How to Compare Infinities

There are just as many numbers between 0 and 1 as there are between 0 and 2. Though infinity may behave counterintuitively at first, everything makes more sense as we reexamine the topic from the basics.

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

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[BeautyMarkRush]

"How can we be confident we don't accidentally MISCOUNT- * count appears *"

My pun tolerance has just been exceeded

[UnbornAssassin]

Wow, well made. Hope your channel grows! Hopefully someone picks you up

[dontreadnames]

I just did basic calculus and linear algebra, so I'm no expert in math. Feel free to correct me.

If we take any number between 0 and 1, and add +1 to each of this number, we can find its friend between 1 and 2. So we can conclude that 0-1 and 1-2 have the same amount of number. If we add both sets of number, we could say the set [0, 2] is twice as big as the set [0, 1] (i know my notation is wrong but i think you understand).

[wayland]

I love how he used quotes from The Fault in Our Stars

[MultiKB]

This is excellent, you should be having way more subs than this! Keep making these types of videos!

[almightysapling]

(3) isn't *necessarily* wrong. "Big" and "size" are not well-defined mathematical terms. Discussing cardinality, you are correct. But if m is the Lebesgue measure, then we see m([0, 1]) = 1 < 2= m([0, 2]), and with the name "measure" and the value being measured frequently being called "volume", we can see that this is a perfectly good correct interpretation of (3).

[jiqbal]

"yes we can!" lol. this video makes math so fun

[amirbaer]

Terrific videos! What software are you using for the presentation?

[MathCuriousity]

What about the natural numbers and the numbers between 0 and 1?

Also why does it seem like your def of bijection is a bit diff from the one I learned in elementary set theory ?

[eric]

thank you for making these concept easy to understand ^_^

[TheVman]

Assuming the first three statements in "Two truths and a lie" consist of only one lie, not knowing much about infinities one can come to the conclusion that 3 is false. The one thing one would have to assume about infinities is that the set of numbers between 0 and 1 and the set between 0 and 2 are similar, specifically that if the set of numbers between 0 and 1 is infinite then the set between 0 and 2 is also infinite.
Now with this assumption and pure logic, assume statement 2 is false. Then the set between 0 and 2 is also not infinite by the assumption. Therefore statement 3 is false, but this contradicts the fact that there is only one false statement. Assume statement 1 is false. Then all infinities are the same size. This makes statement 3 impossible as it says there is an infinite set is bigger than anothet making statement 3 false. This once again contradicts the fact that there is only one lie. Therefore statement 3 must be false.

[liuhaixiang]

Isn't that both ways of mapping are bijection? The counter-example of creating a label [0, 0, 0] for the oranges are not the same, because it is not a bijection mapping.

[ZantetsukenGilgamesh]

Call me stupid but would the infinity between 0 and 2 always be twice as large as the infinity between 0 and 1 or between 1 and 2? For every number between 0 and 1, I can just add 1 if I’m counting between 1 and 2.

If there are infinite dimensions with infinite amounts of infinities, wouldn’t some infinite infinities be more infinite than others?

[alanbarnett]

Bijection. I like it. It sounds filthy...

[robertwilsoniii]

It seems to me that any discrete counting set is always going to be smaller than some continuous counting set. I think this is what makes this hard for people to understand, and now that I realize it, I think it's universally poorly explained and written in all textbooks on real analysis (perhaps because the authors of these books didn't understand it well enough to write and explain clearly).

Oh math, the only subject where no one knows what they're talking about.

[whatno]

Cantor: there are as many numbers between 0 and 2 as there are 0 and 1
Lebesgue: there are twice as many numbers between 0 and 2 than there are 0 and 1
Finitist: sTOP

[MikeRosoftJH]

I'm going to intentionally misunderstand the statement at the beginning, and say that obviously statement 2 is not true: every number between 0 and 1 is finite in magnitude (for example, they are all less than 2), so there are no infinite numbers between 0 and 1. (On the other hand, it's very likely that when realizing real numbers as sets in set theory, some or all of them will be realized as infinite sets. For example, real numbers can be defined as sets of rational numbers - Dedekind cuts, or using sequences of rational numbers - as Cauchy equivalence classes; and in both cases a real number will be realized as an infinite set. So in this sense the statement 2 is true. :-)

[matthewvicendese]

Spanish and English ... La tortuga beben leche!
Who else has Duo lingo?

[Sumit-MMA]

I don't think so because infinity is not finite

[dylanslingsby]

I didn't understand the one with decimals
Good video though