag about it ?
@@NuclearSmoores No, they're giving a compliment from the perspective of someone who already knows the material. If you didn't, that's not inherently bad but also not our problem.
@@NuclearSmoores Did you intend for your comment to sound so insecure?
Is Godel's Incompleteness Theorem somehow related to this?
@@NuclearSmoores I wouldn't say mentioning that you've studied something is bragging. Maybe if he'd also said he found it mind-numblingly easy...
I really love this series, I was learning group theory in school and wanted to investigate more and found this. I love this part of math where it starts from an empty frame and it's like a jigsaw puzzle witch makes a beatiful and very compact picture and explains everything beautifully.(I know I'm late)
Thanks! One day, I will make a group theory video!
Which, you poor thing
This is a great video! Best of luck building your channel, I’m hoping for great things!
I love your strong appeal to first principles in your explanation. It's a wonderful breath of fresh air from all the teachers who say "This is the way it is just accept it"
it was presented very well but the focus on first principles is kinda inherent to the topic
I came across this definition of the numbers when writing my thesis, because I had to deal with "the different infinites", and I still think about it sometimes. You've explained in a really intuitive way some of the basic concepts that I find hardest to fully understand as a pure mathematician. Thank you
Excellent video. One thing I must be ultra nerdy about. Red is a psychophysical dimension that helps the creation of navigatable reality from the physical senses. It's not necessarily evoked by wavelength, although the color dimension is used as a ruler to add information about wavelength (a magnificent ruler). In experiences like synaesthesia or psychedelic experiences, red can indicate a whole mess of other information. So red is some kind of essential psychophysical entity that only by correlation happens to usually coincide with wavelength. For example, is redness produced by an optical illusion that exhausts green cones in your mind and forces you to experience red caused by a certain wavelength? #pedantics #unnecessary_and_unhelpful_additions
For sure. We need two definitions of red. But psychophysical dimensions could include other things perhaps, and we can hallucinate a lot of things, such as archetypal psychotropic "spiders". When we ask about "purple" it becomes simpler to define the two distinctions of colour though. In fact he gave an erroneous definition of "red" and defined "spectral red". Spectral colours are a pure wavelength. You can produce a colour using secondary wavelengths that don't contain a wavelength, such that adding green wave to your red one makes "yellow" but contains no yellow wavelength. Purple cannot contain a purple wavelength, even, ita entirely produced this way.
I've been studying the von Neumann hierarchy lately. I love the dichotomy you draw between 3 as an property (adjectival) and 3 as a noun
First time I feel like I've really understood the structure of the nested sets. Been mildly interested in the subject for over a decade now. But this was easily the best visualization and introductory walkthrough of the logic supporting it I've ever seen. This content feels like just the right mix of Matt Parker and 3B1B.
By definition, a definition must be "something" that defines a definable.
@@gualbertomicolano8130 boop.
The subtle extension of the text box reading “extensionality” at 27:00 as you say “there’s a subtle addition… an extensionality extension” was top notch. Great video!
I see it! What a clever, hidden joke.
What are the random numbers and letters that pop up in the background, like 68 (5:58), 45 (15:36), 4C (31:44), and 70 (43:06), for? I thought there'd be a message at the end that was like "Did you catch these? I've hidden a message throughout the video! Find it and let me know in the comments!" Also, I'm kind of surprised you didn't touch on the modern definition of the meter, which is the distance light travels in one 299,792,458th of a second. Anyway, the quality of this video is insane given that you haven't been on KZhead very long. Keep up the great work!
Well. I wouldn't want to give toooo much away now would I...?
@@AnotherRoof If you translate the hex into ASCII you get 'hELp'. Honestly now I'm looking forward to the next video also to see if it'll have something hidden in it as well...
@@NICO_THE_PRO I'm looking forward to releasing the next video very soon! It might interest you to know that you're not quite done with this one, though...
This video was recommended to me, and after watching the first minute or two I honestly expected a good few hundred thousand subscribers. Absolutely shocked to see less than a thousand subs, glad to join the party and watch your channel grow to where it should be. Great work!
I had these exact same thought! The video was so well produced that I just assumed I would see at least 100k subs. What a hidden gem of a channel!
It's practically his first video, it's doing quite good
That's because nowadays, to make KZhead videos, you need high quality(especially these kind of informative videos)most of the time. It's the trend and standards set by the viewers and big creators. So it's not surprising to see all these underrated channels with high production quality. You're definitely gonna attract less people if you record something with a low quality compared to most creators.
must have taken an unearthly amount of time to put this together, really informative investigation of ZF axioms and naive set theory
0:55 - It's not a very concrete definition for "what is red", because it doesn't state whether "red" means anything that reflects / emits light in that range of wavelengths or only things that (in addition to doing that), also *don't* emit light in other ranges of wavelengths. Because, if emitting light in that range is enough, then white things are red. And that is not what "red" means to most people. In fact, even most green (or blue, etc.) objects emit _some_ light in that range. Or the fact that you can make people see (for example) yellow without exposing them to _any_ "yellow" wavelengths, because our visual system can't identify multiple simultaneous spectral peaks, and instead merges them into an "average" hue (ex., red light close to green light = we see yellow; that's how most monitors work). Except it's not _really_ an average, because we're trichromats, so it's an average along a sort of "ring" that excludes the opposite side (i.e., if you see red and blue, the "average" wavelength would be green... but you _don't_ see a mix of red and blue as green; instead, your visual system says "the average would be green, but since I do have a green detector and can't detect any _actual_ green, I'll say this is a fictitious colour (ex., magenta)". Which doesn't even correspond to a physical wavelength! So, colour (and colour perception, and colour naming) is actually one of the fuzziest, least "objective" areas of perception and meaning you could have picked for that analogy. 😛
it was an analogy after all, it need not be concrete in all sense rather convey the meaning the context of the topic for which it has been used and i guess 99% people got gist of the topic. and your criticism is vaild and fair but i guess it's a overkill in this condition 🙂 but thanks for the extra dimensions you added here
@@vikaspoddar9427 - His point was that the definition of red was straightforward and objective (by contrast with the definition of a number, which was more complex). But it really, really isn't. Colour perception is an incredibly complicated field once you look (ha-ha) into it. People who think the definition of a colour is a simple and objective thing (ex., that it boils down to a single number - like a wavelength) probably just haven't looked into it, and those people probably _also_ think the definition of a number is very simple. So that analogy was just misleading in regard to how different _colour perception_ is from the concept of _an individual wavelength,_ and people who "got the gist", as you say, probably now think the definition of colour is "very concrete", when it isn't.
okay, now i am getting your point
maybe a better analogy he could've used was matter states, we have a pretty clear definition of them, and there's not much confusion over whether water is a liquid or a solid
@@circumplex9552 - Although some very "slow" liquids (ex., pitch/resin) can appear solid at short time scales, it would certainly have been a better example than colour _perception_ (which isn't even a physical concept; it didn't take long for physicists studying light to figure out that we can see the same colour when exposed to different mixes of wavelengths).
Your logical progression, examples, subtle wit, and timing with edits was incredible. I could feel my brain being pushed towards the end when it was all finally coming together.
This is a criminally good video for how few views it has. One criticism: the transition from intuition-defined sets to more rigorously defined sets ended up feeling a little unsatisfying, because the impetus for the transition: the paradoxical set that contains all sets that don't contain themselves, ends up still just being defined to not exist by axiomatic fiat in ZF.
I think you have a misunderstanding. This isn't just determined by fiat. There is no law saying "the set containing all sets does not exist". Rather the language of ZF set theory simply does not allow you to describe something such as "the set of all sets". This sentence cannot be interpreted with the rules of ZF set theory. ZF set theory allows you to describe sets only with a very restricted list of rules. This restricted grammar prevents you from making paradoxical statements.
@@Kurushimi1729 sorry, I was being unclear. I probably also shouldn't have used the term ZF, because this isn't a criticism of ZF itself. What I mean is the presentation of regularity given in the video deliberately glosses over the more basic foundation of regularity, which normally would be fine because it's a little complicated why it disallows sets that contain themselves, but I feel like it should have gotten a deeper treatment because this was the whole reason we transitioned in the first place.
To be fair, the channel seems to have only just started out and this is the first video, which is probably why it didn't have many views. However, people's recommendation algorithm brought them to this in the past 3 days. Like me, from my KZhead home page.
@@dougthayer5829 yea, I personally dont understand how regularity in this situation prevents sets from containing themselves. Based on what he said "for a set to be legitimate, it must contain at least one element which contains nothing in common with the set itself." But this at first glance doesnt prevent a set from containing itself as long as the other elements in the set dont have anything in common. For example, a set containing itself and the number 3 would be a valid set under this definition. The number 3 contains nothing in common with the main set, so this would satisfy the rule. So im not sure if he just said the rule slightly wrong or if im interpreting something wrong, but this rule doesnt seem to prevent sets from containing themselves at least with the way he explained it.
@@eragon78 This wouldn’t work. You’re imagining a set that looks like {3,{3,{3,…}}}. And you’re right, since 3 is not a set it doesn’t contain an element in common with {3,{3,{3,…}}}, and so this doesn’t violate the rule. The problem is what if I contain that entire set within another surrounding set called B (which I can do using elementary set logic): B = {{3,{3,{3,…}}}}. B must contain an element which contains nothing in common with B. However B only contains {3,{3,{3,…}}} which contains {3,{3,{3,…}}} which is in an element it shares in common with B. This means that allowing your construction would implicitly violate the law of regularity using our atomic set operations, even if it appears not to at first and therefore isn’t an allowed set. Hope that makes sense. (Bear in mind, I’m not a mathematician so I hope this gets the general idea across but my wording is anything but precise).
Relevant to the difficulty of creating rigorous definitions is that the definitions of the SI base units were changed a few years ago due to various shortcomings of the original definitions (consistency, usefulness, etc). The mass of Le Grand K changed over time, which meant the kilogram itself was changing over time. The shape of the Earth changed over time, which altered the meter. The temperature at which water freezes depends on a number of factors. Etc.
The meter was redifined to 3x10^-8 of the distance light travels in a second
And 1 Kelvin was redefined as 7.25*10^23 (boltsmann constant) of the temperature of gas with an average kenetic energy of 1J
Dude. This video was so well put together! It was clear and well spoken. It furthered my understanding of the basics of set theory. And I was shocked at the end when you said this was your first proper video. If you keep going eith content like this you'll go far! And help alot of people! Great job.
I wish the youtube algorith gives me more recomendations like this one. This videos are gold
Praise be to the algorithm! I hope you enjoy my other videos 🙂
As someone who is taking a real analysis course right now, this was a really great video as an introduction to set theory
Never once ever thought about any of this before in my life. Extremely interesting and funny to boot! You explained this really well especially with those boxes representing the empty sets.
Love your videos DeSinc
but can you do an accelerated backhop off that box?
Gauss boosting only possible because tau cannon breaks Russel's paradox confirmed.
@@marblepants probably
Amazing video, I can see this channel has a great potential. I particularly like the use of physical objects when making a point, as it helps to focus and understand the subject better. This, along with the way the information is framed and presented, step by step, such that it feels not just that I'm learning something, but also doing a kind of investigation on the theory of numbers and discovering more and more along the way.
What a fantastic video! Really cant believe you're a new channel with how skillful this investigation was presented. Absolutely supernatural how much time you spent on this. Nice job, man
Shout out to the Professor James "singing banana" Grime for introducing me to another great mathematics educator! Whilst you didn't teach me anything I didn't know, the video wasn't intended to do that! I still watched the whole thing because you were entertaining, and that's good because that was intended! I dropped out of mathematical learning when I got to University so I've only learned some of this stuff through people like you taking the time to educate and inform, and I find it helpful to have multiple ways of thinking about things, and different people reinforcing core concepts. I hope you keep it up!
Thanks so much for your comment. Reaching out to those who stopped their formal study of maths but remained interested was the whole point of this video -- so glad you found value in the content even if it was regarding a topic about which you were already familiar. Would love to make more videos so watch this space! Also, James Grime
@@AnotherRoof I don't know if you saw it but he tweeted about your video so I hope there's an uptick in views. Certainly it's where I came from!
@@phyphor Yeah I saw, what a guy! His Enigma Machine videos on Numberphile originally inspired me all those years ago. It's taken me so long to actually make something but his tweet made my day!
I'm glad - mutual respect and sharing knowledge are things that a KZhead community allows for. Good on you for actually getting your video made, no matter how long it took it's worth it!
This is great! I studied mathematics in grad school and I've seen this nested set, equivalence class definition of numbers before, but it never really clicked as a useful and meaningful definition before. Thanks!
So glad this video popped up in my recommended, it was quite the mathematical treat. Congrats on surpassing pi thousand subscribers, that number certainly deserves to be higher!
Great video on set theory! I think it's also fascinating to note that we could not have done this without making axiomatic assumptions. That's the essence of Gödel's incompleteness theorem. It's wild that we can create a rigorous definition of numbers through sets, but that we still ultimately have to make unprovable assumptions about the behavior of sets to do so!
Interesting. Such as?
holy shit this is insanely well made
You have achieved perfection on your first attempt at KZhead!
Every single second of this kept my full attention and interest. I love your content! This video is absolutely wonderful, and has made me want to start my own investigation into numbers and set theory. I've been given a new lens to look through mathematics with. Thank you so much!
As someone who is absolutely fascinated by the intersection of math and philosophy, i've read quite a lot about sets, Bertrand Russell, Gottleib Frege, Kurt Gödel, etc. However, this is my favorite explanation of ZF so far. Well done, and you've now a new subscriber :-)
Really hope this channel blows up as much as other math channels of similar production quality (like stand-up maths, for example). The narrative is great and I really wish there was a channel like this before I had to figure it all by myself from what my teachers told me. The theme of this video is something I haven't really seen on YT and I wish there were more videos like this.
Thank you so much for this comment! You've literally expressed my ambition (a successful channel like Stand-Up Maths) and USP (explaining degree-level maths / philosophy of maths in an accessible way) in one go. So glad you found the video useful and enjoyable -- much more to come to I hope you've subscribed!
@@AnotherRoof I can’t wait to see your next video!
I can't believe this is your first video, you explained way better then many of numberphile's guests I'm a freshman in maths, and the topic that most frightened me was set theory, i thought i'd never get a good grip on ZF, and i'm just so happy i get it now. You made my day
does zf stand for zet feory
No lol
@@nescirian Yes.
This is a PHENOMENAL video. Love the style, love your jokes, and love how well you described things. Made it very easy for me to follow.
Great video - thanks! I only recently discovered your KZhead channel so look forward to watching your other maths investigation videos. A lot of effort obviously went into filming this. Brilliant!
I'm busy with my first year studying a BSc in maths. This was so much fun and was a great revision exercise for how sets work. Thank you 😊
I learnt this concept in Topology but I was having a hard time fully understanding it. Your lecture is awesome and now I mostly understand numbers!
I wish my uni algebra classes went into this before going into set theory as a whole. Understanding the 'Why' behind set theory would have made me look at it so much differently. This knowledge didn't give me new math skills or anything but it just made the reasoning behind the structure of what I learned so clear. Thank you so much! -A computer engineer
Great video and clearly represented. The pace was good to keep up interest even though the video was long. Cheers and look forward to seeing more.
Hey. KZhead suggested your video and i really enjoyed it. You explain things very well and the creative use of props and the well timed funny quips were great. I hope you keep at it. KZhead success is a mystery but you certainly have a very interesting presentation style to attract people. How many times when you are learning something you think .. man how many cakes did this guy buy 😂
...so glad a number was explained. Now I can meaningfully say that this presentation is a ten. Out of ten of course. Cheers
Building all the props probably took a while, but they really helped visualize the operations involved. I had *seen* the nested sets before, but this is the first time I *understand* how those are numbers.
The best description I've seen of Paeno's book. Looking forward to your next video
You seem to have a great understanding of both, the type and placement of humor fitting for this kind of video. I'll definitely be keeping up with your journey on YT
This is a very good video, very impressive for a new channel! I'd suggest continuing to aim for quality (rather than quantity). I think your channel has the potential to become a "standard video reference" for certain math topics, just like 3Blue1Brown's calculus series for example (I know several professors that encourage their students to watch his videos). Curious for the next video! I'd be very interested to see how simple operations like addition can be derived from these foundations.
I've only seen two videos of this channel and I already love it ! I should go on an Investigation to find other channels like these... To Focus on a subject and give intuition about it is the best way to learn !
I LOVE the way you explain this concept. I hope you do many more
Great video, loved it… an in-depth investigation into integers! Excited to see what your next mysterious maths topic will be!
Man I am _so glad_ I found this channel. As someone who's studying physics but who was never really into the whole, like, Lab Investigation Research side of it, always preferred the theory aspect, I _love_ learning more about exactly how to take that mental picture that one would have of the world and make it more rigorous, defining mathematical concepts in specific ways and seeing _how_ all that comes together. I can't wait for more videos!
Holy crap is it possible that you and I somehow share the same brain because your comment here is the most precise description of my OWN interests and way of thinking I’ve ever heard. 😳 Also, If you haven’t cracked the code yet on the neurocognitive underpinnings of our unique way of thinking you’re in for an unexpected and fascinating ride. I’ve amassed an inappropriate amount of material about it over the past couple of years so I’ve got tons of reading material for you on that topic if you’re interested!
Holy crap is it possible that you and I somehow share the same brain because your comment here is the most precise description of my OWN interests and way of thinking I’ve ever heard. 😳 Also, If you haven’t cracked the code yet on the neurocognitive underpinnings of our unique way of thinking you’re in for an unexpected and fascinating ride. I’ve amassed an inappropriate amount of material about it over the past couple of years so I’ve got tons of reading material for you on that topic if you’re interested!
One of the best videos I have seen this year! Keep it up man! Some day the channel will blow up!
Fantastic video and thorough explanation of set theory that has broken down the concepts better than any of my previous investigation into the topic. I look forward to your future videos. The ghosts of Zermelo and Fraenkel would be proud of your spirited teaching style.
did this ARG ever go anywhere? It's been 10 months but the page that say "I'll add more to this page as the investigation proceeds" hasn't been updated.
I suggest that our use of the word 'number' as a _noun_ leads us to an unwarranted reification. If we want to define the noun 'number' (at least in the sense of natural number) U suggest this: *A number is an answer to a 'how many?' question.* This can be generalized beyond English. _Three_ and _trois_ are the same number (though 'three' and 'trois' are different numerals.) Cf Wittgenstein.
I'd love to see a continuation in a similar style with the axiom of choice.
"This is false" is a truthfull statement. We concluded that already. You really are living on the edge...
Holy cow. You are probably the best math/science communicator I've seen! Thank you. This makes so much sense now.
This supposedly rigorous definition still uses the concept of a unit. I really think this is the one time maths has gone to a great effort with zero improvement in our understanding. This is no stronger or more rigorous than just defining 1, adding, and subtracting.
Not sure where you see this "unit" being used... the only concepts that are needed are "set" and "element" (within a set). Also, can you be more rigorous about that "just defining 1, adding, and subtracting"? ;-)
Technically, there's no unit. It's just the definition of numbers themselves in relation to nothing.
This definition is just "the one empty set" and then using it recursively with arbitrary rules. You could do the same thing with *any* made up concept that is singular. It's not that this doesn't work, just that it's exactly as rigorous and has just as many assumptions as any of infinite alternatives. Personally I think we'd be better off just actually defining what it means to count, and admit that the verb form will always be vague as you are defining what's being counted abritrarily.
@@tristanridley1601 But i thought "having some existing number and getting to its successor" *_WAS_* the process of counting. :-) This just needs to start _somewhere_ , and defining zero as the (unique) empty set works. Also doesn't need the idea of "one" yet (in case you used "the _one_ empty set" at the start there to reference the number one instead of the concept that it seems to be unique).
@@TheEnmineer in relation to *sets, not quite nothing. Almost, though, since the main part of that relation is the _empty_ set, but that's still something. :-)
This is a very good video especially for such a small channel
Didn't actually watch the whole thing as I already know set theory but I can see that you've put a great deal of time into this and it's well put together. I've liked the video and left it running so the algorithm is happy and you get the views you deserve.
i am really surprised to see this is the first informational video you made, since it was so well explained and so based and refused mostly to stick with the easy path. cant wait to see what more would you make. honestly, i am wiating to see. this was very interesting and inspiring to learn more about mathematics on high level. thank you
Could you do a video on how we define things like addition and multiplication using this theory?
Hoping to do that in video #3 so get yourself subscribed!
When you create your sixth rule, the first thing you say is "Let's take a duplicate of that thing" - I don't remember "Duplicating items" to be a thing that we're allowed to do by the existing rules. So this new rule is dependent on us having an ability that seems to have come out of nowhere. Did I miss something?
In math, it's not like objects have a notion of count and by using one, you run out. Have you ever run out of fives? I really hate it when I'm trying to calculate 5+3-2-1 but I ran out of fives so I can't get an answer f(x) = x + x, omg where did that second x come from? we only got one as input!
@@Double-Negative now what was the point of being so sarcastic?
@@monicarenee7949 because rigor is boring and jokes are fun
lol "just press control w." I not only think in terms of lbs and ounces, but I also use a Mac! Very impressed with your video quality. Very well done, and you are kicking ass for such a young channel.
the greatest Investigation into set theory i have ever seen, now I see it trough an entirely different lens!
Haven't heard about set theory before but this was a very interesting video and I'm looking forward to more. So if I understand things correctly then: 0 = { } 1 = { { } } 2 = { {{ }} , { } } 3 = { { {{ }} { } } , {{ }} , { } } I'm very interested in how to do something like addition with sets.
The proper way to do addition is using the succ function on numbers. The succ function is a basic building block of the piano axioms which then define everything else. The succ function is basically the function x+1, for the set representation it is defined as S-> {S} U S. (The U is the union symbol). You can see that applying this to 1 gives us 2. Next we define addition recursively, if given x,y as numbers and asked to evaluate x + y we first check if y is zero(aka {}), if it is then x+y=x. If it is not then by the way we defined numbers we must have y=succ(z) for some other number z. We then define x+y = succ(x+z). One would then prove that this definition has all the lovely properties of addition, specifically commutivity and associativity.
@@yakov9ify nitpick: it is spelled “Peano”, not “piano”
@@drdca8263 Indeed it is, mb.
Do the writings in "digital chalk", that sparsely appear throughout the video, mean anything? Loved the video, by the way, crazy that it is your first one! Congrats!!
See the comment above that cracked the code :)
WOW! you are awesome man. I loved this video. You really went to the bare bones of the question and build it from scratch. Already subscribed and will binge all videos in your channel!
Thank you! You really made us embark on a thrilling investigation into the paranormal realm of numbers. This will definitely leave me both mesmerised and intellectually stimulated.
I upvoted for the anti-Imperialist system joke at @4:20 Jokes aside, I really appreciate this video. I had wondered about the real definition of numbers in the past, and your video helped me to understand that concept through a new lens. I appreciate your thorough investigation and write-up on the matter.
Although that may alienate the US audience. And by the frequent use of the word "math" in the comments, they are among us!
@@patmcgibbon7263I don't know, as a USian I loved it
While the presentation is brilliant and entertaining, I'd to point that the number construction showed is one implementation of the concept of number in set theory. There are other ways to embed this concept in either set theory or other foundational background (logic or category theory for example). The critical fact is that all these implementations would agree on how the numbers behave or said more precisely, they all will be isomorphic.
In my first abstract algebra class in grad school, the professor started off day 1 with: "How do you know that your 1 and my 1 are the same?" He then spent the next week going through set theoretic constructions of the natural numbers, integers, and rational numbers. But then he mentioned that you could do this other ways too. And he finished that sequence of lessons with, "So how do we know that your 1 and my 1 are the same? We don't, but as long as our natural numbers are isomorphic, it doesn't matter." That was one of the most memorable sequence of lessons from grad school.
The empty set, like for example the point (length=0) can't be found in nature. Those entities live in our minds. But they certainly don't have their origin in our minds. Therefore, I think that it is perfectly justified to call entities such as the point or empty set *divine entities* indeed - or platonic, if you prefer.
I am so impressed by this video. High quality and really informative. If you keep up with this kind of conten you will become a huge hit. Instantly subbed and shared, wishing you the best.
Fantastic presentation and explanation, however there were a few leaps that I think 99% of people would have had to pause for a few years to finish a set theory course before understanding. Luckily I've already done that, so I loved this. Keep up the great content and we'll have another awesome channel among the great community of KZhead mathemeticians
I think assuming that there is a set that contains nothing is a better assumption then assuming there's at least one thing which you can put in a set and make a subset with nothing out of that. That justification didn't really make sense to me
When you were doing the "successor" procedure, how can you duplicate it? Is it allowed by the rules, or did I miss some part of the video that allowed this? Anyways, awesome video. Suprised to see this doesn't have as much views at this deserved to be. I also love the effort of using carboards for visual instead of animations. Earned a new subscriber. 😊
i guess you don't need a rule for that, you just are able to assume that: If this thing exists, it can exist two times without a problem. And from there on you can just put two of these things that exist in a set etc.
the two sets are the same set; nothing actually happens. 'duplication' is just a weird physical metaphor since he is stuck using the actual physical boxes
This is similar to the question I had: if duplication is a legitimate procedure which says “If you have a thing you can duplicate it” then why bother with the rest? Surely at that point you have already defined numbers (or assumed them silently) when doing the duplication. 1) A thing exists, the empty set. 2) I duplicate the empty set. 3) “Two” is the set of all of the sets that are here after I duplicated the empty set. I just don’t understand how it remains necessary to define numbers further if you are allowing ‘duplication’ in the procedure. Duplication is taking one thing and creating of it one thing and another thing, which you can simply call two things.
i am 7 minutes into the video and i am just captivated by your way of presenting things! Keep up the good work!!!
Thank you KZhead! Because of it, we have creators like you Sir (from up north). Kudos. Happy to support your work.
Kids, that's why you'll need some math in real life since 90+20 is 100
When you said (30:30), "If you're like [...] you know that at least something exists," I rather felt a bit uncomfortable. You said at 23:30 that we were pretending we knew nothing. I wish you'd added just one more sentence: "So, let's add, [the empty set exists] to our list of assumptions." I agree with you at 41:30 that you justified that this axiom exists. However, that justification was not based on the initial assumptions, so it feels rather like cheating. Given the way you justified the video topic, I feel like avoiding intuition should take priority over parsimony (or rather, that any use of intuition should be called out as being an explicit assumption).
idk... we can't really pretend that we know "nothing" -- we should at least be confident that KNOWLEDGE exists. ;-)
Difficult topic, clearly presented, and the delivery maintains interest. That's a winning formula. Well done.
Absolutely amazing video. Happy to find another maths geek Northerner ;) Best of luck on all your future videos!!
I'm curious about the successor definition as it seemed surprisingly complicated to me. I expected after you said duplicate the set and put it in another set that that would be enough to create a unique set. What do we gain by doing the union operation afterwards?
The “Russian doll” sets each only have one element. The union lets us create sets with additional elements. It’s a little easier in the mathematical sense to say “how many boxes would I see when I open this box (not caring what’s inside them)” than “how many times do I have to unbox what’s inside before I get nothing.”
loved it!
Thanks for checking it out Tom, glad you enjoyed it!
Yes! This is exactly what i was looking for! I've always relied on extensive periods of time searching through Wikipedia, asking ChatGPT, or finding some occasional videos on topics relating to the fundamentals of mathematics (mathematic theories, mathematical logic, all of that). But, now I actually found a channel that's all about that! Thank you so much. Most of the things you talk about in your videos I have already learned about myself through extensive studies. Of course I will still continue to search though the internet, but this channel will be very good for me. I wish I found it earlier 😅.
Definitely relatable. A 40minute in depth explanation without being boring. You can end up being a less corny Matt Parker =)
PS fully enjoy Matt's vids, but you gotta admit, he is corny.
Brilliant! I've thought about this for a long time. This dives into some of the more obvious problems that are too often swept under the rug. I'm still convinced that the axioms were chosen in order to recreate the number system we already accepted based on intuition. We didn't independently discover these axioms, we constructed them in order to get the results we were looking for. Obviously this is very useful in modern mathematics and countless other fields, but it does not reveal metaphysical truth. I always had a big problem with the axiom "The Empty Set exists". I call it existence by decree. We never really get past this big "IF", we just kick it over to one of the axioms. Who says that pairing is true? What allows us to conjure up a third thing every time we see two things? This decree is equally as bold (or vapid) as declaring that the Empty Set must exist. Notice also that "pairing" is defined as something we do with TWO things, but then we boldly state that this is sorta-kinda the same as two simultaneous instances of the same thing. How do we justify this? Well, it allows us to construct the number system we're trying to construct. So we're working backward from intuition, and creating arbitrary axioms as a support structure.
They did the same thing with the definitions of the metric units. They started with something imprecise then went looking for something rigorous that targets it. The result is still a rigorous definition even though it's based on something arbitrary.
We absolutely did discover independent axioms. Numbers are not contingent on our observations of them in order to exist. 2+2=4 is true whether we know what two is or understand the concept of equation or addition.
It gets even worse than that, Godel proved that if these axioms are consistent, then they are not complete and if they are complete, then they are not consistent. Trying to axiomatically define the numbers is a fool's errant. Better approach is merely to accept their existence, be it on account of intuition or faith or revelation or whatever, and then just be content to work from there; that's really what mathematics does, in practice, anyways.
@@costakeith9048 Accept their existence based on faith, huh... Most people aren't ready for that conversation.
@@somebonehead In reality, that's what people are already doing anyways; some people just like to try to deceive themselves. There is no objective basis for suggesting that the axioms of mathematics are consistent and complete and, as Godel proved, there never will be.
I’m really disappointed this didn’t win, there were a lot of other strong submissions but this one was easily my favorite out of the many, many videos I watched. Regardless, this channel has an incredibly bright future ahead as you’ve somehow released three successive videos that captivated me just as much as this one, if not more. I’d love a video discussing the axiom of choice and the source of its contentiousness within the mathematical community. I can’t wait to see what awesome videos you make next! :D
Win what????
Here in India, when I search what is a number on KZhead this is the second video in the results after numberphile's.
@@trappedcosmos SoME2, a competition of maths videos that happened last year. Keep an eye out for SoME3 videos that start appearing! This competition is organized by 3b1b every year since 2021
In addition to being an educational, interesting, and witty video, I found the background electronica absolutely hypnotizing.
Thanks for the kind words! The music is a little divisive -- I get comments both thanking me and condemning me for it. But when I watch my videos without music it doesn't feel right -- I drastically prefer the atmosphere with music!
You deserve more subs
Thanks! Share my video far and wide, it'll be doing me a huge favour :)
This is an absolutely amazing video, love how much you put into it Any and all issues I have were either addressed in your extra stuff/afterword or come down to philosophy of math, which I am a nerd of, and I can put aside to recognize the amazing pedagogical use this series can have, so I very much will send this to students/friends I would put out that the initial analogies are somewhat strained and I can't tell if that's me being smartass wjth a different philosophy of if they are strained in general
It's great to see a quality explanation of the maths in set theory. Best explanation I've seen.
Great video! Although I would have found it better if you didn't stress that much that this "the one true definition" of the natural numbers. For example you can also construct the natural numbers in (a typed) lambda calculus. (Oh I guess you said at the end that axiomatic set theory is only one foundational system for mathematics)
Yeah, kinda just goes with overstressing how definitions "should" be objective. Like, yeah, it can definitely be useful, but honestly saying "red" is a certain wavelength might not be the best definition for your purposes.
Especially when there already is a universal definition of the natural numbers - Peano's axioms - that applies to every formulation of the natural numbers, from the set-theoretic encoding, to the lambda calculus encoding, to natural number objects in category theory, to the data type formulation in functional programming. They all have a zero and a successor function you can do recursion on, and the settings with some way of working with predicates have induction. More of a focus on the Peano axioms, or at least on the concept of zero and successors, which get a mention, but not as broadly as I'd like, as the fundamental intuition that every encoding is based on would have been nice, rather than just treating the set-theoretic encoding as the one "real" definition, not even regarding it as an encoding, rather than just the most common encoding in classical math.
"Definitions shouldn't depend on how we feel in our gut, they should be robust" - for math/science, yes. For discussing emotions, less so. There are some things which literally *have* to be defined on how people feel about them, because they *are* the feelings those people are feeling. That's kind of off-topic for this video, but I feel it's worth mentioning as a side note from the main discussion. Which, for the record, is amazing and I'm glad I watched. Also France is involved all the time because that's where the metric system originated. It was proposed and started by a church vicar, on which note... it's interesting how involved church leaders were in early science when so many of them deny science now. And in another fun story, USA was one of the earliest adopters of metric, and actively pushed other nations to adopt it as well. They're largely responsible for the rest of the world using the system they promptly threw out like hipsters because it got too mainstream.
6:01 This sort of trivia sticks with you for life. Also, this humour is A+
Your boxes were *so* much easier to understand than the numerous times I've come across this written down...
10 mins into the video, I feel the need to mention that, you cannot prove a definition, you can only choose a definition. so while saying that "3 is the number after 2" is an incomplete definition, and I'd agree if you say it's flawed, I still would consider it a definition regardless of how logical it is
Both are wrong. While you and I would agree that we cannot prove a definition, only draw a conclusion on top of presuppositions and assumptions, I would disagree that we choose definitions. That is simply an unstable worldview. If we open Pandora's Box and leave definitions up to our own choosing, we will eventually and inevitably decide to shape the world around us, instead of shaping ourselves around the world. Think about everything theoughout history, racism, colonialism, et cetera, that came about because people chose to define certain groups of people as intrinsically inferior. It may seem like a leap to go from numbers to colonialism but that's the kind of thinking you invite when you open the floor to that logic.
@@somebonehead I don't understand how choosing a definition leads to an unstable worldview, mostly because if you don't choose it and can't prove it either, what is the third option. but even then, for example, let's say I choose the definition that "if you are playing a singleplayer game, and you beat it slower than the time on the site 'how long to beat' you are bad at the game". now, completionists and people who stay afk in games a bunch, or are looking for easter eggs, they will probably all disagree. and what happens then? just because I choose a definition, doesn't mean others will follow it. if I am having a discussion about free will, and me and the interlocutor have vastly different base definitions, we cannot go anywhere with the topic. having an incompatible definition with someone else just prevents meaningful discussion until you address the difference in definitions. notice that I didn't mention what do you choose your definition on. if someone picks a definition that is racist, and people around also agree with choosing that definition, well, they will be racist until they meet part of the world where it doesn't allow that. The concept of picking your definition doesn't say you cannot argue which definition is better or moral, it also won't prevent nor promote good nor bad concepts, I would say it's completely neutral edit: I re-read the start and I feel the need to address "only draw a conclusion on top of presuppositions and assumptions" . this sounds pretty much "choose a definition, but attempt to be educated about it" which is obviously "better" than picking a definition without that, and it feels like cheating, because the third proposed option is just a variation of the first option, not an actually different option