Видео

me when im rating something to 1 from 10 and i choose 4: (the middle finger)

Thank you! You explained this very well

Another reason "real" numbers are anything but.

모든게 무한하다면 무한히 겹치는 것도 무한하지 않을까요?

if you have infinite money then there is no purpose in making a profit.

Awesome youtuber

Pick the first on the left!

What you're describing here isn't the axiom of choice. That you can pick an arbitrary element from any one set is a principle of first order logic. The axiom of choice is only needed when you need to pick an element from each of an infinite collection of sets.

Mathematically this is correct!

Basically, for any nonempty set, there exists a function that can choose an element from the set to make a new set. I’ve also heard it as a choice function for the power set of a set to make a set from one element of each set in the power set. Must exist*

"Choosing" an element from a non-empty set doesn't require the axiom of choice. Having a "uniform" choice or a "simultaneous" choice over infinitely many sets does require. One can prove the finite axiom of choice. If you have n-many non-empty sets, then there exists a uniform choice of elements from the sets. Equivalently, any surjective function between finite sets has a section.

if A is good A^c isnt always bad, imagine the rule is A is good if A^c = {n + 1 : n exists in A, 1 ... m -1 : m is the lowest number in A}, then if A is the set of odd numbers, both A and A^c are good sets, unless ive typed the definition of A^c incorrectly this will disprove that (1 ... m means all the numbers 1 up to m, im not good with set theory notation )

Just pick your favorite one, how is this hard?

Good job

3 persons you , Alice & Bob ,So there have to be atleast 2 off the same kind to her but she doesnt know you ,so she tells not the truth but is kinda guessing therefore a knave bob is the Knight

Thank you so much. This video helps me so muh 🫶

Absolutely wonderful explanation. Thank you, brother.

Rule: Label each continuous sub-interval in the evidently discontinuous set of real numbers you have been asked to consider with a pair of labels I and S, corresponding to the infimum and supremum of that sub-interval. Take the mean of the smallest such infimum label and its associated supremum label, then choose that number. Job done. . . . The axiom of choice is bullshit, by the way.

Axiom of Choice is a scam.

간단함.우리의 뇌는 이미 양자역학적으로 돌아가고 인간의 뇌는 양자컴퓨터임.풍부한 감정 옳은 의지 좋은 환경 고난 좋은 사람들 사이에 있으면 필연적 우연으로 성취할 수 있음.스메일 그로텐딕 허준이씨처럼

I really didn't want to know about McNugget Numbers ...

Infinite money isn't possible. So the expectancy is still zero, but not rvenly distributed..you win a dollar most of the time and you tlose a lot with very small probably. The thing is they balance each other

“Just pick the smallest element” And this is why the AoC is equivalent to the Well-ordering principle!

The Choice Theorem is born of the inconsistency (not incompleteness) of Arithmetic. The theorem poses the undefined "plus one" of Addition againat the definable prospects of Multiplication.

Wow I love it

jojo refrence

The animations showing internal/external rotation force when using a barbell were the best I've ever seen in a fitness video.

Hang on. Wikipedia says that the axiom of choice says that, given some indexing set I and a family of non-empty sets, {S_i for each i in I}, it's possible to construct an indexed set {x_i for each i in I}, where x_i\in S_i. No restriction on the cardinality of I. But you say that the axiom of choice needs to be invoked in order to pick one element out of just *one* non-empty set of reals? That's certainly not true --- if I is countable, then the axiom of countable choice is enough.

At 2:39 , where you got F(z) = -z/(z²+z-1), what if we input z = 1...? Since F(z) = 0z⁰ + 1z¹ + 1z² + 2z³...., F(1) should be equal to the sum of first n ( as n-->inf) fibonacci numbers, And if we compute F(1) from the formula you deduced, we get -1/(1²+1-1), which equals -1, But how is this possible, F(1) by definition should be the summation of the fibonacci series, how can it be negative 1...? Ik this video was posted 3 years ago and i have no hopes of getting a reply, but by chance if anybody sees this, plz reply so that i can be reminded of this video sometime in the future, I am still a student so maybe by then i would have gotten the mathematical maturity to be able to solve this by myself... Also, if i made a mistake plz feel free to correct me in the comments..!!

This is the best explanation of the axiom of choice.

6:39 alice wins. Since the harmonic series diverges she can always color enough squares to equal (exceed) 1 at every step, no matter how much bob colors before. 1+1+1+1+1...=infinity

ok - Alice and Bob colors tape on which some TM with infinite tape is run - if TM stops then Alice wins... it is uncomptuable :)

can we encode turing machine number this why and say that first person which TM halts wins ? this is clearly uncomputable...

I found the proof challenging, what’s the easiest intuitive proof for the Catalan numbers formula?

Pretty sure the problem is the tokenization, not neural networks per say.

It also intakes text tokenistically. Since patterns like "123" can be single tokens, it creates a lot of confusion when it tries to process data with unique numbers in it. It's much harder to train it and make it predict digital interactions properly when there are just random variations introduced like that

So the Dyck Paths were very funny but this was also a pretty helpful video so thanks I guess but I still am pretty sure the Dyck Paths were the best part idk

Alice is a knave and Bob is a knight

Can you win a game without an end? Going to infinity is always impossible. The game has to have an end to have a winner at the end

If Alice is a knave why did she say she was indeed a knave?🤔

It's often told that math with axiom of choice is weird because it produces weird things like the famous Banarch Tarski paradoxon (which isn't a mathematical paradox afaik), but there are a lot more weird consequences in maths without the axiom of choice

I know nothing about math. Why would anyone pick the smallest element?

the 🤘 means 18 when you use the finger counting technique, 18 is 6x3, means 666

2:26 I have a counter example to the first player could have always made a move to make the same game state as after the second player's first move. If instead of the top right I go one to the left of the top right and then the second player plays one below the top right, the resulting game shape is an inverted L removed from the top right. There is no way to make an inverted L shape with one move...?

Thank you for this great explanation. I wish I had seen it when I was a kid. I used to be smart, now I'm 57 and forget my own phone number!! I have a question for you, similar, maybe you could do the video on it. If the formula for the surface area of a circle is πr2, why isn't the formula for the volume of a sphere πr3? Doing the visuals would require some CGI though, still, it would be good. Cheers. 🙂

I like your magic words Keep going!

my head hurts

Wtf

super cool

bababa