The diagonalisation argument, Part 1

[@travisschnider]

The way that you explain that each decimal place determines that the new number is unique made this click for me. Thank you for taking the time to explain this.

[@furansart]

Now I get it! The whole point of this argument is that, since one digit is enough to make a different number (as you pointed out at the very beginning), if you make a number that has at least one different digit from every other number, then you have a number which is different from all of the others. Since we use the diagonal (which is infinite) we can create an infinite number that is different from them all! It's basically a parametrized rule to establish difference as you go to infinity. Since the pool of the parameters (i and j, the rows and the columns) are infinite, the diagonalization creates an infinite answer that fulfills the condition of having at least one digit different from that of another number for every digit!! This argument was making me really frustrated hahahah, I'm happy that I finally got it. Thank you very much! That simple sentence you said made it all make sense.

[@calebmolina1143]

John, you are a lifesaver!

[@jsgames6843]

beautiful explanation, thanks!

[@ThatKGKid]

Wow what a beautiful explanation! Just started reviewing this in my Real Analysis class and didn't quite understand the argument that Cantor was forming. This really helped me. Thank you!!

[@vavavarken]

Finally understand this from 7:33, other videos just skipped this part...

[@bernard6260]

Yeah, because many things are hidden from us , people.

[@antoniocortijo-rodgers75]

Totally agree, really made the proof comprehensive

[@aaryanandaryakhot4422]

Thanks a lot 😊

[@marischan2869]

A perfect explanation!!! thank you

[@chevlonmacguinstudios]

Sounds like c++ memory management

[@jamestagge3429]

This business about the infinite string of real numbers being larger than that of natural numbers is discussed and considered in a context in which it is ignored that there is no such thing as infinity in material reality for it defies the means and manner of existence which is that anything that does exist must be distinct, delineable and quantifiable. This understanding includes the products of the realm of the abstract as well in that there is no abstraction which is not ultimately the product of the material, contextual referents in reality, that context from which they arise. For example, the abstraction of a pink flying elephant is one formed of the fusion of the material colour pink, the material phenomenon flying and the material entity, elephant. What mathematicians such as Cantor have done is employ the most general understanding of infinity as a concept but ignore the inevitable contradictions which arise, muddying the waters of the context in which their propositions are formulated and presented. 1. Consider that the infinite string of natural numbers is a progression, that which extends outward in length into infinity. Each unit member is a value, the progression advancing by that value plus 1 each time. However, that to which it is being compared, i.e., the infinite set of real numbers is structurally the opposite in the proposition. • In the infinite string or natural numbers, the span between any two unit members is ignored and the line proceeds from each value to the next. • In the proposed infinite string of real numbers, the unit members from the first to any other which might be identified is itself infinite. For this reason, the string cannot exist beyond its consideration as a line segment which is still problematic, its overall length a value arbitrarily assigned but finite. So, in the case of the real numbers, the infinite line of unit members would be contained within two designated units with infinite points between. The string of numbers does NOT extend outward but rather within itself. This is comparing apples to oranges. - There could be no list of real numbers for the designation of the very first in the list would never be completed or would just be impossible for it would be infinite in length. None of the real numbers could be designated and thus, nor could the list. This is not unlike the problems that arise with line segments in which it is claimed that they are composed of infinite points, yet they cannot be because if of finite length, each end would have to be designated by a point beyond which there was no other which by definition would mean that those points would have to have scope and dimension which would mean that there could not be infinite points composing the line segment. However, if they had scope and dimension, what would that be? If 10x, why not 5x then why not 1x, ad infinitum. Thus, the line segment could NOT be composed of infinite points but at the same time would have to be, demonstrating that infinity cannot be paired with material concepts due to the inevitable contradictions. • What then would be the measure by which the string of real numbers was determined to be larger than that of natural numbers? Would it be that the string of real numbers is bigger by means of the number of unit members between the two designated points? It certainly could not be considered bigger for its conceptual length by which the string of natural numbers would be considered, it being the only way that it could be considered. This proposition of Cantor’s seems to be very sloppy in its disregard for the true nature of these concepts of infinity he employs.

[@Daveboymagic]

Cantor really confused ya'll with decimals. Just forget them and answer this. What is bigger infinite A or (infinite B + 1)? The answer is they are the same. It doesnt matter if I change numbers in infinite B as infinite A will always catch up. What you are doing is suddenly acting like infinite A ends, when you are saying the new dog is not on the "list", as if the "list" is finite. Keep on going and it will be there. If you keep changing the new dog, you must also keep adding collars.

[@mzohaibmathematician1200]

Very beautiful ❤️

[@antdgar]

If you are creating new dogs, what is stopping you from creating new collars as well?

[@johnlevine2909]

Just to clarify: I'm not actually creating a new dog here - I'm just finding the name (real number) of a dog that already exists but which has clearly has no collar under my suggested mapping.

[@abagel5968]

Oh my gosh, thank you so much! This helped me for my homework!!

[@swaroopsahoo214]

The video just folds open the crux of this concept so crystal clear that isn't done by anyone on KZbin as of now in my exploration.

[@everythingmaths6147]

Great explanation!

[@farmerjohn6526]

Did you ever think of diagonalizing the natural numbers too? Thus the natural numbers on the left also exist that dont have a dog. So both sides are uncountable...duh. On another note, nothing stops me from creating an infinite number of dogs thst all match the diagonal, and thus the dogs and collars are all countable in that infinite set. So what does that mean,? So even if there are an infinite number of numbers i cannot match there are an infinite number i can match...so there are an infinite number of sets that havd an infinite number of numbers that do match..howvis that possible??.

[@MikeRosoftJH]

That doesn't work, because every natural number is finite in magnitude (and that's by definition; a set is finite, if its number of elements is equal to some natural number); and so its decimal representation has finitely many digits. If you try to apply the diagonal procedure to a sequence of all natural numbers, you get a sequence containing infinitely many non-zero digits, and that doesn't represent any natural number.

[@beautifulpollution]

Why even use the diagonal if you're going to randomize the number?

[@aboogiewithdahoodie]

The diagonal is used to guarantee that the new number is not already listed, as the one number changed in each diagonal guarantees that the new number was not alreayd there before.

[@charlesscobb]

I’m not getting it (yet). If this list of numbers is infinite, how do you know that your new number is not further down on your infinite list of numbers?

[@jeffjo8732]

Further down than what? The "new number" uses every number in the infinite list, so there is no "further down."

[@thebardlydm]

Every number in the set is having one number changed (that's why diagonal), so you just keep changing one number diagonally, so you are guaranteed to have one number different from all other listed numbers

[@gravity6316]

wow

[@raginibhayana8305]

thank you so much, this really helps

[@jandraor]

Thank you so much. Finally, I understood it. Life is much better with dogs.

[@nothingham4742]

beautifully explained, brits have always been good with MATH

[@saurabhk96]

brilliant explanation

[@pendaranroberts4350]

It seems that the argument is that because for any natural number no matter how large there are real numbers not yet listed that this implies that there are more real numbers than natural numbers. However, I don't see how it follows that there are more real numbers than natural numbers, because there are infinitely more natural numbers bigger than the one you chose and so infinite space on the list for the infinite real numbers.

[@MartinPoulter]

If you think the proof involves choosing a natural number, you're not understanding the argument. It does not involve such a choice. The argument assumes that all natural numbers are mapped to a real number.

[@pendaranroberts4350]

It assumes they are all mapped. But it doesn’t show they aren’t. It’s just sophistry. There are so many problems with the supposed proof. I mean for one how can you take the diagonal of an infinite list? Why should our reasoning regarding diagonals on finite lists extend to infinite lists? If the diagonal isn’t on the list then add it at the top. If it’s listable now how could it not have been before? Magic? I guess so!

[@MartinPoulter]

@@pendaranroberts4350 Which is more probable: 1) that all qualified mathematicians and most students of mathematics are wrong about something which is taught in introductory courses, failing to notice multiple errors in a theorem regarded as having been proven for a century. 2) that you haven't quite understood the structure of the argument yet? Evidence for 2) comes from your suggestion that we can add the diagonal number to the list: that's a tacit admission that the list is incomplete, the bijection is impossible and hence that Cantor's reductio achieves what it intends to. You're showing the argument is sound while claiming that it's just sophistry.

[@pendaranroberts4350]

If you look through history the majority has been wrong over and over. Have you heard of the pessimistic induction? In reply to your other point, you dismiss the argument I’m making without proper engagement. The diagonal supposedly isn’t on the list and so not all the reals are listable. But add it to the list. It is now listed and so is listable. If it’s listable why wasn’t it on the list? We assumed all the real numbers were listed. So shouldn’t every listable real be listed? If it wasn’t listable before why is it now?

[@MartinPoulter]

@@pendaranroberts4350 I'm well aware of the pessimistic counterinduction (to give it its *correct* name). I'm aware it doesn't say that the sum total of qualified experts are more likely to be wrong than one guy in KZbin comments. I'm also aware of cranks and crackpots, and it's a characteristic move of those people to discover that almost every qualified expert disagrees with them and conclude from that those experts must be wrong. A load of rhetorical questions does not make a logical argument. A list of every real number (in a finite interval) is not possible. The Cantor argument demonstrates this with deductive logic and you have no counter-argument. The idea of whether a number is or is not "listable" is your own invention and is nothing to do with Cantor.

[@jamesshelton3827]

Woohoo guess who just won the infinity lottery ticket

[@jamesshelton3827]

OK your problem is that your new number is kind of not really new but actually very very special and on top of that there are actually an infinity of them

[@jamesshelton3827]

OK Think for a moment homeboy

[@jamesshelton3827]

youve got pretty much everything right but one very simple yet rather extremely elusive thing

[@jamesshelton3827]

ok so youve figured out that your number is actually unique and not equal to any number in that first diagram you got there

[@jamesshelton3827]

so where you gonna put that number in your one to one and onto list homeboy?

[@tbnman]

Wouldn't this be easier to visualize by mapping dogs as elements of the natural numbers (since dogs are easier to see as countable) to collars as elements of the reals?

[@vincentrusso4332]

I don't understand the illustration and don't see how its applicable in the real world.

[@bernard6260]

This works only in mathematica universe

[@EntirelyPointlessContent]

It exists in the real world and is applicable in new ways of thinking, the most important thing we can do in the real world