This guy's vibe is a fifth year UCSC student who got stoned before class one day, accidentally wandered into a graduate number theory seminar and got HOOKED.
Firefighters: Why did your house burn? Combo: ... And various axioms could be inserted to support ... Firefighters: Sir, are you okay? Combo: ... With a bowl where my cat can experience infinity, or a multitude of infinities ... Firefighters:
@@chri-kThat depends on CPU architecture and data type. There's no difference between 0 and -0 in integer math on any computer you're likely to find running today.
dude, I just looked and I'm shocked you're sat just below 50k subs. you produce some of the best content out of all the science communicators I consume. Complex topics, easy to consume, great jokes. love your shit.
Hey Combo Lords, sorry that some recent episodes have taken a while to finish. I’ve been in the process of semi-moving houses and other personal things (see the livestreams on my @Domotro channel for some details). My next episode may also take a few weeks, since it’ll be a long one about what I consider to be the most underrated concept in number theory. After that, I can hopefully get back to a quicker release schedule. Anyway, thank for watching! Here are some timestamps of different parts (and check the description for more info and links) 0:00 - Intro 1:37 - How Many Square Numbers Are There? 2:51 - Surjective/Injective/Bijective Cats 6:03 - How Many Integers Are There? 7:28 - How Many Fractions Are There? 12:46 - How Many Real Numbers Are There? 13:36 - Cantor's Diagonal Argument 19:54 - The Hierarchy of Aleph Numbers 21:28 - The Continuum Hypothesis 24:51 - Outro
I hope your new neighbors are as tolerant of your chaos as your current ones! Also I hope to see you feeding squirrels from your hands again in a future episode, I loved seeing that in one of your previous ones!
Whoa. That diagonalisation made me feel odd and giddy. I can't explain it, it just makes me feel good. Like I'd just read a really good novel. I've got to try this math thing some more. Didn't think this would be how I fall off the wagon...
It's a smart argument, but I still feel it's flawed. There is no such thing as the "complete" diagonal, so it does not make any sense to argue whether it's in the list or not. But you certainly can expand it one digit at a time, adding the partial diagonal to the list. And since there's no upper limit where it would stop, you could argue that it works for the whole diagonal. The fact that I can't write down the finite row number of the diagonal does not mean the diagonal is missing in the list.
You bring up some good points, particularly around expanding the diagonal digit one at a time. It reminds me of partial sums of a series and similar methods of dealing with infinity. However, Combo’s argument isn’t related to the fact that you can’t write down the infinite list or the infinite diagonal. The argument for why the rational numbers and the natural numbers are the same size proves this. Combo doesn’t have to write down every rational number, because he’s able to provide a rule for how you would list each rational number. There is no rule anyone can provide that lists every number between 0 and 1.
You are mad, and I think I love it, such a unique style, yet you explain advanced math concepts very well. You're a mystery, but I really like those videos!
16:16 - If we take one of the 8digit calculators, start with 0 and add 1 every second, then after 1/PI century you would reach overflow, sounds like an infinity to me. PI is defined here as 3,14 - keep that number as CONST type so you could redefine PI in the future in case the required for you PI value changes ;)
I look a bit differently at infinities. A 3D cube has an infinite ammount of layers of 2D planes stacked together. So a 3D cube is an infinite times larger than a 2D plane. A 2D plane has an infinite ammount of layers of 1D lines stacked together. So a 2D plane is an infinite times larger than a 1D line. But a 3D cube is an infinite^2 times larger than a 1D line. To make matters a bit more ocmplicated. If we have, let's say 7 2D planes. Then these are 7*infinite times larger than a 1D line. But a 3D cube is infinite/7 times larger than these 7 2D planes. The relative size difference between the 7 2D planes and the 1D line is now bigger than the size differnce between the 7 2D planes and the 3D cube. Now for a tricky question. How do the 2 relative size differences relate to each other? Is it 7, 14 or 49?
This is what I've noted far as relating countable and uncountable infinities in life: Completable tasks are in effect the equivalent of countable infinity. Simple example being bringing your hands together from an arbitrary distance apart, if looked at from the mathematical perspective you could turn such a task into sequential or countably infinite task of partitioning the overall goal into descending half steps which in effect "can't end", but in reality this infinity "collapses" and the process does experientially terminate (not going to go down the electron sphere repulsion rabbit hole and defining "touching"). For most of you, you'd recognize such a concept as a variant Zeno's paradox, but the importance of a paradox is it's a demonstration of a misconstruing of details and concepts to make something appear contradictory to reality/daily experience. As for uncountable infinity, the most simple and understandable concept would be to recreate any action or behavior in exactness to a prior occurrence. While our human minds eventually disregard precision after some arbitrary amount of "sig figs" in truth we never actually repeat the same action to exactness. This inherent inability to truly reenact actions in their totality can be analogized and explicitly be demonstrated with the simple task of asking a person to point to pi on a labeled number line. Even if Pi is clear labeled, the nature of our physical existence would prevent us from doing such a thing, partially because a graph/number line is a mental abstraction so the graphical representation would be purely demonstrative and not physically representational, but also because of the inherent degeneracy/inaccuracy of our neuromuscular coordination (again our mind end's up saying "good enough").
absolutely not a criticism, because I understand the diagonal argument was an explanation and not a rigorous proof, and it was a great explanation btw. But I would like to point out that some care has to be taken when choosing the values from the diagonal digits when doing a rigorous proof. If all diagonal digits in the enumeration were 8, the number that was supposed to be missing would be 0.999... which is not in the open interval 0,1
@@chri-k agreed. There are loads of ways of doing the argument formally without this issue coming up. Another way is always using 4, except when the digit is 4, in which case you use a 7.
Is this a valid argument (assuming you're adding 1 mod 10 to pick digits) if your infinitely long list happens to do this at any point: ... 0.900000..., 0.990000..., 0.999000... ... Then you'd be producing something that looks like 0.00000...xyz, which just isn't a number ( I don't think ordering the reals like this is a valid thing to do, but we assumed the reals are countably infinite as part of the proof so it should be fine )
If you take all the real numbers 0 to 1 and represent them in binary, where a 0 means not a member, and a 1 means a member, then real numbers are identical to taking subsets of the countable numbers. For example, in decimal, having 1/3 = 0.333..., that would correspond to a binary real number, and also a set where all the odd numbers (where the half's place is labeled as zero, and you keep going from there) are included in this subset. (0.01010101...)
There is unfortunately a complication here from the fact that 0.01111111… = 0.10000000… in binary, but yes, you can almost always match up a binary number with a subset of the naturals in this way.
Countability of the number of different ways of arranging a 52 card deck is mind-boggling. If we deal one arrangement every second, and then after a billion years we take a 1cm step along the equator, and every time we reach the Pacific ocean we empty out a 5mL medicine spoon, and when the Pacific is dry we place a piece of paper on a pile, and when the pile reaches the moon, we count 'one', and repeat until we get to 'a million' then we are still only 90% of the way through dealing all those different card arrangements.(and the universe is long dead) Cantor was bonkers. Plus between any two real numbers you can always find a rational number, and between any two rational numbers you can find a real number, but somehow they don't alternate (isn't that illogical?). And there is no matching inverse of infinity, unless you allow positive zero, negative zero and double zero. It totally screws up physics because every exponential function need to 'start' at a different point in time when that first infinitesimal is added!
The number that Cantor would make adding 1 to each decimal diagonally would occur roughly at a place that starts with first half of the digits being 0 and the last half the represented the number made, being the decimals represent whole numbers infinite its just a glitch making it equal too the whole and included, and that would have to be tallied prior to the manipulation to be seen because it's change would cause it to tend to 0 on the number line ending in 0,1 or 2 infinitesimal if repeated for an infinite number of times
Something really funny about this: All the real numbers you've EVER dealt with are part of the set of Computable Numbers, and that set is still just a countable infinity! Pi, e, every rational, the roots of the rationals, every possible solution to any solvable equation. All of them. Because the list of every possible computation is countable: {0, 1, 00, 01, 10, 11, 000, 001...}. It doesn't have to be binary either. Any other language you use to represent computations, the number of strings in that language is still only countably infinite. To get the set of Real Numbers, you must also include the uncountably infinite quantity of Uncomputable Numbers! Yes that does mean that almost every real number is impossible to compute. Approximately 100%.
If there are different sizes of infinity, how can it be that 0.999... equals 1? For instance, if we can say there are infinitely many numbers between 0.999... and 1, such as 1/ℵ₁ or 1/ℵ₂, doesn’t this contradict the equality?
1/(an aleph number) is not a defined quantity on the real number line. You can't take the reciprocal of an aleph number in the same way as we are used to doing with a real number. We typically assume that every number non-zero (n) has a version (1/n) in its system, but aleph numbers aren't the same. Some branches of math allow that sort of thing, but those systems are different and lose many typical properties of arithmetic we take for granted. When proving things about .999 repeating we are typically discussing real numbers, not systems that allow infinitesimals of that form.
@@ComboClass Here is proof that ℵ0 ∈ N. The cardinality of the set {1} is 1, the cardinality of the set {1, 2} is 2, and the cardinality of the set {1, 2, 3, ..., N} is N. and the cardinality of the set of natural numbers is ℵ0 (aleph-null). Thus, ℵ0 is a natural number. we know that all natural number are real number . Because ℵ0 is a number in the real number system, 1/ℵ0 is a real number as well ( not zero) . Since ℵ0 < ℵ1, it follows that 1/ℵ0 > 1/ℵ1, which implies that the claim of no gap in the real numbers is false. Therefore, there must be a gap between 0.99... and 1; thus, they cannot be equal in the real number system. So, if this video is correct, which is my opinion, it is 100% accurate and the best one I have seen in years. It proves that 0.99... cannot be equal to 1 in the real number system, although that may not be your intention but it did.
There is a widespread misconception about the aleph numbers. It is known that the cardinality of the real numbers is the same as 2^(Aleph_0). Cantor himself proved this. The Continuum Hypothesis is the claim that Aleph_1 = 2^(Aleph_0). This is the statement which is independent from the standard axioms of set theory, so we can't say that it is true, and we can't say that it is false. But the misconception is what Aleph_1 means and what 2^(Aleph_0) means. By definition, Aleph_1 is the smallest cardinal number greater than Aleph_0. And by definition, 2^(Aleph_0) is the size of the power set of the natural numbers. But the popular misconception is that these things have the opposite meaning. It is commonly _wrongly_ believed that Aleph_1 is, by definition, the cardinality of the set of real numbers, and 2^(Aleph_0) is the next cardinality after Aleph_0. But again, those are _wrong_ and got popularized for some reason. Apparently, there was an old book that _assumed_ the continuum hypothesis (without saying that it was assuming it) and caused the widespread misconception.
I like to think that Cantor didn't *prove* there are more real numbers than natural numbers; rather he *defined* a way to enumerate an infinite set. After that, the proof was easy.
Around 20:45 you kind of suggest that in ZFC there could be infinite cardinalities that aren't a member of the ℵ (aleph) hierarchy. That's not true: they're all alephs. (But it could becone true if we drop the axiom of choice and just work in ZF.)
I guess I misphrased it by referring to all of them as ZFC and not clarifying about the axiom of choice. I left that ending part of the episode very simplified to be understandable to non-mathematicians but you are correct about those names. I’ll add a clarification in the description
@@ComboClass Don't get me wrong, your sentence I referred to is literally true; in set theory they certainly study infinite sets that don't have the cardinality of an aleph. It's just that people who listen to the mention of ZFC later towards the end might get the wrong idea.
Yeah I appreciate the thoughts. I'll clarify some things in the description later. And someday in a future grade I may make an episode explaining set theory concepts/models more clearly, since it was tricky to fit it all in this episode.
I'm very intrigued about this uncountable infinity being 2^ℵ0 possibly being Aleph 1. I was under the impression that there were a series of aleph infinities but that uncountable infinity was bigger than all of them. How it could have been proven to be 2^ℵ0 has me very much intrigued. As does the reason people think it might be ℵ1.
16:45 this is the issue with cantors diagonal argument "I wrote random digits", is not how you construct real numbers. None of what you had written down was actually a number, they were just finite sequences of digits followed by "..." they literally have no more meaning than that. If you wrote down every "real number defined by a finite string", and diagonalize on those, then you would get an infinite word, which is not a valid wff. There are not "more" real numbers than there are natural numbers. Though it is still the case that a bijection doesnt exist in ZFC, but interpreting "no bijections" as "different sizes" for infinite sets is incorrect.
If there are bigger infinites, are there smaller infinites? And what's an example of a set with cardinality aleph 2? aleph 3?... etc. Sorry if these are dumb questions
If S and T are infinite sets, and S is bigger than T, then T is smaller than S. You can ask if there are smaller infinities than the set of natural numbers N, and there are not. To prove this, we can use the well-ordering principle, the fact that every nonempty set of natural numbers has a smallest element. Then given any infinite subset S of N, we can show there is a bijection from N to S. So suppose S is an infinite subset of N. Then S is nonempty, so by the well-ordering principle, there is a smallest member of S. Define f(1) to be this smallest member of S. Then, we continue recursively. Assuming f(1), ..., f(n) have already been constructed, consider the set S\{f(1), ..., f(n)} of all elements of S with the exception of f(1), ..., f(n). Since S is infinite but {f(1), ..., f(n)} is finite, S\{f(1), ..., f(n)} is nonempty. Hence, by the well-ordering principle, S\{f(1), ..., f(n)} has a smallest member, which we take to be f(n+1). First, we will show that f is a valid function. The only way it could fail to be a valid function is if some natural number n doesn't have a defined f(n). But by how f is constructed, this is only possible if S has at most n-1 members, contradicting that S is infinite. Next, it is clear from construction that f is injective/one-to-one, since f(n+1) is taken from a set specifically chosen not to include f(1), ..., f(n). Finally, we show that f is surjective/onto. Since S is a subset of N, every element of S is a natural number. So an element of S is some natural number m. But then, by how f is constructed, m must be f(1) or f(2) or .... or f(m). There are only m natural numbers less than or equal to m, so there are at most m elements of S less than or equal to m, so m is hit within the first m outputs, for sure. Now, if T is _any_ infinite set with cardinality less than or equal to N, then T injects into N. So T is then in bijection with the range of that injection, which is a subset of N, which has the same cardinality as N, by the above proof. "And what's an example of a set with cardinality aleph 2? aleph 3?... etc." Great questions! We can't provide many examples because the continuum hypothesis is independent from our axioms of set theory. We can say that the collection of all ordinal numbers of size aleph 0 is a set of size aleph 1. And we can say that the collection of all ordinal numbers of size aleph 1 is a set of size aleph 2. But we can't say much else. We don't know which aleph number is the cardinality of the set of real numbers. So we can't say anything definitively. These are _definitely_ not dumb questions.
To note, most of my mathematical knowledge was not from going into the field in a formal way. I just enjoy reading, discussing, and investigating mathematical topics :)
@@ComboClassI say this as someone who knows a thing or two about maths and has finally woken up to the nonsense of it. Take the black pill and check out John Gabriel's work, like I did. Don't be brainwashed by centuries of mathematicians with mental issues. These people were unstable & prone to self destruction.
If we take generalized continuum hypothesis, then Aleph_1 is the cardinality of the set of real numbers, and Aleph_2 is the cardinality of the set of all functions on real numbers (or of the set of all subsets of real numbers).
9:48 🥴 Fractions: Count of things per count of others eg. a/b with _numbers_ {a,b}. Proper fraction: a/b with {a,b} coprime. 0: Not a number, but lack of number, just a placeholder - could use " " instead. => Not a fraction 👉 0/1. Explanation: 0 things is uncountable b/c there's nothing there, so you can't have it in a ratio or fraction. 0⁺ and 0⁻ are just synonymous with the fake infinitesimals of Mainstream Calculus. Newton used something like this in his works, even writing 0/0 as if that meant something... 😕 If you were going to use 0⁺ and 0⁻ in a way that makes more sense, you certainly can't say that they have the same property as 0 proper, namely wiping out all your calculations when you multiply but 0 and contributing nothing to you calculations, like politicians who tax you and sometimes give you your own money back as "benefits" or the famous "net zero" idea of some bright spark who wanted to screw us of for some nefarious globalist agenda. So, if you were thinking of using 0⁺ or 0⁻, you may as well just use ±1/N with some really big natural number N. Infinity: Not a number, uncountable. a/infinity with a=number: Not a fraction. The general concept of "real numbers", or more specifically "irrational reals" requires a/infinity to exist as part of a real for any base, when this in reality can never be achieved. Even fractions such as ⅓~0.333... can be described in base closed form for base 3, i.e. ⅓ = 0.1 (base 3). _Irrationals_ can't be represented in closed form for _any_ number base, which means they are not numbers of any kind, just incomplete algorithms that we assign names & symbols to. If irrationals are incomplete, then they cannot be placed on a number line - only successive rational approximations.
It was a nickname that emerged from a joke with friends and I decided to keep, because I like having a "mononym" where people can search a single word and find me
"after continuing this forever, we could create a new number, which is not on the list". There's no "after" infinity. You have created an artificial singularity (Infinite diagonal) and inserted it in the middle. With properly defined set membership for Infinite vectors, there would be no contradiction. Any finite subset of the diagonal can be appended at the end of the list, without affecting the diagonal.
Why have mathematics become so dull. It used to be about the magnificence and mystery of the world and logical(aka geometrical) structure of things. And now this.
Very interesting topic and nicely presented and explained. By the way, I recently discovered an amazingly simple algorithm that generates all positive rational numbers in a way that never creates any duplicates and such that every fraction is already reduced to lowest terms. Here's how it works: Start with 1/1 at the top of a tree. Now for each m/n in the tree, create two children m/(m+n) and (m+n)/n. So, 1/1 would generate 1/2 and 2/1 as its children. 1/2 would generate 1/3 and 3/2, 2/1 would generate 2/3 and 3/1, etc. With this tree, you can simply scan each row from left to right and create the bijection with the natural numbers. I've forgotten where I learned about this algorithm or what it's called, but I thought I'd share it here. Anyway, kieep up the good work.
@@legendgames128 Of course, and zero. Easy to do: assign the above to all the positive even integers, and the corresponding negatives to the odd positive integers.
There aren't different sizes of infinity. Infinite sets don't have sizes. That's the definition of infinity. Countability is a property of some infinite sets, but that doesn't relate to "size" which is not applicable for infinite sets.
