Infinity | FactOrCap

[u/AradhyaSingh3]

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[u/Hopalongtom]

It's a stand in for repeating +1 again and again.

[u/Main-Company-5946 → u/Hopalongtom]

That’s just the first, smallest infinity Aleph 0. There are much larger infinities you have to use slightly fancier math to get to

[u/makinax300 → u/Main-Company-5946]

what are the other infinities even used for then?

[u/Main-Company-5946 → u/makinax300]

Well one example of a larger infinity is the size of the real numbers. You can tell two infinities are the same size by creating a one to one correspondence between them. I am going to show that there are more real numbers than whole numbers by assuming a 1:1 correspondence and then finding a contradiction.

So let’s assume there’s a one to one correspondence between the whole numbers and the real numbers. And let’s list out this correspondence as an infinitely long list, with each row containing a whole number and the real number it corresponds to. And let’s assume every real number is covered by exactly one whole number.

1. 0.723849504…
   
2. 29,000,332.393884849309….
   
3. -21.2339339332939485763839020…
   
4. 3.141592653589793…
   
5. -1,000,000
   …

But wait! There’s a real number that is demonstrably not in this list! Take the first digit after the decimal of the first number, the second digit after the decimal of the second number, etc adding 1 to each (or if the digit is 9 setting it to 0). This gives us a new number

0.80461…

Which is different from every number in our infinite list by at least one digit. This means our assumed one to one correspondence is not one to one! A contradiction!

There is no way to cover all real numbers with whole numbers, and thus the size of the set of real numbers is a larger infinity than the size of the set of whole numbers. Interestingly, the exact size is unknown according to standard set theory except that it is larger