Some Infinities ARE Bigger Than Other Infinities (Diagonalization)

With so many infinities being the same, just which infinities are bigger, and how can we prove it?

Created by: Cory Chang
Produced by: Vivian Liu
Script Editors: Justin Chen, Brandon Chen, Elaine Chang, Zachary Greenberg

---

[divyamohan]

This is high-key the only video online that made me understand Cantor's Diagonalization. It wasn't even the Pokemon pictures, it was just the really clear animations that put this together. Thank you so much <3

[Sirgreen]

Thank you sincerely. I just spent hours watching lectures from other professors on youtube and reading math forum posts to understand this, and you explained it brilliantly in less than 7 minutes. This has been a fantastic help.

[nohjrd]

Brilliant video. That's easily the best explanation of this proof I've ever seen.

[sander]

This Math is so discrete even I'm not picking up on it.

[hundertzwoelf]

Those are very well-explaining videos! Thanks for doing this. I wonder why you don't have more views/subscribers yet.

[yourkingdomcomeyourwillbedone]

When you look at a fraction of a real number, it's like you're zooming-in on one point of the numberline Infinitely.

When you look at a never-ending set of integers, it's like you're zooming-out to see the ever-increasing numberline.

[streetleveltech]

I first learned about bigger and smaller infinities when I was a junior high school kid and read George Gamow's "One, Two, Three, Infinity." I still haven't recovered from the shock.

[chanm]

There's lots of videos on YT about Georg Cantor's diagonalization proof re. the cardinality of the infinite set of all rational numbers vs the infinite set of all real numbers... but, not gonna lie, I picked this one because it had Pokemon in the thumbnail instead of some old guy in front of a black board. And I actually understood it this time!

Man, just found your channel. It's definitely amazing! How do you animate your videos?
Love you content and editing style!

[davidjpfeiffer]

The videos on your channel are really impressive. You're creating high quality content without a large number of viewers, but I suspect that will change quickly. Great stuff, keep up the great work!

[DouglasZwick]

Some infinities' mothers are bigger than other infinities' mothers

[eric]

thank your for clearing up the concepts, I was having a hard time understanding these infinities concept from automata theory class. :D

[mr.spinoza]

By far the clearest video on the topic. I finally understand it.

[TWPO]

I wish I had found your channel while I was taking my Discrete Mathematics course! Great videos

[presterjohn]

I think it has clicked with this video. You're all basically saying because the angular number (that is infinitely long) is derived from the infinite list, for a change in all numbers (whatever the rules you use to change it, the only rule: no number matches the number it originally was), no number in that infinite list could possibly match the new number because it is derived from an angle of all the numbers (hence infinitely long). The list could contain any possible number including the one you made up... including until you form a number from the angle of those numbers. This number would be there, but a change to it would make it not possibly there even in all infinity.

[noylevi]

this is the best video i've eve seen for a mathematical proof! well done with the animation and simple explaination

[razaranyi]

i've been looking for an explanation for that for days, this on is BY FAR the best

[ChrisRoxDuhh]

Finally a video that explained this so clearly!

[jamestagge]

I think that all would find this interesting. I would greatly appreciate comments…………The means which Cantor employed in his proposition of diagnalization, i.e., 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 none 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 (forever). 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 within the boundaries of 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, extending out forever.
• In the proposed infinite string of real numbers, the list of unit members from the first unit member designated to the next, any other which might be identified (e.g., 1 to 2 or perhaps 1 to etc.) is itself infinite. For this reason, the string cannot exist beyond its consideration as a line segment which is still problematic for reasons I point out below, 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 and could extend beyond that. 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 have infinite digits. 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 these end points 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 such 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? Here we see that the string of natural numbers was being considered by Cantor as such per its unending length, that length forced by the denial of consideration of the span between specified unit members, e.g., 1 to 2 to 3, etc. However the string of real numbers could not be judged in its size in comparison to the string of natural numbers because it would have infinite members within the span of the first two unit members specified. What this means is that both must be considered in terms of unit members only and not by the abstractions of their lengths, as if each at any specified length would have different quantities of unit members. Instead, the string of natural numbers could not be considered in terms of the span between two designated members and the string of real numbers must be considered in and only in that manner. Since length of the string then is not a consideration, we are left to consider only and compare the number of unit members in which case they are equal, their “quantity” being infinite.
• I would venture that because of the above, we can only conclude that the list of natural numbers which is infinite, “stretched” along an infinite line could be “aligned” with the real numbers which are infinite while the “quantity” of them is contained within a finite distance, i.e., the length of a line segment arbitrarily defined. So, the comparison of the one quantity with the other is apart from the means of containment of each.
This proposition of Cantor’s seems to be a bad analogy to make a mathematical point and is very sloppy in its disregard for the true nature of these concepts of infinity he employs.

[alexroman]

Hey man, love the videos! I just have a question about the logic here. How is it ok to just alter a list like that. That list WOULD'VE existed if you didn't pull it out and alter it. The only reason it can't exist now is because of the external parameters you put on it. I feel like it's analogous to questioning where your dinner went after you just ate it. :/