Integers & Rationals are both infinite but is it the SAME infinity?

What does it mean for two infinite sets to have the same size? For instance, are the Integers and the Rationals (numbers like 2/3) the same size? They are certainly both infinite, but the question is whether infinite really only represents one concept or whether it divides into multiple concepts.

In this video we investigate a notion called "countable" which extends from the basic way we count finite sets. According to that notion, the integers and the rationals have the same size.

In the next video we will compare the integers and the REAL numbers.

Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

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[jsnmad]

Excellent break down. He did not take anything for granted. Gradually building by using clear visual presentations. Concepts in mathematics can be easier understood when visualized. Nice work.

[itskathan]

Just nailed it, Solved by countability doubt in Theory of Computation. Love from India 💕

[MyOneFiftiethOfADollar]

I wonder if Cantor had anyone he could hug when he first noticed this. What a lonely moment if not! He would shake your hand if he could on your rendering of his famous diagonal argument

[RuichenZhao]

The diagram with a zigzag path is so illustrative

[aenimosity]

Great video, you're a great teacher. Thanks!

[jingyiwang]

Thank you so much for this amazing and patient explanation! It is really helpful to me!

[shortpianocovers]

You are such a great teacher 😊.Thank you so much...❤

[alainrogez]

I like also the Stern-Brocot tree. It is quite elegant.

[BlaqueT]

This was an excellent video. I struggled with this proof for so long, but you made it clear in a very short amount of time. I do have a question though, can you map from the naturals to the negative integers, since the naturals have no negative elements by definition?

[half-soul]

Great video. However, I didn't understand how would you biject the negative and positive together..?

[shayhan]

I don't get why that method of elimination is justified isn't the 1/1, 2/2 being the same just matter of encoding the numbers in that format?

[jimmyandtheband]

You are such an inspiring teacher. Thank you for your passion and time to break things down!! This was eye-opening!!! Love the vids!!! :D

[MisterrLi]

Yes, you can show that the set of Rationals is listable. It's like taking the full set of Natural numbers, and
add more natural numbers to it. Sure, all the positions of the Natural numbers are filled up, but you can
always find more empty positions by moving values around. For example, you can relabel all Natural
numbers to the position plus 1 to fit in one extra number, and move numbers to position times 2 to
fit in an infinity of new numbers, and so on.

You could also pair up the Natural numbers with the same natural number values in the Rational number set.
That way you can see that the actual infinite number of rational numbers is bigger, since it contains infinitely
many values that are not found in the set of natural numbers. This method works when the two sets have the
same elements, so that these can be paired off, and then you simply compare the rest of the elements.



For Rationals, they then get a different number of infinite elements, about N^2, depending on the definition
of them, and if you remove numbers with similar values.

[rryan]

Anyway I could reach out to you to show you a method I came up with that would suggest all real numbers are countable? It’s based off this model with some tweaks!

[henriquenunes]

Dear Dr. Bazett. Topologically speaking the interior of Q (rationals - an aleph0 set) is empty. Lots of people tell me that the interior of R\Q is also an empty set. I believe that the interior of R\Q is at least an aleph0 set. I hope this is not a Godel/Cohen issue, but it just you share some thoughts please? Thanks very much.

[eppssilon]

ngl, it almost feels like our math is flawed if we consider them the same size, one is an infinite line and the other is an infinite square that can be zigzagged into a line
Especially in my case, I am trying to find a function from Q (rationals) to N (naturals or positive integers) and it needs to be injective, meaning each element from the starting group Q should have a unique image in the landing group N, no two elements should share the same image, (if f(a) = f(b), it should mean that a = b, because no two have the same image), if we consider the two groups the same size, we should be able to assign them in this way, but I just can't seem to visualize it at all (abstractly or graphically), the explanation of the zigzag is pretty cool to visualize that its countable or "listable", but there is no real logic beyond visualization, there is no way you could tell me what's the Nth number is unless you count by hand, which yeah, isn't really practical if you ask me
maybe I am missing something but this feels like a paradox and that there is no such function

[swarnendumaity]

I think irrational no. sets is bigger than rational no. because we can never line them up...there are infinite no. of digits in a single irrational no. itself.

[abdulmateen]

I think irrationals have bigger 'size' than integers

[ShinyLP]

7:13 Bruh you just spent 7 mins proving that this is not true

[SachinAggarwal-pg]

Well thats not the case that every positive infinite set has the same size, as in case of set of real number we cannot map it with integers..