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  @shaharudinhamidun2117Күн бұрын

  I wonder how the solution above has eluded the great mathematicians for 350 years, namely Hilbert, Gauss, Euler. Hmmmmm

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  @sttngiscoolКүн бұрын

  How can you study a thinker for years and not know how to correctly pronounce his name…

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  @alexanderramos327310 сағат бұрын

  it is not standard academic culture to pronounce as if a native speaker or trying to do an accent. If a British person was named Johnson it would sound different than an American Johnson - and the phonetic spelling would change and may warrant a textual inflection or ‘accent’ in writing. Professional communication is about rolling through these simple details and not “pronouncing his name correcrly” just to sound smart speaking two different languages in the same sentence.

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  @spr1ngcactu52 күн бұрын

  It’s a lot easier to get into America now, just have feet.

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  @yb36048 күн бұрын

  the quote at the end gave me the chills lovely video mate stay healthy

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  @spacebuddy53399 күн бұрын

  This video is too short!! Please is there another one that's longer?

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  @Music_Creativity_Science10 күн бұрын

  The physical explanation emanates from Newtons third law, action & reaction of force. If a force operates (creates acceleration) in one direction, the same amount of force always operates (creates acceleration) in the opposite direction. No exceptions in this universe. Mathematical explanations/derivations only contain the created kinetic energy in one of the directions, but the total created kinetic energy from F = m • a is m • v^2 (without the 1/2). Nothing strange about it! Much more interesting to reflect on is that release of absolute energy in the universe, for example in an explosion (release of binding energy in molecules), or release of atomic energy, results in relative kinetic energy. Kinetic energy is always only relative. So, where did the real absolute energy go (it can't have disappeared) ? Relative kinetic energy is no exchange/compensation of energy to the universe as a whole, from the absolute energy (E = m • c^2). The answer is, the absolute energy was used to slow down time, to slow down the rate of change in all the accelerated atoms involved in the process. Time dilation is physical, and happens during acceleration phases of matter through (or against) the quantum vacuum in space. Physical time dilation does not happen during constant relative motions (acceleration is an absolute motion). It takes energy to slow down physical time, how a brain thinks about time (psychological time) is something completely different.

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  @Fedeposter11 күн бұрын

  There is a mistake in the first order statement, it should say [... Union of A forall A in X (f(A) in A)]

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  @wolphramjonny775112 күн бұрын

  obviously the well ordering theorem is wrong. Well maybe not all infinite dimensional spaces have a basis, and so what? may be the ones we care about do have one, so please forget about this retarded theorem,. it might or might not be true for specific cases, who cares, they all belong to infinity, that is, nothing than can actually be built. It is the same as asking if angels have erections, who cares, no one will ever see and angel, we don't even know if they exist.

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  @SanderBessels16 күн бұрын

  The concept on infinity leads to very strange paradoxes. If you want all of mathematics to make sense, you should keep within the safe realm of finite sets and stay away from the AoC. Another problem arises from the axiom of the “excluded middle” (all statements are true or false). I really think you should do a video on “constructionism” or “intuitionism”, which is the mathematics without the excluded middle. Only direct proofs are allowed (A -> B -> C etc.), or put differently, to prove C, it’s not enough to prove that not C leads to a contradiction. You have to prove that C follows from the axioms. Every existence proof therefore also leads to a construction of the element.

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  @Tata-ps4gy16 күн бұрын

  For axiom of choice all the way 🎉

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  @user-mp9um5qj3u19 күн бұрын

  I am on no side... I am on the side of correct mathematics

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  @compucademy20 күн бұрын

  Why can't it just be added to the axioms?

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  @stevewhitt910920 күн бұрын

  I am on the side of anti-trump cards

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  @octopusjjsnook21 күн бұрын

  Please edit out the unecessary, mindless, unmusical, distracting and utterly infuriating background drivelling noise. Ask anyy educational cognitive psycchologst about attention and retention. Educationally V, poor so fail! Best wishes.

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  @writerightmathnation948124 күн бұрын

  Your discussion of the well ordering principle leaves a lot to be desired. It’s so misleading in terms of its relationship to the axiom of choice. It’s amazing the well ordering principle does not say anything at all about the usual ordering of the real numbers. Thus you have totally misrepresented the reason for the concerns about the axiom of choice and how one resolves such concerns. In fact, you failed to mention how they can be resolved, which is a misrepresentation of the issue at hand it’s sad that people who consider themselves mathematicians, take this viewpoint. No one has ever said that the axiom of choice implies that the usual ordering of the real numbers is a little ordering, so your claim that it’s a contradiction is absolutely false.

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  @NSINGHK24 күн бұрын

  I also created a theorem yesterday, Its formula for sum of square of first n numbes, The formula is I simplifed to 1 summation, so its: SUMMATION[i=0...n] (n-i)(2i+1) Thats it. Lets say 4. Summation[0...4], lets start with For i=0 (4-0) (For i=3) (4-1) + (4-1)+ (4-1) For i=2 (4-2)+(4-2)+(4-2)+(4-2)+(4-2) For i=3 (4-3)+(4-3)+(4-3)+(4-3)+(4-3)+(4-3)+(4-3) For i=4 (4-4) This become: 4 + 9 + 8 + 7 + 0 = 30 Now do it in normal way, you will get 30.

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  @azertyQ27 күн бұрын

  Isn't the whole thing about axioms is that they're assumed true? If you could prove an axiom from the other axioms, it wouldn't be an axiom but an outcome of those axioms.

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  @azertyQ27 күн бұрын

  Also Gödel wants you to say hi to your mom for him

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  @annaclarafenyo818527 күн бұрын

  This axiom is not of "indisputable importance", because the cases where it is used in practical applications, it isn't the set-theoretic axiom of choice, but the second-order arithmetic axiom of choice, which is not controversial.

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  @mostexcellentlordship28 күн бұрын

  You have a number line. You decide to place an "origin" somewhere - where this line "ends" - and on the "other side" the same number line is reflected and you call it "negative numbers". This is all so arbitrary. Reflected around what axis precisely? Introducing this arbitrary "origin" is completely equivalent to introducing another orthogonal axis without which this weird "reflection" argument makes no sense. You create a 2D space with this move. In fact, we should be weirded out by negative numbers alone. How can you just make this move? Reflect a line in "another" direction? Which direction? How can you just do this? And what do you do with this remaining dimension you added? Complex number on the other hand just make perfect sense. I now challenge you to apply the "complex plane" to conceptual space: if at 1 is "existing" and at -1 is "non-existing", what does a 90degree CCW rotation away from 1 represent? If you'll applaud in silence I will go gently on you during my Nobel acceptance speech.

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  @randomstrategy767929 күн бұрын

  An ordering where every open set of reals has a smallest element can actually be constructed as follows without the axiom of choice: Given x and y: 1) If x and y are both irrational, x<y if and only if x<y in the classical sense. 2) if x is rational and y is irrational, then x<y (and vice-versa) 3) If x and y are both rational, add up their respective numerator and denominators in reduced form; smallest sum is smallest; if the sums are equal, smallest denominator is smallest. And then if those are equal, then the negative version is smaller than the positive version. Then since every open interval contains a rational, you can find the smallest rational in our ordering, which will be the smallest element of the entire open interval. This ordering we constructed is of course not a well-ordering, but does still satisfy the property we wanted.

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  @VeteranVandalАй бұрын

  Complex numbers can be easily visualised at least. Now quaternions can't.

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  @faresalahdАй бұрын

  أنا أعتقد أن بديهية الاختيار تشبه المسلمة الموازية لإقليدس، هناك أنظمة متسقة بافتراض أنها صحيحة وأنظمة متسقة أخرى بافتراض أنها خاطئة ولكن لم نوجد تلك الأنظمة بعد

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  @hanna8399Ай бұрын

  For 4:00, there seems to be an easy definition to make it well-ordered, just let x ∈ (a,b), and the less-or-equal-than is defined by comparing abs(x1 - (a+b)/2) <= abs(x2 - (a+b)/2). Then x0 = (a+b)/2 is the "smallest" element. Please point out where I am wrong...

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  @acortisАй бұрын

  Wonderful explanation! Please keep them coming!

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  @santerisatama5409Ай бұрын

  I'm Intuitionist and happily reject all axiomatics. :)

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  @pauldruhg2992Ай бұрын

  I'm siding with 1-naturals!

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  @AchrononmasterАй бұрын

  @4:20 isn't that a bit of anti-ZFC propaganda? 🤣 Our intuition does not apply, since a "well-ordering" could be a very crazy thing (most likely is), like the Banach-Tarski construction -- which is also anti-ZFC propaganda LOL. You cannot use physical intuitions in pure mathematics _all the time_ only _some of the time._ Also, @3:50 the Banach-Tarski Thm does not just depend on AC, it requires a definition of the reals. It is a composite result of several axioms. You could call it one of the first results inn "complexity theory" of mathematics? Things in the platonic realm are not always what they seem, and that is a *_good_* thing. If mathematics were just physics + computation life would be boring, literally (no mental qualia).

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  @AchrononmasterАй бұрын

  (*) and you never know which of the time the physical intuition will be valid! (q.v., the dart situation in the supposed proof the Continuum Hypothesis is false, due to Freiling & Brown. NB: the CH is probably not false --- refer to Ultimate-L --- so it's your dart throw intuition that sucks, or to put it another way, probability theory is the hardest subject in all of mathematics LOL).

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  @comic4reliefАй бұрын

  2:08 'Independence' is misspelled.

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  @vincentvandergoes444Ай бұрын

  I despise the Axiom of Choice. Although Gödel proved that it is consistent with the other ZF axioms of standard set theory, his proof relies on a significant loophole in the axiom of Powerset. The axiom of Powerset asserts that every set has a powerset, but it does not clearly define the powerset of an infinite set, such as N. Consequently, the ZF axioms permit the construction of a set theory model where the "powerset" of an infinite set like N includes only the subsets with finite descriptions. This model is known as the Constructible Universe. The Constructible Universe has an extremely sparse notion of the powerset, where all sets are essentially countable. (Although the powerset of N isn't technically countable within the model, as the diagonal argument still applies.) Unsurprisingly, the Axiom of Choice holds true in the Constructible Universe, making it consistent with ZF. However, this only indicates that the ZF axioms allow for pathological models of set theory. Many people mistakenly believe that the Axiom of Choice has been proven consistent with the intuitive notion of sets and powersets, but this is not the case.

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  @simonwillover4175Ай бұрын

  1:45 this is missing parenthesis!

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  @canaldoprof.victorsennaАй бұрын

  Try to use Game Theory, the first situation you are basically describing what is called 'dominant strategy' for white and extending your second idea for the whole game, it gives us the intuition for why the Nash Equilibrium of the game of chess should be a draw (or one of many ways of possible draws). Supercomputers are almost "proving" that this could be true.

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  @maxv7323Ай бұрын

  A lot of people have pointed out a variety of mistakes in this video, but in my opinion the biggest error - one that I see made very often - is the idea that choice is something that we should decide is valid or not. In reality, there is no problem with the current state of affairs where we study mathematics both with and without the axiom of choice. There is no conflict here, simply two different systems both being studied.

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  @MetaMathsАй бұрын

  Why do you see this as an error ? The idea of this video is to present this decision as a philosophical challenge.

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  @aidanmokalla7601Ай бұрын

  3:16 if two theorems are equivalent, how can one be weaker than the other?

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  @paulbloemen7256Ай бұрын

  I am of the side that is true, combined with the side that works. If something is obviously true, and it works, thus has practical use, then it is fine with me. You should try it in daily life too.

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  @narendra672Ай бұрын

  😊

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  @BoyKhongklaiАй бұрын

  That's still no valid reason to wear old newspapers on one's head

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  @HUEHUEUHEPonyАй бұрын

  imagine unironically believing in the HAXIOM of choice

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  @kimcrowe-bo5tbАй бұрын

  Prime #’s on $20 bills worth anything? And to whom

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  @petrosthegooberАй бұрын

  didnt mention Hausdorffs maximality principle smh

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  @MetaMathsАй бұрын

  Well, you just did !

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  @lllevokelllАй бұрын

  Do not treat non-computable things AS IF they were computable. Formal math claims things like large cardinals and unmeasurable infinite sets ARE computable in the limited sense that the algorithms of pure logic (e.g deduction, excluded middle), still work when given such things as inputs. But this is lunacy, and blatantly mistaken, because things like Banach-Tarski are EXPLICIT EXAMPLES of that very machinery breaking down and throwing errors when you do that. The Axiom of Choice is guilty of this error. The "for every" in its formula goes infinite, and treats noncomputable things illegally "as if" they were computable. It's a flawed axiom.

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  @emilianovalzacchi1737Ай бұрын

  Where does the cover from the video comes from ???

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  @aniksamiurrahman6365Ай бұрын

  I think the Axiom of Choice needs some nerfing so that Well ordering appears only where it makes sense. Well, nothing new to Set theory. ZFC did the same for Set membership rule in Naive Set theory.

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  @STF413Ай бұрын

  This property is NOT unique to (regular) hexagons. It's also true to squares. If I'm not mistaken, this is true to ALL regular 2n-gons or 4n-gons. I haven't got the time to check this part though.

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  @aaronchow2366Ай бұрын

  What’s the background music?

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  @cougar2013Ай бұрын

  If you pick a side, that’s a….choice!

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  @micktheman6Ай бұрын

  This makes zero sense to me

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  @jimnewton4534Ай бұрын

  question. there is an axiom claiming the existence of the empty set. If we discard this axiom, what does the axiom of choice look like? I.e., does the axiom of choice depend on the existence of the empty set?

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  @kylecow1930Ай бұрын

  the existance of the empty set follows from the axiom ∃ x (x=x) and the comprehension schema. Essentially once anything exists, so does the emptyset

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  @liijioАй бұрын

  Proving continuum hypothesis , proving inconsistency in ZFC , constructing ZFC from naive set specification , resolving Russell's paradox , constructing infinite number system , construct and ensure overall consistent mathematical universe and developing arithmetic system - edition 8 May 2024 DOI: 10.13140/RG.2.2.21713.75361 LicenseCC BY-NC-ND 4.0

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  @samalthaus1497Ай бұрын

  Of course not....I see,.... circular

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  @samalthaus1497Ай бұрын

  Primes...plural