A Hierarchy of Infinities | Infinite Series | PBS Digital Studios

Hey all! Just wanted to give a clarifying not. There is no general consensus in math as to whether zero is a natural number or not. To precisely distinguish them, one can use the phrases "non-negative integers" or "positive integers." However (!) in set theory, which is the branch of mathematics being discussed in this episode, there is a consensus: Zero is a natural number. The set theoretic constructions of the natural numbers (e.g. in the Peano Axioms) includes zero.

What do you mean we can't count to infinity? Chuck Norris did it.....twice

I don't believe you.

Cantor's stuff is wrong

Or we need to listen to our intuitions and define these infinities differently...change the math!

Is the infinity between 0 and 0.1 the same as the infinity between 0 and infinity?

:) i like her style and voice ❤

Serious question, how is this useful?

the easiest way to explain real numbers vs natural numbers for me was:
0 with 0
1 with 0.1
2 with 0.01
3 with 0.001
and so on, you will never even get to 0.2, you'll just keep adding zeros to infinity, so you can't pair the collections. you can't even really count real numbers, because after zero you get 0.000000000000000000000000000000000000000000000....∞......1, you'll never get there.

We didn't see a tower. We just saw two sets.

what about complex numbers? do they fall above the real set?

"All Brontosaures are thin at one end, much much thicker in the middle, and then thin again at the far end." - Anne Elk

These aren't different sizes of infinity 😆 because infinity has no size. These are just different types.

The reason why I love when a women explain something compared to a man because they have a softer and more nurturing voice and loving voice as a human being ...being cared by a mother that comes naturally but at the same time hate it also because they make some weird sounds and noise..listen very clearly and closely once you catch it you cant avoid it anymore 😅even this lady after every sentence she makes a noise sound that is similar to like when your cooking something sticky ..like that niaak niaak sound 😂 or something and most of the women uses their nose while talking like ..when our nose are blocked . like all those weird noise ...it's very small to detect but once you do you can't un hear it

Ok but why

I don't have an opinion about this.

Infinity means forever. so no amount of word play can change that. It's like saying one room has less nothing in it then the other room with nothing it in. By nothing I mean absolute zero.

IT DOESN'T MATTER !

It still seems to me that counting numbers are the same infinity as the real 0 to 1.
The bijection just needs to be flipped to compare the first number.

X/1 vs 1/x then x-1/1 vs 1/x-1, then x-2/1 and 1/x-2 and so on

Assigning infinity to X, the counting infinity is equal to the real number infinity.

Do I win a prize? Haha.

I have always disagreed with the way bijection is used to prove infinities equal to one another because it seems to me more like an illusion caused by the sheer magnitude of infinity and our inability to fully comprehend it all.

For example if you take that circle and label all natural numbers and then try to match all integers, bijection would tell you that somehow they’re both able to be matched perfectly and are therefore equal but in actuality that’s an impossibility. To understand why just match every number to itself and you’ll find you can only actually fit an amount equal to the natural numbers and there’s no space for the negatives left.

This even works with the infinite hotel paradox.

You have an infinite hotel and it’s completely full but one more person walks in and you need to find a room for them so you have the guest in room 1 move to the 2nd room the guest in the 2nd move to the 3rd and so on… this seems to free up a room for the new guest and there seems to always be a room available for the preexisting guests to move into, thus the conclusion is that even though you added a new customer that the infinite quantity hasn’t changed at all its still the same size. But if you consider the fact that there’s always a customer in transit between rooms you realize there’s always at least 1 person who’s left without a room so this doesn’t work either.

im infinite like math and i will reflect for ever like 2 mirrors facing each other ♾

Wow!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

I understand what this video is saying.. but I guess I’m stuck on how is conceptualizing infinity into different made up groups changes infinity. I feel we can make sub groups but we are only changing the idea of it not in a way making a bijecture measurement. I should probably think of a more literate way to explain that

If the continuum hypothesis cannot be disproven, that means that it's not possible to construct such a set whose size would be between the natural numbers and the reals, right? Because if it was possible to construct such a set, it would be possible to disprove the continuum hypothesis. Given that it's not possible to think of such a set, I'd say that we might as well say that it doesn't exist and that the continuum hypothesis is true.

Math is so freaking cool that I hate it infinitely :D

So I'm having a question with another person and I am wondering if I am correct, The best way I can explain it is to imagine you have a hotel with infinite rooms all of which are full, now one person checks out. would the set of hotel rooms be a larger infinity than the set of people in rooms? as you would have an infinite number of full rooms still but you would also have one empty room.

Rationals: am I a joke for you?

Is complex number a subset of real numbers?

The beginning. There are "Bigger infinities". It sounds like semantics for just saying, "Infinity is Infinitely bigger than itself."

When talking about infinity do they know the difference between a point, line, spiral, circle, sphere, and building those... or as in dimensions if maybe possible of infinite amount of dimensions...

So if there are infinite kinds of infinities, does that mean aleph null kinds of inifnities or what?

Om Puri

Om Puri

Natural numbers are the smallest kind of infinity - Is this a definition in the infinity hierarchy or is there a logical proof for this fact?

I gotta watch this video 20 more x.

Bijection is a stupid concept

Worst field of math

what about set of complex no.,🤔 isn't that bigger than real no. 🤔

This is just mental masturbation, there isn't a larger infinity. This statement is a paradox and they are putting glitz and glammer to distract that

"this blew my mind" 8th Grade Algebra Class

So if set theory cant solve the continuum hypothesis, what is math missing? A new branch of math? More axioms?

So are there larger infinities than the real numbers? A “hierarchy” with just two members isn’t that noteworthy.

People who don't understand this yet disagree are funny lol

there are obviously not different sizes of infinity. that contradicts the definition of infinity.

There are N0(Aleph Null) natural numbers.
There are N1(Aleph One) real numbers.
And so on...

Woah the first infinity is 1

This is closely related to the construction of the Set theory

At first we have set, then classes then conglomerates, etc.

You did not show us why there are an infinite number of infinities.

the 5 fingers on both hands analogy for the bijection is different than trying to pair things with infinity, it seems like an awful big leap of faith to just assume you can do that with infinite sets. I am still not convinced the diagonal method works, adding a "new" number to a finite sample-list (a number that would be covered by infinite numbers anyway) doesn't seem like proof.....that is just my own option though