(0,1) is uncountable

The Cantor diagonalization proof is a famous example of a proof that the set of real numbers between 0 and 1 is uncountably infinite. This means that there are infinitely many real numbers in this range, and they cannot be counted using the natural numbers (1, 2, 3, etc.).

The proof is based on the idea of constructing a list of all of the real numbers between 0 and 1, and then showing that there must be at least one real number that is not on the list.

To do this, we can start by creating a list of all of the real numbers between 0 and 1 with a finite decimal expansion. For example, the numbers 0.1, 0.01, 0.001, etc. would all be on the list.

Next, we can create a new number by taking the first digit of the first number on the list, the second digit of the second number on the list, the third digit of the third number on the list, and so on. For example, if the first number on the list is 0.12345 and the second number is 0.01234, then our new number would be 0.12341.

Since our new number is different from every number on the original list, it must not be on the list. This means that there are infinitely many real numbers between 0 and 1, and they cannot be counted using the natural numbers.

The Cantor diagonalization proof is a powerful example of how infinite sets can behave differently from finite sets, and it has had a significant impact on the development of modern mathematics. So, (0,1) is uncountably infinite.

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

[leosong]

i love how you said "change" each digit. most over vids say add 1 unless its 9 in which case subtract 1. ur way gets straight to the point and doesn't make things more complicated

[Ninja]

Technical phrase would be that natural numbers are a countable infinity while real numbers are an uncountable infinity.

In fact, integers, rationals, and even algebraic numbers are all countable.

[kinshuksinghania]

“Some infinities are bigger than other infinities”

We can take all natural numbers and find a one to one image for all these natural numbers in the set (0, 1] and still there would be an infinite number of elements in the set that won’t be one to one image of any of the natural numbers.
The simplest relation we can use is N to 1/N

[azz]

Thanks bro, U literally made me understand

[yarik]

There is a some kind of a theorem that states that the quantity of numbers in any, even infinitely small interval, is uncountable.

[maxhagenauer]

There are an infinite amount of irrational numbers between 0 and 1. Remember there are way more irrational numbers than rational numbers. If you picked a random positive integer, what's the probability the square root of it still rational? Pretty small right? The only ones that work are 1, 4, 9, 16... and it keeps on going so there are an infinite amount but there is a much larger infinite amount that don't work.

[rainbowloom]

Can you not use the same method with the natural numbers?

[scottleung]

I watched the Nova episode "Zero to Infinity", and it used this exact same proof.

[mrunmaybehere]

What is we start with pairing 1 with 0.1, then 0.2 with 2. When we reach 10, we do 0.01. 100 would be 0.001. 2300 would be 0.0023.

[qbish_]

Plz reply can we say we have infinite natural numbers in one dimension but there are 2 dimensionally infinite numbers between 0and 1 so they are ♾️ ²

[takodachich.]

This is correct, you can learn more by reading up Discrete Mathematics/Structures, on the topic of Cardinality.

[Blaqjaqshellaq]

The set of real numbers between 0 and 1 is isomorphic to the power set of natural numbers! (That's the set of all sets of natural numbers...)

[pabloasenjo]

But that happens with fractions too, not just real numbers, right?

[albertarias]

finally a short I needed commenting so the algorithim blesses me again

[ChatwithSam]

According to Vsause..
(1, 2) can be called as an uncountable infinity as we can't start nor we can end counting this interval...

[Qreator]

Why can't we do the same for integers, except to the left and get new integers not on our list of all integers like you did for numbers between 0 and 1

[marcusscience]

Why can’t list out all the real numbers between 0 and 1? Because you just can’t…or’s diagonal proof.

[KingGisInDaHouse]

I wish I had a dollar for every number between (0, 1). I already do if you only count whole numbers.

[silverbunny]

If you want to count natural numbers, you go 0, 1, 2, etc.
You can't count rational numbers in order
If you get
The will always be a

[shinsena]

damn po-shen loh literally went to our school and did this exact same proof lol.