How to Count 

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. Couldn't you use [this definition] of ordered pairs? Yes -- the one I use in the video is attributed to Hausdorff is a little non-standard. The Kuratowski definition (a,b):={{a},{a,b}} is more widely used and accepted. I try to make these videos accessible to as many people as possible, and I think the Hausdorff "tagging" description is the most accessible. That's just my opinion -- I make a lot of decisions like this for pedagogical reasons. When teaching a concept for the first time, it's not always best to take the most efficient approach. 2a. "If A is a set, I can always find some element not contained in A" -- what if A contains everything? A fantastic question and I glossed over this because I didn't want to get too bogged down. But the axiom of regularity has a consequence which says that all sets cannot contain themselves. So for any set A, there exists at least one element not contained in A, namely A itself! 2b. So couldn't you use A itself as the "tag" instead of 1 -- yes, a great idea! 3. Why can't we define a function which maps one element to two different things? A "function" is a thing invented by mathematicians to serve a purpose, and it turns out it's just more useful in serving that purpose when it's defined such that no element can map to two different things. Try removing that stipulation and see what sort of theory develops! 4. There exists a bijection from 'n' to 'k' implies n = k. How do we know? Well spotted. Well done to those who called me out on this -- this is a statement which requires proof, and I did record an explanation of this but I felt it was getting too bogged down, especially for viewers unfamiliar with the material. Another hand-wavey pedagogical choice!

Currently teaching my 1 year-old how to count. So far we've got to 3, but I guess we should scrap that progress and restart at Extensionality!

Really excited about this channel and where it's going. This video and your previous one are such a good starting point for people interested in "proper math"

I love the mathematical bricks. Building knowledge one brick at a time. Also opens the door for some good jokes about getting hit by hard truths, where those truths are things like the brick of fermat's last theorem

I'm an elementary school teacher, so I'm always thinking about how to unpack things that are intuitive to me but new to my students such that I can teach them clearly and effectively. I won't exactly be sharing this video with any elementary school students learning how to count, but I can absolutely appreciate how easy you make it look to get complex ideas across clearly and engagingly. Kudos!

This kind of "step by step proving something we took for granted" is something I absolutely LOVE about math. Would be great to see this turned into a series, where you prove even more basic mathematics from the ground up!

I know this channel is new but it's amazing to see such a small channel with great production quality, clear explanations, and a bit of humour and personality. Really excited to see where you go.

For the curius ones, a good way of encoding ordered pairs is (x,y) = {x, {x, y}} The set {x, {x, y}} contains the pair {x, y} and the first element of the ordered pair (the element x). In this way you don't have to worry about which labels you should choose.

Two things: (1) I love the wall-building visual because that's literally what you're doing: Starting with the very fundamentals (What is a number? What is counting?) to build a solid mathematical understanding from the ground up. As a high school math teacher, I see constantly that my students struggle with complex math almost entirely because they don't understand the fundamentals, and that we can't explain it to them properly because WE don't fully understand it either. This is beautiful content and I look forward to much more of it. (2) I also love that you're using inexpensive, everyday objects for your visuals. Many teachers depend too much on technology or kits or fancy purchased manipulatives (all of which are FANTASTIC in their place!!) and struggle to teach without them. It's ALSO important for us to be able to communicate concepts to students using materials that are familiar and accessible to them so the focus is on the concept and not on the fancy toy. I love to see it!

As a programmer, the first few minutes were like living my life. Explicit instructions must be given

I love the idea that this episode builds on the previous one. It's not just a discussion about some mathematical topic, but an extention of the previous one and that's honestly something I don't see often on RU-vid. It would be awesome to see another part of this journey!

I had such a hard time in foundations courses in college because so many things "just worked" already. My intuition, especially when it's correct, is a huge crutch. Your examples and counter examples are so reasonable in a way I never heard in school and I love the building blocks as a constant visual reference.

Is this channel going to build mathematics from the ground up? I LOVE IT

Love this type of stuff. Math is so pedantic and thats why its great. Please do more videos how we build math from the ground up.

I can't believe the quality of your videos. One would think you'd been making videos for years and years. This is incredible.

This is the dopest mathematical channel of the last 3 years.

I really like how you never take a step without explaining why it's justified, while at the same time avoiding getting caught in the weeds. You strike a great balance of being thorough, concise, and entertaining throughout. Also, I'm commenting in hopes that the almighty algorithm will shine on you once more. Your channel has some serious potential and I hope more people see what you're making here because it's truly great.

Another amazing video! I love seeing things about ‘foundational’ math, it forces you to confront your intuition, then rewards you at the end by rigorously justifying it. Very satisfying! And speaking of satisfying, that moment at 27:25 was great. The tiles sliding out, demonstrating the actual, real, no-analogy-it-really-is equality between counting and the actual number… so exciting!

Man, your videos are fascinating! As someone who loves math but doesn't really know where to go to learn more, this is an amazing resource. Thank you for making this

Wouldn’t an efficient “tag” for 1 and 2 be A and B itself so we would have {{c,A},{c,B}} since by regularity our elements a,b,c can’t be A themselves. I know we aren’t using regularity in this video but in this sense it helps us use a ‘best’ tag. Still in middle of the video but loving it so far!

disliked. you didn't finish telling everyone about me. i was knighted in 1705 for discovering what we now know as sir jections

this channel is going to blow up

RU-vid algorythm has blessed me today. Looking forward to all the future videos!

Another video this fast n just as long?? I'm going to love this channel

You've like immediately became one of my favorite channels, looking forward to your future work!

I lost it when he brought out the building blocks, literally. This is brilliant. Couldn't have enough of it.

You are so damn good at this! This is what the internet was made for: the most talented and likeable teacher for the world to learn from. Can't wait for the next one.

What a channel man. This dude is such a perfectionist in a good way. I am pretty sure that he has optimized the quality of math lessons, great job, I have no words

Watching your videos takes me back to watching Open University broadcasts on BBC2 in the 1980s

I love the highly abstract and low-level bits of math and the visualisations you use. In university I had a series of lectures leading to the definition of a function and when I understood it, I was amazed at how beautiful and general the definition of something seemingly so obvious and basic is.

Underrated channel, your videos are so exciting.

I didn't get maths at school but I like it now (from a distance). I watch other channels but they always go beyond my understanding at some point. Your last video and this one are absolutely brilliant - really accessible, but thorough. I've watched them a few times to get them straight in my head. Fantastic work.

I had to pause this at several points to digest what was happening, and finally getting that “aha!” was truly satisfying. I appreciate you making this so easy to understand and logical. I’m self studying Mathematics, and have found it confusing at some points; This video has helped me to finally understand some of the concepts that eluded me. Thank you!!

I really like this series of videos, just to add that the best way to tag the element of A is probably to use {a,A}, and the fact that A can't be an element of itself!

Absolutely love this. I've always wanted to understand maths from the axioms but assumed it was too complicated. You explain it so clearly!

I really enjoy the conversational tone and tactile nature of your videos. I especially love the bricks in this one. Looking forward to more!

These foundational math topics are so fascinating, and I love your delivery and humor. Looking forward to the next one!

These videos are wonderful!

The production value of these is remarkably high for someone who has made as few videos as you. Looking forward to the next one!

I've been studying mathematics and computation casually for a while and every problem I found (others have found) in our usage of computation, writing or using or distributing or understanding software (universal algorithms), for instance, all eventually leads to the knowledge in your videos. Regardless of which angle. All roads lead to how to count I guess. And your videos do a wonderful job of highlighting these concepts from their own merit.

Great video, just like the previous one! Happy new French subscriber here. At 15:52, the nitpicker in me would say that you need to prove that there is a universal rule to find those tags and identify which signifies "first" and which signifies "second" :P It would seem reasonable that you could make one, so fair enough, but I haven't actually ever seen that concretely written down. The construction I usually take is to define (a, b) := { a, {a, b} }. This is good enough to prove the usual characterization of pairs, i.e. (a, b) = (c, d) if and only if [a = c and b = d]. But as I mentioned this is mere nitpicking, the content is obviously solid and the explanation is crystal clear. Excited for the next videos!

These two videos (What is 3 and this one) are some of the best representations of explanatory logical thinking I've ever seen. You've got a good sense of humor, and it adds a pleasant spice, but you're not shrieking comedy into your videos in a forced way. You're not only knowledgable and passionate about the subject material, you've also come up with a great, simple way to express all of your ideas. There's an element of what you're doing that reminds me of a magic show. And I think math should be somewhat like that, building and building until the cord is pulled and you see what you've been building this whole time and it sharpens into focus and you find yourself laughing at the a-ha moment. It's very good, very powerfully simple stuff. Well done on both counts, if you keep going like this you're going to build an impressive channel. I watch videos on physics and maths for fun, but as much as I love them it's easy for the eyes to glaze over at certain points, and that literally never happened once with your videos. I dunno just top marks, man, well done.

The clarity of these explanations is fantastic - I just stumbled into this channel and now I'm learning set theory, which is something I never thought I'd say!

I think I see where this channel is headed, and I love it! New favourite ❤

I already learned this in university, but you're such a good teacher I was still entertained!

These videos are so fantastic. I feel like I learn so much and yet for a while, as the video goes on I feel like I know less and less. Everything I thought I knew gets gradually stripped away. Then it gets rebuilt in a new way that is much more interesting and stronger than before.

This is the most intuitive explanation of Sets, Relations and Functions I've seen!

Nice explanation! The way I think about disproving the |Z| > |2Z| thing is doing it in reverse. Make a function Z -> 2Z where we multiply every integer by 4, thus hitting just "half" the elements in 2Z. By the same reasoning this would mean |Z| < |2Z|. Clearly |Z| > |2Z| and |Z| < |2Z| can't both be true, and these functions show instead that |Z| >= |2Z| and |Z|

I’m in top set at my school. I never thought a video on how to count would confuse me.

I learned so much from this 2 part series! Definitely something I don't want to learn in a school setting (It'll be hell), but perfect as an edutainment video like this.

I'm so pleased to have found your channel so early. Your content is great and leverages things I've already learned to introduce new concepts. Love it! Very clearly explained. Keep going!

Yes. Yeeeeessss. YASSSSSSS! MORE! Thanks bud! Super stoked for this channel.

I absolutely love, love, LOOOOOOOOOOOOOOOOOVE this channel. Manim is fantastic, and I love the animations people make with along with of course 3b1b's videos, but your unique physical style is so playful. Your humor is charming, and you really embrace the exploratory and fun of entering higher math. Great content, and I'm so happy you make these videos!

I absolutely adore your channel. Please keep it up for years to come.

I feel you on the not wanting to make erronious assumptions even as a kid. I remember when we learned addition but it was 3 rows high instead of two (5+3+8=). Other kids had no problem, but for me it was TOTALLY different! Since much of math is presentation, I think it is legitimate to consider a change in presentation a change in the rules.

This video very clear and focused! No noise, like in Wikipedia about the subject! Thanks!

I'm in my 70's and I enjoy watching math videos of subjects I have studied, or perhaps should have. One thing that came to mind, and I see you have listed a correction, is sets. I recall counting numbers being defined as follows: 0 = {}, 1 = {0}, 2 = {1} ... I don't recall if this is an important difference from what you present, or was demonstrating something I have forgot.

As a back-end developer with a strong SQL background I now understand the intricacies of the theory behind my job better, THANK YOU!

Glad I stumbled on this new channel. It's people like you that make math fun again.

Great respect for all the teaching aids, it must have taken you so much time to cut all the cardboard and organize all other materials. keep it up!!

This is what we learned in class 11 but definitely not this way. It was an awesome experience. And you made a masterpiece imo.

Can't wait to see what you bring us in future episode's. It is often good to relearn the primoral basics of any subject.

Fantastic stuff. Loved the little sleight of hand bits mixed in, added to the entertainment without detracting from the lesson. Thanks for the refresher!

Excellent job so far! Love the clear explanations, the use of props and the right amount of deadpan British humor! Keep it up and you're gonna go far!

I am a very big fan of math channels am am always glad to find more, it's great that you are covering interesting basic things you never think of to. Very well done, Congratulations on giving RU-vid another great math channel.

Woah, this is such an underrated channel! As a fellow educational RU-vidr, I understand how much work must go into this- amazing job!! Liked and subscribed :)

WOW, this was amazing, I wish I saw this when I was younger. The props you use in the problem make it more real (like an engineering or computer problem). Very excited to see how far you can take this concept. Best way i've seen this explained by far.

Your explanations are really compelling. The props and gentle humour are really engaging.

That demo at the beginning was fantastic!

I hope your students appreciate how well you explain things. I sure do!

As always, your explanations are the perfect balance of satirical and didactic

Great video. For the first time, I feel like I have an intuitive grasp on where the different sized infinites come from.

These videos are so good! Feels like something completely new, different. Congrats and please keep it up!

Amazing video, this explained all of the assumptions taken in my college algebra class that I didn't even think to question

It's so nice to have a educational video that focuses on the core mechanisms of the subject before anything else. A flowing ground up build with simple props that neatly represent nothing more and nothing less than they need. Clear description of the tools available and their relevance and the limits and variations of the functions. Most the videos I see do a quick outline of the principal or logic bullet-points before focusing on the jazzy conclusions. To me logic requires presentable proofs otherwise it's just clever or overcooked conjecture -which can be very entertaining but don't provide solid answers let alone an introduction to usable tools that I can use and prove through related applications in my own self learning. I look forward to your future videos

Loved the video. Your ability to describe these concepts in a way that is easy to understand is fascinating.

It took me a few beats to get the Sir Jection joke 😂💀 it's a testament to your teaching skill that you guided my intuition to the correct conclusions so many times. Your videos are excellent!!

I'll jump on the train and say your videos are fantastic and to keep it up. Great presentation!

Big fan of the various examples of how natural candidates for definitions fail. Keep this up! Excited to see how far you go into set theory.

so I started watching this video, then you convinced me to watch the previous. then about half way thru this video I said to myself "this was a lot of rigorous math, I need a small break" so I took a break. then I come back, resume the video and its the part where you say "you should take a break here" that was perfectly timed

I love your channel. Such high quality, it's incredible!

I love your props; first time in pushing 2 decades that I feel like I actually get wtf set theory is.

Thank you so much, this series is being enlightening to me.

Love this video, I'm trash at math and I understood this so easily.

Ah yes, my favorite song. It's just another brick in the body of mathematical knowledge by Pink Function

This is sooo goood!! 😱🤩 I can't believe you just started this channel! Even though I don't generally subscribe to a lot of channels I subscribed to support your channel growing!

Imagine my surprise when I realized I couldn't count. Can't wait to start teaching the kids set theory in order to help them understand how to count. Thanks! :)

One thing I learned from French counting (and Rewboss In Germany). They have individual names for 11-16 and then they say ten seven, ten eight and ten nine! English count the other way round (three ten, four ten,...).

Better then the lectures i had in university on this subject! Great Video!

Your two videos are beautifull, as a mathmatitian myself I love how you are explaining this and could be use for people who are studing math or are just curious. I wish I had this when I was taking number theory. I hope you can make a video about the Axion of Choice, and you can continue to construct the Whole Numbers, Rational and Irrational Numbers. Great work.

Your first 2 videos are unmatched, amazing teacher. Keep building and your channel will grow with the math ^^

It's neat seeing how many assumptions are inherent in even very basic exercises like counting and comparing the sizes are two very small sets. It makes me really appreciate how amazing our brains are that they can automatically recognize all the reasonable assumptions to make without our conscious minds even being aware of it. It saves so much time and effort compared to if we actually had go through all this logic all the time -- imagine having to take all the steps he went through in this video every time you had to count something! Of course, this often causes trouble, since our subconscious minds don't always make the _correct_ assumptions, and sometimes make assumptions where _no_ assumptions should be made, or generalize things too far, such as automatically applying stereotypes to individuals without even realizing it. And intuition can certainly lead us astray sometimes, like he talked about in the prior video. But even so, while our brains our far from perfect and our pattern recognized skills sometimes take things too far and cause problems, our brains are still pretty darn amazing.

20:53 « If you wanna take a break ». I actually just decided to take a quick break at that very moment. This man is not just a very educational mathematician, he is actually a wizard

Your way of explaining these concepts is incredibly good 😮

You deserve more recognition, you're redoing all math only using the axioms

I find your videos incredibly interesting. You do a great job at explaining things!

Really amazing! Thank you for this!

I’m a high school math teacher & I’m learning so much from your videos. I’m a big fan of rigorously scrutinizing our first principles. A request, could you make a video on regularity principle & it’s connections to Gödel’s two theorems?

Love the channel! I am hoping you will slowly and rigorously build maths from the axioms up only to finally throw Godel at us and give us some sweet Lovecraftian existential dread.

First of all, amazing channel, just found this stuff, love it, hope you keep doing more math fundamentals because it is a fascinating subject. Also this topic makes me think of that one part in the Wayside School series where a teacher is trying to tell him he doesn't know how to count, so, to prove her wrong, he counts to 3. "A thousand, a million, three." "That's wrong." "But teacher, he got to the right number!"