I’ve been diving a bit into infinity with this online class. Part of trying to understand this is realizing that I’ve held certain concepts in my head there were wrong.

I had this idea that infinity is the biggest you can get to. But that’s not really what infinity is. If I have an infinite amount of natural number (1,2,3,4,5 . . . ), I still have constraints of what that infinity means. There is an infinite amount of natural numbers, yes. But I’m still just dealing with natural numbers.

There are also an infinite amount of rational numbers (think fractions/decimals) between 0 and 1.

Some infinities are bigger than other infinities. You can compare infinities by doing a bijection of one set to another. If you can connect each entry in one set to a corresponding entry in another set and vice versa, then the sets are the same size. To prove whether infinities are the same or different size, you just have to create a rule or process to form a bijection.

If I have infinite rational numbers, then this is bigger than infinite natural numbers. I can count infinite natural numbers. But infinities do not have to be countable.

There is a diagonal proof that basically says this: try to write a list of all the rational numbers in decimal form. Now take the first digit of the first number and change that digit. Then take the second digit of the second number and change that. The third number, change the third digit. And if you keep changing the digits on for infinity, then the number you end up with will not be on your list. Thus, the rational numbers are not countable and are bigger than the set of natural numbers.

There are also an infinite amount of infinities that are larger and larger than another.

I had a concept of infinity that was incorrect. But it was useful to me, until I wanted to learn more and I had to change what I thought.