Godel’s second incompleteness basically says: “completeness (all true statements are provable), consistency (only true statements are provable), and arithmetic-pick two”
A correction. Fermat's last theorem was not just for third powers, that had been known for a long time and for quite high exponents. Wiles' achievement was to prove it for literally all positive integer exponents.
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Just a small correction, AFAIK Wiles did not show that FLT follows from Taniyama-Shimura, that had been known for a long time and isn’t that hard. Also proving Taniyama-Shimura was an extremely important result for mathematics, so proving FLT was more of an icing on the cake.
To correct this correction: Taniyama-Shimura WAS in fact proven by Wiles (and one other), so it wasn't "known for a long time", and neither is it not "that hard". It was regarded as a terribly difficult problem.
@@fysher3316The Taniyama-Shimura conjecture was not proven by Wiles. He proved a specific case of it (semistable elliptic curves) that included Fermat's last theorem (there is an amazing video by Aleph 0 on the topic). Using his work from 1995 on that proof a group of mathematicians finally proved the whole conjecture in 2001.
Hey, in 2:38 you used an image of a painter called Richard Hamilton from London. However, the actual mathematician is called Richard Streit Hamilton and lives in Ohio.
10:38 my understanding from taking discrete math years ago is that godels incompleteness theorm wasnt: “any math system has true statements that cannot be proven true and also cant prove that it isn’t inconsistent” but more so “any math system that doesn’t have true statements that can’t be proven true is inconsistent and any consistent math system has true statements that cant be proven true.” like it’s one or the other. A math system can only be useless (inconsistent and unprovable truths), have unprovable truths, or consistent. Is that wrong?
One way to think of it: 1. A system is complete 2. A system is consistent 3. A system is recursively enumerable 4. A system can express basic arithmetic You can only pick 3. A system can be both complete and consistent, say Presburger arithmetic. It is strictly weaker (can’t even express multiplication) than Peano arithmetic, which is subject to Gödel incompleteness. Tarski even devised a complete axiomization of geometry, but it too fails to satisfy the hypothesis of Godel’s incompleteness theorem like Presburger arithmetic. The hypothesis of Godel incompleteness is that it it can express arithmetic such as PA, once it reaches that threshold it can no longer be both complete and consistent. Edit: #3 also makes it so this only applies to first order logic, as second order logic is not recursively enumerable.
One correction: "algebraic groups" are a concept from algebraic geometry (certain representable functors into the category of groups). What you mean during the classification of simple groups are just "groups"
So, for the 4 color theorem, it’s actually not true that 4 colors can color a map without same color elements sharing a border on maps of any complexity. There are some rules that the map has to follow. Say, for example, that there is a set of 4 countries that form a circular border, and each country occupies one quadrant of this circle. These four countries could not then be enclaves inside of a fifth country that encircles all of them, because no matter which of the four colors that the fifth country is colored in, it would match up with and border one of the quadrants of the same color.
I'm actually working on x⁵-x-1=0 right now. I have a hunch that while it cannot be solved algebraically (by radicals), it can be solved transcendentally (something containing e=2.718...). Even if I could do that, it would be short of a full explanation of higher-degree polynomials. It also might still be impossible to have a single formula for all quintics, but it's a step in the right direction.
That the solutions to equations like x^5-x-1 = 0 are transcendental is not a bad guess at first brush, but actually cannot be true by definition. Transcendental numbers are defined to be numbers that cannot be expressed as the solution to a polynomial with rational coefficients. So, for example, there is no polynomial with rational coefficients that gives e or pi as a solution. There is actually a general formula for the solution of quintic and higher degree finite polynomials, in terms of hypergeometric functions. The output of these functions is not radical (cannot be written as a rational power of a rational number), so there is no contradiction with Galios' result. However, these numbers are still not transcendental, since they are solutions to rational polynomial equations. In essence though, your intuition is correct: the general space of numbers that solves these equations is necessarily a larger group than just radicals. This set of numbers is actually called Algebraic Numbers, because they solve algebraic equations.
from educational pov i would add euclids parallel postulate before continuum hypothesis. or at least mention it as easy to understand analogy. im not sure if its ever stated as "an unsolved problem" however its solved as an axiom of choice.
That quintic equation solution impossibility is a cursed one! Galois died in his 20, also Niels Henrik Abel died at 25! Abel provided the first formal proof of that! Gauss of course beat them to it but he never published it formally, for him it was a near sure guess which people read in his notebook after his death!
Haken: pronounced HAH Ken. One son is a composer and another a computer scientist I think, associated like him with the University of Illinois at Urbana-Champaign (at least early on).
5:59 why does it say "can't color this with 4 colors"? You clearly can - just make the purple bit blue, and the little blue nubbin one green (or red or yellow)
@@tupoibaran3706 the main issue is that 4 color theorem is about contiguous planes, so case presented is invalid from the theorem perspective. Theorem is not about real appliances, when countries may have separate territories somewhere else.
@@em.1633 Nah, his sum of natural numbers= -1/12 is the biggest lie which people who want to learn about Maths believe it's true. I mean, saying that 1+2+3+4+...=-1/12 sounds pretty elegant once you see how he found that sum, but you need to dig further to understand that the sum of n from 1 to infinity diverges and that's on period, and since it diverges, there's a property which , by doing partial sums of the original sum, we can get different convergences (which proves, again, the big series diverges). But no one explains this to the newbies in math, they take the well-know value for the sum and get the wrong idea of Analysis.
Fermat's last theorem states that for all n>2 there are no integer solution to the equation aⁿ+bⁿ=cⁿ, what you presented in the video is just a specific case
ffs, so many statements are presented wrong. fermat last theorem said about any nth power bigger than 2, not just 3. 3rd power was prove impossible long before Wiles.
Insolvability of the quintic equation was actually first proved by Abel and Ruffini, Galois only later generalized the theorem and simplified the proof
You didn't state Fermat's last theorem correctly. The case of 3 as the exponent was proved shortly after Fermat's death. So was exponent 4. But the theorem said there was no equation for any integer exponent greater than 2.
I think you're underselling Grigori's contribution to the Poincaré conjecture in the way you bring up his use of Hamilton's work, he always admitted this and when he declined the prize he said it was because Hamilton's work had been equal to his own.
You need a pop stop for your mic badly, but other than that great vid. Also since I'm being a ballbreaker I might as well add this critique: you should speak more naturally, and less with the generic "youtuber giving lecture" monotonous tone
He butchers the last name too lol. Can't be helped since he's American but I wish he would just stick to an Anglican pronunciation so that it at least doesn't sound annoyingly pretentious.
Unfortunately, there are quite a few mistakes in this. Just to name two: Niels Henrik ABEL proved that the quintic is generally insoluble, not Galois. Fermat's Last Theorem is for n > 2, not n = 3. EULER proved the n = 3 statement long before Wiles.
As a cuber, I am very confused how algebra is related to cubing. I mean, we use a completely different type of notation and there is no mathematical relation besides the R2s and stuff
There are 6 "basic" moves that can be performed on a Rubik's cube. These are the 90 degrees clockwise rotations of each of the 6 faces. (This is assuming we keep the cube in a fixed orientation, so the centre squares of each face do not move.) Each move is a rearrangement of the coloured squares on the cube. Moves can also be composed (i.e. performed in sequence) to further rearrange the squares. Moves can also be reversed, since each basic move can be undone by performing the corresponding counter-clockwise rotation. Each configuration of the squares on the cube can be described by a sequence of moves that takes the cube from the solved position to that particular position. (Although such a sequence of moves is not unique; for instance, RRLRR gives the same configuration as L.) In mathematics, a "group" is a collection of things that can be composed and reversed. The set of possible configurations of a Rubik's cube is a group. Group theory is a subfield of algebra. This is why Rubik's cubes can be studied using algebra. I presume you thought that "algebra" meant "equation solving" like one learns in high school. This is *part* of algebra, but in mathematics, algebra is a hundred times bigger than that. (And it is unfortunate that so few people know this.) Algebra involves the study of groups, rings, fields, modules, lattices, monoids, and possibly categories, depending on who you ask. These are all in the same vein as a group, in the sense that they are collections of things that can be "put together" somehow. (For instance, a monoid is like a group, but without the requirement that its elements be invertible.) I think the original meaning of the word "algebra" (or rather, the Arabic word which became "algebra" when borrowed into English) was actually something like "put together" or "broken apart". If you are wondering what it "looks like" to study the Rubik's cube group, Google "Rubik's cube group".
Imagine 5 (or 25 or as much as you want) countries meet at the pole. Then you can't use 4 colors, you have to use as many as there are those countries.
It also assumes that exclaves are treated as separate entities. Otherwise you can easily make 5 mutually bordering countries. I’m surprised that he even shows it in the graphic at 5:59 but doesn’t comment on it
the poincare conjecture isnt really about what the most general shape is. The way you formulated it in this video makes it seem that the circle is a more "general" shape than the square, which is kind of exactly what topology is not about. Im sure you know this, just wanted to point out that the formulation is super misleading for someone who doesnt know about topology.