Comparing the Sizes of Sets in Different Ways (pt 2, (0,1), [0,1), and the real numbers)

In the last video we developed the Lebesgue measure and compared the size of N and Z. Now we will compare two other pairs of sets, and I'll offer some closing remarks.

(as well as previews, behind-the-scenes, and more content).

Special thanks to my supporters on Patreon:
cb, Dru Vitale

Music by Sakis Strigas
(Undertow, Eulogy, Lateralus - Tool)

My current equipment for making videos (affiliate links):

[EpicMathTime]

damn, ain't nobody watching this video! 😂 Spooky Halloween specials coming soon.

[blackpenredpen]

Ahhh very nice! Brings back my good old memories!

[tobiasthrien]

The notion of size we developed for finite sets simply isn't generalizable to infinite sets in an obvious/trivial way. We cannot preserve all it's familiar properties and therefore a certain freedom of choice arises for our more general definition.
Nevertheless certain definition might turn out to be more valuable than others.

In my opinion there are two central colliding ideas that cause the struggle probably every math enthusiast has to go through when first encountering this apparent paradoxon.
The first one is that a set containing another set has to be bigger than it, which is totally obvious for finite sets and therefore hard to give up on.
The other one is that sets are of equal size if you can find a one-to-one function (bijection) between them, which means that you can identify each element of one set with exactly one element of the other set (in both ways so no element is missed). Essentialy this is in no way different from renaming every element of a set, which shouldn't change it's size.

Even though both ideas have there legitmacy there exists a key difference between them.
The "problem" with the set containment argument is that it's a statement about the size of a set which inherently depends on the identity/properties of it's members. You can therefore exchange one element by another one and loose your proposition about the size of this set.
In contrast the second idea embraces the fact that the notion of size of a set should be independent of the specific properties of it's members.



I hope I made myself clear and didn't make too much grammatical or spelling errors since this isn't my native language :)

[alephnull]

Petition to rename infinite measures and quantities to "damn long"

[MathManMcGreal]

Ah yes the classic problem left as an exercise trick... I did that on a review key for ONE PROBLEM and kids were not happy! 🤣

[nathangrant]

i can't help but think that someone was behind the camera pulling funny faces and you are spending the whole video trying not to laugh. :D

[MrYoung-tyhn]

This made no sense to my math 106 brain, but I still loved it. Keep up the great work Epic!

[nicolastorres]

Imbeddings when equipped with certain structur

[willnewman]

For Z and N, it would also be instructive to look at density. In this case, we can say N is half the size of Z.

[chemistro]

*I'm ready for my absolutely glorious mathbrah fix*

[MrNygiz]

A, B measurable A subset of B then B=(B/A)UA

[MasterHigure]

1:05 Why no strict subset symbol? It's not only appropriate, but carries the meaning you actually want to convey (i.e. that one set is a subset of another, and they are NOT equal). Same at 3:25.

[benjamincolson]

Is there any set that has a bigger cardinality than the integers, but a smaller cardinality than the reals?

[peterosudar]

Do you really believe in the real numbers? Or just use them to work through a mathematical narrative? i.e. produce enjoyable and thoughtful YouTube content? (That of which I appreciate and enjoy! fyi)