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@AllExistence Says:
TL;DR: You can assume something is true if that allows you to prove whatever you trying to prove.
@AllExistence Says:
Ball thing sounds like nonsense tbh.
@AllExistence Says:
I don't get it. But is axiom of choice even real?
What if it doesn't work? I also can come up with something and call it an axiom.
@PTurtle Says:
At the very beginning, the number I thought of was 37. Seeing you say "I can choose any random number like *37* or 42-" as your *first guess* caught me so off guard.
@TheLuminaryCollective Says:
Infinity doesn't exist. These problems only exist on paper, never in the real world because this math is imaginative, not actual.
@johnnybueti Says:
I understand it's proof by contradiction and we're _assuming_ that such a surjective function exists that maps natural numbers to this arbitrary real number... but what just doesn't sit right with me is how arbitrary such function _WOULD_ be.
If we assume that such function exists in the first place, then Cantor's conclusion could be likely true, but otherwise to me it looks like a conjecture.
I don't get it. 😵💫 What am I missing?
@Zenith-TheMachine Says:
Me watching this video who understands nothing about math: I like your funny words magic man!
@drsingingeagle Says:
This is EXACTLY what drove me insane in college math classes. If there are infinite numbers between 1 and 2, between 2 and 3, between 3 and 4, how do we even determine that numbers are even real? Where do we draw the line between 1 and 2? What if _two apples_ are actually illusions? 🤯
So I went into the “soft” sciences and got one of my PhDs in anthropology instead. 😅
@billx4266 Says:
this is like chinese to me
@MitsuhashiTakashi Says:
isn't this just schizophrenia.
@Memessssss Says:
WWWW
@rainyfeathers9148 Says:
Dayum...They couldn't let the man cook? Nice save Zerms🫡
@jonpatchmodular Says:
A true axiom is a true axiom. There is no such thing as "half a true axiom".
- "Henry", probably
@gabriele5107 Says:
4:06 I have a question: what if, instead of using the natural numbers in crescent order, we map both the real numbers and natural numbers in any order? For the real number, we select any real number in the table. For the natural number, we select the same digits from the real number, just selecting the the decimal part (just removing the "0," before). For example:
Natural | Real
1234... | 0,1234...
2345... | 0,2345...
8452... | 0,8452...
What I am missing here?
@marktester5799 Says:
If Calvin and Arminius were mathematicians.
@topherwagner3032 Says:
This is like the argument of fate being real or not. Do we have a choice or hand in our existence. Can we change anything by making a choice? Or are all choices and consequences already pre laid out. According to this its our choice whether or not we want to accept that or not. But sort of like saying i dont fit into a category, is in itself a category. So did they not answer the question all along, because if you are choosing to not make a choice, you are still choosing something, that is your choice.
@somene_v1 Says:
what if infinity is the friends we made along the way
@lukeduffey3459 Says:
jill
@unnameduser1994 Says:
14:55
So the solution is to find more unique indices for a set than the set contains unique numbers. But that's only possible for countable infinites, right, not for uncountable?
@Nerex7 Says:
Thinking about the paradoxical nature of infinity is always fun (and terrifying). It's like two waterfalls of different sizes that go on forever. You can say the bigger waterfall has more water coming down, but both are infinite, so the water they make is infinite as well, yet one will have a bigger pool than the other.
@James-qv1dr Says:
I thought this was going to be a Terrance Howard troll video haha
@Wines2023 Says:
At 3:12, all the squares after 15^2 are incorrect
@psyskeptic9979 Says:
Not a fruitful area of mathematics. What is the point!
@jahblohnsteron Says:
countable infinitiy is less than uncountable infinity by virutue of their comparitive GROWTH RATE. despite both being undending one will ALWAYS be behind the other. @grok: am I losing it here or please opine on this matter
@Persephone_Falasteen Says:
Not a scientific charlatan 😂
@modestasgrazys5547 Says:
This does not seem like a solved problem after all. The freedom to choose whether you like axiom of choice or not should not impact a solid rule-based system: math. If after all this mystery, the answer to it is "well, it depends", the natural question arises: "how it depends" -> "on what rule of generalized system this dependence is based"?
@skfalpink123 Says:
Many thanks for producing these bloody brilliant videos!
@thehorizontries4759 Says:
If yall really understood how math and proofs and science work you wouldn’t be saying that mathematicians just make stuff up or any of these jokes about math being built on assumptions and stuff
@mike814031 Says:
5:09 wait a moment, but is it really not on the list if you consider it would eventually appear on the list if the list is truly infinite? That seems a little unreasonable. Could that be a flaw with his reasoning in his diagonalisation proof? And if it is would it disprove it? There maybe be an easier way to consider what I’m saying so it’s easier to visualize, instead of using random numbers what if you use them in a numerical order? Or does it matter either way? it almost seems like his proof relies on visualizing numbers that are more in a random order like the ones you chose to use to visualize it with? then would his proof still apply? I would think that you have to consider when you’re dealing with infinities you will at some point still have that number appear even if it hasn’t shown up yet, just because you came up with a seemingly different one with your diagonal addition trick it doesn’t necessarily mean it wouldn’t have appeared at some point and that seems like a very flimsy argument to rely on for his diagonalisation proof. Do you agree? I would argue they are both simply infinite sets and neither one is greater than the other. It’s simply either infinite or it’s not. And it’s hard to believe some infinities are bigger than others if they never end, then how can they actually be bigger if they never stop and therefore are not even countable? It also seems like they depend largely on reasoning & logic and are therefore easily confused because people can have flawed & biased reasoning. And i strongly believe his reasoning was flawed when coming up with that conjecture (it seems more like a conjecture to me than a proof) and i do have a great admiration & respect for cantor. I can’t help but strongly believe there are not such things as different sized infinities, because you can’t prove there are if you can’t count them. And you can too order them 1 to 1, but it never ends if both sets never end, so why try to argue which is bigger if they both go forever?! It would be insanity to try to argue differently, or to argue there is such a thing as an infinity that is bigger than another. You can’t count them if they don’t end, there’s nothing else to it, it’s as simple as that. If you can’t count it, you can’t show it’s any different from another infinity you can’t count. Things are either infinite or they aren’t. Period. That example earlier was extremely flimsy because it relies on a tiny example and it doesn’t satisfy what you’re trying to suggest or prove. My point is Galileo was exactly right to begin with, there only one size of infinity and more or less than don’t apply there. We shouldn’t make it overly complicated. It seems like Cantor was missing the point because it’s irrelevant if they map 1-1 or not if they never end, they never end. There is no end result to compare if you count forever. Cantor was a bit like an obsessive madman almost like someone with obsessive compulsive disorder, like he wasn’t satisfied with just accepting the basic truth about infinities, so he focused on a flimsy idea that was more of a whataboutism that didn’t legitimately disprove anything Galileo established.
@vladislavurumov538 Says:
In conclusion, you are given the choice whether or not to use the Axiom of Choice
@JJamahJamerson Says:
I picked 37 and then he says 37 and I’m like “wait….”
@kezeal Says:
I lost the plot early in the video but something kept me watching despite me not understanding half of it.
@PatrickHughes-u5q Says:
It is adding two when it shows natural and squared numbers so 1-4 is 3 and 4-9 is 5 and so on and so forth
@T_Mo271 Says:
About halfway through this presentation, I lost track of why it was important. But oddly, this is the 2nd video I've watched in the last two weeks that discussed the Axiom of Choice. On neither occasion did I understand why.
@IloveCaeeeeeeee Says:
It didn't break Cantor, it broke all the viewers too
@drewwilliams6016 Says:
Infinity must continue on counting lmao why break a perfect circle 🤣🤣🤣
@zinoudzoumani9384 Says:
all these sientists created all these rools to prove what they think is real and now we are stuck in because we must learn all of it to get a job
@Baalaaxa Says:
I'm not a mathematician, so I don't know about the Axiom of Choice, but at least you always have a choice if you want to use the Axiom of Choice, and live with the consequences. I call it the Axiom of Choice of Axiom of Choice. Since there is always a choice, it must be true.
@AkezhChannel Says:
27:51
@Larry26-f1w Says:
Mathematics disproved the official narrative of 9/11. Statistics rule out a body recovery rate of less than 10% of a building’s occupants no matter the cause of collapse. Powerful stuff these numbers!
@JackDespero Says:
What is fascinating about this historical videos is that you learn all these names, and they feel like titles. Cantor's cardinality, Zermelo-Frankel set theory, Kronecker delta.
It is fascinating to see them attach to the very real people from which those titles came.
In a sense, it helps a scientist like me to feel a part of this magnificent tapestry that is science. One may look around today and think that all the great things in science have already been done. "Oh surely in the times of Einstein and Planck they were not bickering around this petty details, they were always talking about the grandest concepts".
And then you read history and you learn that science and scientist have changed so much, and yet so little. And how else could it be, as we are all humans?
@musiclife7251 Says:
Friendly reminder to never use ChatGPT. Their whistle blower died under extremely suspicious circumstances. Best Wishes ❤
@timgonzales2891 Says:
Why does this matter. I don't think I'm smart enough to get it.
@NateROCKS112 Says:
It's often said that a well-ordering of the reals can't be constructed, but it sort of depends on what you mean by "construct." In the classical sense of the word, it often just means a formula that uniquely describes a well ordering. It turns out that this is independent of ZFC: if V = L holds, then you can give a formula that describes a well-ordering of the reals, but if you add in omega_1 "random reals" to a universe satisfying V = L, then there is no definable well-ordering. See MathOverflow answer 6597 by Joel Hamkins.
If by "constructive" we mean _constructivism_ (math without excluded middle, the axiom of choice, and a host of related principles), then if we don't assume countable choice, it is even consistent that ℝ is countable (by a proof of Bauer and Hanson). Even assuming countable choice, in -the effective topos (basically, where everything is realized by a computer program),- the topos realized by infinite-time Turing machines, ℝ is uncountable (no surjection from ℕ onto it) but -is also -_-subcountable-_- (- there is an injection ℝ → ℕ -)- . I believe this injection also defines a well-order in the constructive sense: say x < y iff f(x) < f(y). See MO answer 453340 by Andrej Bauer. (Edit: Note, as with most things in a constructive setting, that you have to be careful when defining what a well order is: here, it means that (<) is an extensional and transitive relation such that you can do induction on it. Importantly, induction doesn't guarantee _existence_ of a minimal element for _every_ subset, avoiding constructive gripes about classical existence.)
In sum, math is weirder than we might think. Also, disclaimer: I don't know too much about the topic, and I am absolutely clueless on predicative mathematics (to which Kronecker subscribed). I've just skimmed a few MathOverflow answers and papers.
@DreamersDisease88 Says:
Do math is not for me I am completely lost at the park with you and the white paper board
@JSLEnterprises Says:
When you're just so smart, you butcher Georg ("Gay-org"), instead of understanding its just "George" with a "Je" sound given his ethnic background. Remember this is the same ethnic background that spawned Johann (pronounced Yo-Han, with an emphasized n), thus phonetically his name is pronounced 'Jorge'
@andyvitz Says:
you know Infinity is 0.01 right you know why that is take 0.01 off the human DNA we are all don't exist do you think some alien race is going to know what that is sideways 8 yeah right
@DickWaggles Says:
this is how the federal government determines how much money to print
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